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Fixed point theorem for weakly Chatterjea-type cyclic contractions

Sumit Chandok1 and Mihai Postolache2*

Author Affiliations

1 Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar, 143001, India

2 Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independenţei, Bucharest, 060042, Romania

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Fixed Point Theory and Applications 2013, 2013:28  doi:10.1186/1687-1812-2013-28


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/28


Received:28 November 2012
Accepted:27 January 2013
Published:11 February 2013

© 2013 Chandok and Postolache; 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 article, we introduce the notion of a Chatterjea-type cyclic weakly contraction and derive the existence of a fixed point for such mappings in the setup of complete metric spaces. Our result extends and improves some fixed point theorems in the literature. Example is given to support the usability of the result.

MSC: 41A50, 47H10, 54H25.

Keywords:
fixed point; cyclic contraction mapping

1 Introduction and preliminaries

It is well known that the fixed point theorem of Banach, for contraction mappings, is one of the pivotal results in analysis. It has been used in many different fields of mathematics but suffers from one major drawback. More accurately, in order to use the contractive condition, a self-mapping T must be Lipschitz continuous, with the Lipschitz constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M1">View MathML</a>. In particular, T must be continuous at all points of its domain.

A natural question arises:

Could we find contractive conditions which will imply the existence of a fixed point in a complete metric space but will not imply continuity?

Kannan [1,2] proved the following result giving an affirmative answer to the above question.

Theorem 1.1If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2">View MathML</a>is a complete metric space and the mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M3">View MathML</a>satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M4">View MathML</a>

(1.1)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M5">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M6">View MathML</a>, thenThas a unique fixed point.

The mappings satisfying (1.1) are called Kannan-type mappings.

A similar type of contractive condition has been studied by Chatterjea [3]. He proved the following result.

Theorem 1.2If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2">View MathML</a>is a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M3">View MathML</a>satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M9">View MathML</a>

(1.2)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M5">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M6">View MathML</a>, thenThas a unique fixed point.

In Theorems 1.1 and 1.2, there is no the requirement for the continuity of T.

Alber and Guerre-Delabriere [4] introduced the concept of weakly contractive mappings and proved the existence of fixed points for single-valued weakly contractive mappings in Hilbert spaces. Thereafter, in 2001, Rhoades [5] proved the fixed point theorem which is one of the generalizations of Banach’s contraction mapping principle because the weakly contractions contain contractions as a special case, and he also showed that some results of [4] are true for any Banach space. In fact, weakly contractive mappings are closely related to the mappings of Boyd and Wong [6] and of Reich types [7].

Fixed point problems involving different types of contractive type inequalities have been studied by many authors (see [1-24] and the references cited therein).

In [22], Kirk et al. introduced the following notion of a cyclic representation and characterized the Banach contraction principle in the context of a cyclic mapping.

Definition 1.1[22]

Let X be a non-empty set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M12">View MathML</a> be an operator. By definition, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M13">View MathML</a> is a cyclic representation of X with respect to T if

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M14">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M15">View MathML</a> are non-empty sets;

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M16">View MathML</a>.

It is the aim of this paper to introduce the notion of a cyclic weakly Chatterjea-type contraction and then derive a fixed point theorem for such cyclic contractions in the framework of complete metric spaces.

2 Main results

To state and prove our main results, we will introduce our notion of a Chatterjea-type cyclic weakly contraction in a metric space. In this respect, let Φ denote the set of all monotone increasing continuous functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M17">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M18">View MathML</a>, if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M19">View MathML</a>, and let Ψ denote the set of all lower semi-continuous functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M20">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M21">View MathML</a>, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M22">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M23">View MathML</a>.

Definition 2.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2">View MathML</a> be a metric space, m be a natural number, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M25">View MathML</a> be non-empty subsets of X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M26">View MathML</a>. An operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M27">View MathML</a> is called a Chatterjea-type cyclic weakly contraction if

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M28">View MathML</a> is a cyclic representation of Y with respect to T;

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M29">View MathML</a>

for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M31">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M32">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M34">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M35">View MathML</a>.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M38">View MathML</a>be non-empty closed subsets ofXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M39">View MathML</a>. Suppose thatTis a Chatterjea-type cyclic weakly contraction. ThenThas a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M40">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M41">View MathML</a>. We can construct a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M42">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M43">View MathML</a> .

If there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M45">View MathML</a>, hence the result. Indeed, we can see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M46">View MathML</a>.

Now, we assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M47">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M48">View MathML</a> . As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M49">View MathML</a>, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M50">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M51">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M52">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M53">View MathML</a>. Since T is a Chatterjea-type cyclic weakly contraction, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M54">View MathML</a>

(2.1)

Since μ is a non-decreasing function, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M55">View MathML</a> , we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M56">View MathML</a>

(2.2)

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M57">View MathML</a>. Thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M58">View MathML</a> is a monotone decreasing sequence of non-negative real numbers and hence is convergent. Therefore, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M59">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M60">View MathML</a>. Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M61">View MathML</a> in (2.2), we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M62">View MathML</a>.

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M61">View MathML</a> in (2.1) and using the continuity of μ and lower semi-continuity of ψ, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M64">View MathML</a>. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M65">View MathML</a>, hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M66">View MathML</a>. Thus we have proved that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M67">View MathML</a>

Now, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68">View MathML</a> is a Cauchy sequence. For this purpose, we prove the following result first.

Lemma 2.1For every positiveϵ, there exists a natural numbernsuch that if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M69">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M70">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M71">View MathML</a>.

Proof Assume the contrary. Thus there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M72">View MathML</a> such that for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M73">View MathML</a>, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M74">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M75">View MathML</a> satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M76">View MathML</a>.

Now, we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M77">View MathML</a>. Then, corresponding to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M78">View MathML</a>, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M79">View MathML</a> such that it is the smallest integer with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M80">View MathML</a> satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M75">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M82">View MathML</a>. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M83">View MathML</a>. By using the triangular inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M84">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85">View MathML</a> and using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M86">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M87">View MathML</a>

(2.3)

Again, by the triangular inequality,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M88">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85">View MathML</a> and using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M86">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M91">View MathML</a>

(2.4)

Consider

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M92">View MathML</a>

(2.5)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M93">View MathML</a>

(2.6)

On taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85">View MathML</a> in inequalities (2.5) and (2.6), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M95">View MathML</a>

(2.7)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M96">View MathML</a>

(2.8)

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M97">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M98">View MathML</a> lie in different adjacently labeled sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M99">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M100">View MathML</a> for certain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M101">View MathML</a>, using the fact that T is a Chatterjea-type cyclic weakly contraction, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M102">View MathML</a>

(2.9)

On taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85">View MathML</a> in (2.9), using (2.7) and (2.8), the continuity of μ and lower semi-continuity of ψ, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M104">View MathML</a>

Consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M105">View MathML</a>, which is contradiction with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M106">View MathML</a>. Hence the result is proved. □

Now, using Lemma 2.1, we will show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68">View MathML</a> is a Cauchy sequence in Y. Fix <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M72">View MathML</a>. By Lemma 2.1, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M110">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M111">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M112">View MathML</a>

(2.10)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M113">View MathML</a>, we can also find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M114">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M115">View MathML</a>

(2.11)

for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M116">View MathML</a>.

Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M117">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M118">View MathML</a>. Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M119">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M120">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M121">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M122">View MathML</a>. So, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M123">View MathML</a>

(2.12)

Using (2.10), (2.11) and (2.12), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M124">View MathML</a>

(2.13)

Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68">View MathML</a> is a Cauchy sequence in Y. Since Y is closed in X, then Y is also complete and there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M126">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M127">View MathML</a>.

Now, we will prove that x is a fixed point of T.

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M128">View MathML</a> is a cyclic representation of Y with respect to T, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68">View MathML</a> has infinite terms in each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M99">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M131">View MathML</a>. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M133">View MathML</a> and we take a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M134">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M68">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M136">View MathML</a>. By using the contractive condition, we can obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M137">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M85">View MathML</a> and using the continuity of μ and lower semi-continuity of ψ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M139">View MathML</a>

which is a contradiction unless <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M140">View MathML</a>. Hence x is a fixed point of T.

Now, we will prove the uniqueness of the fixed point.

Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M141">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M142">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M143">View MathML</a>) are two fixed points of T. Using the contractive condition and the continuity of μ and lower semi continuity of ψ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M144">View MathML</a>

which is a contradiction unless <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M145">View MathML</a>. Hence the main result is proved. □

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M146">View MathML</a>, then we have the following result.

Corrollary 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M38">View MathML</a>be non-empty closed subsets ofXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M150">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M151">View MathML</a>is an operator such that

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M28">View MathML</a>is a cyclic representation ofYwith respect toT;

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M153">View MathML</a>

for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M31">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M156">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M33">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M35">View MathML</a>. ThenThas a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M40">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M160">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M161">View MathML</a>, we have the following result.

Corrollary 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M38">View MathML</a>be non-empty closed subsets ofXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M150">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M151">View MathML</a>is an operator such that

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M28">View MathML</a>is a cyclic representation ofYwith respect toT;

(2) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M161">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M169">View MathML</a>

for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M31">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M156">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M33">View MathML</a>. ThenThas a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M174">View MathML</a>.

3 Applications

Other consequences of our results, for mappings involving contractions of integral type, are given in the following. In this respect, denote by Λ the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M175">View MathML</a> satisfying the following hypotheses:

(h1) μ is a Lebesgue-integrable mapping on each compact of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M176">View MathML</a>;

(h2) for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M72">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M178">View MathML</a>.

Corrollary 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M38">View MathML</a>be non-empty closed subsets ofXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M150">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M151">View MathML</a>is an operator such that

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M28">View MathML</a>is a cyclic representation ofYwith respect toT;

(2) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M161">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M186">View MathML</a>

for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M30">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M31">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M189">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M33">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M191">View MathML</a>. ThenThas a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M40">View MathML</a>.

If we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M193">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M194">View MathML</a>, we obtain the following result.

Corrollary 3.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M2">View MathML</a>be a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M3">View MathML</a>be a mapping such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M197">View MathML</a>

for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M198">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M161">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M200">View MathML</a>. ThenThas a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M40">View MathML</a>.

Example 3.1 Let X be a subset in ℝ endowed with the usual metric. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M202">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M203">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M204">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M27">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M206">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M207">View MathML</a>. It is clear that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M208">View MathML</a> is a cyclic representation of Y with respect to T. Furthermore, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M17">View MathML</a> is given as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M210">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M211">View MathML</a> is given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M212">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M35">View MathML</a>.

Now, we prove that T satisfies the inequality of Chatterjea-type cyclic weakly contraction, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M214">View MathML</a>. To see this fact, we examine three cases.

Case 1. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M215">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M216">View MathML</a>

(3.1)

and

(3.2)

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M218">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M219">View MathML</a>

Hence, the given inequality is satisfied.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M220">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M221">View MathML</a>

Hence the given inequality is satisfied.

Case 2. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M222">View MathML</a>. Then from (3.1) and (3.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M223">View MathML</a>

Hence the given inequality is satisfied.

Case 3. Finally, suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M224">View MathML</a>. Then from (3.1) and (3.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M225">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M226">View MathML</a>

Hence the given inequality is satisfied.

Therefore, all the conditions of Theorem 2.1 are satisfied, and so T has a fixed point (which is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/28/mathml/M227">View MathML</a>).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors are thankful to learned referee(s) for suggestions.

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