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Cyclic generalized contractions and fixed point results with applications to an integral equation

Hemant Kumar Nashine1, Wutiphol Sintunavarat2 and Poom Kumam2*

Author Affiliations

1 Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur, Chhattisgarh, 492101, India

2 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand

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Fixed Point Theory and Applications 2012, 2012:217  doi:10.1186/1687-1812-2012-217

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/217


Received:13 June 2012
Accepted:12 November 2012
Published:28 November 2012

© 2012 Nashine et al.; licensee Springer

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

Abstract

We set up a new variant of cyclic generalized contractive mappings for a map in a metric space and present existence and uniqueness results of fixed points for such mappings. Our results generalize or improve many existing fixed point theorems in the literature. To illustrate our results, we give some examples. At the same time as applications of the presented theorems, we prove an existence theorem for solutions of a class of nonlinear integral equations.

MSC: 47H10, 54H25.

Keywords:
fixed point; cyclic generalized <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1">View MathML</a>-contraction; integral equation

1 Introduction and preliminaries

All the way through this paper, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M2">View MathML</a>, we designate the set of all real nonnegative numbers, while ℕ is the set of all natural numbers.

The celebrated Banach’s [1] contraction mapping principle is one of the cornerstones in the development of nonlinear analysis. This principle has been extended and improved in many ways over the years (see, e.g., [2-5]). Fixed point theorems have applications not only in various branches of mathematics but also in economics, chemistry, biology, computer science, engineering, and other fields. In particular, such theorems are used to demonstrate the existence and uniqueness of a solution of differential equations, integral equations, functional equations, partial differential equations, and others. Owing to the magnitude, generalizations of the Banach fixed point theorem have been explored heavily by many authors. This celebrated theorem can be stated as follows.

Theorem 1.1 ([1])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3">View MathML</a>be a complete metric space andTbe a mapping ofXinto itself satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M4">View MathML</a>

(1)

wherekis a constant in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M5">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M6">View MathML</a>.

Inequality (1) implies the continuity of T. A natural question is whether we can find contractive conditions which will imply the existence of a fixed point in a complete metric space but will not imply continuity.

On the other hand, cyclic representations and cyclic contractions were introduced by Kirk et al.[6]. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M7">View MathML</a> is called cyclic if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M8">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M9">View MathML</a>, where A, B are nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3">View MathML</a>. Moreover, T is called a cyclic contraction if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M11">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M12">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M13">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M14">View MathML</a>. Notice that although a contraction is continuous, a cyclic contraction need not to be. This is one of the important gains of this theorem.

Definition 1.1 (See [6,7])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3">View MathML</a> be a metric space. Let p be a positive integer, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M16">View MathML</a> be nonempty subsets of X, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M17">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18">View MathML</a>. Then Y is said to be a cyclic representation of Y with respect to T if

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M19">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20">View MathML</a> are nonempty closed sets, and

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M21">View MathML</a>.

Following the paper in [6], a number of fixed point theorems on a cyclic representation of Y with respect to a self-mapping T have appeared (see, e.g., [3,7-15]).

In this paper, we introduce a new class of cyclic generalized <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1">View MathML</a>-contractive mappings, and then investigate the existence and uniqueness of fixed points for such mappings. Our main result generalizes and improves many existing theorems in the literature. We supply appropriate examples to make obvious the validity of the propositions of our results. To end with, as applications of the presented theorems, we achieve fixed point results for a generalized contraction of integral type and we prove an existence theorem for solutions of a system of integral equations.

2 Main results

In this section, we introduce two new notions of a cyclic contraction and establish new results for such mappings.

In the sequel, we fixed the set of functions by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M23">View MathML</a> such that

(i) ℱ is nondecreasing, continuous, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M24">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M25">View MathML</a>;

(ii) ψ is nondecreasing, right continuous, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M26">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M27">View MathML</a>.

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M28">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M29">View MathML</a>.

We state the notion of a cyclic generalized <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M30">View MathML</a>-contraction as follows.

Definition 2.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3">View MathML</a> be a metric space. Let p be a positive integer, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M16">View MathML</a> be nonempty subsets of X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M17">View MathML</a>. An operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M34">View MathML</a> is said to be a cyclic generalized <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1">View MathML</a>-contraction for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38">View MathML</a> if

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39">View MathML</a> is a cyclic representation of Y with respect to T;

(b) for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20">View MathML</a> (with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42">View MathML</a>),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M43">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M44">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M45">View MathML</a>

Our first main result is the following.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48">View MathML</a>be nonempty closed subsets ofX, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49">View MathML</a>. Suppose<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M50">View MathML</a>is a cyclic generalized<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1">View MathML</a>-contraction mapping for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M52">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a>. ThenThas a unique fixed point. Moreover, the fixed point ofTbelongs to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M55">View MathML</a> (such a point exists since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M56">View MathML</a>). Define the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M57">View MathML</a> in X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M58">View MathML</a>

We shall prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M59">View MathML</a>

(2)

If, for some k, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M60">View MathML</a>, then (2) follows immediately. So, we can suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M61">View MathML</a> for all n. From the condition (a), we observe that for all n, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M62">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M63">View MathML</a>. Then, from the condition (b), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M64">View MathML</a>

(3)

On the other hand, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M65">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M66">View MathML</a>

Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M67">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M68">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M69">View MathML</a>, so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M70">View MathML</a>

a contradiction. Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M71">View MathML</a>

and thus

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M72">View MathML</a>

(4)

Similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M73">View MathML</a>

(5)

Thus, from (4) and (5), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M74">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M75">View MathML</a>. Now, from

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M76">View MathML</a>

and the property of ψ, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M77">View MathML</a>, and consequently (2) holds.

Now, we shall prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M57">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3">View MathML</a>. Suppose, on the contrary, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M57">View MathML</a> is not a Cauchy sequence. Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M81">View MathML</a> for which we can find two sequences of positive integers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M82">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M83">View MathML</a> such that for all positive integers k,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M84">View MathML</a>

(6)

Further, corresponding to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M85">View MathML</a>, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M86">View MathML</a> in such a way that it is the smallest integer with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M87">View MathML</a> satisfying (6). Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M88">View MathML</a>

(7)

Using (6), (7), and the triangular inequality, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M89">View MathML</a>

Thus, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M90">View MathML</a>

Passing to the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M91">View MathML</a> in the above inequality and using (2), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M92">View MathML</a>

(8)

On the other hand, for all k, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M93">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M94">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M95">View MathML</a> (for k large enough, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M96">View MathML</a>) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M97">View MathML</a> lie in different adjacently labeled sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M98">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M99">View MathML</a> for certain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M100">View MathML</a>. Using (b), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M101">View MathML</a>

(9)

for all k. Now, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M102">View MathML</a>

(10)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M103">View MathML</a>

(11)

for all k. Using the triangular inequality, we get

which implies from (8) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M105">View MathML</a>

(12)

Using (2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M106">View MathML</a>

(13)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M107">View MathML</a>

(14)

Again, using the triangular inequality, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M108">View MathML</a>

Passing to the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M109">View MathML</a> in the above inequality, using (14) and (12), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M110">View MathML</a>

(15)

Similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M111">View MathML</a>

Passing to the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M109">View MathML</a>, using (2) and (12), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M113">View MathML</a>

(16)

Similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M114">View MathML</a>

(17)

Now, it follows from (12)-(16) and the continuity of φ that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M115">View MathML</a>

(18)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M116">View MathML</a>

(19)

Passing to the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M109">View MathML</a> in (9), using (17), (18), (19), and the condition (ii), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M118">View MathML</a>

which is a contradiction. Thus, we proved that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M57">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3">View MathML</a> is complete, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M6">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M123">View MathML</a>

(20)

We shall prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M124">View MathML</a>

(21)

From the condition (a), and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M55">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M126">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M127">View MathML</a> is closed, from (20), we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M128">View MathML</a>. Again, from the condition (a), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M129">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M130">View MathML</a> is closed, from (20), we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M131">View MathML</a>. Continuing this process, we obtain (21).

Now, we shall prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M132">View MathML</a> is a fixed point of T. Indeed, from (21), since for all n there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M133">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M134">View MathML</a>, applying (b) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M135">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M136">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M137">View MathML</a>

(22)

for all n. On the other hand, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M138">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M139">View MathML</a>

Passing to the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M140">View MathML</a> in the above inequality and using (20), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M141">View MathML</a>

(23)

Passing to the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M140">View MathML</a> in (22), using (23) and (20), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M143">View MathML</a>

Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M144">View MathML</a>. In this case, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M145">View MathML</a>

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M146">View MathML</a>

a contradiction. Then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M147">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M132">View MathML</a> is a fixed point of T.

Finally, we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M132">View MathML</a> is the unique fixed point of T. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M150">View MathML</a> is another fixed point of T, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M151">View MathML</a>. From the condition (a), this implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M152">View MathML</a>. Then we can apply (b) for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M135">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M154">View MathML</a>. We obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M155">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M132">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M150">View MathML</a> are fixed points of T, we can show easily that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M158">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M159">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M160">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M161">View MathML</a>

a contradiction. Then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M162">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M163">View MathML</a>. Thus, we proved the uniqueness of the fixed point. □

In the following, we deduce some fixed point theorems from our main result given by Theorem 2.1.

If we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M165">View MathML</a> in Theorem 2.1, then we get immediately the following fixed point theorem.

Corollary 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M3">View MathML</a>be a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M167">View MathML</a>satisfy the following condition: there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M171">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M172">View MathML</a>. ThenThas a unique fixed point.

Remark 2.1 Corollary 2.1 extends and generalizes many existing fixed point theorems in the literature [1,16-21].

Corollary 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48">View MathML</a>be nonempty closed subsets ofX, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18">View MathML</a>. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a>such that

(a′) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39">View MathML</a>is a cyclic representation ofYwith respect toT;

(b′) for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20">View MathML</a> (with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42">View MathML</a>),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M184">View MathML</a>

ThenThas a unique fixed point. Moreover, the fixed point ofTbelongs to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54">View MathML</a>.

Remark 2.2 Corollary 2.2 is similar to Theorem 2.1 in [7].

Remark 2.3 Taking in Corollary 2.2 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M186">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M11">View MathML</a>, we obtain a generalized version of Theorem 1.3 in [6].

Corollary 2.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48">View MathML</a>be nonempty closed subsets ofX, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18">View MathML</a>. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a>such that

(a′) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39">View MathML</a>is a cyclic representation ofYwith respect toT;

(b′) for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20">View MathML</a> (with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42">View MathML</a>),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M199">View MathML</a>

ThenThas a unique fixed point. Moreover, the fixed point ofTbelongs to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54">View MathML</a>.

Remark 2.4 Taking in Corollary 2.3 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M186">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M11">View MathML</a>, we obtain a generalized version of Theorem 3 in [13].

Corollary 2.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48">View MathML</a>be nonempty closed subsets ofX, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18">View MathML</a>. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a>such that

(a′) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39">View MathML</a>is a cyclic representation ofYwith respect toT;

(b′) for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20">View MathML</a> (with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42">View MathML</a>),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M214">View MathML</a>

ThenThas a unique fixed point. Moreover, the fixed point ofTbelongs to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54">View MathML</a>.

Remark 2.5 Taking in Corollary 2.4 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M186">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M11">View MathML</a>, we obtain a generalized version of Theorem 5 in [13].

Corollary 2.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M46">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M47">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M48">View MathML</a>be nonempty closed subsets ofX, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M49">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M18">View MathML</a>. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a>such that

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M39">View MathML</a>is a cyclic representation ofYwith respect toT;

(b) for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M40">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M20">View MathML</a> (with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M42">View MathML</a>),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M229">View MathML</a>

ThenThas a unique fixed point. Moreover, the fixed point ofTbelongs to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M54">View MathML</a>.

We provide some examples to illustrate our obtained Theorem 2.1.

Example 2.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M231">View MathML</a> with the usual metric. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M232">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M233">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M234">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M235">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M236">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M237">View MathML</a>. It is clear that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M238">View MathML</a> is a cyclic representation of Y with respect to T. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M36">View MathML</a> be defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M240">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a> of the form <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M242">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M243">View MathML</a>. For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M244">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M246">View MathML</a>

So, T is a cyclic generalized <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M1">View MathML</a>-contraction for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38">View MathML</a>. Therefore, all conditions of Theorem 2.1 are satisfied (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M249">View MathML</a>), and so T has a unique fixed point (which is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M250">View MathML</a>).

Example 2.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M231">View MathML</a> with the usual metric. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M252">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M253">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M234">View MathML</a>. Define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M255">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M256">View MathML</a>

Clearly, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M257">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M258">View MathML</a>. Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M127">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M130">View MathML</a> are nonempty closed subsets of X. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M238">View MathML</a> is a cyclic representation of Y with respect to T.

Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M262">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M263">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M264">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M265">View MathML</a>

On the other hand, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M266">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M267">View MathML</a>

Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M268">View MathML</a>

Define the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M269">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M270">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a> of the form <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M242">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M243">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M275">View MathML</a>

(24)

Moreover, we can show that (24) holds if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M276">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M277">View MathML</a>. Similarly, we also get (24) holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M278">View MathML</a>.

Now, all the conditions of Theorem 2.1 are satisfied (with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M279">View MathML</a>), we deduce that T has a unique fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M280">View MathML</a>.

3 An application to an integral equation

In this section, we apply the result given by Theorem 2.1 to study the existence and uniqueness of solutions to a class of nonlinear integral equations.

We consider the nonlinear integral equation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M281">View MathML</a>

(25)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M282">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M283">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M284">View MathML</a> are continuous functions.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M285">View MathML</a> be the set of real continuous functions on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M286">View MathML</a>. We endow X with the standard metric

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M287">View MathML</a>

It is well known that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M288">View MathML</a> is a complete metric space.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M289">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M290">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M291">View MathML</a>

(26)

We suppose that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M293">View MathML</a>

(27)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M294">View MathML</a>

(28)

We suppose that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M295">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M296">View MathML</a> is a decreasing function, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M297">View MathML</a>

(29)

We suppose that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M298">View MathML</a>

(30)

Finally, we suppose that, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M295">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M300">View MathML</a> with (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M301">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M302">View MathML</a>) or (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M303">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M304">View MathML</a>),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M305">View MathML</a>

(31)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M269">View MathML</a> is a nondecreasing function that belongs to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M307">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M38">View MathML</a>.

Now, define the set

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M309">View MathML</a>

We have the following result.

Theorem 3.1Under the assumptions (26)-(31), problem (25) has one and only one solution<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M310">View MathML</a>.

Proof Define the closed subsets of X, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M127">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M130">View MathML</a>, by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M313">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M314">View MathML</a>

Define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M315">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M316">View MathML</a>

We shall prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M317">View MathML</a>

(32)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M318">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M319">View MathML</a>

Using condition (29), since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M320">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M321">View MathML</a>, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M322">View MathML</a>

The above inequality with condition (27) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M323">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292">View MathML</a>. Then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M325">View MathML</a>.

Similarly, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M326">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M327">View MathML</a>

Using condition (29), since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M320">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M321">View MathML</a>, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M330">View MathML</a>

The above inequality with condition (28) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M331">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292">View MathML</a>. Then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M333">View MathML</a>. Finally, we deduce that (32) holds.

Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M334">View MathML</a>, that is, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M336">View MathML</a>

This implies, from condition (26), that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M338">View MathML</a>

Now, using conditions (30) and (31), we can write that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M292">View MathML</a>, we have

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M341">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M37">View MathML</a> of the form <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M343">View MathML</a>. Using the same technique, we can show that the above inequality holds also if we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M344">View MathML</a>.

Now, all the conditions of Theorem 2.1 are satisfied (with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M279">View MathML</a>), we deduce that T has a unique fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M346">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/217/mathml/M310">View MathML</a> is the unique solution to (25). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). Moreover, the third author was supported by the Commission on Higher Education (CHE), the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) (Grant No. MRG5580213).

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