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Existence of a solution of integral equations via fixed point theorem

Selma Gülyaz1, Erdal Karapınar2*, Vladimir Rakocević3 and Peyman Salimi4

Author Affiliations

1 Department of Mathematics, Cumhuriyet University, Sivas, Turkey

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

3 Faculty of Sciences and Mathematics, University of Nis, Visegradska 33, Nis, 18000, Serbia

4 Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran

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Journal of Inequalities and Applications 2013, 2013:529  doi:10.1186/1029-242X-2013-529

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/529


Received:14 May 2013
Accepted:18 October 2013
Published:11 November 2013

© 2013 Gülyaz et al.; licensee Springer.

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

Abstract

In this paper, we establish a solution to the following integral equation:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M2">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M3">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M4">View MathML</a> are continuous functions. For this purpose, we also obtain some auxiliary fixed point results which generalize, improve and unify some fixed point theorems in the literature.

MSC: 47H10, 54H25.

Keywords:
cyclic representation; fixed point

1 Introduction and preliminaries

Fixed point theory is one of the most efficient tools in nonlinear functional analysis to solve the nonlinear differential and integral equations. The existence/uniqueness of a solution of differential/integral equations turns into the existence/uniqueness of a (common) fixed point of the operators which are obtained after suitable substitutions and elementary calculations; see, e.g., [1-14].

In this paper, we first obtain some fixed point theorems to solve the integral equation mentioned above. For the sake of completeness, we recollect some basic definitions and elementary results. Let X be a nonempty set and T be a self-mapping on X. Then, the set of all fixed points of T on X is denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M5">View MathML</a>. Let Ψ be the set of all functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M6">View MathML</a> satisfying the following conditions:

(1) ψ is continuous,

(2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M7">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M8">View MathML</a>,

(3) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M9">View MathML</a>.

Cyclic mapping and cyclic contraction were introduced by Kirk-Srinavasan-Veeramani to improve the well-known Banach fixed point theorem. Later, various types of cyclic contraction have been investigated by a number of authors; see, e.g., [6,15-17].

Definition 1.1[18]

Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M10">View MathML</a> is a metric space and T is a self-mapping on X. Let m be a natural number and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M12">View MathML</a>, be nonempty sets. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M13">View MathML</a> is called a cyclic representation of X with respect to T if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M14">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M15">View MathML</a>.

Definition 1.2[17]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M16">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M17">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M18">View MathML</a> be two functions. We say that T is r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-admissible if

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M21">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M22">View MathML</a>,

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M23">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M24">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M25">View MathML</a>.

Definition 1.3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M26">View MathML</a> be a metric space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M27">View MathML</a> be a self-mapping, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M13">View MathML</a> is a cyclic representation of Y with respect to T. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M29">View MathML</a> be two functions. An operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M27">View MathML</a> is called:

• a cyclic weak r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-C-contractive mapping of the first kind if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M32">View MathML</a>

(2)

holds for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M33">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M34">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M35">View MathML</a>.

• a cyclic weak r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-C-contractive mapping of the second kind if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M37">View MathML</a>

(3)

such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M38">View MathML</a> holds for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M33">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M34">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M35">View MathML</a>.

2 Auxiliary fixed point results

We state the main result of this section as follows.

Theorem 2.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M26">View MathML</a>be a complete metric space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M43">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M44">View MathML</a>be nonempty closed subsets of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M10">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M46">View MathML</a>. Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M47">View MathML</a>is a cyclic weakr-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-C-contractive mapping of the first kind such that

(i) Tisr-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-admissible;

(ii) there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M51">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M52">View MathML</a>;

(iii) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a>is a sequence inYsuch that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M54">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M55">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57">View MathML</a>as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60">View MathML</a>.

ThenThas a fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M61">View MathML</a>. Moreover, if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M63">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M65">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M66">View MathML</a>, thenThas a unique fixed point.

Proof Let there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M50">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M51">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M52">View MathML</a>. Since T is r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-admissible, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M71">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M72">View MathML</a>. Again, since T is r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-admissible, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M74">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M75">View MathML</a>. By continuing this process, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M76">View MathML</a>

(4)

On the other hand, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M50">View MathML</a>, there exists some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M78">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M79">View MathML</a>. Now <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M80">View MathML</a> implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M81">View MathML</a>. Thus there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M82">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M83">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M84">View MathML</a>. Similarly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M85">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M86">View MathML</a>. Hence, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M87">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M88">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M86">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M90">View MathML</a>. In case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M91">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M92">View MathML</a> , then it is clear that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M93">View MathML</a> is a fixed point of T. Now assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M94">View MathML</a> for all n. Hence, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M95">View MathML</a> for all n. Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M96">View MathML</a>. We shall show that the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M97">View MathML</a> is non-increasing. Due to (2) with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M98">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M99">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M100">View MathML</a>

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M101">View MathML</a>

(5)

and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M102">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56">View MathML</a>. Then there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M105">View MathML</a>. Suppose, on the contrary, that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M106">View MathML</a>. Also, taking limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M107">View MathML</a> in (5), we deduce

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M108">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M109">View MathML</a>

(6)

Taking limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a> in (5) and using (6), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M111">View MathML</a>

Consequently, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M112">View MathML</a>, which yields <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M113">View MathML</a>. Hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M114">View MathML</a>

(7)

We shall show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> is a Cauchy sequence. To reach this goal, first we prove the following claim:

(K) For every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M117">View MathML</a> such that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M118">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M119">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M120">View MathML</a>.

Suppose, to the contrary, that there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116">View MathML</a> such that for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M117">View MathML</a>, we can find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M123">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M124">View MathML</a> satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M125">View MathML</a>

(8)

Now, we take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M126">View MathML</a>. Then, corresponding to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M127">View MathML</a>, one can choose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M128">View MathML</a> in such a way that it is the smallest integer with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M129">View MathML</a> satisfying <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M124">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M131">View MathML</a>. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M132">View MathML</a>. By using the triangular inequality,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M133">View MathML</a>

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M134">View MathML</a> in the last inequality, keeping (7) in mind, we derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M135">View MathML</a>

(9)

Again,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M136">View MathML</a>

Taking (7) and (9) into account, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M137">View MathML</a>

(10)

as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M134">View MathML</a> in (9).

Also we have the following inequalities:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M139">View MathML</a>

(11)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M140">View MathML</a>

(12)

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M134">View MathML</a> in (11) and (12), we derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M142">View MathML</a>

(13)

Again, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M143">View MathML</a>

(14)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M144">View MathML</a>

(15)

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M134">View MathML</a> in (14) and (15), we conclude that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M146">View MathML</a>

(16)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M147">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M148">View MathML</a> lie in different adjacently labeled sets <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M150">View MathML</a> for certain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M151">View MathML</a>, using the fact that T is a cyclic weak r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-C-contractive mapping of the first kind, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M153">View MathML</a>

(17)

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M154">View MathML</a>

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a> in the inequality above and keeping the expressions (7), (9), (10), (13), (16) in mind, we conclude that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M156">View MathML</a>

Thus, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M157">View MathML</a>, which yields that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M158">View MathML</a>. Hence, (K) is satisfied.

We shall show that the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> is Cauchy. Fix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116">View MathML</a>. By the claim, we find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M161">View MathML</a> such that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M162">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M163">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M164">View MathML</a>

(18)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M165">View MathML</a>, we also find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M166">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M167">View MathML</a>

(19)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M168">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M169">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M170">View MathML</a>. Then, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M171">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M172">View MathML</a>. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M173">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M174">View MathML</a>. So, we have, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M175">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M176">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M177">View MathML</a>

By (18) and (19) and from the last inequality, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M178">View MathML</a>

This proves that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> is a Cauchy sequence. Since Y is closed in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M180">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M181">View MathML</a> is also complete, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M182">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M183">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M181">View MathML</a>. In what follows, we prove that x is a fixed point of T. In fact, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M185">View MathML</a> and, as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M186">View MathML</a> is a cyclic representation of Y with respect to T, the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> has infinite terms in each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M189">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M190">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M191">View MathML</a> and we take a subsequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M192">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M194">View MathML</a>. Now from (iii) we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60">View MathML</a>. By using the contractive condition, we can obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M197">View MathML</a>

(20)

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M198">View MathML</a>

Passing to the limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M199">View MathML</a> in the last inequality, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M200">View MathML</a>

which implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M201">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M202">View MathML</a>. Finally, to prove the uniqueness of the fixed point, suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M203">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M63">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M65">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M208">View MathML</a>. The cyclic character of T and the fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M209">View MathML</a> are fixed points of T imply that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M210">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M208">View MathML</a>. That is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M212">View MathML</a>. Using the contractive condition, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M213">View MathML</a>

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M214">View MathML</a>

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M215">View MathML</a> and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M216">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M217">View MathML</a>, which is a contradiction. This finishes the proof. □

Example 2.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M218">View MathML</a> with the metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M219">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M220">View MathML</a>. Suppose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M221">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M222">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M223">View MathML</a>. Define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M224">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M225">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M226">View MathML</a>

Also, define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M227">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M228">View MathML</a>. Clearly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M229">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M230">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M231">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M232">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M233">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M234">View MathML</a>. On the other hand, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M235">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M236">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M237">View MathML</a>. Similarly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M238">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M239">View MathML</a>. Therefore, T is an r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-admissible mapping. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> be a sequence in X such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M242">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M243">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M246">View MathML</a>. So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M234">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M248">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M249">View MathML</a>.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M250">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M251">View MathML</a>. Now, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M252">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M253">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M254">View MathML</a>. Also, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M255">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M256">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M257">View MathML</a>. That is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M258">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M250">View MathML</a> and all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M251">View MathML</a>. Hence,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M261">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M250">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M251">View MathML</a>. Then T is a cyclic weak r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-C-contractive mapping of the first kind. Therefore all the conditions of Theorem 2.1 hold and T has a fixed point in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M265">View MathML</a>. Here, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M266">View MathML</a> is a fixed point of T.

Theorem 2.3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M26">View MathML</a>be a complete metric space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M43">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M44">View MathML</a>be nonempty closed subsets of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M270">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M46">View MathML</a>. Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M47">View MathML</a>is a cyclic weakr-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-C-contractive mapping of the second kind such that

(i) Tisr-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-admissible;

(ii) there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M51">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M52">View MathML</a>;

(iii) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a>is a sequence inYsuch that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M54">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M55">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57">View MathML</a>as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60">View MathML</a>.

ThenThas a fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M61">View MathML</a>. Moreover, if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M63">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M65">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M66">View MathML</a>, thenThas a unique fixed point.

Proof By a similar method as in the proof of Theorem 2.1, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M292">View MathML</a>

(21)

We shall show that the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M293">View MathML</a> is non-increasing. Due to (3) with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M98">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M99">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M296">View MathML</a>

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M297">View MathML</a>

(22)

and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M102">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56">View MathML</a>. Then there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M105">View MathML</a>. We shall show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M113">View MathML</a> by the method of reductio ad absurdum. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M106">View MathML</a>. By letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a> in (22), we deduce

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M305">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M306">View MathML</a>

(23)

Taking limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a> in (22) and using (23), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M308">View MathML</a>

Thus, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M112">View MathML</a> and hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M113">View MathML</a>, which is a contradiction. Consequently, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M311">View MathML</a>

(24)

We shall show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> is a Cauchy sequence. To reach this goal, first we prove the following claim:

(K) For every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M117">View MathML</a> such that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M118">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M119">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M120">View MathML</a>.

Suppose, to the contrary, that there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116">View MathML</a> such that for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M117">View MathML</a> we can find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M123">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M124">View MathML</a> satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M322">View MathML</a>

(25)

Following the related lines in Theorem 2.1, we deduce

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M323">View MathML</a>

(26)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M324">View MathML</a>

(27)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M325">View MathML</a>

(28)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M326">View MathML</a>

(29)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M147">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M148">View MathML</a> lie in different adjacently labeled sets <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M150">View MathML</a> for certain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M151">View MathML</a>, using the fact that a cyclic weak r-<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M19">View MathML</a>-C-contractive mapping of the second kind, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M333">View MathML</a>

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M334">View MathML</a>

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a> in the inequality above and by applying (24) (26), (27), (28), (29), we deduce that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M336">View MathML</a>

Consequently, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M157">View MathML</a>, and hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M158">View MathML</a>. As a result, we conclude that (K) is satisfied. We assert that the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> is Cauchy. Fix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M116">View MathML</a>. By the claim, we find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M161">View MathML</a> such that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M162">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M163">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M344">View MathML</a>

(30)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M165">View MathML</a>, we also find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M166">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M347">View MathML</a>

(31)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M168">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M169">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M170">View MathML</a>. Then there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M171">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M172">View MathML</a>. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M173">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M174">View MathML</a>. So, we have, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M175">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M176">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M357">View MathML</a>

By (30) and (31) and from the last inequality, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M358">View MathML</a>

This proves that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> is a Cauchy sequence. Since Y is closed in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M180">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M181">View MathML</a> is also complete, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M182">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M183">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M181">View MathML</a>. In what follows, we prove that x is a fixed point of T. In fact, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M185">View MathML</a> and, as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M186">View MathML</a> is a cyclic representation of Y with respect to T, the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> has infinite terms in each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M11">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M189">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M190">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M191">View MathML</a> and we take a subsequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M192">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M194">View MathML</a>. Now from (iii) we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60">View MathML</a>. By using the contractive condition, we can obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M377">View MathML</a>

(32)

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M378">View MathML</a>

Passing to the limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M199">View MathML</a> in the last inequality, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M380">View MathML</a>

which implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M201">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M202">View MathML</a>. Finally, to prove the uniqueness of the fixed point, suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M203">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M63">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M60">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M65">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M208">View MathML</a>. The cyclic character of T and the fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M389">View MathML</a> are fixed points of T imply that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M210">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M208">View MathML</a>. That is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M212">View MathML</a>. Using the contractive condition, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M393">View MathML</a>

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M394">View MathML</a>

Hence, we obtain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M215">View MathML</a>, which implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M216">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M217">View MathML</a> a contradiction. □

3 Existence of solutions of an integral equation

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M2">View MathML</a>, we denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M399">View MathML</a> the set of real continuous functions on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M400">View MathML</a>. We endow X with the metric

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M401">View MathML</a>

It is evident that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M402">View MathML</a> is a complete metric space.

Consider the integral equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M403">View MathML</a>

(33)

(1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M3">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M4">View MathML</a> are continuous functions.

(2) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M406">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M407">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M408">View MathML</a>

(34)

Assume that for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M410">View MathML</a>

(35)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M411">View MathML</a>

(36)

Let for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M412">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M413">View MathML</a> be a decreasing function, that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M414">View MathML</a>

(37)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M415">View MathML</a>. There exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M416">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M417">View MathML</a> such that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M419">View MathML</a> with (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M420">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M421">View MathML</a>) or (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M422">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M423">View MathML</a>), then for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M424">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M425">View MathML</a>

(38)

(3) Assume that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M426">View MathML</a>

(39)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M427">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418">View MathML</a>. Suppose that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M429">View MathML</a>

(40)

(4) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> is a sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M431">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M432">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M433">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418">View MathML</a>.

(5) There exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M437">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M438">View MathML</a>.

Theorem 3.1Under assumptions (1)-(5), integral equation (33) has a solution in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M439">View MathML</a>.

Proof Define the closed subsets of X, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M440">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M441">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M442">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M443">View MathML</a>

Also define the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M444">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M445">View MathML</a>

Let us prove that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M446">View MathML</a>

(41)

Suppose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M447">View MathML</a>, that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M448">View MathML</a>

Applying condition (37), since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M449">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M450">View MathML</a>, we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M451">View MathML</a>

The above inequality with condition (35) imply that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M452">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409">View MathML</a>. Then we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M454">View MathML</a>.

Similarly, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M455">View MathML</a>, that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M456">View MathML</a>

Using condition (37), since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M449">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M458">View MathML</a>, we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M459">View MathML</a>

The above inequality with condition (36) imply that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M460">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409">View MathML</a>. Then we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M462">View MathML</a>. Also, we deduce that (41) holds.

Now, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M463">View MathML</a>, that is, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M465">View MathML</a>

This implies from condition (34) that for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M409">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M467">View MathML</a>

Now, by conditions (39) and (38), we have, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M412">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M469">View MathML</a>

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M470">View MathML</a>

Define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M471">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M472">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M473">View MathML</a>. Further, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M474">View MathML</a>.

Hence,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M475">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M463">View MathML</a>. By a similar method, we can show that the above inequality holds if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M477">View MathML</a>. Now, all the conditions of Theorem 2.1 hold and T has a fixed point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M478">View MathML</a> in

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M479">View MathML</a>

That is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M480">View MathML</a> is the solution to (33). □

Example 3.2 In this example, we denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M481">View MathML</a> the set of real continuous functions on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M482">View MathML</a>. We endow X with the metric

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M483">View MathML</a>

Consider the following continuous functions:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M484">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M485">View MathML</a>

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M486">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M487">View MathML</a>. Then, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M488">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M489">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M490">View MathML</a>

Also, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M491">View MathML</a>. Define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M492">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M493">View MathML</a>

Clearly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M494">View MathML</a>. Also, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M495">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M496">View MathML</a>. On the other hand,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M497">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M498">View MathML</a>. That is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M499">View MathML</a>. Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M501">View MathML</a>.

Assume <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M502">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M503">View MathML</a> with (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M504">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M421">View MathML</a>) or (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M422">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M507">View MathML</a>). Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M508">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M509">View MathML</a>, which implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M510">View MathML</a>. That is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M511">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M512">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M416">View MathML</a>. Further,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M514">View MathML</a>

and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M515">View MathML</a>

Assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M53">View MathML</a> is a sequence in X such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M432">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M56">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M57">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M58">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M521">View MathML</a>. So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M522">View MathML</a>. That is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M418">View MathML</a>.

Therefore, all of the conditions of Theorem 3.1 are satisfied. Then the integral equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M524">View MathML</a>

has a solution in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M525">View MathML</a>. Here, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M526">View MathML</a> is a solution.

But if we chose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M527">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M528">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M529">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M530">View MathML</a>. That is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M531">View MathML</a>

Also,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M532">View MathML</a>

and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/529/mathml/M533">View MathML</a>

That is, Theorem 3.1 of [6] cannot be applied to this example.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors thank the anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper. The third author (V Rakocević) is supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.

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