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Fixed Point Theory and Applications

, 2009:917175 | Cite as

Fixed Point Theorems for Contractive Mappings in Complete Open image in new window -Metric Spaces

Open Access
Research Article

Abstract

We prove some fixed point results for mappings satisfying various contractive conditions on Complete Open image in new window -metric Spaces. Also the Uniqueness of such fixed point are proved, as well as we showed these mappings are Open image in new window -continuous on such fixed points.

Keywords

Basic Concept Differential Geometry Arbitrary Point Applied Science Point Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Metric spaces are playing an increasing role in mathematics and the applied sciences.

Over the past two decades the development of fixed point theory in metric spaces has attracted considerable attention due to numerous applications in areas such as variational and linear inequalities, optimization, and approximation theory.

Different generalizations of the notion of a metric space have been proposed by Gahler [1, 2] and by Dhage [3, 4]. However, HA et al. [5] have pointed out that the results obtained by Gahler for his Open image in new window metrics are independent, rather than generalizations, of the corresponding results in metric spaces, while in [6]the current authors have pointed out that Dhage's notion of a Open image in new window -metric space is fundamentally flawed and most of the results claimed by Dhage and others are invalid.

In 2003 we introduced a more appropriate and robust notion of a generalized metric space as follows.

Definition 1.1 ([7]).

Let X be a nonempty set, and let Open image in new window be a function satisfying the following axioms:

() Open image in new window if Open image in new window ,

() Open image in new window ,

() Open image in new window ,

() Open image in new window (symmetry in all three variables),

() Open image in new window , for all Open image in new window , (rectangle inequality).

Then the function Open image in new window is called a generalized metric, or, more specifically a Open image in new window -metric on Open image in new window , and the pair Open image in new window is called a Open image in new window -metric space.

Example 1.2 ([7]).

We now recall some of the basic concepts and results for Open image in new window -metric spaces that were introduced in ([7]).

Definition 1.3.

Let Open image in new window be a Open image in new window -metric space, let Open image in new window be a sequence of points of Open image in new window , we say that Open image in new window is Open image in new window -convergent to Open image in new window if Open image in new window ; that is, for any Open image in new window there exists Open image in new window such that Open image in new window , for all Open image in new window (throughout this paper we mean by Open image in new window the set of all natural numbers). We refer to Open image in new window as the limit of the sequence Open image in new window and write Open image in new window .

Proposition 1.4.

Let Open image in new window be a Open image in new window -metric space then the following are equivalent.

(1) Open image in new window is Open image in new window -convergent to Open image in new window .

(2) Open image in new window , as Open image in new window .

(3) Open image in new window , as Open image in new window .

Definition.

Let Open image in new window be a Open image in new window -metric space, a sequence Open image in new window is called Open image in new window -Cauchy if given Open image in new window , there is Open image in new window such that Open image in new window for all Open image in new window that is if Open image in new window as Open image in new window .

Proposition 1.6.

In a Open image in new window -metric space Open image in new window , the following are equivalent.

(1)The sequence Open image in new window is Open image in new window -Cauchy.

(2)For every Open image in new window there exists Open image in new window such that Open image in new window for all Open image in new window .

Definition 1.7.

Let Open image in new window and Open image in new window be Open image in new window -metric spaces and let Open image in new window be a function, then Open image in new window is said to be Open image in new window -continuous at a point Open image in new window if given Open image in new window , there exists Open image in new window such that Open image in new window ; Open image in new window implies Open image in new window . A function Open image in new window is Open image in new window -continuous on Open image in new window if and only if it is Open image in new window -continuous at all Open image in new window .

Proposition 1.8.

Let Open image in new window , Open image in new window be Open image in new window -metric spaces, then a function Open image in new window is Open image in new window -continuous at a point Open image in new window if and only if it is Open image in new window -sequentially continuous at Open image in new window ; that is, whenever Open image in new window is Open image in new window -convergent to Open image in new window , Open image in new window is Open image in new window -convergent to Open image in new window .

Proposition 1.9.

Let Open image in new window be a Open image in new window -metric space, then the function Open image in new window is jointly continuous in all three of its variables.

Definition 1.10.

A Open image in new window -metric space Open image in new window is said to be Open image in new window -complete (or a complete Open image in new window -metric space) if every Open image in new window -Cauchy sequence in Open image in new window is Open image in new window -convergent in Open image in new window .

2. The Main Results

We begin with the following theorem.

Theorem 2.1.

Let Open image in new window be a complete Open image in new window -metric space and let Open image in new window be a mapping which satisfies the following condition, for all Open image in new window ,

where Open image in new window . Then Open image in new window has a unique fixed point (say Open image in new window ) and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

Suppose that Open image in new window satisfies condition (2.1), let Open image in new window be an arbitrary point, and define the sequence Open image in new window by Open image in new window , then by (2.1), we have
But, by (G5), we have
So, (2.3)becomes
So, it must be the case that
which implies
Let Open image in new window , then Open image in new window and by repeated application of (2.7), we have
Then, for all Open image in new window we have by repeated use of the rectangle inequality and (2.8) that
taking the limit as Open image in new window , and using the fact that the function Open image in new window is continuous on its variables, we have Open image in new window , which is a contradiction since Open image in new window . So, Open image in new window . To prove uniqueness, suppose that Open image in new window is such that Open image in new window , then (2.1) implies that Open image in new window , thus Open image in new window again by the same argument we will find Open image in new window , thus
and we deduce that
but (G5) implies that

and (2.13) leads to the following cases,

(1) Open image in new window ,

(2) Open image in new window

(3) Open image in new window

In each case take the limit as Open image in new window to see that Open image in new window and so, by Proposition 1.4, we have that the sequence Open image in new window is Open image in new window -convergent to Open image in new window , therefor Proposition 1.8 implies that Open image in new window is Open image in new window -continuous at Open image in new window .

Remark 2.2.

If the Open image in new window -metric space is bounded (that is, for some Open image in new window we have Open image in new window for all Open image in new window ) then an argument similar to that used above establishes the result for Open image in new window .

Corollary 2.3.

Let Open image in new window be a complete Open image in new window -metric spaces and let Open image in new window be a mapping which satisfies the following condition for some Open image in new window and for all Open image in new window :

where Open image in new window , then Open image in new window has a unique fixed point (say Open image in new window ), and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

From the previous theorem, we have that Open image in new window has a unique fixed point (say u), that is, Open image in new window . But Open image in new window , so Open image in new window is another fixed point for Open image in new window and by uniqueness Open image in new window .

Theorem 2.4.

Let Open image in new window be a complete Open image in new window -metric space, and let Open image in new window be a mapping which satisfies the following condition for all Open image in new window

where Open image in new window , then Open image in new window has a unique fixed point (say Open image in new window ), and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

Suppose that Open image in new window satisfies the condition (2.16), let Open image in new window be an arbitrary point, and define the sequence Open image in new window by Open image in new window , then by (2.16) we get
since Open image in new window , then it must be the case that
but from (G5), we have
so (2.18) implies that
let Open image in new window , then Open image in new window and by repeated application of (2.20), we have
Taking the limit as Open image in new window , and using the fact that the function Open image in new window is continuous in its variables, we get
since Open image in new window , this contradiction implies that Open image in new window .To prove uniqueness, suppose that Open image in new window such that Open image in new window , then
so we deduce that Open image in new window . This implies that Open image in new window and by repeated use of the same argument we will find Open image in new window . Therefor we get Open image in new window , since Open image in new window , this contradiction implies that Open image in new window . To show that Open image in new window is Open image in new window -continuous at Open image in new window let Open image in new window be a sequence such that Open image in new window in Open image in new window , then
Thus, (2.25) becomes
but by (G5) we have Open image in new window , therefor (2.26) implies that Open image in new window and we deduce that

Taking the limit of (2.27) as Open image in new window , we see that Open image in new window and so, by Proposition 1.8, we have Open image in new window which implies that Open image in new window is Open image in new window -continuous at Open image in new window .

Corollary 2.5.

Let Open image in new window be a complete Open image in new window -metric space, and let Open image in new window be a mapping which satisfies the following condition for some Open image in new window and for all Open image in new window

where Open image in new window , then Open image in new window has a unique fixed point (say Open image in new window ), and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

The proof follows from the previous theorem and the same argument used in Corollary 2.3.

Theorem 2.6.

Let Open image in new window be a complete Open image in new window -metric space, and let Open image in new window be a mapping which satisfies the following condition, for all Open image in new window

where Open image in new window , then Open image in new window has a unique fixed point, say Open image in new window , and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

Suppose that Open image in new window satisfies the condition (2.29). Let Open image in new window be an arbitrary point, and define the sequence Open image in new window by Open image in new window , then by (2.29), we have
But by (G5) we have
taking the limit as Open image in new window , and using the fact that the function Open image in new window is continuous in its variables, we obtain Open image in new window . Since Open image in new window this is a contradiction so, Open image in new window . To prove uniqueness, suppose that Open image in new window is such that Open image in new window , then
thus Open image in new window and we deduce that
By the same argument we get

therefore, (2.38) implies two cases.

Case 1.

Open image in new window .

Case 2.

Open image in new window .

But, by (G5) we have Open image in new window , so case 2 implies that Open image in new window In each case taking the limit as Open image in new window , we see that Open image in new window and so, by Proposition 1.8, we have Open image in new window which implies that Open image in new window is Open image in new window -continuous at Open image in new window .

Corollary 2.7.

Let Open image in new window be a complete Open image in new window -metric spaces, and let Open image in new window be a mapping which satisfies the following condition for some Open image in new window and for all Open image in new window

where Open image in new window , then Open image in new window has a unique fixed point, say Open image in new window , and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

The proof follows from the previous theorem and the same argument used in Corollary 2.3. The following theorem has been stated in [8] without proof, but this can be proved by using Theorem (2.6) as follows.

Theorem 2.8 ([8]).

Let Open image in new window be a complete Open image in new window -metric space and let Open image in new window be a mapping which satisfies the following condition, for all Open image in new window

where Open image in new window , then Open image in new window has a unique fixed point, say Open image in new window , and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

Setting Open image in new window in condition (2.40), then it reduced to condition (2.29), and the proof follows from Theorem (2.6).

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Copyright information

© Z. Mustafa and B. Sims. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsThe Hashemite UniversityZarqaJordan
  2. 2.School of Mathematical and Physical SciencesThe University of NewcastleAustralia

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