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Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property

Ravi P Agarwal12, Wutiphol Sintunavarat3* and Poom Kumam3*

Author Affiliations

1 Department of Mathematics, Texas A&M University Kingsville, Kingsville, TX, 78363, USA

2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

3 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Kru, Bangkok, 10140, Thailand

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Fixed Point Theory and Applications 2013, 2013:22  doi:10.1186/1687-1812-2013-22

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/22


Received:7 July 2012
Accepted:15 January 2013
Published:31 January 2013

© 2013 Agarwal et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we prove the coupled coincidence point theorems for a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible mapping in partially ordered cone metric spaces over a solid cone without the mixed g-monotone property. In the case of a totally ordered space, these results are automatically obvious under the assumption given. Therefore, these results can be applied in a much wider class of problems. We also prove the uniqueness of a common coupled fixed point in this setup and give some example which is not applied to the existence of a common coupled fixed point by using the mixed g-monotone property but can be applied to our results.

MSC: 47H10, 54H25.

Keywords:
cone metric spaces; common coupled fixed point; coupled coincidence point; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible mappings; mixed g-monotone property

1 Introduction

The famous Banach contraction principle states that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3">View MathML</a> is a complete metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M4">View MathML</a> is a contraction mapping (i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M5">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6">View MathML</a>, where α is a non-negative number such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M7">View MathML</a>), then T has a unique fixed point. This principle is one of the cornerstones in the development of nonlinear analysis. Fixed point theorems have applications not only in the various branches of mathematics, but also in economics, chemistry, biology, computer science, engineering, and others. Due to the importance, generalizations of Banach’s contraction principle have been investigated heavily by several authors.

Following this trend, the problem of existence and uniqueness of fixed points in partially ordered sets has been studied thoroughly because of its interesting nature. In 1986, Turinici [1] presented the first result in this direction. Afterward, Ran and Reurings [2] gave some applications of Turinici’s theorem to matrix equations. The results of Ran and Reurings were further extended to ordered cone metric spaces in [3-5]. In 2005, Nieto and Rodríguez-López [6] extended Ran and Reurings’s theorems for nondecreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

The notion of coupled fixed points was introduced by Guo and Lakshmikantham [7]. Since then, the concept has been of interest to many researchers in metrical fixed point theory. In 2006, Bhaskar and Lakshmikantham [8] introduced the concept of a mixed monotone property (see further Definition 2.4). They proved classical coupled fixed point theorems for mappings satisfying the mixed monotone property and also discussed an application of their result by investigating the existence and uniqueness of a solution of the periodic boundary value problem. Following this result, Harjani et al.[9] (see also [10,11]) studied the existence and uniqueness of solutions of a nonlinear integral equation as an application of coupled fixed points. Very recently, motivated by the work of Caballero et al.[12], Jleli and Samet [13] discussed the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M8">View MathML</a>

(1.1)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M9">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M10">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M11">View MathML</a> is the Riemann-Liouville fractional derivative and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M12">View MathML</a> is continuous, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M13">View MathML</a> (f is singular at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M14">View MathML</a>) for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M15">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M16">View MathML</a> is nondecreasing with respect to the first component and decreasing with respect to its second and third components.

Since their important role in the study of the existence and uniqueness of a solution of the periodic boundary value problem, a nonlinear integral equation, and the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem, a wide discussion on coupled fixed point theorems aimed the interest of many scientists.

In 2009, Lakshmikantham and Ćirić [14] extended the concept of a mixed monotone property to a mixed g-monotone mapping and proved coupled coincidence point and common coupled fixed point theorems which are more general than the result of Bhaskar and Lakshmikantham in [8]. A number of articles on coupled fixed point, coupled coincidence point, and common coupled fixed point theorems have been dedicated to the improvement; see [15-30] and the references therein.

On the other hand, in 2007, Huang and Zhang [31] have re-introduced the concept of a cone metric space which is replacing the set of real numbers by an ordered Banach space E. They went further and defined the convergence via interior points of the cone by which the order in E is defined. This approach allows the investigation of cone spaces in the case when the cone is not necessarily normal. They also continued with results concerned with the normal cones only. One of the main results from [31] is the Banach contraction principle in the setting of normal cone spaces. Afterward, many authors generalized their fixed point theorems in cone spaces with normal cones. In other words, the fixed point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is non-normal but just has interior that is nonempty. In this case only, proper generalizations of results from the ordinary metric spaces can be obtained. In 2011, Janković et al.[32] gave some examples showing that theorems from ordinary metric spaces cannot be applied in the setting of cone metric spaces, when the cone is non-normal.

Recently, Nashine et al.[33] established common coupled fixed point theorems for mixed g-monotone and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible mappings satisfying more general contractive conditions in ordered cone metric spaces over a cone that is only solid (i.e., has a nonempty interior) which improve works of Karapınar [34] and Shatanawi [35]. This result is an ordered version extension of the results of Abbas et al.[36].

In this work, we show that the mixed g-monotone property in common coupled fixed point theorems in ordered cone metric spaces can be replaced by another property due to Ðorić et al.[37]. This property is automatically satisfied in the case of a totally ordered space. Therefore, these results can be applied in a much wider class of problems. Our results generalize and extend many well-known comparable results in the literature. An illustrative example is presented in this work when our results can be used in proving the existence of a common coupled fixed point, while the results of Nashine et al.[33] cannot.

2 Preliminaries

In this section, we give some notations and a property that are useful for our main results. Let E be a real Banach space with respect to a given norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M18">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M19">View MathML</a> be a zero vector of E. A nonempty subset P of E is called a cone if the following conditions hold:

1. P is closed and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M20">View MathML</a>;

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M21">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M22">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M23">View MathML</a>;

3. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M24">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M25">View MathML</a>.

Given a cone <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M26">View MathML</a>, a partial ordering <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M27">View MathML</a> with respect to P is naturally defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M28">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M29">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M30">View MathML</a>. We will write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M31">View MathML</a> to indicate that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M32">View MathML</a> but <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M33">View MathML</a>, while <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M34">View MathML</a> will stand for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M35">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M36">View MathML</a> denotes the interior of P.

The cone P is said to be normal if there exists a real number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M37">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M38">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M39">View MathML</a>

The least positive number K satisfying the above statement is called a normal constant of P. In 2008, Rezapour and Hamlbarani [38] showed that there are no normal cones with a normal constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M40">View MathML</a>.

In what follows, we always suppose that E is a real Banach space with cone P satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M41">View MathML</a> (such cones are called solid).

Definition 2.1 ([31])

Let X be a nonempty set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M42">View MathML</a> satisfy

1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M43">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M45">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M46">View MathML</a>;

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M47">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6">View MathML</a>;

3. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M49">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M50">View MathML</a>.

Then d is called a cone metric on X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3">View MathML</a> is called a cone metric space.

Definition 2.2 ([31])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3">View MathML</a> be a cone metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M53">View MathML</a> be a sequence in X, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M54">View MathML</a>.

1. If for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M55">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M56">View MathML</a>, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M57">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M58">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M59">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M53">View MathML</a> is said to converge to x. This limit is denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M61">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M62">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M63">View MathML</a>.

2. If for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M55">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M56">View MathML</a>, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M57">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M67">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M68">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M53">View MathML</a> is called a Cauchy sequence in X.

3. If every Cauchy sequence in X is convergent in X, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3">View MathML</a> is called a complete cone metric space.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3">View MathML</a> be a cone metric space. Then the following properties are often used (particularly when dealing with cone metric spaces in which the cone need not be normal):

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M72">View MathML</a>) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M73">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M74">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M75">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M76">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77">View MathML</a>) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M78">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M79">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M80">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M81">View MathML</a>) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M82">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M83">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M84">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M85">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M86">View MathML</a>) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M87">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M88">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M89">View MathML</a>, then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M90">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M91">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M92">View MathML</a>.

Definition 2.3 Let X be a nonempty set. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M93">View MathML</a> is called an ordered cone metric space if

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3">View MathML</a> is a cone metric space,

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M95">View MathML</a> is a partially ordered set.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M95">View MathML</a> be a partially ordered set. By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M97">View MathML</a>, we mean <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M98">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6">View MathML</a>. Elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6">View MathML</a> are called comparable if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M101">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M102">View MathML</a> holds. A mapping f is said to be g-nondecreasing (resp., g-nonincreasing) if, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M104">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M105">View MathML</a> (resp., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M106">View MathML</a>). If g is the identity mapping, then f is said to be nondecreasing (resp., nonincreasing).

Definition 2.4 ([8,14])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M107">View MathML</a> be a partially ordered set and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M108">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M109">View MathML</a>. The mapping F is said to have a mixedg-monotone property if F is monotone g-nondecreasing in its first argument and monotone g-nonincreasing in its second argument, that is, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M110">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M111">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M112">View MathML</a>

(2.2)

hold. If in the previous relations g is the identity mapping, then it is said that F has a mixed monotone property.

Definition 2.5 ([8,14])

Let X be a nonempty set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M113">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M114">View MathML</a>. An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115">View MathML</a> is called

(C1) a coupled fixed point of F if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M116">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M117">View MathML</a>;

(C2) a coupled coincidence point of mappings g and F if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M118">View MathML</a>

and in this case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M119">View MathML</a> is called a coupled point of coincidence;

(C3) a common coupled fixed point of mappings g and F if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M120">View MathML</a>

Definition 2.6 ([36])

Let X be a nonempty set. Mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M121">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M122">View MathML</a> are called

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M123">View MathML</a>) w-compatible if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M124">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M125">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M126">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M127">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M129">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M130">View MathML</a>.

It is easy to see that w-compatible implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible. The following example shows that the converse of the above argument is not true.

Example 2.7 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M132">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M135">View MathML</a>

It is easy to see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M136">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M137">View MathML</a>, but <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M138">View MathML</a>. Hence, F and g are not w-compatible.

However, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M139">View MathML</a> is possible only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M140">View MathML</a> and for all points in this case, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M141">View MathML</a>. Therefore, F and g are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible.

For elements x, y of a partially ordered set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M95">View MathML</a>, we will write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M144">View MathML</a> whenever x and y are comparable (i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M145">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M98">View MathML</a> holds).

Next, we give a new property due to Ðorić et al.[37].

Let X be a nonempty set and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133">View MathML</a>. We will consider the following condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M149">View MathML</a>

(2.3)

In particular, when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M150">View MathML</a>, it reduces to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M151">View MathML</a>

(2.4)

Remark 2.8 We obtain that the conditions (2.3) and (2.4) are trivially satisfied if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M95">View MathML</a> is the totally ordered.

The following examples show that the condition (2.3) ((2.4), resp.) may be satisfied when F does not have the mixed g-monotone property (monotone property, resp.).

Example 2.9 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M153">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M154">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M155">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M157">View MathML</a> but <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M158">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156">View MathML</a>, the mapping F does not have the mixed g-monotone property. But it has property (2.3) since

(1) For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M161">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M162">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M163">View MathML</a>.

(2) For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M165">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M166">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M163">View MathML</a>.

Example 2.10 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M168">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M154">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M170">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M172">View MathML</a> but <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156">View MathML</a>, the mapping F does not have the mixed monotone property. But it has property (2.4) since

(1) For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M176">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M177">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M163">View MathML</a>.

(2) For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M156">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M180">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M181">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M163">View MathML</a>.

(3) The other two cases are trivial.

3 Coupled coincidence point theorems lacking the mixed g-monotone property

In this section, we give the existence of coupled coincidence point theorems in ordered cone metric spaces lacking the mixed g-monotone property. Our first main result is the following theorem.

Theorem 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183">View MathML</a>be an ordered cone metric space over a solid conePand let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133">View MathML</a>. Suppose that the following hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187">View MathML</a>is a complete subspace ofX;

(ii) gandFsatisfy property (2.3);

(iii) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M189">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M190">View MathML</a>;

(iv) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M191">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M192">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M193">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M195">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M196">View MathML</a>,

(3.1)

holds;

(v) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198">View MathML</a>when<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>inX, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200">View MathML</a>fornsufficiently large.

Then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M202">View MathML</a>

that is, Fandghave a coupled coincidence point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115">View MathML</a>.

Proof Starting from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M204">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M205">View MathML</a> (condition (iii)) and using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186">View MathML</a> (condition (i)), we can construct sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208">View MathML</a> in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M209">View MathML</a>

(3.2)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210">View MathML</a>. By (iii), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M211">View MathML</a>, and the condition (ii) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M212">View MathML</a>

Proceeding by induction, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213">View MathML</a> and, similarly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210">View MathML</a>. Therefore, we can apply the condition (3.1) to obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M216">View MathML</a>

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M217">View MathML</a>

(3.3)

Similarly, starting with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M218">View MathML</a> and using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M222">View MathML</a>

(3.4)

Combining (3.3) and (3.4), we obtain that

(3.5)

Now, starting from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M224">View MathML</a> and using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M227">View MathML</a>, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M228">View MathML</a>

Similarly, starting from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M229">View MathML</a> and using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M227">View MathML</a>, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M233">View MathML</a>

Again adding up, we obtain that

(3.6)

Finally, adding up (3.5) and (3.6), it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M235">View MathML</a>

(3.7)

with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M236">View MathML</a>

(3.8)

since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M193">View MathML</a>.

From the relation (3.7), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M238">View MathML</a>

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M239">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M240">View MathML</a> is a coupled coincidence point of F and g. So, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M241">View MathML</a>.

For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M242">View MathML</a>, repeated use of the triangle inequality gives

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M243">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M244">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M245">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M246">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M63">View MathML</a>.

From (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M86">View MathML</a>), we have for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M79">View MathML</a> and large n,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M250">View MathML</a>

By (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M81">View MathML</a>), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M252">View MathML</a>

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M253">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M254">View MathML</a>

then by (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M81">View MathML</a>), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M256">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M257">View MathML</a> for n large enough. Therefore, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208">View MathML</a> are Cauchy sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187">View MathML</a>. By completeness of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M262">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M263">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M264">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>.

By (v), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M266">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M267">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M268">View MathML</a>. Now, we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M271">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M272">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M273">View MathML</a>, from (3.1) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M274">View MathML</a>

which further implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M275">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M263">View MathML</a>, then for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M79">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M278">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M279">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M280">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M281">View MathML</a>

Now, according to (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77">View MathML</a>), it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M283">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269">View MathML</a>. Similarly, we can prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286">View MathML</a> is a coupled coincidence point of the mappings F and g.

So, we suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M287">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M273">View MathML</a>. Using (3.1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M289">View MathML</a>

which further implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M290">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M263">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M264">View MathML</a>, then for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M79">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M278">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M295">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M296">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M297">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M280">View MathML</a>. Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M299">View MathML</a>

Now, according to (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77">View MathML</a>), it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M283">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269">View MathML</a>. Similarly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286">View MathML</a> is a coupled coincidence point of the mappings F and g. □

Remark 3.2 In Theorem 3.1, the condition (ii) is a substitution for the mixed g-monotone property that has been used in most of the coupled coincidence point theorems so far. Therefore, Theorem 3.1 improves the results of Nashine et al.[33]. Moreover, it is an ordered version extension of the results of Abbas et al.[36].

Corollary 3.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183">View MathML</a>be an ordered cone metric space over a solid conePand let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133">View MathML</a>. Suppose that the following hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187">View MathML</a>is a complete subspace ofX;

(ii) gandFsatisfy property (2.3);

(iii) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M189">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M190">View MathML</a>;

(iv) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M313">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M314">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M195">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M196">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M318">View MathML</a>

(3.9)

holds;

(v) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198">View MathML</a>when<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>inX, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200">View MathML</a>fornsufficiently large.

Then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M323">View MathML</a>

that is, Fandghave a coupled coincidence point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115">View MathML</a>.

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M150">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M326">View MathML</a> is the identity mapping from X into X in Theorem 3.1, we get the following corollary.

Corollary 3.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183">View MathML</a>be an ordered cone metric space over a solid conePand let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133">View MathML</a>. Suppose that the following hold:

(i) Xis complete;

(ii) gandFsatisfy property (2.4);

(iii) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M330">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M331">View MathML</a>;

(iv) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M191">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M192">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M193">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M336">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M337">View MathML</a>,

(3.10)

holds;

(v) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198">View MathML</a>when<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>inX, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200">View MathML</a>fornsufficiently large.

Then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M343">View MathML</a>

that is, Fhas a coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115">View MathML</a>.

Our second main result is the following.

Theorem 3.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183">View MathML</a>be an ordered cone metric space over a solid coneP. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M121">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M122">View MathML</a>be mappings. Suppose that the following hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187">View MathML</a>is a complete subspace ofX;

(ii) gandFsatisfy property (2.3);

(iii) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M189">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M190">View MathML</a>;

(iv) there is some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M353">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M195">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M196">View MathML</a>, there exists

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M357">View MathML</a>

such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M358">View MathML</a>

(v) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198">View MathML</a>when<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>inX, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200">View MathML</a>fornsufficiently large.

Then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M363">View MathML</a>

that is, Fandghave a coupled coincidence point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115">View MathML</a>.

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M186">View MathML</a> (condition (i)), we can start from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M204">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M205">View MathML</a> (condition (iii)) and construct sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208">View MathML</a> in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M370">View MathML</a>

(3.11)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210">View MathML</a>. From (iii), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M211">View MathML</a> and the condition (ii) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M373">View MathML</a>

By repeating this process, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213">View MathML</a>. Similarly, we can prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M213">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M214">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M379">View MathML</a>, from (iv), we have that there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M353">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M381">View MathML</a>

such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M382">View MathML</a>

Similarly, one can show that there exists

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M383">View MathML</a>

such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M384">View MathML</a>

Now, denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M385">View MathML</a>. Since the cases <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M386">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M387">View MathML</a> are trivial, we have to consider the following four possibilities.

Case 1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M388">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M389">View MathML</a>. Adding up, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M390">View MathML</a>

Case 2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M388">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M392">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M393">View MathML</a>

Case 3. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M394">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M392">View MathML</a>. This case is treated analogously to Case 1.

Case 4. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M394">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M389">View MathML</a>. This case is treated analogously to Case 2.

Thus, in all cases, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M398">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M210">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M400">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M401">View MathML</a>

and by the same argument as in Theorem 3.1, it is proved that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208">View MathML</a> are Cauchy sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187">View MathML</a>. By the completeness of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M187">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M262">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M263">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M264">View MathML</a>.

From (v), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M266">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M267">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M268">View MathML</a>. Now, we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M271">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M272">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M273">View MathML</a>, from (iv) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M417">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M418">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M419">View MathML</a> be fixed. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M420">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M421">View MathML</a>, then for n sufficiently large, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M422">View MathML</a>

By property (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77">View MathML</a>), it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M425">View MathML</a>, then we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M426">View MathML</a>

Now, it follows that for n sufficiently large,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M427">View MathML</a>

Therefore, again by property (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77">View MathML</a>), we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269">View MathML</a>. Similarly, we can prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286">View MathML</a> is a coupled point of coincidence of F and g.

Then, we suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M432">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M268">View MathML</a>. For this, consider

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M434">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M418">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M419">View MathML</a> be fixed. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M437">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M438">View MathML</a>, then for n sufficiently large, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M439">View MathML</a>

By property (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77">View MathML</a>), it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M425">View MathML</a>, then we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M443">View MathML</a>

Now, it follows that for n sufficiently large,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M444">View MathML</a>

Thus, again by property (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M77">View MathML</a>), we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M269">View MathML</a>.

Similarly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M270">View MathML</a> is obtained. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286">View MathML</a> is a coupled point of coincidence of the mappings F and g. □

Remark 3.6 It would be interesting to relate our Theorem 3.5 with Theorem 2.1 of Long et al.[39].

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M150">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M326">View MathML</a> is the identity mapping from X into X in Theorem 3.5, we get the following corollary.

Corollary 3.7Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M183">View MathML</a>be an ordered cone metric space over a solid coneP. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M121">View MathML</a>be mappings. Suppose that the following hold:

(i) Xis complete;

(ii) Fsatisfies property (2.4);

(iii) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M188">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M330">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M331">View MathML</a>;

(iv) there is some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M353">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M336">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M337">View MathML</a>, there exists

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M460">View MathML</a>

such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M461">View MathML</a>

(v) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M198">View MathML</a>when<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>inX, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M200">View MathML</a>fornsufficiently large.

Then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M44">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M466">View MathML</a>

that is, Fhas a coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M115">View MathML</a>.

4 Common coupled fixed point theorems lacking the mixed monotone property

Some questions arise naturally from Theorems 3.1 and 3.5. For example, one may ask if there are necessary conditions for the existence and uniqueness of a common coupled fixed point of F and g?

The next theorem provides a positive answer to this question with additional hypotheses to Theorems 3.1 and 3.5.

For the given partial order ⪯ on the set X, we will denote also by ⪯ the order on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M468">View MathML</a> given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M469">View MathML</a>

(4.1)

Theorem 4.1In addition to the hypotheses of Theorem 3.1, suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M470">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M471">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M472">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M473">View MathML</a>

IfFandgare<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible, thenFandghave a unique common coupled fixed point, that is, there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M475">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M476">View MathML</a>

Proof From Theorem 3.1, the set of coupled coincidence points of F and g is nonempty. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M286">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M478">View MathML</a> are coupled coincidence points of F, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M479">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M480">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M481">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M482">View MathML</a>. We will prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M483">View MathML</a>

(4.2)

By assumption, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M484">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M485">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M486">View MathML</a>

Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M487">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M488">View MathML</a> and choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M489">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M490">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M491">View MathML</a>. Then, similarly as in the proof of Theorem 3.1, we can inductively define sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M492">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M493">View MathML</a> with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M494">View MathML</a>

for all n. Further, set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M495">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M496">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M497">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M498">View MathML</a> and, in a similar way, define the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M207">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M208">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M501">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M502">View MathML</a>. Then it is easy to show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M503">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M504">View MathML</a>

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M505">View MathML</a>.

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M506">View MathML</a>

we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M507">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M508">View MathML</a>. It is easy to show that, similarly,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M509">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M510">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M511">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M512">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M510">View MathML</a>. Thus, from (3.1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M514">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M515">View MathML</a>

In the same way, starting from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M516">View MathML</a>, we can show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M517">View MathML</a>

Thus,

(4.3)

In a similar way, starting from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M519">View MathML</a>, resp. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M520">View MathML</a>, and adding up the obtained inequalities, one gets that

(4.4)

Finally, adding up (4.3) and (4.4), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M522">View MathML</a>

(4.5)

where λ is determined as in (3.8), and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M523">View MathML</a>.

By inequality (4.5) n time, we have

It follows from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M525">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M63">View MathML</a> that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M527">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M419">View MathML</a> and large n. Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M529">View MathML</a>

it follows by (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M81">View MathML</a>) that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M531">View MathML</a> for large n, and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M532">View MathML</a> when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>. Similarly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M534">View MathML</a> when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>. By the same procedure, one can show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M536">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M537">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M199">View MathML</a>. By the uniqueness of the limit, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M539">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M540">View MathML</a>, i.e., (4.2) is proved. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M119">View MathML</a> is the unique coupled point of coincidence of F and g.

Note that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M119">View MathML</a> is a coupled point of coincidence of F and g, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M543">View MathML</a> is also a coupled point of coincidence of F and g. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M544">View MathML</a> and therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M545">View MathML</a> is the unique coupled point of coincidence of F and g.

Next, we show that F and g have a common coupled fixed point. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M546">View MathML</a>. Then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M547">View MathML</a>. Since F and g are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M549">View MathML</a>

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M550">View MathML</a> is a coupled point of coincidence of F and g. By the uniqueness of a coupled point of coincidence of F and g, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M551">View MathML</a>. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M552">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M553">View MathML</a> is a common coupled fixed point of F and g.

Finally, we show the uniqueness of a common coupled fixed point of F and g. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M554">View MathML</a> be another common coupled fixed point of F and g. So,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M555">View MathML</a>

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M550">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M557">View MathML</a> are two common coupled points of coincidence of F and g and, as was previously proved, it must be <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M558">View MathML</a>, and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M559">View MathML</a>. This completes the proof. □

Next, we give some illustrative example which supports Theorem 4.1, while the results of Nashine et al.[33] do not.

Example 4.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M560">View MathML</a> be ordered by the following relation:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M561">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M562">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M563">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M564">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M565">View MathML</a>

It is well known (see, e.g., [40]) that the cone P is not normal. Let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M566">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M6">View MathML</a>, for a fixed <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M568">View MathML</a> (e.g., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M569">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M570">View MathML</a>). Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M3">View MathML</a> is a complete ordered cone metric space over a non-normal solid cone.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M134">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M133">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M574">View MathML</a>

Consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M575">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M576">View MathML</a>, we have for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M577">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M578">View MathML</a>, but

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M579">View MathML</a>

So, the mapping F does not satisfy the mixed g-monotone property. Therefore, Theorems 3.1 and 3.2 of Nashine et al.[33] cannot be used to reach this conclusion.

Now, we show that Theorem 4.1 can be used for this case.

Take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M580">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M581">View MathML</a>. We will check that the condition (3.1) in Theorem 3.1 holds.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M194">View MathML</a> satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M583">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M584">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M585">View MathML</a>

Next, we show that F and g are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible. We note that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M587">View MathML</a>, then we get only one case, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M588">View MathML</a>, and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M589">View MathML</a>

Therefore, F and g are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible.

Moreover, other conditions in Theorem 4.1 are also satisfied. Now, we can apply Theorem 4.1 to conclude the existence of a unique common coupled fixed point of F and g that is a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M591">View MathML</a>.

The following uniqueness result corresponding to Theorem 3.5 can be proved in the same way as Theorem 4.1.

Theorem 4.3In addition to the hypotheses of Theorem 3.5, suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M470">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M471">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M594">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M595">View MathML</a>

IfFandgare<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible, thenFandghave a unique coupled common fixed point, that is, there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M475">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M598">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank Professor Hichem Ben-El-Mechaiekh and the referee for valuable comments. The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the third author would like to thank the Commission on Higher Education, the Thailand Research Fund and KMUTT under Grant No. MRG5580213 for financial support during the preparation of this manuscript.

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