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Cyclic contractions via auxiliary functions on G-metric spaces

Nurcan Bilgili1 and Erdal Karapınar2*

Author Affiliations

1 Department of Mathematics, Institute of Science and Technology, Gazi University, Ankara, 06500, Turkey

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

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Fixed Point Theory and Applications 2013, 2013:49  doi:10.1186/1687-1812-2013-49

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/49


Received:24 October 2012
Accepted:21 February 2013
Published:8 March 2013

© 2013 Bilgili and Karapınar; licensee Springer

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

Abstract

In this paper, we prove the existence and uniqueness of fixed points of certain cyclic mappings via auxiliary functions in the context of G-metric spaces, which were introduced by Zead and Sims. In particular, we extend, improve and generalize some earlier results in the literature on this topic.

MSC: 47H10, 54H25.

Keywords:
fixed point; G-metric space; cyclic maps; cyclic contractions

1 Introduction and preliminaries

It is well established that fixed point theory, which mainly concerns the existence and uniqueness of fixed points, is today’s one of the most investigated research areas as a major subfield of nonlinear functional analysis. Historically, the first outstanding result in this field that guaranteed the existence and uniqueness of fixed points was given by Banach [1]. This result, known as the Banach mapping contraction principle, simply states that every contraction mapping has a unique fixed point in a complete metric space. Since the first appearance of the Banach principle, the ever increasing application potential of the fixed point theory in various research fields, such as physics, chemistry, certain engineering branches, economics and many areas of mathematics, has made this topic more crucial than ever. Consequently, after the Banach celebrated principle, many authors have searched for further fixed point results and reported successfully new fixed point theorems conceived by the use of two very effective techniques, combined or separately.

The first one of these techniques is to ‘replace’ the notion of a metric space with a more general space. Quasi-metric spaces, partial metric spaces, rectangular metric spaces, fuzzy metric space, b-metric spaces, D-metric spaces, G-metric spaces are generalizations of metric spaces and can be considered as examples of ‘replacements’. Amongst all of these spaces, G-metric spaces, introduced by Zead and Sims [2], are ones of the interesting. Therefore, in the last decade, the notion of a G-metric space has attracted considerable attention from researchers, especially from fixed point theorists [3-25].

The second one of these techniques is to modify the conditions on the operator(s). In other words, it entails the examination of certain conditions under which the contraction mapping yields a fixed point. One of the attractive results produced using this approach was given by Kirk et al.[26] in 2003 through the introduction of the concepts of cyclic mappings and best proximity points. After this work, best proximity theorems and, in particular, the fixed point theorems in the context of cyclic mappings have been studied extensively (see, e.g., [27-43]).

The two upper mentioned topics, cyclic mappings and G-metric spaces, have been combined by Aydi in [22] and Karapınar et al. in [36]. In these papers, the existence and uniqueness of fixed points of cyclic mappings are investigated in the framework of G-metric spaces. In this paper, we aim to improve on certain statements proved on these two topics. For the sake of completeness, we will include basic definitions and crucial results that we need in the rest of this work.

Mustafa and Sims [2] defined the concept of G-metric spaces as follows.

Definition 1.1 (See [2])

Let X be a nonempty set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M1">View MathML</a> be a function satisfying the following properties:

(G1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M2">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M3">View MathML</a>,

(G2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M4">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M5">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M6">View MathML</a>,

(G3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M7">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M8">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M9">View MathML</a>,

(G4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M10">View MathML</a> (symmetry in all three variables),

(G5) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M11">View MathML</a> (rectangle inequality) for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M12">View MathML</a>.

Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a> is called a G-metric space.

Note that every G-metric on X induces a metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M14">View MathML</a> on X defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M15">View MathML</a>

(1)

For a better understanding of the subject, we give the following examples of G-metrics.

Example 1.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M16">View MathML</a> be a metric space. The function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M17">View MathML</a>, defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M18">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M19">View MathML</a>, is a G-metric on X.

Example 1.2 (See, e.g., [2])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M20">View MathML</a>. The function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M21">View MathML</a>, defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M22">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M19">View MathML</a>, is a G-metric on X.

In their initial paper, Mustafa and Sims [2] also defined the basic topological concepts in G-metric spaces as follows.

Definition 1.2 (See [2])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24">View MathML</a> be a G-metric space, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a> be a sequence of points of X. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a> is G-convergent to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M27">View MathML</a> if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M28">View MathML</a>

that is, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M31">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M32">View MathML</a>. We call x the limit of the sequence and write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M33">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M34">View MathML</a>.

Proposition 1.1 (See [2])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24">View MathML</a>be aG-metric space. The following are equivalent:

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a>isG-convergent tox,

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M37">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M38">View MathML</a>,

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M39">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M38">View MathML</a>,

(4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M41">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M42">View MathML</a>.

Definition 1.3 (See [2])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24">View MathML</a> be a G-metric space. A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a> is called a G-Cauchy sequence if, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M47">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M48">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M49">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M50">View MathML</a>.

Proposition 1.2 (See [2])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24">View MathML</a>be aG-metric space. Then the following are equivalent:

(1) the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a>isG-Cauchy,

(2) for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M30">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M55">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M56">View MathML</a>.

Definition 1.4 (See [2])

A G-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a> is called G-complete if every G-Cauchy sequence is G-convergent in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>.

Definition 1.5 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a> be a G-metric space. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M60">View MathML</a> is said to be continuous if for any three G-convergent sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M62">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M63">View MathML</a> converging to x, y and z respectively, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M64">View MathML</a> is G-convergent to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M65">View MathML</a>.

Note that each G-metric on X generates a topology <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M66">View MathML</a> on X whose base is a family of open G-balls <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M67">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M68">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M27">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29">View MathML</a>. A nonempty set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M71">View MathML</a> is G-closed in the G-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M73">View MathML</a>. Observe that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M74">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29">View MathML</a>. We recall also the following proposition.

Proposition 1.3 (See, e.g., [36])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M24">View MathML</a>be aG-metric space andAbe a nonempty subset ofX. The setAisG-closed if for anyG-convergent sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a>inAwith limitx, we have<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M78">View MathML</a>.

Mustafa [5] extended the well-known Banach contraction principle mapping in the framework of G-metric spaces as follows.

Theorem 1.1 (See [5])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be a completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M80">View MathML</a>be a mapping satisfying the following condition for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M81">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M82">View MathML</a>

(2)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M83">View MathML</a>. ThenThas a unique fixed point.

Theorem 1.2 (See [5])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be a completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M80">View MathML</a>be a mapping satisfying the following condition for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M86">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M87">View MathML</a>

(3)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M83">View MathML</a>. ThenThas a unique fixed point.

Remark 1.1 We notice that the condition (2) implies the condition (3). The converse is true only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M89">View MathML</a>. For details, see [5].

Lemma 1.1 ([5])

By the rectangle inequality (G5) together with the symmetry (G4), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M90">View MathML</a>

(4)

A map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M91">View MathML</a> on a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M16">View MathML</a> is called a weak ϕ-contraction if there exists a strictly increasing function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M93">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M94">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M95">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M5">View MathML</a>. We notice that these types of contractions have also been a subject of extensive research (see, e.g., [44-49]). In what follows, we recall the notion of cyclic weak ψ-contractions on G-metric spaces. Let Ψ be the set of continuous functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M93">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M98">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M99">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M100">View MathML</a>. In [36], the authors concentrated on two types of cyclic contractions: cyclic-type Banach contractions and cyclic weak ϕ-contractions.

Theorem 1.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102">View MathML</a>be a family of nonemptyG-closed subsets ofXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M103">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M104">View MathML</a>be a map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M105">View MathML</a>

(5)

Suppose that there exists a function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M106">View MathML</a>such that the mapTsatisfies the inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M107">View MathML</a>

(6)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M108">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M109">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M110">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M111">View MathML</a>

(7)

ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>.

The following result, which can be considered as a corollary of Theorem 1.3, is stated in [36].

Theorem 1.4 (See [36])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102">View MathML</a>be a family of nonemptyG-closed subsets ofX. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M115">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M104">View MathML</a>be a map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M117">View MathML</a>

(8)

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M118">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M119">View MathML</a>

(9)

holds for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M108">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M109">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M110">View MathML</a>, thenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>.

In this paper, we extend, generalize and enrich the results on the topic in the literature.

2 Main results

We start this section by defining some sets of auxiliary functions. Let ℱ denote all functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M124">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M125">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M126">View MathML</a>. Let Ψ and Φ be the subsets of ℱ such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M127">View MathML</a>

Lemma 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a>be a sequence inXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M130">View MathML</a>is nonincreasing,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M131">View MathML</a>

(10)

If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a>is not a Cauchy sequence, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29">View MathML</a>and two sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M134">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M135">View MathML</a>of positive integers such that the following sequences tend toεwhen<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M136">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M137">View MathML</a>

(11)

Proof

Due to Lemma 1.1, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M138">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M139">View MathML</a> regarding the assumption of the lemma, we derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M140">View MathML</a>

(12)

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a> is not G-Cauchy, then, due to Proposition 1.2, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29">View MathML</a> and corresponding subsequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M143">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M144">View MathML</a> of ℕ satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M145">View MathML</a> for which

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M146">View MathML</a>

(13)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M147">View MathML</a> is chosen as the smallest integer satisfying (13), that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M148">View MathML</a>

(14)

By (13), (14) and the rectangle inequality (G5), we easily derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M149">View MathML</a>

(15)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150">View MathML</a> in (15) and using (10), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M151">View MathML</a>

(16)

Further,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M152">View MathML</a>

(17)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M153">View MathML</a>

(18)

Passing to the limit when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150">View MathML</a> and using (10) and (16), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M155">View MathML</a>

(19)

In a similar way,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M156">View MathML</a>

(20)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M157">View MathML</a>

(21)

Passing to the limit when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150">View MathML</a> and using (10) and (16), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M159">View MathML</a>

(22)

Furthermore,

(23)

and

(24)

Passing to the limit when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150">View MathML</a> and using (10) and (16), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M163">View MathML</a>

(25)

By regarding the assumptions (G3) and (G5) together with the expression (13), we derive the following:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M164">View MathML</a>

(26)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150">View MathML</a> in the inequality above and using (12) and (16), we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M166">View MathML</a>

(27)

 □

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102">View MathML</a>be a family of nonemptyG-closed subsets ofXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170">View MathML</a>be a map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M171">View MathML</a>

(28)

Suppose that there exist functions<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M172">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M173">View MathML</a>such that the mapTsatisfies the inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M174">View MathML</a>

(29)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M178">View MathML</a>

(30)

ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>.

Proof First we show the existence of a fixed point of the map T. For this purpose, we take an arbitrary <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M180">View MathML</a> and define a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a> in the following way:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M182">View MathML</a>

(31)

We have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M180">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M184">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M185">View MathML</a>, … since T is a cyclic mapping. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M186">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M187">View MathML</a>, then, clearly, the fixed point of the map T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M188">View MathML</a>. From now on, assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M189">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M190">View MathML</a>. Consider the inequality (29) by letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M191">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M192">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M193">View MathML</a>

(32)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M194">View MathML</a>

(33)

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M195">View MathML</a>, then the expression (32) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M196">View MathML</a>

(34)

So, the inequality (34) yields <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M197">View MathML</a>. Thus, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M198">View MathML</a>

This contradicts the assumption <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M199">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M190">View MathML</a>. So, we derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M201">View MathML</a>

(35)

Hence the inequality (32) turns into

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M202">View MathML</a>

(36)

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M203">View MathML</a> is a nonnegative, nonincreasing sequence that converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M204">View MathML</a>. We will show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M205">View MathML</a>. Suppose, on the contrary, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M206">View MathML</a>. Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M207">View MathML</a> in (36), we derive that

(37)

By the continuity of ψ and the lower semi-continuity of ϕ, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M209">View MathML</a>

(38)

Then it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M210">View MathML</a>. Therefore, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M205">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M212">View MathML</a>

(39)

Lemma 1.1 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M191">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M214">View MathML</a> implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M215">View MathML</a>

(40)

So, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M216">View MathML</a>

(41)

Next, we will show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a> is a G-Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>. Suppose, on the contrary, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a> is not G-Cauchy. Then, due to Proposition 1.2, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M29">View MathML</a> and corresponding subsequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M143">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M144">View MathML</a> of ℕ satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M145">View MathML</a> for which

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M224">View MathML</a>

(42)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M147">View MathML</a> is chosen as the smallest integer satisfying (42), that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M226">View MathML</a>

(43)

By (42), (43) and the rectangle inequality (G5), we easily derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M227">View MathML</a>

(44)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M150">View MathML</a> in (44) and using (39), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M229">View MathML</a>

(45)

Notice that for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M230">View MathML</a> there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M231">View MathML</a> satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M232">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M233">View MathML</a>

(46)

Thus, for large enough values of k, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M234">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M235">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M236">View MathML</a> lie in the adjacent sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M237">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M238">View MathML</a> respectively for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M239">View MathML</a>. When we substitute <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M240">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M241">View MathML</a> in the expression (29), we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M242">View MathML</a>

(47)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M243">View MathML</a>

(48)

By using Lemma 2.1, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M244">View MathML</a>

(49)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M245">View MathML</a>

(50)

So, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M246">View MathML</a>

(51)

So, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M247">View MathML</a>. We deduce that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M248">View MathML</a>. This contradicts the assumption that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M249">View MathML</a> is not G-Cauchy. As a result, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M25">View MathML</a> is G-Cauchy. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a> is G-complete, it is G-convergent to a limit, say <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M252">View MathML</a>. It easy to see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M253">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M180">View MathML</a>, then the subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M255">View MathML</a>, the subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M256">View MathML</a> and, continuing in this way, the subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M257">View MathML</a>. All the m subsequences are G-convergent in the G-closed sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M237">View MathML</a> and hence they all converge to the same limit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M253">View MathML</a>. To show that the limit w is the fixed point of T, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M260">View MathML</a>, we employ (29) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M191">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M262">View MathML</a>. This leads to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M263">View MathML</a>

(52)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M264">View MathML</a>

(53)

Passing to limsup as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M265">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M266">View MathML</a>

(54)

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M267">View MathML</a> and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M268">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M260">View MathML</a>.

Finally, we prove that the fixed point is unique. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M270">View MathML</a> is another fixed point of T such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M271">View MathML</a>. Then, since both v and w belong to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>, we set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M273">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M262">View MathML</a> in (29), which yields

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M275">View MathML</a>

(55)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M276">View MathML</a>

(56)

On the other hand, by setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M277">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M278">View MathML</a> in (29), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M279">View MathML</a>

(57)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M280">View MathML</a>

(58)

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M281">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M282">View MathML</a>. Indeed, by definition, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M283">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M282">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M285">View MathML</a>, then by (56) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M286">View MathML</a> and by (55),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M287">View MathML</a>

(59)

and, clearly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M288">View MathML</a>. So, we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M282">View MathML</a>. Otherwise, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M290">View MathML</a>. Then by (58), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M291">View MathML</a> and by (57),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M292">View MathML</a>

(60)

and, clearly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M293">View MathML</a>. So, we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M282">View MathML</a>. Hence the fixed point of T is unique. □

Remark 2.1 We notice that some fixed point result in the context of G-metric can be obtained by usual (well-known) fixed point theorems (see, e.g., [50,51]). In fact, this is not a surprising result due to strong relationship between the usual metric and G-metric space (see, e.g., [2,3,5]). Note that a G-metric space tells about the distance of three points instead of two points, which makes it original. We also emphasize that the techniques used in [50,51] are not applicable to our main theorem.

To illustrate Theorem 2.1, we give the following example.

Example 2.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M295">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M296">View MathML</a> be given as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M297">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M298">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M299">View MathML</a>. Define the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M300">View MathML</a> as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M301">View MathML</a>

(61)

Clearly, the function G is a G-metric on X. Define also <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M302">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M303">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M304">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M305">View MathML</a>. Obviously, the map T has a unique fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M306">View MathML</a>.

It can be easily shown that the map T satisfies the condition (29). Indeed,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M307">View MathML</a>

which yields

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M308">View MathML</a>

(62)

Moreover, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M309">View MathML</a>

(63)

We derive from (63) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M310">View MathML</a>

(64)

On the other hand, we have the following inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M311">View MathML</a>

(65)

By elementary calculation, we conclude from (65) and (64) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M312">View MathML</a>

(66)

Combining the expressions (62) and (65), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M313">View MathML</a>

(67)

Hence, all conditions of Theorem 2.1 are satisfied. Notice that 0 is the unique fixed point of T.

For particular choices of the functions ϕ, ψ, we obtain the following corollaries.

Corollary 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102">View MathML</a>be a family of nonemptyG-closed subsets ofXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170">View MathML</a>be a map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M318">View MathML</a>

(68)

Suppose that there exists a constant<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M319">View MathML</a>such that the mapTsatisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M320">View MathML</a>

(69)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M324">View MathML</a>

(70)

ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>.

Proof The proof is obvious by choosing the functions ϕ, ψ in Theorem 2.1 as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M326">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M327">View MathML</a>. □

Corollary 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102">View MathML</a>be a family of nonemptyG-closed subsets ofXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170">View MathML</a>be a map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M332">View MathML</a>

(71)

Suppose that there exist constantsa, b, c, dandewith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M333">View MathML</a>and there exists a function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M173">View MathML</a>such that the mapTsatisfies the inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M335">View MathML</a>

(72)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M338">View MathML</a>. ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>.

Proof

Clearly, we have

(73)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M341">View MathML</a>

(74)

By Corollary 2.1, the map T has a unique fixed point. □

Corollary 2.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102">View MathML</a>be a family of nonemptyG-closed subsets ofXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170">View MathML</a>be a map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M346">View MathML</a>

Suppose that there exist functions<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M172">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M173">View MathML</a>such that the mapTsatisfies the inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M349">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M353">View MathML</a>

(75)

ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>.

Proof The expression (75) coincides with the expression (30). Following the lines in the proof of Theorem 2.1, by letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M191">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M356">View MathML</a>, we get the desired result. □

Cyclic maps satisfying integral type contractive conditions are amongst common applications of fixed point theorems. In this context, we consider the following applications.

Corollary 2.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102">View MathML</a>be a family of nonemptyG-closed subsets ofXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170">View MathML</a>be a map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M361">View MathML</a>

Suppose also that there exist functions<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M172">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M173">View MathML</a>such that the mapTsatisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M364">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M365">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177">View MathML</a>. ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>.

Corollary 2.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M13">View MathML</a>be aG-completeG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M102">View MathML</a>be a family of nonemptyG-closed subsets ofXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M169">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M170">View MathML</a>be a map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M374">View MathML</a>

Suppose also that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M375">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M319">View MathML</a>and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M377">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M176">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M177">View MathML</a>. ThenThas a unique fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M112">View MathML</a>.

Proof The proof is obvious by choosing the function ϕ, ψ in Corollary 2.4 as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M326">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/49/mathml/M327">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

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