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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Fixed point results for ( α ψ , β φ ) -contractive mappings for a generalized altering distance

Maher Berzig1 and Erdal Karapınar2*

Author Affiliations

1 Tunis College of Sciences and Techniques, Tunis University, 5 Avenue Taha Hussein, Tunis, Tunisia

2 Department of Mathematics, Atilim University, Incek, Ankara, 06836, Turkey

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Fixed Point Theory and Applications 2013, 2013:205  doi:10.1186/1687-1812-2013-205

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/205


Received:24 April 2013
Accepted:23 July 2013
Published:29 July 2013

© 2013 Berzig and Karapınar; licensee Springer

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

Abstract

In this manuscript, we extend the concept of altering distance, and we introduce a new notion of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1">View MathML</a>-contractive mappings. We prove the existence and uniqueness of a fixed point for such mapping in the context of complete metric space. The presented theorems of this paper generalize, extend and improve some remarkable existing results in the literature. We also present several applications and consequences of our results.

1 Introduction and preliminaries

Fixed point theory is one of the core research areas in nonlinear functional analysis since it has a broad range of application potential in various fields such as engineering, economics, computer science, and many others. Banach contraction mapping principle [1] is considered to be the initial and fundamental result in this direction. Fixed point theory and hence the Banach contraction mapping principle have evidently attracted many prominent mathematicians due to their wide application potential. Consequently, the number of publications in this theory increases rapidly; we refer the reader to [2-19].

In this paper, by introducing a new notion of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M3">View MathML</a>-contractive mappings, we aim to establish a more general result to collect/combine a number of existing results in the literature.

We start by recalling the notion of altering distance function introduced by Khan et al.[12] as follows.

Definition 1.1 A function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M4">View MathML</a> is called an altering distance function if the following properties are satisfied:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M5">View MathML</a> is continuous and nondecreasing.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M6">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M7">View MathML</a>.

Now, we present a definition, which will be useful later.

Definition 1.2 Let X be a set, and let ℛ be a binary relation on X. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8">View MathML</a> is an ℛ-preserving mapping if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M9">View MathML</a>

In the sequel, let ℕ denote the set of all non-negative integers, let ℝ denote the set of all real numbers.

Example 1.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M10">View MathML</a> and a function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M11">View MathML</a> defined as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M12">View MathML</a>.

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M13">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M14">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M15">View MathML</a>

Define the first binary relation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M17">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M18">View MathML</a>, and define the second binary relation by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M19">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M20">View MathML</a>. Then, we easily obtain that T is simultaneously <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a>-preserving and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a>-preserving.

Definition 1.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M23">View MathML</a>. We say that ℛ is N-transitive on X if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M24">View MathML</a>

The following remark is a consequence of the previous definition.

Remark 1.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M23">View MathML</a>. We have:

(i) If ℛ is transitive, then it is N-transitive for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M23">View MathML</a>.

(ii) If ℛ is N-transitive, then it is kN-transitive for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M27">View MathML</a>.

Definition 1.4 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a> be a metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a> two binary relations on X. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M32">View MathML</a>-regular if for every sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M34">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M35">View MathML</a>, and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M36">View MathML</a>

there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M38">View MathML</a>

Definition 1.5 We say that a subset D of X is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M39">View MathML</a>-directed if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M40">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M41">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M42">View MathML</a>

2 Main results

Before we start the introduction of the concept <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M43">View MathML</a>-contractive mappings, we introduce the notion of a pair of generalized altering distance as follows:

Definition 2.1 We say that the pair of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M44">View MathML</a> is a pair of generalized altering distance where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M45">View MathML</a> if the following hypotheses hold:

(a1) ψ is continuous;

(a2) ψ is nondecreasing;

(a3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M46">View MathML</a>.

The condition (a3) was introduced by Popescu in [15] and Moradi and Farajzadeh in [14].

Definition 2.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a> be a metric space, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8">View MathML</a> be a given mapping. We say that T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1">View MathML</a>-contractive mappings if there exists a pair of generalized distance <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M50">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M51">View MathML</a>

(1)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M52">View MathML</a>.

In the sequel, the binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a> are defined as following.

Definition 2.3 Let X be a set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M55">View MathML</a> are two mappings. We define two binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a> on X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M58">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M59">View MathML</a>

Now we are ready to state our first main result.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M61">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8">View MathML</a>be an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M3">View MathML</a>-contractive mapping satisfying the following conditions:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64">View MathML</a>isN-transitive for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a>;

(ii) Tis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64">View MathML</a>-preserving for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a>;

(iii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M68">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M69">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a>;

(iv) Tis continuous.

Then, Thas a fixed point, that is, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M71">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M72">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M73">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M69">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a>. Define the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> in X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M77">View MathML</a>

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M78">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M79">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M80">View MathML</a> is a fixed point T. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M81">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M79">View MathML</a>. From (ii) and (iii), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M83">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M84">View MathML</a>

By induction, from (ii) it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M85">View MathML</a>

(2)

and, similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M86">View MathML</a>

(3)

Substituting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M87">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M88">View MathML</a> in (1), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M89">View MathML</a>

So, by (2) and (3) it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M90">View MathML</a>

(4)

Using the monotone property of the ψ-function, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M91">View MathML</a>

It follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M92">View MathML</a> is monotone decreasing, and, consequently, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M93">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M94">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M35">View MathML</a> in (4), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M96">View MathML</a>

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M97">View MathML</a>, then by (a3) we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M98">View MathML</a>

(5)

On the other hand, by (2) and (i), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M99">View MathML</a>

(6)

Similarly, by (3) and (i), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M100">View MathML</a>

(7)

Now, for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M101">View MathML</a>, substituting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M102">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M103">View MathML</a> in (1), where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M104">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M105">View MathML</a>

So, by (6) and (7), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M106">View MathML</a>

(8)

Using the monotone property of the ψ-function, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M107">View MathML</a>

It follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M108">View MathML</a> is monotone decreasing and consequently, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M109">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M110">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M111">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M112">View MathML</a>

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M113">View MathML</a>, then by (a3) we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M114">View MathML</a>

(9)

Next, we claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> is a Cauchy sequence. Suppose if we obtain a contradiction, that T is not a Cauchy sequence. Then, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M116">View MathML</a>, for which we can find subsequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M117">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M120">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M121">View MathML</a>

(10)

Further, corresponding to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M122">View MathML</a>, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M123">View MathML</a> in such a way that it is the smallest integer with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M124">View MathML</a> and satisfying (10). Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M125">View MathML</a>

(11)

Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M126">View MathML</a>

(12)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127">View MathML</a> and using (5),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M128">View MathML</a>

(13)

Furthermore, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M129">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M130">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M131">View MathML</a>. Hence, by (11) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M132">View MathML</a>

(14)

Again, letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127">View MathML</a> and using (5),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M134">View MathML</a>

(15)

On the other hand, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M135">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127">View MathML</a> in the above inequalities, using (5), (9) and (15), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M137">View MathML</a>

(16)

By setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M138">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M139">View MathML</a>, in (1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M140">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M141">View MathML</a>

Now, using (6) and (7), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M142">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127">View MathML</a>, using (15), (16) and the continuity of ψ and φ, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M144">View MathML</a>

(17)

which implies by (a3) that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M145">View MathML</a>, a contradiction with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M146">View MathML</a>. Hence, our claim holds, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> is a Cauchy sequence. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a> is a complete metric space, then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M71">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M150">View MathML</a>

From the continuity of T, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M151">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M152">View MathML</a>. Due to the uniqueness of the limit, we derive that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M153">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M154">View MathML</a> is a fixed point of T. □

Theorem 2.2In Theorem 2.1, if we replace the continuity ofTby the<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M155">View MathML</a>-regularity of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156">View MathML</a>, then the conclusion of Theorem 2.1 holds.

Proof Following the lines of the proof of Theorem 2.1, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> is a Cauchy sequence. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a> is a complete metric space, then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M71">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M160">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M161">View MathML</a>. Furthermore, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> satisfies (2) and (3), that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M163">View MathML</a>

Now, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M155">View MathML</a>-regular, then there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M168">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M170">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M171">View MathML</a> for all k. By setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M172">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M173">View MathML</a>, in (1), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M174">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M175">View MathML</a>

Using the monotone property of the ψ-function, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M176">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M127">View MathML</a> in the above inequality, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M178">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M179">View MathML</a>. □

Theorem 2.3Adding to the hypotheses of Theorem 2.1 (respectively, Theorem 2.2) thatXis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M155">View MathML</a>-directed, we obtain uniqueness of the fixed point ofT.

Proof Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M154">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M182">View MathML</a> are two fixed points of T. Since X is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M155">View MathML</a>-directed, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M41">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M185">View MathML</a>

(18)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M186">View MathML</a>

(19)

Since T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64">View MathML</a>-preserving for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a>, from (18) and (19), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M189">View MathML</a>

(20)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M190">View MathML</a>

(21)

Using (20), (21) and (1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M191">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M192">View MathML</a>

(22)

Using the monotone property of the ψ-function, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M193">View MathML</a>

It follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M194">View MathML</a> is monotone decreasing and consequently, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M93">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M196">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M35">View MathML</a> in (22), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M198">View MathML</a>

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M199">View MathML</a>, then by (a3) we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M200">View MathML</a>

(23)

Similarly, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M201">View MathML</a>

(24)

Using (23) and (24), the uniqueness of the limit gives us <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M202">View MathML</a>. □

3 Some corollaries

In this section, we derive new results from the previous theorems.

3.1 Coupled fixed point results in complete metric spaces

Let us recall the definition of a coupled fixed point introduced by Guo and Lakshmikantham in [5].

Definition 3.1 (Guo and Lakshmikantham [5])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M203">View MathML</a> be a given mapping. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M204">View MathML</a> is a coupled fixed point of F if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M205">View MathML</a>

Lemma 3.1A pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M206">View MathML</a>is a coupled fixed point ofFif and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M206">View MathML</a>is a fixed point ofTwhere<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M208">View MathML</a>is given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M209">View MathML</a>

(25)

Definition 3.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a> be a metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M211">View MathML</a> be a given mapping. We say that F is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1">View MathML</a>-contractive mappings if there exists a pair of generalized distance <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M50">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M214">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M215">View MathML</a>.

In this section, we define two binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M216">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M217">View MathML</a> as follows.

Definition 3.3 Let X be a set, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M216">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M217">View MathML</a> be two binary relations on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M221">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M222">View MathML</a>

Definition 3.4 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a> be a metric space. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M224">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225">View MathML</a>-biregular if for all sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M226">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M229">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M35">View MathML</a>, and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M231">View MathML</a>

there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M232">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M233">View MathML</a>

Definition 3.5 We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225">View MathML</a>-bidirected if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M236">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M237">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M238">View MathML</a>

We have the following result.

Corollary 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a>be a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M240">View MathML</a>be an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1">View MathML</a>-contractive mapping satisfying the following conditions:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M242">View MathML</a>isN-transitive for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M244">View MathML</a>);

(ii) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M245">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M246">View MathML</a>

(iii) There exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M247">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M248">View MathML</a>

(iv) Fis continuous, or<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M224">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225">View MathML</a>-biregular.

Then, Fhas a coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M251">View MathML</a>. Moreover, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225">View MathML</a>-bidirected, then we have the uniqueness of the coupled fixed point.

Proof By Lemma 3.1, a pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M206">View MathML</a> is a coupled fixed point of F if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M206">View MathML</a> is a fixed point of T. Now, consider the complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M256">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M257">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M258">View MathML</a>

From (iv), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M259">View MathML</a>

(26)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M260">View MathML</a>

(27)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M261">View MathML</a> is nondecreasing, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M262">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M263">View MathML</a>. Hence, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M264">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M265">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M266">View MathML</a> are the functions defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M267">View MathML</a>

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M268">View MathML</a> is given by (25). We shall prove that T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M269">View MathML</a>-contractive mapping.

Define two binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M272">View MathML</a>

First, we claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M273">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M274">View MathML</a> are N-transitive. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M275">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M276">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M277">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M278">View MathML</a>

By definitions of a and b, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M279">View MathML</a>

or

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M280">View MathML</a>

Hence, by (i), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M281">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M282">View MathML</a>

or

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M283">View MathML</a>

Then our claim holds.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M284">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M285">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M286">View MathML</a>. Using condition (ii), we obtain immediately that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M287">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M288">View MathML</a>. Then T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M273">View MathML</a>-preserving for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M274">View MathML</a>. Moreover, from condition (iii), we know that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M291">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M292">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M274">View MathML</a>. If F is continuous, then T also is continuous. Then all the hypotheses of Theorem 2.1 are satisfied. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M294">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225">View MathML</a>-biregular, then we easily have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M294">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M39">View MathML</a>-regular. Hence, Theorem 2.2 yields the result. We deduce the existence of a fixed point of T that gives us from (25) the existence of a coupled fixed point of F. Now, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225">View MathML</a>-bidirected, one can easily derive that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M39">View MathML</a>-directed by regarding Lemma 3.1 and Definition 3.5. Finally, by using Theorem 2.3, we obtain the uniqueness of the fixed point of T, that is, the uniqueness of the coupled fixed point of F. □

3.2 Fixed point results on metric spaces endowed with N-transitive binary relation

In [18], Samet and Turinci established fixed point results for contractive mappings on metric spaces, endowed with an amorphous arbitrary binary relation. Very recently, this work has been extended by Berzig in [2] to study the coincidence and common fixed points.

In this section, we establish a fixed point theorem on metric space endowed with N-transitive binary relation .

Corollary 3.2LetXbe a non-empty set endowed with a binary relation. Suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156">View MathML</a>is complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8">View MathML</a>satisfy the-weakly<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M44">View MathML</a>-contractive conditions, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M308">View MathML</a>

whereψandφare altering distance functions. Suppose also that the following conditions hold:

(i) isN-transitive (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M244">View MathML</a>);

(ii) Tis a-preserving mapping;

(iii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M68">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M313">View MathML</a>;

(iv) Tis continuous or<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156">View MathML</a>is-regular.

ThenThas a fixed point. Moreover, ifXis-directed, we have the uniqueness of the fixed point.

Proof In order to link this theorem to the main result, we define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M317">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M318">View MathML</a>

(28)

and we define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M319">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M320">View MathML</a>

(29)

Next, by using (28), (29) and Definition 2.3, the conclusion follows directly from Theorems 2.1, 2.2 and 2.3. □

3.3 Fixed point results for cyclic contractive mappings

In [13], Kirk et al. have generalized the Banach contraction principle. They obtained a new fixed point results for cyclic contractive mappings.

Theorem 3.1 (Kirk et al.[13])

For<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M322">View MathML</a>be a nonempty closed subsets of a complete metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M324">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M325">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M327">View MathML</a>;

(ii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M328">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M329">View MathML</a>

ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M330">View MathML</a>.

Let us define the binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a>.

Definition 3.6 Let X be a nonempty set and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M322">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a> be nonempty closed subsets of X. We define two binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M335">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M336">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M337">View MathML</a>

Now, based on Theorem 2.2, we will derive a more general result for cyclic mappings.

Corollary 3.3For<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M322">View MathML</a>be nonempty closed subsets of a complete metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M324">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M325">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M327">View MathML</a>;

(ii) there exist two altering distance functionsψandφsuch that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M345">View MathML</a>

ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M330">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M347">View MathML</a>. For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a>, we have by assumption that each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M322">View MathML</a> is nonempty closed subset of the complete metric space X, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M350">View MathML</a> is complete.

Define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M351">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M352">View MathML</a>

and define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M353">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M354">View MathML</a>

Hence, Definition 2.3 is equivalent to Definition 3.6.

We start by checking that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M355">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M356">View MathML</a> are N-transitive. Indeed, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M357">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M358">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M359">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M360">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M361">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M362">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M363">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M364">View MathML</a>

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M365">View MathML</a>. Hence, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M366">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M367">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M368">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M369">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M355">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M356">View MathML</a> are N-transitive.

Next, from (ii) and the definition of α and β, we can write

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M372">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M373">View MathML</a>. Thus, T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M3">View MathML</a>-contractive mapping.

We claim next that T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M355">View MathML</a>-preserving and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M356">View MathML</a>-preserving. Indeed, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M377">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M378">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M379">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M380">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M381">View MathML</a>; hence, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M383">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M384">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M385">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M386">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M387">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M388">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M389">View MathML</a>. Hence, our claim holds.

Also, from (i), for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M390">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M392">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M393">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M394">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M395">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M396">View MathML</a>.

Now, we claim that Y is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M397">View MathML</a>-regular. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> be a sequence in Y such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M399">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M152">View MathML</a>, and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M401">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M402">View MathML</a>

It follows that there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M403">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M404">View MathML</a>

so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M405">View MathML</a>

By letting

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M406">View MathML</a>

we conclude that the subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37">View MathML</a> satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M408">View MathML</a>

hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M409">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M410">View MathML</a> for all k, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M411">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M412">View MathML</a>, which proves our claim.

Hence, all the hypotheses of Theorem 2.2 are satisfied on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M413">View MathML</a>, and we deduce that T has a fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M154">View MathML</a> in Y. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M415">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M417">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M321">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M419">View MathML</a>.

Moreover, it is easy to check that X is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M397">View MathML</a>-directed. Indeed, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M377">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M383">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M423">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M424">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M425">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M426">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M427">View MathML</a>. Thus, X is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M428">View MathML</a>-directed.

Finally, the uniqueness follows by Theorem 2.3. □

4 Related fixed point theorems

In this section, we show that many existing results in the literature can be deduced from our results.

4.1 Classical fixed point results

Corollary 4.1 (Dutta and Choudhury [4])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a>be a complete metric space, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8">View MathML</a>be a self-mapping satisfying the inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M431">View MathML</a>

whereψandφare altering distance functions. ThenThas a unique fixed point.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M432">View MathML</a> be the mapping defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M433">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M434">View MathML</a>. Then T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1">View MathML</a>-contractive mappings. It is easy to show that all the hypotheses of Theorems 2.1 and 2.2 are satisfied. Consequently, T has a unique fixed point. □

Corollary 4.2 (Rhoades [17])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a>be a complete metric space, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8">View MathML</a>be a self-mapping satisfying the inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M438">View MathML</a>

whereφis an altering distance functions. ThenThas a unique fixed point.

Proof Following the lines of the proof of Corollary 4.1, by taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M439">View MathML</a>, we get the desired result. □

4.2 Fixed point results in partially ordered metric spaces

We start by defining the binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a> and the concept of ≤-directed.

Definition 4.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M442">View MathML</a> be a partially ordered set.

1. We define two binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a> on X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M445">View MathML</a>

2. We say that X is ≤-directed if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M446">View MathML</a> there exists a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M447">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M448">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M449">View MathML</a>.

Corollary 4.3 (Harjani and Sadarangani [8])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M442">View MathML</a>be a partially ordered set anddbe a metric onXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a>is complete. Suppose that the mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M8">View MathML</a>is weakly contractive, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M453">View MathML</a>

whereψandφare altering distance functions. Suppose also that the following conditions hold:

(i) Tis a nondecreasing mapping;

(ii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M73">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M455">View MathML</a>;

If either:

(iii) Tis continuous or,

(iii′) for every sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a>inXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M457">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a>is a nondecreasing sequence, there exists a subsequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M460">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M129">View MathML</a>.

ThenThas a fixed point. Moreover, ifXis ≤-directed, we have the uniqueness of the fixed point.

Proof Using Definition 2.3, we can define the binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a> by the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M464">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M465">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M466">View MathML</a>

In case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M467">View MathML</a>, the functions α and β are well defined, because the altering functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M468">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M469">View MathML</a> are null only, and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M470">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M471">View MathML</a> which is not the case.

We can verify easily that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M22">View MathML</a> are 1-transitive.

Next, we claim that T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M1">View MathML</a>-contractive mappings. Indeed, in case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M475">View MathML</a>, we easily get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M476">View MathML</a>

and in case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M477">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M478">View MathML</a>

hence, our claim holds.

Moreover, from the monotone property of T, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M479">View MathML</a>

and similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M480">View MathML</a>

Thus T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M64">View MathML</a>-preserving for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a>. Now, if condition (iii) is satisfied, that is, T is continuous, the existence of a fixed point follows from Theorem 2.1. Suppose now, that the condition (iii′) is satisfied, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M33">View MathML</a> be a nondecreasing sequence in X, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M484">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M485">View MathML</a> for all n. Suppose also that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M486">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M152">View MathML</a>. From (iii′), there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M37">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M460">View MathML</a> for all k. This implies from the definition of α and β that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M490">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M491">View MathML</a> for all k, which implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M492">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M65">View MathML</a> and for all k. In this case, the existence of a fixed point follows from Theorem 2.2.

To show the uniqueness, suppose that X is ≤-directed, that is, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M446">View MathML</a> there exists a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M447">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M448">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M449">View MathML</a>, which implies from the definition of α and β that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M498">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M499">View MathML</a>. Hence, Theorem 2.3 gives us the uniqueness of this fixed point. □

4.3 Coupled fixed point theorems

Next, in order to prove a coupled fixed point results in partially ordered set, we need to define an order relation on the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M501">View MathML</a> be a partially ordered set endowed with a metric d, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M502">View MathML</a> be a given mapping. We endow the product set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a> with the partial order:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M504">View MathML</a>

Definition 4.2 Let X be a set and binary relations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M505">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M336">View MathML</a> on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M508">View MathML</a>

Corollary 4.4 (Harjani et al.[6])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156">View MathML</a>be a partially ordered set and suppose that there exists a metricdinXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M28">View MathML</a>is a complete metric space. Suppose that the mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M511">View MathML</a>is weakly<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M44">View MathML</a>-contractive, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M513">View MathML</a>

whereψandφare altering distance functions. Suppose also that the following conditions hold:

(i) Fis a mixed monotone mapping;

(ii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M514">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M515">View MathML</a>;

(iii) Fis continuous or<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M294">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225">View MathML</a>-biregular.

ThenFhas a coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M518">View MathML</a>. Moreover, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M220">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M225">View MathML</a>-bidirected, we have the uniqueness of the fixed point.

Proof The conclusions then follows directly from Corollary 3.1. □

4.4 Fixed point results for cyclic contractive mappings

In this section, we will derive from our results the fixed point theorem of Karapınar and Sadarangani [11] for cyclic weak <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M521">View MathML</a>-contractive mappings.

Definition 4.3 (Păcurar and Rus [16])

Let X be a nonempty set, m a positive integer and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M522">View MathML</a> an operator. By definition, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M523">View MathML</a> is a cyclic representation of X with respect to T if

1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M524">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M525">View MathML</a> are nonempty sets;

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M526">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M527">View MathML</a>.

Definition 4.4 (Karapınar and Sadarangani [10])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156">View MathML</a> be a metric space, let m be a positive integer, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M529">View MathML</a> be closed non-empty subsets of X, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M530">View MathML</a>. An operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M531">View MathML</a> is called a cyclic weak <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M521">View MathML</a>-contraction if

1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M533">View MathML</a> is a cyclic representation of Y with respect to T, and

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M534">View MathML</a> is an altering distance function such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M535">View MathML</a>

(30)

for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M536">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M537">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M538">View MathML</a>.

Corollary 4.5 (Karapınar and Sadarangani [10])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M156">View MathML</a>be a complete metric space, letmbe a positive integer, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M540">View MathML</a>be closed non-empty subsets ofXand let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M541">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M542">View MathML</a>is an altering distance function, andTis a cyclic weakφ-contraction, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M541">View MathML</a>is a cyclic representation ofYwith respect toT. Then, Thas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/205/mathml/M544">View MathML</a>.

Proof The proof follows immediately from Corollary 3.3. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors thank Professor Mircea-Dan Rus for his remarkable comments, suggestion and ideas that helped to improve this paper.

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