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Fixed point theorems for generalized contractions on GP-metric spaces

Nurcan Bilgili1, Erdal Karapınar2* and Peyman Salimi3

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

3 Department of Mathematics, Sahand University of Technology, Tabriz, Iran

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Journal of Inequalities and Applications 2013, 2013:39  doi:10.1186/1029-242X-2013-39


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/39


Received:8 November 2012
Accepted:23 January 2013
Published:6 February 2013

© 2013 Bilgili et al.; licensee Springer

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

Abstract

In this paper, we present two fixed point theorems on mappings, defined on GP-complete GP-metric spaces, which satisfy a generalized contraction property determined by certain upper semi-continuous functions. Furthermore, we illustrate applications of our theorems with a number of examples. Inspired by the work of Jachymski, we also establish equivalences of certain auxiliary maps in the context of GP-complete GP-metric spaces.

MSC: 47H10, 54H25.

Keywords:
fixed point; partial metric space; GP-metric space

1 Introduction and preliminaries

In 1922, Stefan Banach [1] stated his celebrated theorem on the existence and uniqueness of a fixed point of certain self-maps defined on certain metric spaces for the first time. Specifically, this elegant theorem, also known as the Banach contraction mapping principle, can be formulated as follows: any mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M1">View MathML</a> has a unique point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M3">View MathML</a> provided that there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M4">View MathML</a> satisfying the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M5">View MathML</a> for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M7">View MathML</a> is a complete metric space. A mapping T for which the inequality mentioned above holds is called a contraction.

Since its first appearance, the Banach contraction mapping principle has become the main tool to study contractions as they appear abundantly in a wide array of quantitative sciences. Its most well-known application is in ordinary differential equations, particularly, in the proof of the Picard-Lindelöf theorem which guarantees the existence and uniqueness of solutions of first-order initial value problems. It is worth emphasizing that the remarkable strength of the Banach principle originates from the constructive process it provides to identify the fixed point. This notable strength further attracted the attention of not only many prominent mathematicians studying in many branches of mathematics related to nonlinear analysis, but also many researchers who are interested in iterative methods to examine the quantitative problems involving certain mappings and space structures required in their work in various areas such as social sciences, biology, economics, and computer sciences.

Indeed, in 1994, Matthews, a computer scientist who is an expert on semantics, announced in [2] an analog of Banach’s principle in a new space he called a partial metric space. Matthews’s innovative approach was quickly adopted and improved by fixed point theorists (see, e.g., [3-27]) with the aim of discovering analogs of Banach’s principle in the context of partial metric spaces to broaden its applications and enrich the fixed point theory as a result.

A closer look to the work of these distinguished mathematicians after Matthews’s studies reveals that their discoveries can be categorized in terms of the techniques implemented to produce the analogs of Banach’s principle. The first technique is to introduce new space structures with certain properties which guarantee the existence and/or uniqueness of fixed points of contractions. In addition to Matthews’s investigations, cone metric spaces, D-metric spaces, and G-metric spaces (see, e.g., [28-44]) constitute a few of the examples to the first approach. The second technique is to introduce mappings defined on metric spaces satisfying certain new contractive conditions. For example, cyclic contractions and weak ϕ contractions can be listed as a few.

As another example to the first approach mentioned above, Zand and Nezhad [43] recently introduced GP-metric spaces which are a combination of the notions of partial metric spaces and G-metric spaces. Then they proved a number of fixed point theorems on these new spaces for certain type of contractions. In this paper, we exercise the second approach by using the space structure they initiated to prove certain fixed point theorems for generalized contractions. First, we review the necessary notation, definitions, and fundamental results produced on GP-metric spaces that we will need in this work.

Definition 1.1[43]

Let X be a non-empty set. A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M8">View MathML</a> is called a GP-metric if the following conditions are satisfied:

(GP1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M9">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M10">View MathML</a>;

(GP2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M11">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M12">View MathML</a>;

(GP3) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M13">View MathML</a> , symmetry in all three variables;

(GP4) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M14">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M15">View MathML</a>.

Then the pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M16">View MathML</a> is called a GP-metric space.

Example 1.1[43]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M17">View MathML</a> and define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M12">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20">View MathML</a> is a GP-metric space.

Proposition 1.1[43]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>be aGP-metric space, then for anyx, y, zand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M22">View MathML</a>, it follows that

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M23">View MathML</a>;

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M24">View MathML</a>;

(iii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M25">View MathML</a>;

(iv) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M26">View MathML</a>.

Proposition 1.2[43]

EveryGP-metric space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20">View MathML</a>defines a metric space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M28">View MathML</a>where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M29">View MathML</a>

Definition 1.2[43]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20">View MathML</a> be a GP-metric space and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31">View MathML</a> be a sequence of points of X. A point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M32">View MathML</a> is said to be the limit of the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M34">View MathML</a> if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M35">View MathML</a>

Proposition 1.3[43]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20">View MathML</a>be aGP-metric space. Then, for any sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M37">View MathML</a>inXand a point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2">View MathML</a>, the following are equivalent:

(A) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31">View MathML</a>isGP-convergent tox;

(B) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M40">View MathML</a>as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M41">View MathML</a>;

(C) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M42">View MathML</a>as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M43">View MathML</a>.

Definition 1.3[43]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> be a GP-metric space.

(S1) A sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31">View MathML</a> is called a GP-Cauchy if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M46">View MathML</a> exists (and is finite);

(S2) A GP-partial metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> is said to be GP-complete if and only if every GP-Cauchy sequence in X is GP-convergent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M32">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M49">View MathML</a>.

Now, we introduce the following.

Definition 1.4 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> be a GP-metric space.

(M1) A sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M31">View MathML</a> is called 0-GP-Cauchy if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M52">View MathML</a>;

(M2) A GP-metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> is said to be 0-GP-complete if and only if every 0-GP-Cauchy sequence in XGP-converges to a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M32">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M55">View MathML</a>.

Example 1.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M56">View MathML</a> and define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M12">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20">View MathML</a> is a GP-complete GP-metric space. Moreover, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M60">View MathML</a> (where ℚ denotes a set of rational numbers), then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20">View MathML</a> is a 0-GP-complete GP-metric space.

Lemma 1.1 (See [45])

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M20">View MathML</a>be aGP-metric space. Then

(A) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M63">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M9">View MathML</a>;

(B) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M65">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M66">View MathML</a>.

In the rest of this paper, we will denote the positive natural numbers by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M67">View MathML</a> and the natural numbers by ℕ.

2 Main results

In this section, we present our findings on fixed point theorems on 0-GP-complete GP-metric spaces. We first start with the following definition.

Definition 2.1 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> be a GP-metric space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M69">View MathML</a> be a map. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M70">View MathML</a> denote the value

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M71">View MathML</a>

(1)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72">View MathML</a>.

Lemma 2.1If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>is aGP-metric space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M74">View MathML</a>is a map, then, for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M76">View MathML</a>

(2)

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2">View MathML</a>. Then

(3)

The proof is complete. □

Lemma 2.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>be aGP-metric space and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M74">View MathML</a>be a map such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M81">View MathML</a>

(4)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83">View MathML</a>is a function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M84">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2">View MathML</a>satisfies<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M87">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>, then the following hold:

(a) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M89">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>;

(b) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M91">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M92">View MathML</a>.

Proof

(a) From Lemma 2.1, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M93">View MathML</a>

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M87">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>, then by Lemma 1.1(B), we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M96">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. Consequently, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M98">View MathML</a>. Now by condition (4), we deduce that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M99">View MathML</a>

(5)

that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M100">View MathML</a>. Hence, (a) holds.

Clearly, (b) follows from (4), (a), and the fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M101">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. □

Definition 2.2 A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83">View MathML</a> is called upper semi-continuous from the right if for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M104">View MathML</a> and each sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M105">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M106">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M107">View MathML</a>, the equality holds <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M108">View MathML</a>.

Theorem 2.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>be aGP-completeGP-metric space and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110">View MathML</a>be a map such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M111">View MathML</a>

(6)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83">View MathML</a>is an upper semi-continuous function from the right such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M84">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117">View MathML</a>.

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2">View MathML</a>. If there is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M120">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M121">View MathML</a> is a fixed point of T and the uniqueness of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M121">View MathML</a> follows as in the last part of the proof below. Hence, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M87">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. Put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M125">View MathML</a> and construct the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M126">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M127">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M129">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M130">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M131">View MathML</a>. Define the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M132">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M133">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M131">View MathML</a>. From Lemma 2.2(b) we know that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M132">View MathML</a> is a non-increasing sequence. Hence, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M136">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M137">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M138">View MathML</a>. We will show that c must be equal to 0. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M139">View MathML</a>. By taking limitsup as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M138">View MathML</a> in condition (b) of Lemma 2.2, we get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M141">View MathML</a>

(7)

and so, by upper semi-continuity from the right of the function ϕ, we deduce

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M142">View MathML</a>

(8)

which is a contradiction. Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M143">View MathML</a>. Consequently, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M144">View MathML</a>. Next we show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M145">View MathML</a>. Assume the contrary. Then there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M146">View MathML</a> and sequences <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M147">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M148">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M67">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M150">View MathML</a> and such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M151">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M152">View MathML</a>. From the fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M153">View MathML</a>, we can suppose, without loss of generality, that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M154">View MathML</a>. For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M152">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M156">View MathML</a>

(9)

and hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M157">View MathML</a>. Now, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M158">View MathML</a> be such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M159">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M160">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M161">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M162">View MathML</a>

(10)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M161">View MathML</a>. So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M164">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M165">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M166">View MathML</a> and ϕ is upper semi-continuous from the right, we deduce that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M167">View MathML</a>. On the other hand, for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M152">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M169">View MathML</a>

(11)

so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M170">View MathML</a>, a contradiction because <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M171">View MathML</a>. Consequently, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M172">View MathML</a> and thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M126">View MathML</a> is a Cauchy sequence in the GP-complete GP-metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>. Hence, there is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M176">View MathML</a>

(12)

We show that z is a fixed point of T. To this end, we first note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M177">View MathML</a>, so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M178">View MathML</a>. On the other hand, since for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M180">View MathML</a>, it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M181">View MathML</a>

(13)

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M182">View MathML</a> and thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M183">View MathML</a>. Finally, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M184">View MathML</a> be such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M185">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M186">View MathML</a>

(14)

Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M187">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M188">View MathML</a>. This concludes the proof. □

Definition 2.3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M189">View MathML</a> be a given function. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M190">View MathML</a> will denote the value

(15)

Then we obtain the following statement.

Corollary 2.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>be aGP-completeGP-metric space and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110">View MathML</a>be a map such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M194">View MathML</a>

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M195">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83">View MathML</a>is an upper semi-continuous from the right function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M84">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117">View MathML</a>.

Proof Clearly, by taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M201">View MathML</a> in the hypothesis, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M202">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6">View MathML</a>. Then the conditions of Theorem 2.1 hold. This concludes the proof. □

Example 2.1 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M17">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M205">View MathML</a> be defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M206">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> is a GP-complete GP-metric space. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M208">View MathML</a> be defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M209">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M210">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M211">View MathML</a>.

Proof Without loss of generality, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M212">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M213">View MathML</a>

Then the condition of Theorem 2.1 holds and T has a unique fixed point 0 in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M214">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M215">View MathML</a>. □

Example 2.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M17">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M205">View MathML</a> be defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M206">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> is a GP-complete GP-metric space. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M220">View MathML</a> be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M221">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M222">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M223">View MathML</a>.

Proof To prove this example, we need to consider the following cases:

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M224">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M225">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M226">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M227">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M228">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M229">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M230">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M231">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M232">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M230">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M234">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M235">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M236">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M234">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M238">View MathML</a>

Then the condition of Theorem 2.1 holds and T has a unique fixed point 0 in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M214">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M215">View MathML</a>. □

Lemma 2.3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83">View MathML</a>be nondecreasing and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M243">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M84">View MathML</a>.

Theorem 2.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>be aGP-completeGP-metric space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110">View MathML</a>be a map such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M247">View MathML</a>

(16)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M248">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M250">View MathML</a>is a nondecreasing function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M243">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117">View MathML</a>.

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M2">View MathML</a>. If there is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M120">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M121">View MathML</a> is a fixed point of T and the uniqueness of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M121">View MathML</a> follows as in the last part of the proof below. Hence, we will assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M87">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. Put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M125">View MathML</a> and construct the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M126">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M127">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M129">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M130">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. By Lemma 2.2(b),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M269">View MathML</a>

(17)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. Then, since ϕ is nondecreasing, we deduce that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M271">View MathML</a>

(18)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M88">View MathML</a>. Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M273">View MathML</a>. Now, choose an arbitrary <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M146">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M275">View MathML</a>, it follows from Lemma 2.3 that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M171">View MathML</a>, so there is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M277">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M278">View MathML</a>

(19)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M279">View MathML</a>. Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M280">View MathML</a>

(20)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M279">View MathML</a>. So,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M282">View MathML</a>

(21)

and following this process,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M283">View MathML</a>

(22)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M279">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M285">View MathML</a>. Consequently,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M286">View MathML</a>

(23)

and thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M287">View MathML</a> is a Cauchy sequence in the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M288">View MathML</a>-complete <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M288">View MathML</a>-metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>. Hence, there is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M292">View MathML</a>

(24)

We show that z is a fixed point of T. Assume the contrary. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M293">View MathML</a>. For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M294">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M295">View MathML</a>

(25)

From our assumption that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M293">View MathML</a>, it easily follows that there is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M297">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M298">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M299">View MathML</a>. So,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M300">View MathML</a>

(26)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M299">View MathML</a>. Taking limits as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M43">View MathML</a>, we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M303">View MathML</a>

(27)

a contradiction. Consequently, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M183">View MathML</a>. Finally, the uniqueness of z follows as in Theorem 2.1. □

Example 2.3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M305">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M306">View MathML</a> be defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M206">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> is a GP-complete GP-metric space. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M220">View MathML</a> be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M310">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M311">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M223">View MathML</a>.

Proof Without loss of generality, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M212">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M314">View MathML</a>

And so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M315">View MathML</a>

Then the condition of Theorem 2.2 holds and T has a unique fixed point 0 in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M316">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M215">View MathML</a>. □

Example 2.4 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M17">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M205">View MathML</a> be defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M206">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a> is a GP-complete GP-metric space. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M220">View MathML</a> be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M323">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M222">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M223">View MathML</a>.

Proof To prove this example, we need to examine the following cases:

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M326">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M327">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M328">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M329">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M327">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M331">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M332">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M333">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M334">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M335">View MathML</a>

Then the condition of Theorem 2.2 holds and T has a unique fixed point 0 in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M316">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M215">View MathML</a>. □

Corollary 2.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>be aGP-completeGP-metric space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110">View MathML</a>be a map such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M340">View MathML</a>

(28)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M341">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M6">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83">View MathML</a>is a nondecreasing function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M344">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117">View MathML</a>.

Similarly, we have the corollary below.

Corollary 2.3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>be aGP-completeGP-metric space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110">View MathML</a>be a map such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M350">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M351">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M195">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M83">View MathML</a>is a nondecreasing function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M344">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M116">View MathML</a>. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M117">View MathML</a>.

In [46], Jachymski proved the equivalence of auxiliary functions (see Lemma 1). Inspired by the results from this remarkable paper of Jachymski, we finish this paper by stating the following theorem.

Theorem 2.3 (See [46])

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M21">View MathML</a>be aGP-completeGP-metric space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M110">View MathML</a>be a self-mapping. Assume that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M360">View MathML</a>

Then the following statements are equivalent:

(i) there exist functions<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M361">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M362">View MathML</a>

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72">View MathML</a>;

(ii) there exists a function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M364">View MathML</a>such that for any bounded sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M365">View MathML</a>of positive reals, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M366">View MathML</a>implies<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M367">View MathML</a>and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M368">View MathML</a>

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72">View MathML</a>;

(iii) there exists a continuous function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M370">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M371">View MathML</a>and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M372">View MathML</a>

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72">View MathML</a>;

(iv) there exist a function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M374">View MathML</a>and a nondecreasing, right-continuous function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M375">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M376">View MathML</a>and for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M378">View MathML</a>

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72">View MathML</a>;

(v) there exists a continuous and nondecreasing function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M380">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M376">View MathML</a>and for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M85">View MathML</a>with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M383">View MathML</a>

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/39/mathml/M72">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

References

  1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math.. 3, 133–181 (1922)

  2. Matthews, SG: Partial metric topology. Proc. 8th Summer Conference on General Topology and Applications. 183–197 (1994)

  3. Ćirić, LJ, Samet, B, Aydi, H, Vetro, C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput.. 218, 2398–2406 (2011). Publisher Full Text OpenURL

  4. Di Bari, C, Milojević, M, Radenović, S, Vetro, P: Common fixed points for self-mappings on partial metric spaces. Fixed Point Theory Appl.. 2012, Article ID 140 (2012)

  5. Ilić, D, Pavlović, V, Rakocević, V: Some new extensions of Banach’s contraction principle to partial metric space. Appl. Math. Lett.. 24, 1326–1330 (2011). Publisher Full Text OpenURL

  6. Karapinar, E, Erhan, IM: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett.. 24, 1900–1904 (2011). Publisher Full Text OpenURL

  7. Karapinar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl.. 2011, Article ID 4 (2011)

  8. Karapinar, E: A note on common fixed point theorems in partial metric spaces. Miskolc Math. Notes. 12, 185–191 (2011)

  9. Karapinar, E: Some fixed point theorems on the class of comparable partial metric spaces on comparable partial metric spaces. Appl. Gen. Topol.. 12, 187–192 (2011)

  10. Karapinar, E: Ćirić types nonunique fixed point theorems on partial metric spaces. J. Nonlinear Sci. Appl.. 5, 74–83 (2012)

  11. Karapinar, E: Weak ϕ-contraction on partial metric spaces. J. Comput. Anal. Appl.. 14, 206–210 (2012)

  12. Karapinar, E, Shobkolaei, N, Sedghi, S, Vaezpour, SM: A common fixed point theorem for cyclic operators on partial metric spaces. Filomat. 26, 407–414 (2012). Publisher Full Text OpenURL

  13. Karapinar, E, Erhan, I, Ulus, AY: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci.. 6(1), 239–244 (2012)

  14. Nashine, HK, Kadelburg, Z, Radenović, S: Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces. Math. Comput. Model. doi:10.1016/j.mcm.2011.12.019 (2011)

  15. Kopperman, R, Matthews, SG, Pajoohesh, H: What do partial metrics represent. Spatial Representation: Discrete vs. Continuous Computational Models, Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Wadern (2005)

  16. Oltra, S, Valero, O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste. 36, 17–26 (2004)

  17. O’Neill, SJ: Two topologies are better than one. Technical report, University of Warwick, Coventry (1995)

  18. Romaguera, S: Fixed point theorems for generalized contractions on partial metric spaces. Topol. Appl.. 159, 194–199 (2012). Publisher Full Text OpenURL

  19. Romaguera, S: Matkowski’s type theorems for generalized contractions on (ordered) partial metric spaces. Appl. Gen. Topol.. 12, 213–220 (2011)

  20. Romaguera, S, Schellekens, M: Partial metric monoids and semivaluation spaces. Topol. Appl.. 153, 948–962 (2005). Publisher Full Text OpenURL

  21. Romaguera, S, Valero, O: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci.. 19, 541–563 (2009). Publisher Full Text OpenURL

  22. Schellekens, MP: A characterization of partial metrizability: domains are quantifiable. Theor. Comput. Sci.. 305, 409–432 (2003). Publisher Full Text OpenURL

  23. Abedelljawad, T, Karapınar, E, Taş, K: Existence and uniqueness of common fixed point on partial metric spaces. Appl. Math. Lett.. 24, 1894–1899 (2011). Publisher Full Text OpenURL

  24. Altun, I, Sola, F, Simsek, H: Generalized contractions on partial metric spaces. Topol. Appl.. 157, 2778–2785 (2010). Publisher Full Text OpenURL

  25. Altun, I, Erduran, A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl.. 2011, Article ID 508730 (2011)

  26. Aydi, H: Some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci.. 2011, Article ID 647091 (2011)

  27. Aydi, H: Some fixed point results in ordered partial metric spaces. J. Nonlinear Sci. Appl.. 4(2), 1–15 (2011)

  28. Mustafa, Z, Sims, B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal.. 7(2), 289–297 (2006)

  29. Mustafa, Z, Sims, B: Some remarks concerning D-metric spaces. Valencia, Spain, July 2003. (2003)

  30. Mustafa, Z: A new structure for generalized metric spaces with applications to fixed point theory. Ph.D. Thesis, The University of Newcastle, Callaghan, Australia (2005)

  31. Mustafa, Z, Obiedat, H, Awawdeh, F: Some fixed point theorem for mappings on complete G-metric spaces. Fixed Point Theory Appl.. 2008, Article ID 189870 (2008)

  32. Mustafa, Z, Shatanawi, W, Bataineh, F: Existence of fixed point results in G-metric spaces. Int. J. Math. Math. Sci.. 2009, Article ID 283028 (2009)

  33. Mustafa, Z, Obiedat, H: A fixed point theorem of Reich in G-metric spaces. CUBO. 12(1), 83–93 (2010). Publisher Full Text OpenURL

  34. Mustafa, Z, Sims, B: Fixed point theorems for contractive mappings in complete G-metric spaces. Fixed Point Theory Appl.. 2009, Article ID 917175. doi:10.1155/2009/917175 (2009)

  35. Mustafa, Z, Khandagjy, M, Shatanawi, W: Fixed point results on complete G-metric spaces. Studia Sci. Math. Hung.. 48(3), 304–319 (2011)

  36. Mustafa, Z, Awawdeh, F, Shatanawi, W: Fixed point theorem for expansive mappings in G-metric spaces. Int. J. Contemp. Math. Sci.. 5(49-52), 2463–2472 (2010). PubMed Abstract OpenURL

  37. Mustafa, Z, Aydi, H, Karapınar, E: On common fixed points in G-metric spaces using (E.A) property. Comput. Math. Appl. doi:10.1016/j.camwa.2012.03.051 (2012)

  38. Mustafa, Z, Aydi, H, Karapınar, E: Mixed g-monotone property and quadruple fixed point theorems in partially ordered metric spaces. Fixed Point Theory Appl.. 2012, Article ID 71 (2012)

  39. Saadati, R, Vaezpour, SM, Vetro, P, Rhoades, BE: Fixed point theorems in generalized partially ordered G-metric spaces. Math. Comput. Model.. 52, 797–801 (2010). Publisher Full Text OpenURL

  40. Shatanawi, W: Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces. Fixed Point Theory Appl.. 2010, Article ID 181650 (2010)

  41. Shatanawi, W: Some fixed point theorems in ordered G-metric spaces and applications. Abstr. Appl. Anal.. 2011, Article ID 126205 (2011)

  42. Valero, O: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Topol.. 6(2), 229–240 (2005)

  43. Zand, MRA, Nezhad, AD: A generalization of partial metric spaces. J. Contemp. Appl. Math.. 24, 86–93 (2011)

  44. Abbas, M, Khan, AR, Nazir, T: Coupled common fixed point results in two generalized metric spaces. Appl. Math. Comput.. 217, 6328–6336 (2011). Publisher Full Text OpenURL

  45. Aydi, H, Karapinar, E, Salimi, P: Some fixed point results in GP-metric spaces. J. Appl. Math.. 2012, Article ID 891713 (2012)

  46. Jachymski, J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal.. 74, 768–774 (2011). Publisher Full Text OpenURL