Abstract and Applied Analysis

Volume 2012 (2012), Article ID 815870, 19 pages

doi:10.1155/2012/815870

Research Article

Some Fixed-Point Results for a -Weak Contraction in -Metric Spaces

Wasfi Shatanawi¹ and Mihai Postolache²

¹Department of Mathematics, Hashemite University, Zarqa, Jordan

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

Received 7 August 2012; Accepted 4 September 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Wasfi Shatanawi and Mihai Postolache. 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.

Abstract.

We introduce the concepts of a -weak contraction mapping of types and and we establish some fixed point theorems for a -weak contraction mapping of types and in complete -metric spaces. Our results generalize several well-known comparable results in the literature.

1. Introduction

As it is well known, one of the most useful theorems in nonlinear analysis is the Banach contraction principle [1]. Many authors generalized this famous result in different ways. In their outstanding article [2], Mustafa and Sims introduced a new notion of generalized metric space, called -metric space and gave a modification to the contraction principle of Banach. After then, several authors studied various fixed and common fixed point problems for adequate classes of contractive mappings in generalized metric spaces (for an up to date reference list, see [3–28]).

Despite these important advances in nonlinear analysis, our problem, imposed by theoretical and practical reasons as well, had not been studied so far. In this paper, we introduce the concepts of a -weak contraction mapping of types and and we establish some fixed point theorems for a -weak contraction mapping of types and in complete -metric spaces. Our study is encouraged by its possible application, especially in discrete models for numerical analysis, where iterative processes are extensively used due to their versatility for computer simulation. These models play an essential role in applied mathematical studies of certain nonlinear processes in relation with economics, biology, numerical physics, and tribology. Also, we are motivated by several scientists, who pay attention to such kind of problems. On one hand, Berinde [29–31], Berinde and Pcurar [32], and Pcurar [33, 34] studied many existing results for almost contraction mappings in metric spaces. On other hand, Samet and Vetro [35] and Shatanawi [36] studied some existing results for almost contraction mappings in metric spaces.

This paper aims to establish new results for almost contraction mappings on nonlinear analysis on -metric spaces. It is organized as follows. Next, in Section 2 our framework is introduced, in Section 3 our main results are given, while in Section 4 we introduce a nontrivial example and application to support the use ability of our results. Finally, we conclude the paper.

2. Previous Definitions and Results

To make our presentation self-contained, in this section we give basic definitions and previous results, which are used throughout the paper. This background is organized as a whole, given credit to [2] by Mustafa and Sims.

Definition 2.1. Let be a nonempty set and let be a function satisfying the following properties:

(G1) if ,(G2), for all with ,(G3) for all with ,(G4), symmetry in all three variables,(G5) 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. 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. Let be a -metric space. Then the following statements are equivalent:

(1) is -convergent to ;(2) as ;(3) as ;(4) as .

Definition 2.4. 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. Let be a -metric space. Then the following statements are equivalent.

(1)The sequence is -Cauchy.(2)For every , there is such that , for all .

Definition 2.6. A -metric space is called -complete if every -Cauchy sequence in is -convergent in .

Proposition 2.7. Let be a -metric space. Then, for any it follows that:

(i)if , then ;(ii);(iii);(iv);(v);(vi).

3. Main Results

We start with the following definition.

Definition 3.1 (see [37]). The function is called an altering distance function, if the following properties are satisfied.

(1) is continuous and nondecreasing. (2) if and only if .

Let be a -metric space and be a mapping. We set

With this setting, we introduce the following definitions.

Definition 3.2. Let be a -metric space. A mapping is called a -weak contraction of type if and only if there exist two constants and such that

for all .

Definition 3.3. Let be a -metric space and let be altering distance functions. A mapping is called a -weak contraction of type if and only if there exist a constant such that

for all .

Now, we can introduce and prove our main result.

Theorem 3.4. Let be a complete -metric space and let be a -weak contraction of type . Then has a unique fixed point.

Proof. Consider in and define a sequence in such that

If for some , , then , that is, has a fixed point. Thus, we may assume that for all . Now, we finish our proof by the following steps.

Step One. We will show that

Given . Since is -weak contraction, we have

where

From (3.6)-(3.7), we obtain

By , we have

Thus

If for some in ,

then

Thus and hence . Therefore, which is a contradiction. So,

Thus, we get

By (3.14), is a nonincreasing sequence. Hence, there is such that

Taking limit as in (3.15), we obtain

Therefore and hence . Thus

Step Two. We show that is a -Cauchy sequence in . Assume on contrary, then there exists for which we can find two subsequences and of such that is the smallest index for which

This means that

From (3.19), (3.20), and (), we get

Taking limit as and using (3.18), we have

Using again, we obtain

Taking in the above inequality and using (3.18) and (3.22), we arrive at

Again applying (, we have

By using (3.18), (3.22) and on taking limit as in the above inequality, we get

Also,

By taking the limit in the above inequalities and using (3.18), (3.24), we obtain

Now, we have

Letting , using (3.18), (3.22), (3.24), (3.26), (3.28), and the continuity of , , and , we have

Thus and hence , a contradiction. Thus is a -Cauchy sequence in .

Step Three. We show that has a fixed point.

Since is a -Cauchy sequence in the complete -metric space , there is such that

Since is -weak contraction, we have

where

Letting in (3.33) we get

On letting in (3.32) and using the continuity of , , and , we get

Therefore and hence . Thus is a fixed point of .

Step Four. We prove the uniqueness of the fixed point.

In this respect, we proceed by reductio ad absurdum. Suppose that there are two fixed points of , say such that . Since is -weak contraction, we have

where

By (3.36)-(3.37), we obtain

a contradiction. Thus, , and hence . Therefore the fixed point of is unique.

Corollary 3.5. Let be a complete -metric space and let be a -weak contraction of type . Then has a unique fixed point.

Proof. Since is a -weak contraction, there are two constant and such that

Define altering distance functions by and . Then

Thus is -weak contraction. Hence by Theorem 3.4, we conclude that has a unique fixed point.

Corollary 3.6. Let be a complete -metric space and a mapping. Suppose there exists such that

for all . Then has a unique fixed point.

Corollary 3.7. Let be a complete -metric space and let be a mapping. Suppose there exist nonnegative real numbers with such that

for all . Then has a unique fixed point.

Proof. Follows from Corollary 3.6, by noting that

Consider again be a -metric space, and a mapping. We set

Now, we introduce the following definitions.

Definition 3.8. Let be a -metric space. A mapping is called a -weak contraction of type if and only if there exist two constants and such that

for all .

Definition 3.9. Let be a -metric space and let be altering distance functions. A mapping is called a -weak contraction of type if and only if there exist a constant such that

for all .

Following the same arguments as those in the proof of Theorem 3.4, we obtain the following.

Theorem 3.10. Let be a complete -metric space and let be altering distance function. If be a -weak contraction of type , then has a unique fixed point.

The following results are direct consequences of Theorem 3.10.

Corollary 3.11. Let be a complete -metric space and let be a -weak contraction of type . Then has a unique fixed point.

Corollary 3.12. Let be a complete -metric space and let be a mapping. Suppose there exists such that

for all . Then has a unique fixed point.

Corollary 3.13. Let be a complete -metric space and let be a mapping. Suppose there exist nonnegative real numbers with , such that

for all . Then has a unique fixed point.

4. Example and Application

In this section, we introduce an example and application to support the validity of our results.

Example 4.1. Let . Define by

and by

Also, define by and . Then

(a) is a complete nonsymmetric -metric space, (b)for all and , we have

that is, is -weak contraction of type .

Proof. The proof of part (a) follows from [12]. Now, given . We divide the proof of part (b) into the following cases.

Case  1 (). Here, we have and hence

Case  2 (). Here and . Thus

Since

we have

Hence

Case  3 (). Here and .

Subcase 3.1 (). Here

Since

we have

Hence

Subcase 3.2 ( ). Here

Since

we have

Hence

Case  4 (). Here, we have and .

Subcase 4.1 ( ). Here, we have

Hence

Subcase 4.2 (). Here,

Since

we have

Hence

Thus satisfies all the hypotheses of Theorem 3.4. Hence has a unique fixed point. Here is the unique fixed point of .

As an application of our results, we introduce some fixed-point theorems of integral type.

Denote by the set of functions satisfying the following hypotheses:

(1) is a Lebesgue integrable function on each compact subset of , (2)for every , we have .

It is an easy matter, to see that the mapping defined by

is an altering distance function. Now, we have the following result.

Corollary 4.2. Let be a complete -metric space. Let be a mapping. Suppose that there exist such that for , we have

for . Then has a unique fixed point.

Proof. Follows from Theorem 3.4 by taking

Corollary 4.3. Let be a complete -metric space. Let be a mapping. Suppose that there exist such that for , we have

for . Then has a unique fixed point.

Proof. Follows from Theorem 3.10 by taking

5. Conclusion

In this paper we introduced contractive conditions of two kinds, independent of the existing ones, to establish new results on nonlinear analysis on -metric spaces. More accurately, we established fixed point results for two kinds of -weak contraction mappings, in complete -metric spaces. We illustrated our theory with nontrivial example. As applications of our main theorems, we introduced fixed point results for mappings satisfying some contractive conditions of integral type in -metric spaces.

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