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New existence theorems of coincidence points approach to generalizations of Mizoguchi-Takahashi’s fixed point theorem

Ing-Jer Lin* and Tai-Hung Chen

Author Affiliations

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung, 824, Taiwan

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Fixed Point Theory and Applications 2012, 2012:156  doi:10.1186/1687-1812-2012-156


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/156


Received:24 May 2012
Accepted:30 August 2012
Published:19 September 2012

© 2012 Lin and Chen; licensee Springer

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

Abstract

In this paper, we first establish some new existence theorems of coincidence points and common fixed points for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-functions. By applying our results, we obtain some generalizations of Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and the Banach contraction principle. Some examples illustrating our results are also given. Our results generalize and improve some main results in the literature and references therein.

Keywords:
coincidence point; common fixed point; τ-function; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metric; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function; Mizoguchi-Takahashi’s fixed point theorem; Nadler’s fixed point theorem; Banach contraction principle

1 Introduction

In recent years, the celebrated Banach contraction principle (see, e.g., [1]) always plays an essential role in various fields of applied mathematical analysis. The Banach contraction principle has been employed to solve the problems in Banach spaces such as the existence of solutions for nonlinear integral equations and nonlinear differential equations. Also, it has been applied to study the convergence of algorithms in computational mathematics. Additionally, many generalizations of the Banach contraction principle in various different directions have been investigated by several authors in the past; see [1-22]. Because of the importance of the Banach contraction principle, we begin with the theorem as follows.

Theorem BCP (Banach [1])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M6">View MathML</a>be a selfmap. Assume that there exists a nonnegative number<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M7">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M8">View MathML</a>

ThenThas a unique fixed point inX. Moreover, for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M9">View MathML</a>, the iterative sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M10">View MathML</a>converges to the fixed point.

In 1969, Nadler [2] first gave a famous generalization of the Banach contraction principle for multivalued maps, which is as important as the Banach contraction principle.

Theorem NA (Nadler [2])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>be ak-contraction; that is, there exists a nonnegative number<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M13">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M14">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15">View MathML</a>is the class of all nonempty closed bounded subsets ofX. Then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M16">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M17">View MathML</a>.

In 1989, Mizoguchi and Takahashi [3] proved a generalization of Nadler’s fixed point theorem which also gave a partial answer to Problem 9 in Reich [4-6]. It is worth mentioning that the primitive proof of Mizoguchi-Takahashi’s fixed point theorem is difficult. Recently, Suzuki [7] gave a very simple proof of Mizoguchi-Takahashi’s fixed point theorem.

Theorem MT (Mizoguchi and Takahashi [3])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>be a multivalued map. Assume that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M20">View MathML</a>

whereαis a function from<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M21">View MathML</a>into<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M22">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M23">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24">View MathML</a>. Then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M16">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M17">View MathML</a>.

Subsequently, in 2007, Berinde and Berinde [8] proved the following interesting fixed point theorem. That is a generalization of Mizoguchi-Takahashi’s fixed point theorem.

Theorem BB (Berinde and Berinde [8])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>be a multivalued map, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M29">View MathML</a>. Assume that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M30">View MathML</a>

whereαis a function from<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M21">View MathML</a>into<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M22">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M33">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M34">View MathML</a>. Then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M16">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M17">View MathML</a>.

It is obvious that if we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M37">View MathML</a> in Berinde and Berinde’s fixed point theorem, we can obtain Mizoguchi-Takahashi’s fixed point theorem.

Very recently, Du [9] has used a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metric and an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function to establish some new fixed point theorems for nonlinear multivalued contractive maps and generalize the Banach contraction principle, Nadler’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Kannan’s fixed point theorems and Chatterjea’s fixed point theorems for nonlinear multivalued contractive maps in complete metric spaces; see [9] for more detail.

In this paper, we first establish some new existence results of coincidence points and common fixed points for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-functions. By applying our results, we can obtain some generalizations of Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and the Banach contraction principle. Our results generalize and improve some main results in the literature and references therein.

2 Preliminaries

Throughout this paper, we denote the set of positive integers by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M41">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a> be a metric space. For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M44">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M45">View MathML</a>. Also, we denote the class of all nonempty subsets of X by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M46">View MathML</a>, the family of all nonempty closed subsets of X by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M47">View MathML</a>, and the family of all nonempty closed and bounded subsets of X by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15">View MathML</a>. A function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M49">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M50">View MathML</a>

is said to be the Hausdorff metric on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15">View MathML</a> induced by the metric d on X.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a> be a selfmap and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M53">View MathML</a> be a multivalued map. A point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M16">View MathML</a> is called

(i) a fixed point of f if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M55">View MathML</a>;

(ii) a fixed point of T if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M56">View MathML</a>;

(iii) a coincidence point of f and T in X if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M57">View MathML</a>;

(iv) a common fixed point of f and T if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M58">View MathML</a>.

In [9], Sajath and Vijayaraju proved the following theorem.

Theorem 2.1[10]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M60">View MathML</a>be a function such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M61">View MathML</a>for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>satisfy

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M65">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M66">View MathML</a>;

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(c) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX,

thenTandfhave a coincidence point inX.

Remark 2.1 In fact, the condition (a) in Theorem 2.1 should be corrected as

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M69">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M71">View MathML</a>.

Moreover, it is worth mentioning that the proof of Theorem 2.1 is not correct.

The following is the definition of a τ-function which was introduced and studied by Lin and Du.

Definition 2.1[9,11-17]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a> be a metric space. A function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73">View MathML</a> is said to be a τ-function if the following conditions hold:

(τ1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M74">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M75">View MathML</a>;

(τ2) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M77">View MathML</a> in X with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M78">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M79">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M80">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M81">View MathML</a>;

(τ3) For any sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M82">View MathML</a> in X with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M83">View MathML</a>, if there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M77">View MathML</a> in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M85">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M86">View MathML</a>;

(τ4) For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M75">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M88">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M89">View MathML</a> imply <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M90">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73">View MathML</a> be a τ-function. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M92">View MathML</a>.

The following results are crucial and useful in this paper.

Lemma 2.1[9,11,12,14-17]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73">View MathML</a>be any function satisfying (τ3). If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M82">View MathML</a>is a sequence inXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M96">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M82">View MathML</a>is a Cauchy sequence in X.

Recently, Du [5,6] first introduced the concepts of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-functions and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metrics as follows.

Definition 2.2[9,13]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a> be a metric space. A function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73">View MathML</a> is called a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function if it is a τ-function on X with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M103">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M9">View MathML</a>.

Remark 2.3 From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M105">View MathML</a>, if p is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M88">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M108">View MathML</a>.

Definition 2.3[9,13]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a> be a metric space and p be a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function. For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M111">View MathML</a>, define a function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M112">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M113">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M114">View MathML</a>; then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115">View MathML</a> is said to be a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metric on CB(X) induced by p.

Clearly, any Hausdorff metric is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metric, but the reverse is not true.

Definition 2.4[9,15-22]

A function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118">View MathML</a> is said to be an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function (or an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M120">View MathML</a>-function) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M121">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24">View MathML</a>.

Lemma 2.2[9]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118">View MathML</a>be an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M125">View MathML</a>defined by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M126">View MathML</a>is also an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function.

Theorem D[22]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118">View MathML</a>be a function. Then the following statements are equivalent.

(a) φis an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function.

(b) For each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M130">View MathML</a>, there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M131">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M132">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M133">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M134">View MathML</a>.

(c) For each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M130">View MathML</a>, there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M136">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M137">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M138">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M139">View MathML</a>.

(d) For each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M130">View MathML</a>, there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M141">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M142">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M143">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M144">View MathML</a>.

(e) For each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M130">View MathML</a>, there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M146">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M147">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M148">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M149">View MathML</a>.

(f) For any nonincreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M150">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M151">View MathML</a>, we have<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M152">View MathML</a>.

(g) φis a function of a contractive factor[19]; that is, for any strictly decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M153">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M21">View MathML</a>, we have<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M155">View MathML</a>.

It is obvious that if a function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M156">View MathML</a> is nondecreasing or nonincreasing, then it is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function.

3 New coincidence point theorems and a common fixed point theorem

In this section, we generalize Theorem 2.1 which is one of the main results in [10]. Please notice that our proof is quite different from the proof of Theorem 2.1 in [10].

Theorem 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M159">View MathML</a>be a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115">View MathML</a>be a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metric on<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15">View MathML</a>induced bypand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M164">View MathML</a>be an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M166">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>satisfy

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M168">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX,

thenTandfhave a coincidence point inX.

Proof By Lemma 2.2, we can define an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M125">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M126">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M175">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M176">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M178">View MathML</a>. By (ii), there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M179">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M180">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M181">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M182">View MathML</a> which means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M183">View MathML</a> is a coincidence point of T and f in X and we finish the proof. Otherwise, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M184">View MathML</a>, since p is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M186">View MathML</a>. By (i), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M187">View MathML</a>

Hence there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M188">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M189">View MathML</a>. By (ii) again, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M190">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M191">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M192">View MathML</a>

By induction, we can obtain a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M193">View MathML</a> in X satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M194">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M195">View MathML</a>

(3.1)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M196">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M24">View MathML</a>, the inequality (3.1) implies the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M198">View MathML</a> is strictly decreasing in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M21">View MathML</a>. Since κ is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function, by Theorem D, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M201">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M202">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M203">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M204">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M205">View MathML</a>. For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206">View MathML</a>, we have from (3.1) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M207">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M208">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M209">View MathML</a>. We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M210">View MathML</a>. Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M211">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M213">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M214">View MathML</a>, by (3.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M215">View MathML</a>

(3.3)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M203">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M217">View MathML</a> and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M218">View MathML</a>

By Lemma 2.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M219">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>. By the completeness of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M222">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M223">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224">View MathML</a>. From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M225">View MathML</a> and (3.3), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M226">View MathML</a>

(3.4)

So, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M228">View MathML</a>

(3.5)

Therefore, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M229">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M230">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206">View MathML</a>, which implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M232">View MathML</a>. Then, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M233">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M234">View MathML</a>. Moreover, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M223">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M237">View MathML</a>

we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M238">View MathML</a>

which means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M239">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M229">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M243">View MathML</a> is closed, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M244">View MathML</a>, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M245">View MathML</a> is a coincidence point of f and T. The proof is completed. □

Remark 3.1 In Theorem 3.1, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M246">View MathML</a> (the identity map), then we obtain Mizoguchi-Takahashi’s fixed point theorem. So Theorem 3.1 is a generalization of Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and the Banach contraction principle.

Here, we give a simple example illustrating Theorem 3.1.

Example 3.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M247">View MathML</a> with the metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M248">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M249">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M250">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M251">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M252">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M253">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M255">View MathML</a>

for all x, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M256">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M257">View MathML</a>. It is easy to see that p is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function and φ is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function.

Clearly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M260">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a> is a complete subspace of X. We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M262">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70">View MathML</a>. Indeed, we consider the following two possible cases:

Case 1. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M264">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M265">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M266">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M267">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M268">View MathML</a>

Case 2. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M269">View MathML</a>, similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M270">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M271">View MathML</a>

By Cases 1 and 2, we verify that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M262">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70">View MathML</a>. Therefore, all the assumptions of Theorem 3.1 are satisfied. So, we can apply Theorem 3.1 to show that f and T have a coincidence point in X. Actually, 0 is a coincidence point of f and T since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M274">View MathML</a>.

The following result follows immediately from Theorem 3.1.

Corollary 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73">View MathML</a>be a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115">View MathML</a>be a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metric on<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15">View MathML</a>induced bypand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M281">View MathML</a>be a nondecreasing or nonincreasing function. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>satisfy

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M284">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX,

thenTandfhave a coincidence point inX.

In Theorem 3.1, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M288">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M289">View MathML</a> and we have the following corollary.

Corollary 3.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118">View MathML</a>be an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>satisfy

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M295">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX,

thenTandfhave a coincidence point inX.

Corollary 3.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M281">View MathML</a>be a nondecreasing or nonincreasing function. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>satisfy

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M303">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX,

thenTandfhave a coincidence point inX.

Theorem 3.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M159">View MathML</a>be a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115">View MathML</a>be a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metric on<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15">View MathML</a>induced bypand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M164">View MathML</a>be an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M166">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>satisfy

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M317">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M318">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX;

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M321">View MathML</a>ifvis a coincidence point offandT,

thenTandfhave a common fixed point inX.

Proof Following the same argument as in the proof of Theorem 3.1, we can construct two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M82">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M219">View MathML</a> satisfying

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M324">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206">View MathML</a>;

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M326">View MathML</a> is a Cauchy sequence in X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M327">View MathML</a>;

(c) there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M222">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M223">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224">View MathML</a>;

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M244">View MathML</a>;

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M332">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M211">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206">View MathML</a>.

By (c) and (iv), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M335">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M336">View MathML</a>

Therefore, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M337">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M338">View MathML</a>

(3.6)

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M339">View MathML</a>

we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M340">View MathML</a>. By (3.6), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M341">View MathML</a>. By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M233">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M343">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M223">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M224">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M346">View MathML</a>

we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M347">View MathML</a>, which implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M348">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M349">View MathML</a> is closed and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M337">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M206">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M352">View MathML</a>. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M353">View MathML</a>, which means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M354">View MathML</a> is a common fixed point of f and T in X. The proof is completed. □

Remark 3.2 Theorem 3.2 also generalizes and improves Mizoguchi-Takahashi’s fixed point theorem.

Example 3.2 In Example 3.1, we have shown that 0 is a coincidence point of f and T. Clearly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M355">View MathML</a>. So, all the assumptions of Theorem 3.2 are satisfied. By Theorem 3.2, we know that f and T have a common fixed point in X. Actually, 0 is a common fixed point of f and T since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M356">View MathML</a>.

Similarly, we have the following corollary.

Corollary 3.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M73">View MathML</a>be a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-function, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M115">View MathML</a>be a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M2">View MathML</a>-metric on<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M15">View MathML</a>induced bypand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M281">View MathML</a>be a nondecreasing or nonincreasing function. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>satisfy

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M366">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M367">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX;

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M321">View MathML</a>ifvis a coincidence point offandT,

thenTandfhave a common fixed point inX.

Corollary 3.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M118">View MathML</a>be an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M4">View MathML</a>-function. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>satisfy

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M376">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M377">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX;

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M321">View MathML</a>ifvis a coincidence point offandT,

thenTandfhave a common fixed point inX.

Corollary 3.6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M5">View MathML</a>be a metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M281">View MathML</a>be a nondecreasing or nonincreasing function. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M52">View MathML</a>satisfy

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M385">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M70">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M67">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M68">View MathML</a>is a complete subspace ofX;

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/156/mathml/M321">View MathML</a>ifvis a coincidence point offandT,

thenTandfhave a common fixed point inX.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The first author made 80% contribution: problem design, coordination, discussion, revision of the important part, and submission of this paper. The second author made 20% contribution: discussion, responsibility for the important results and typing of this paper.

Acknowledgements

The authors wish to express their hearty thanks to Professor Wei-Shih Du for their valuable suggestions and comments.

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