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Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered dislocated metric space

Muhammad Arshad1, Abdullah Shoaib1 and Ismat Beg2*

Author Affiliations

1 Department of Mathematics, International Islamic University, Islamabad, 44000, Pakistan

2 Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan

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Fixed Point Theory and Applications 2013, 2013:115  doi:10.1186/1687-1812-2013-115

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/115


Received:5 December 2012
Accepted:15 April 2013
Published:29 April 2013

© 2013 Arshad 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

Common fixed point results for mappings satisfying locally contractive conditions on a closed ball in an ordered complete dislocated metric space have been established. The notion of dominated mappings is applied to approximate the unique solution of nonlinear functional equations. Our results improve several well-known conventional results.

MSC: 46S40, 47H10, 54H25.

Keywords:
common fixed point; contractive mapping; closed ball; dominated mapping; dislocated metric space

1 Introduction and preliminaries

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M1">View MathML</a> be a mapping. A point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2">View MathML</a> is called a fixed point of T if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M3">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a> be an arbitrarily chosen point in X. Define a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a> in X by a simple iterative method given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M6">View MathML</a>

Such a sequence is called a Picard iterative sequence, and its convergence plays a very important role in proving the existence of a fixed point of a mapping T. A self-mapping T on a metric space X is said to be a Banach contraction mapping if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M7">View MathML</a>

holds for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M8">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M9">View MathML</a>.

Fixed points results of mappings satisfying a certain contractive condition on the entire domain have been at the center of rigorous research activity (for example, see [1-12]) and they have a wide range of applications in different areas such as nonlinear and adaptive control systems, parameter estimation problems, computing magnetostatic fields in a nonlinear medium and convergence of recurrent networks (see [13-15]).

From the application point of view, the situation is not yet completely satisfactory because it frequently happens that a mapping T is a contraction not on the entire space X but merely on a subset Y of X. However, if Y is closed and a Picard iterative sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a> in X converges to some x in X, then by imposing a subtle restriction on the choice of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>, one may force the Picard iterative sequence to stay eventually in Y. In this case, the closedness of Y coupled with some suitable contractive condition establishes the existence of a fixed point of T. Azam et al.[16] proved a significant result concerning the existence of fixed points of a mapping satisfying contractive conditions on a closed ball of a complete metric space. Recently, many results related to the fixed point theorem in complete metric spaces endowed with a partial ordering ⪯ appeared in literature. Ran and Reurings [17] proved an analogue of Banach’s fixed point theorem in a metric space endowed with a partial order and gave applications to matrix equations. In this way, they weakened the usual contractive condition. Subsequently, Nieto et al.[18] extended this result in [17] for non-decreasing mappings and applied it to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Thereafter, many works related to fixed point problems have also been considered in partially ordered metric spaces (see [17-23]). Indeed, they all deal with monotone mappings (either order-preserving or order-reversing) such that for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M12">View MathML</a>, either <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M13">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M14">View MathML</a>, where f is a self-map on a metric space. To obtain a unique solution, they used an additional restriction that each pair of elements has a lower bound and an upper bound. We have not used these conditions in our results. In this paper we introduce a new condition of partial order.

On the other hand, the notion of a partial metric space was introduced by Matthews in [24]. In partial metric spaces, the distance of a point from itself may not be zero. He also proved a partial metric version of the Banach fixed point theorem. Karapınar et al.[25] have proved a common fixed point in partial metric spaces. Partial metric spaces have applications in theoretical computer science (see [26]). Altun et al.[20], Aydi [27], Samet et al.[28] and Paesano et al.[29] used the idea of a partial metric space and partial order and gave some fixed point theorems for the contractive condition on ordered partial metric spaces. Further useful results can be seen in [28]. To generalize a partial metric, Hitzler and Seda [30] introduced the concept of dislocated topologies and its corresponding generalized metric, named a dislocated metric, and established a fixed point theorem in complete dislocated metric spaces to generalize the celebrated Banach contraction principle. The notion of dislocated topologies has useful applications in the context of logic programming semantics (see [31]). Further useful results can be seen in [32-35]. The dominated mapping, which satisfies the condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M15">View MathML</a>, occurs very naturally in several practical problems. For example, if x denotes the total quantity of food produced over a certain period of time and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M16">View MathML</a> gives the quantity of food consumed over the same period in a certain town, then we must have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M15">View MathML</a>. In this paper, we exploit this concept for contractive mappings [36] to generalize, extend and improve some classical fixed point results for two, three and four mappings in the framework of an ordered complete dislocated metric space X. Our results not only extend some primary theorems to ordered dislocated metric spaces, but also restrict the contractive conditions on a closed ball only. The concept of a dominated mapping has been applied to approximate the unique solution of nonlinear functional equations.

Consistent with [30,32,34] and [35], the following definitions and results will be needed in the sequel.

Definition 1.1 Let X be a nonempty set and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M18">View MathML</a> be a function, called a dislocated metric (or simply <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric), if the following conditions hold for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M20">View MathML</a>:

(i) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M21">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M22">View MathML</a>,

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M23">View MathML</a>,

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M24">View MathML</a>.

The pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M25">View MathML</a> is then called a dislocated metric space. It is clear that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M21">View MathML</a>, then from (i), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M22">View MathML</a>. But if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M22">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M29">View MathML</a> may not be 0.

Recently Sarma and Kumari [34] proved the results that establish the existence of a topology induced by a dislocated metric and the fact that this topology is metrizable. This topology has as a base the family of sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M30">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M31">View MathML</a> is an open ball and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M32">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M34">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M35">View MathML</a> is a closed ball.

Also, Harandi [37] defined the concept of a metric-like space which is similar to a dislocated metric space. Each metric-like σ on X generates a topology <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M36">View MathML</a> on X whose base is the family of open σ-balls

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M37">View MathML</a>

Definition 1.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M38">View MathML</a>, where X is a nonempty set. p is said to be a partial metric on X if for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M20">View MathML</a>:

(P1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M40">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M22">View MathML</a>,

(P2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M42">View MathML</a>,

(P3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M43">View MathML</a>,

(P4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M44">View MathML</a>.

The pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M45">View MathML</a> is then called a partial metric space.

Each partial metric p on X induces a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M46">View MathML</a> topology p on X which has as a base the family of open balls <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M47">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M48">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M34">View MathML</a>.

It is clear that any partial metric is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric. A basic example of a partial metric space is the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M52">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M53">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M54">View MathML</a>. It is also a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric. An example of a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric space which is not a partial metric is given below.

Example 1.3 If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M57">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M58">View MathML</a> defines a dislocated metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a> on X. Note that this metric is not a partial metric as (P2) is not satisfied.

From the examples and definitions, it is clear that any partial metric is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric, whereas a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric may not be a partial metric. We also remark that for those <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metrics which are also partial metrics, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M63">View MathML</a>. Also, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M65">View MathML</a>. Thus it is better to find a fixed point on a closed ball defined by Hitzler in a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric because we restrict ourselves to applying the contractive condition on the smallest closed ball. In this way, we also weaken the contractive condition.

Definition 1.4[30]

A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a> in a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M25">View MathML</a> is called a Cauchy sequence if given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M34">View MathML</a>, there corresponds <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M71">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M72">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M73">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M74">View MathML</a>.

Definition 1.5[30]

A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a> in a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric space converges with respect to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a> if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M78">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M79">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M80">View MathML</a>. In this case, x is called the limit of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a>, and we write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M82">View MathML</a>.

Definition 1.6[30]

A <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a>-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M25">View MathML</a> is called complete if every Cauchy sequence in X converges to a point in X.

In Harandi’s sense, a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a> in the metric-like space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M86">View MathML</a> converges to a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M88">View MathML</a>. The sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M89">View MathML</a> of elements of X is called σ-Cauchy if the limit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M90">View MathML</a> exists and is finite. The metric-like space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M86">View MathML</a> is called complete if for each σ-Cauchy sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M89">View MathML</a>, there is some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M94">View MathML</a>

Romaguera [38] has given the idea of a 0-Cauchy sequence and a 0-complete partial metric space. Using his idea, we can observe the following:

(a) Every Cauchy sequence with respect to Hitzler is a Cauchy sequence with respect to Harandi.

(b) Every complete metric space with respect to Harandi is complete with respect to Hitzler. The following example shows that the converse assertions of (a) and (b) do not hold.

Example 1.7 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M95">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M96">View MathML</a> be defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M58">View MathML</a>. Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M98">View MathML</a> is a Cauchy sequence with respect to Harandi, but it is not a Cauchy sequence with respect to Hitzler. Also, every Cauchy sequence (with respect to Hitzler) in X converges to a point ‘0’ in X. Hence X is complete with respect to Hitzler, but X is not complete with respect to Harandi as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M99">View MathML</a>.

Definition 1.8 Let X be a nonempty set. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100">View MathML</a> is called an ordered dislocated metric space if (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M19">View MathML</a> is a dislocated metric on X and (ii) ⪯ is a partial order on X.

Definition 1.9 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M102">View MathML</a> be a partial ordered set. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M103">View MathML</a> are called comparable if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M104">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M105">View MathML</a> holds.

Definition 1.10[39]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M106">View MathML</a> be a partially ordered set. A self-mapping f on X is called dominated if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M15">View MathML</a> for each x in X.

Example 1.11[39]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M108">View MathML</a> be endowed with the usual ordering and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M109">View MathML</a> be defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M110">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M111">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M112">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2">View MathML</a>, therefore f is a dominated map.

Definition 1.12 Let X be a nonempty set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M114">View MathML</a>. A point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M115">View MathML</a> is called a point of coincidence of T and f if there exists a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M117">View MathML</a>. The mappings T, f are said to be weakly compatible if they commute at their coincidence point (i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M118">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M119">View MathML</a>).

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M120">View MathML</a>, we denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M121">View MathML</a> the set of all limit points of A and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M122">View MathML</a> closure of A in X. We state without proof the following simple facts due to [34].

Lemma 1.13A subset ofAof a dislocated metric space is closed if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M123">View MathML</a>.

Lemma 1.14The topology induced by a dislocated metric is a Hausdorff topology.

Lemma 1.15Every closed ball in a complete dislocated metric space is complete.

We also need the following results for subsequent use.

Lemma 1.16[40]

LetXbe a nonempty set and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M109">View MathML</a>be a function. Then there exists a subset<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M125">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M126">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M127">View MathML</a>is one-to-one.

Lemma 1.17[1]

LetXbe a nonempty set and let the mappings<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M128">View MathML</a>have a unique point of coincidencevinX. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M130">View MathML</a>are weakly compatible, thenS, T, fhave a unique common fixed point.

Theorem 1.18 [[36], p.303]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M131">View MathML</a>be a complete metric space, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M132">View MathML</a>be a mapping, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M133">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>be an arbitrary point inX. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M135">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M136">View MathML</a>

and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M137">View MathML</a>. Then there exists a unique point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M140">View MathML</a>.

2 Fixed points of contractive mappings

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100">View MathML</a>be an ordered complete dislocated metric space, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142">View MathML</a>be dominated maps and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>be an arbitrary point inX. Suppose that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144">View MathML</a>and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M146">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M147">View MathML</a>

(2.2)

If for a non-increasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M149">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150">View MathML</a>implies that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151">View MathML</a>, then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M152">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M153">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M154">View MathML</a>. Also if, for any two pointsx, yin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>, there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M156">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M157">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M158">View MathML</a>, that is, every pair of elements has a lower bound, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a>is a unique common fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>.

Proof Choose a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M161">View MathML</a> in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M162">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M163">View MathML</a>, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M164">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M165">View MathML</a>. Now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M166">View MathML</a> gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M167">View MathML</a>. Continuing this process, we construct a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M168">View MathML</a> of points in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M169">View MathML</a>

First we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M170">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171">View MathML</a>. Using inequality (2.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M172">View MathML</a>

It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M173">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M174">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M175">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M176">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M177">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M178">View MathML</a>. So, using inequality (2.1), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M179">View MathML</a>

(2.3)

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M180">View MathML</a>, then as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M181">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M182">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M183">View MathML</a>). We obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M184">View MathML</a>

(2.4)

Thus from inequalities (2.3) and (2.4), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M185">View MathML</a>

(2.5)

Now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M186">View MathML</a>

Thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M187">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M188">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171">View MathML</a>. It implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M190">View MathML</a>

(2.6)

It implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M191">View MathML</a>

Notice that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M193">View MathML</a>. Therefore there exists a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M194">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M195">View MathML</a>. Also,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M196">View MathML</a>

(2.7)

Now,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M197">View MathML</a>

On taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M80">View MathML</a> and using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M199">View MathML</a> when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M200">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M201">View MathML</a>

By equation (2.7), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M202">View MathML</a>

and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M140">View MathML</a>. Similarly, by using

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M204">View MathML</a>

we can show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M205">View MathML</a>. Hence S and T have a common fixed point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>. Now,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M207">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M208">View MathML</a>

For uniqueness, assume that y is another fixed point of T and S in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a> and y are comparable, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M211">View MathML</a>

This shows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M212">View MathML</a>. Now if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a> and y are not comparable, then there exists a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M214">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M215">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M216">View MathML</a>. Choose a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M217">View MathML</a> in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M218">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M219">View MathML</a>, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M220">View MathML</a>and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M221">View MathML</a>. Now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M222">View MathML</a> gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M223">View MathML</a>. Continuing this process and having chosen <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M224">View MathML</a> in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M225">View MathML</a>

we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M226">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M215">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M216">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M229">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M230">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171">View MathML</a>. We will prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M232">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171">View MathML</a> by using mathematical induction. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M234">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M235">View MathML</a>

It follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M236">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M237">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M175">View MathML</a>. Note that if j is odd, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M239">View MathML</a>

and if j is even, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M240">View MathML</a>

Now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M241">View MathML</a>

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M242">View MathML</a>

Thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M243">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M244">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M215">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M216">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M248">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M249">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M250">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M251">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M253">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M254">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171">View MathML</a>. If n is odd, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M256">View MathML</a>

So, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M212">View MathML</a>. Similarly, we can show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M212">View MathML</a> if n is even. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a> is a unique common fixed point of T and S in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>. □

Theorem 2.1 extends Theorem 1.18 to ordered complete dislocated metric spaces.

Example 2.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M95">View MathML</a> be endowed with the order <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M262">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M263">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M264">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M265">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M266">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M267">View MathML</a>

Clearly, S and T are dominated mappings. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M268">View MathML</a> be defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M269">View MathML</a>. Then it is easy to prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M270">View MathML</a> is a complete dislocated metric space. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M271">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M272">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M273">View MathML</a>

with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M274">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M275">View MathML</a>

Also, for all comparable elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M276">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M277">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M278">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M279">View MathML</a>

So, the contractive condition does not hold on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M280">View MathML</a>. Now if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M281">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M282">View MathML</a>

Therefore, all the conditions of Theorem 2.1 are satisfied. Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M283">View MathML</a> is the common fixed point of S and T. Also, note that for any metric d on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M280">View MathML</a>, the respective condition does not hold on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M285">View MathML</a> since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M286">View MathML</a>

Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M280">View MathML</a> is not complete for any metric d on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M280">View MathML</a>.

Remark 2.3 If we impose a Banach-type contractive condition for a pair of mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142">View MathML</a> on a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M290">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M291">View MathML</a>

then it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M292">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M2">View MathML</a> (that is, S and T are equal). Therefore the above condition fails to find common fixed points of S and T. However, the same condition in a dislocated metric space does not assert that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M294">View MathML</a>, which is seen in Example 2.2. Hence Theorem 2.1 cannot be obtained from a metric fixed point theorem.

Theorem 2.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100">View MathML</a>be an ordered complete dislocated metric space, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M132">View MathML</a>be a dominated map and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>be an arbitrary point inX. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M298">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M299">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M300">View MathML</a>

If, for a non-increasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M149">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150">View MathML</a>implies that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151">View MathML</a>, and also, for any two pointsx, yin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>, there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M306">View MathML</a>such that every pair of elements has a lower bound, then there exists a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a>ofSin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>. Further, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M153">View MathML</a>.

Proof By following similar arguments to those we have used to prove Theorem 2.1, one can easily prove the existence of a unique fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a> of S in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>. □

In Theorem 2.1, condition (2.2) is imposed to restrict condition (2.1) only for x, y in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a> and Example 2.2 explains the utility of this restriction. However, the following result relaxes condition (2.2) but imposes condition (2.1) for all comparable elements in the whole space X.

Theorem 2.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M313">View MathML</a>be an ordered complete dislocated metric space, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142">View MathML</a>be the dominated map and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>be an arbitrary point inX. Suppose that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144">View MathML</a>and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M318">View MathML</a>

Also, if for a non-increasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a>inX, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M320">View MathML</a>implies that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151">View MathML</a>, and for any two pointsx, yinX, there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M322">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M157">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M158">View MathML</a>, then there exists a unique point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a>inXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M326">View MathML</a>. Further, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M153">View MathML</a>.

In Theorem 2.1, the condition ‘for a non-increasing sequence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151">View MathML</a>’ and the existence of z or a lower bound is imposed to restrict condition (2.1) only for comparable elements. However, the following result relaxes these restrictions but imposes condition (2.1) for all elements in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>. In Theorem 2.1, it may happen that S has more fixed points, but these fixed points of S are not the fixed points of T, because a common fixed point of S and T is unique, whereas without order we can obtain a unique fixed point of S and T separately, which is proved in the following theorem.

Theorem 2.6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M331">View MathML</a>be a complete dislocated metric space, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142">View MathML</a>be self-maps and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>be an arbitrary point inX. Suppose that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M334">View MathML</a>and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M336">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M337">View MathML</a>

Then there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M152">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M153">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M154">View MathML</a>. Further, SandThave no fixed point other than<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a>.

Proof By Theorem 2.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M154">View MathML</a>. Let y be another point such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M343">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M344">View MathML</a>

This shows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M212">View MathML</a>. Thus T has no fixed point other than <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a>. Similarly, S has no fixed point other than <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a>. □

Now we apply our Theorem 2.1 to obtain a unique common fixed point of three mappings on a closed ball in an ordered complete dislocated metric space.

Theorem 2.7Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100">View MathML</a>be an ordered dislocated metric space, letS, Tbe self-mappings and letfbe a dominated mapping onXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M349">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M350">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M351">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>be an arbitrary point inX. Suppose that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144">View MathML</a>and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M355">View MathML</a>

(2.8)

for all comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M356">View MathML</a>; and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M357">View MathML</a>

(2.9)

If for a non-increasing sequence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150">View MathML</a>implies that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151">View MathML</a>, and for any two pointszandxin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360">View MathML</a>, there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M361">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M362">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M105">View MathML</a>, that is, every pair of elements in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360">View MathML</a>has a lower bound in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360">View MathML</a>; iffXis a complete subspace ofXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M130">View MathML</a>are weakly compatible, thenS, Tandfhave a unique common fixed pointfzin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369">View MathML</a>.

Proof By Lemma 1.16, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M125">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M126">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M127">View MathML</a> is one-to-one. Now, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M349">View MathML</a>, we define two mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M374">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M375">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M376">View MathML</a>, respectively. Since f is one-to-one on E, then g, h are well defined. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M377">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M378">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M350">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M380">View MathML</a>, therefore g and h are dominated maps. Now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M381">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M382">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M383">View MathML</a>, choose a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M384">View MathML</a> in fX such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M385">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M386">View MathML</a>, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M387">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M388">View MathML</a>. Now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M389">View MathML</a> gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M390">View MathML</a>. Continuing this process and having chosen <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M391">View MathML</a> in fX such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M392">View MathML</a>

then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M393">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M171">View MathML</a>. Following similar arguments of Theorem 2.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M395">View MathML</a>. Also, by inequality (2.9),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M396">View MathML</a>

Note that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M397">View MathML</a>, where fx, fy are comparable. Then by using inequality (2.8), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M398">View MathML</a>

As fX is a complete space, all the conditions of Theorem 2.1 are satisfied, we deduce that there exists a unique common fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M399">View MathML</a> of g and h. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369">View MathML</a>. Now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M401">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M402">View MathML</a>. Thus fz is the point of coincidence of S, T and f. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M403">View MathML</a> be another point of coincidence of f, S and T, then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M404">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M405">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M406">View MathML</a>, a contradiction as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M399">View MathML</a> is a unique common fixed point of g and h. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M408">View MathML</a>. Thus S, T and f have a unique point of coincidence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M409">View MathML</a>. Now, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M130">View MathML</a> are weakly compatible, by Lemma 1.17 fz is a unique common fixed point of S, T and f. □

In a similar way, we can apply our Theorems 2.5 and 2.6 to obtain a unique common fixed point of three mappings in an ordered complete dislocated metric space and a unique common fixed point of three mappings on a closed ball in a complete dislocated metric space, respectively.

In the following theorem, we use Theorem 2.6 to establish the existence of a unique common fixed point of four mappings on a closed ball in a complete dislocated metric space. One cannot prove the following theorem for an ordered dislocated metric space in a way similar to that of Theorem 2.7. In order to prove the unique common fixed point of four mappings on a closed ball in an ordered dislocated metric space, we should prove that S and T have no fixed point other than <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a> in Theorem 2.1.

Theorem 2.8Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M413">View MathML</a>be a dislocated metric space and letS, T, gandfbe self-mappings onXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M414">View MathML</a>. Assume that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>, an arbitrary point inX, and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M334">View MathML</a>and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145">View MathML</a>, the following conditions hold:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M418">View MathML</a>

(2.10)

for all elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M419">View MathML</a>; and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M420">View MathML</a>

(2.11)

IffXis a complete subspace ofX, then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M421">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369">View MathML</a>. Also, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M424">View MathML</a>are weakly compatible, thenS, T, fandghave a unique common fixed pointfzin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360">View MathML</a>.

Proof By Lemma 1.16, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M426">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M427">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M428">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M429">View MathML</a> are one-to-one. Now define the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M430">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M431">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M432">View MathML</a>, respectively. Since f, g are one-to-one on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M433">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M434">View MathML</a>, respectively, then the mappings A, B are well defined. As fX is a complete space, all the conditions of Theorem 2.6 are satisfied, we deduce that there exists a unique common fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M435">View MathML</a> of A and B. Further, A and B have no fixed point other than fz. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369">View MathML</a>. Now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M437">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M438">View MathML</a>. Thus fz is a point of coincidence of f and S. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M439">View MathML</a> be another point of coincidence of S and f, then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M404">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M441">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M442">View MathML</a>, a contradiction as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M409">View MathML</a> is a unique fixed point of A. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M444">View MathML</a>. Thus S and f have a unique point of coincidence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M409">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M129">View MathML</a> are weakly compatible, by Lemma 1.17 fz is a unique common fixed point of S and f. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M447">View MathML</a>, then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M448">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M449">View MathML</a>. Now, as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M450">View MathML</a>, thus gv is the point of coincidence of T and g. Now, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M451">View MathML</a>, a contradiction. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M452">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M424">View MathML</a> are weakly compatible, we obtain gv, a unique common fixed point for T and g. But <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M454">View MathML</a>. Thus S, T, g and f have a unique common fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M435">View MathML</a>. □

Corollary 2.9Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M100">View MathML</a>be an ordered dislocated metric space, letS, Tbe self-mappings and letfbe a dominated mapping onXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M349">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M350">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M351">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>be an arbitrary point inX. Suppose that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144">View MathML</a>and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M463">View MathML</a>

for all comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M356">View MathML</a>; and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M465">View MathML</a>

If for a non-increasing sequence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150">View MathML</a>implies that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151">View MathML</a>, and for any two pointszandxin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360">View MathML</a>, there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M361">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M362">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M105">View MathML</a>; iffXis a complete subspace ofX, thenS, Tandfhave a unique point of coincidence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M409">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M369">View MathML</a>.

In a similar way, we can obtain a coincidence point result of four mappings as a corollary of Theorem 2.8.

A partial metric version of Theorem 2.1 is given below.

Theorem 2.10Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M474">View MathML</a>be an ordered complete partial metric space, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M142">View MathML</a>be dominated maps and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>be an arbitrary point inX. Suppose that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M334">View MathML</a>and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M479">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M480">View MathML</a>

Then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M152">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M482">View MathML</a>. Also, if for a non-increasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M5">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150">View MathML</a>implies that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M486">View MathML</a>, and for any two pointsx, yin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>, there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M156">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M157">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M158">View MathML</a>, then there exists a unique point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M138">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M139">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M154">View MathML</a>.

A partial metric version of Theorem 2.7 is given below.

Theorem 2.11Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M494">View MathML</a>be an ordered partial metric space, letS, Tbe self-mappings and letfbe a dominated mapping onXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M349">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M496">View MathML</a>. Assume that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M4">View MathML</a>, an arbitrary point inX, and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M144">View MathML</a>and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M145">View MathML</a>, the following conditions hold:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M500">View MathML</a>

for all comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M356">View MathML</a>; and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M502">View MathML</a>

If for a non-increasing sequence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M150">View MathML</a>implies that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M151">View MathML</a>, also for any two pointszandxin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360">View MathML</a>, there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M361">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M362">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M105">View MathML</a>; iffXis complete subspace ofXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M509">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M130">View MathML</a>are weakly compatible, thenS, Tandfhave a unique common fixed pointfzin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M360">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/115/mathml/M512">View MathML</a>.

Remark 2.12 We can obtain a partial metric version as well as a metric version of other theorems in a similar way.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

IB gave the idea. MA and AS wrote the initial draft. IB and MA finalized the manuscript. Correspondence was mainly done by IB. All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thank the learned referee for a careful reading and thoughtful comments. The present version of the paper owes much to the precise and kind remarks of three anonymous referees.

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