Open Access

Fixed Point Theory for Contractive Mappings Satisfying -Maps in -Metric Spaces

Fixed Point Theory and Applications20102010:181650

DOI: 10.1155/2010/181650

Received: 23 March 2010

Accepted: 1 June 2010

Published: 14 June 2010

Abstract

We prove some fixed point results for self-mapping in a complete -metric space under some contractive conditions related to a nondecreasing map with for all . Also, we prove the uniqueness of such fixed point, as well as studying the -continuity of such fixed point.

1. Introduction

The fixed point theorems in metric spaces are playing a major role to construct methods in mathematics to solve problems in applied mathematics and sciences. So the attraction of metric spaces to a large numbers of mathematicians is understandable. Some generalizations of the notion of a metric space have been proposed by some authors. In 2006, Mustafa in collaboration with Sims introduced a new notion of generalized metric space called -metric space [1]. In fact, Mustafa et al. studied many fixed point results for a self-mapping in -metric space under certain conditions; see[15]. In the present work, we study some fixed point results for self-mapping in a complete -metric space under some contractive conditions related to a nondecreasing map with for all .

2. Basic Concepts

In this section, we present the necessary definitions and theorems in -metric spaces.

Definition 2.1 (see [1]).

Let be a nonempty set and let be a function satisfying the following properties:

(1)      if ;

(2) ,  for all with ;

(3)   for all with ;

(4) , symmetry in all three variables;

(5)   for all .

Then the function is called a generalized metric, or, more specifically, a -metric on , and the pair is called a -metric space.

Definition 2.2 (see [1]).

Let be a -metric space, and let be a sequence of points of , a point is said to be the limit of the sequence , if , and we say that the sequence is -convergent to or    -converges to .

Thus, in a -metric space if for any , there exists such that for all .

Proposition 2.3 (see [1]).

Let be a -metric space. Then the following are equivalent.

(1) is -convergent to .

(2) as .

(3) as .

(4) as .

Definition 2.4 (see [1]).

Let be a -metric space; a sequence is called -Cauchy if for every , there is such that , for all ; that is, as

Proposition 2.5 (see [3]).

Let be a -metric space. Then the following are equivalent.

(1)The sequence is -Cauchy.

(2)For every , there is such that , for all .

Definition 2.6 (see [1]).

Let and be -metric spaces, and let be a function. Then is said to be -continuous at a point if and only if for every , there is such that and implies . A function is -continuous at if and only if it is -continuous at all .

Proposition 2.7 (see [1]).

Let and be -metric spaces. Then is -continuous at if and only if it is -sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

Proposition 2.8 (see [1]).

Let be a -metric space. Then the function is jointly continuous in all three of its variables.

The following are examples of -metric spaces.

Example 2.9 (see [1]).

Let be the usual metric space. Define by
(2.1)

for all . Then it is clear that is a -metric space.

Example 2.10 (see [1]).

Let . Define on by
(2.2)

and extend to by using the symmetry in the variables. Then it is clear that is a -metric space.

Definition 2.11 (see [1]).

A -metric space is called -complete if every -Cauchy sequence in is -convergent in .

3. Main Results

Following to Matkowski [6], let be the set of all functions such that be a nondecreasing function with for all . If , then is called -map. If is -map, then it is an easy matter to show that

(1) for all ;

(2) .

From now unless otherwise stated we mean by the -map. Now, we introduce and prove our first result.

Theorem 3.1.

Let be a complete -metric space. Suppose the map satisfies
(3.1)

for all . Then has a unique fixed point (say ) and is -continuous at .

Proof.

Choose . Let , . Assume , for each . Claim is a -Cauchy sequence in : for , we have
(3.2)
given , since and , there is an integer such that
(3.3)
Hence
(3.4)
For with , we claim that
(3.5)
We prove Inequality (3.5) by induction on . Inequality (3.5) holds for by using Inequality (3.4) and the fact that . Assume Inequality (3.5) holds for . For , we have
(3.6)
By induction on , we conclude that Inequality (3.5) holds for all . So is -Cauchy and hence is -convergent to some . For , we have
(3.7)
Letting , and using the fact that is continuous on its variable, we get that . Hence . So is a fixed point of . Now, let be another fixed point of with . Since is a -map, we have
(3.8)
which is a contradiction. So , and hence has a unique fixed point. To Show that is -continuous at , let be any sequence in such that is -convergent to . For , we have
(3.9)

Letting , we get . Hence is -convergent to . So is -continuous at u.

As an application of Theorem 3.1, we have the following results.

Corollary 3.2.

Let be a complete -metric space. Suppose that the map satisfies for :
(3.10)

for all . Then has a unique fixed point (say ).

Proof.

From Theorem 3.1, we conclude that has a unique fixed point say . Since
(3.11)

we have that is also a fixed point to . By uniqueness of , we get .

Corollary 3.3.

Let be a complete -metric space. Suppose that the map satisfies
(3.12)

for all . Then has a unique fixed point (say ) and is -continuous at .

Proof.

follows from Theorem 3.1 by taking .

Corollary 3.4.

Let be a complete -metric space. Suppose there is such that the map satisfies
(3.13)

for all . Then has a unique fixed point (say ) and is -continuous at .

Proof.

Define by . Then it is clear that is a nondecreasing function with for all . Since
(3.14)

the result follows from Theorem 3.1.

The above corollary has been stated in [7, Theorem  5.1.7], and proved by a different way.

Corollary 3.5.

Let be a complete -metric space. Suppose the map satisfies
(3.15)

for all . Then has a unique fixed point (say ) and is -continuous at .

Proof.

Define by . Then it is clear that is a nondecreasing function with for all . Since
(3.16)

the result follows from Theorem 3.1.

Theorem 3.6.

Let be a complete -metric space. Suppose that the map satisfies
(3.17)

for all . Then has a unique fixed point (say ) and is -continuous at .

Proof.

Choose . Let , . Assume , for each . Thus for , we have
(3.18)
If
(3.19)
then
(3.20)
which is impossible. So it must be the case that
(3.21)
and hence
(3.22)
Thus for , we have
(3.23)
The same argument is similar to that in proof of Theorem 3.1; one can show that is a -Cauchy sequence. Since is -complete, we conclude that is -convergent to some . For , we have
(3.24)

Case 1.

(3.25)
then we have
(3.26)

Letting , we conclude that , and hence .

Case 2.

(3.27)
then we have
(3.28)

Letting , we conclude that , and hence .

Case 3.

(3.29)
then we have
(3.30)
Letting , we conclude that , and hence . In all cases, we conclude that is a fixed point of . Let be any other fixed point of such that . Then
(3.31)
which is a contradiction since . Therefore, and hence . To show that is -continuous at , let be any sequence in such that is -convergent to . Then
(3.32)

Let , we get that is -convergent to . Hence is -continuous at .

As an application to Theorem 3.6, we have the following results.

Corollary 3.7.

Let be a complete -metric space. Suppose there is such that the map satisfies
(3.33)

for all . Then has a unique fixed point (say ) and is -continuous at .

Proof.

Define by . Then it is clear that is a nondecreasing function with for all . Since
(3.34)

for all the result follows from Theorem 3.6.

Corollary 3.8.

Let be a complete -metric space. Suppose that the map satisfies:
(3.35)

for all . Then has a unique fixed point (say ) and is -continuous at .

Proof.

It follows from Theorem 3.6 by replacing .

Declarations

Acknowledgments

The author would like to thank the editor of the paper and the referees for their precise remarks to improve the presentation of the paper. This paper is financially supported by the Deanship of the Academic Research at the Hashemite University, Zarqa, Jordan.

Authors’ Affiliations

(1)
Department of Mathematics, Hashemite University

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Copyright

© W. Shatanawi. 2010

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