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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 985080, 10 pages

http://dx.doi.org/10.1155/2014/985080
Research Article

The Generalized Weaker -Contractive Mappings and Related Fixed Point Results in Complete Generalized Metric Spaces

1Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 4087, Jeddah 21491, Saudi Arabia

2Department of Applied Mathematics, National Hsinchu University of Education, Taiwan

3Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

4Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia

Received 6 February 2014; Accepted 7 May 2014; Published 26 May 2014

Academic Editor: Qamrul Hasan Ansari

Copyright © 2014 Maryam A. Alghamdi et al. 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 notion of generalized weaker -contractive mappings in the context of generalized metric space. We investigate the existence and uniqueness of fixed point of such mappings. Some consequences on existing fixed point theorems are also derived. The presented results generalize, unify, and improve several results in the literature.

1. Introduction and Preliminaries

In [1], Branciari introduced the notion of generalized metric space by weakening the triangular inequality of metric assumption with quadrilateral inequality. The author [1] characterized and proved the analog of famous Banach fixed point theorem in the setting of generalized metric space. Although the theorem of Branciari [1] is correct, the proofs had gaps [2] since the topology of generalized metric space is not strong enough as the topology of metric space. The disadvantages of generalized metric space can be listed as follows: generalized metric need not be continuous; a convergent sequence in generalized metric space need not be Cauchy; generalized metric space need not be Hausdorff, and hence the uniqueness of limits cannot be guaranteed.Despite the weakness of the topology of generalized metric space, in [3, 4], the authors suggested some techniques to get a (unique) fixed point in such spaces.

On the other hand, Samet et al. [5] introduced the notion of - contraction mappings and proved the existence and uniqueness of such mappings in complete metric space. The results of this paper are very impressive since several existing results derived from the main theorem of Samet et al. [5] quiet easily. Later, a number of authors have appreciated these results and have used this technique to get further generalization via - contraction mappings; see, for example, [610].

In this paper, we introduce the generalized weaker - contraction mappings in the setting of generalized metric spaces. Consequently, we investigate the existence and uniqueness of fixed point by caring the problems mentioned above.

Let us recall basic definitions and notations and interesting results that will be in the sequel.

Let be the family of functions satisfying the following conditions:(i) is nondecreasing;(ii)there exist and and a convergent series of nonnegative terms such that for and any .In the literature such functions are called either Bianchini-Grandolfi gauge functions (see, e.g., [1113]) or -comparison functions (see, e.g., [14]).

Lemma 1 (see, e.g., [14]). If , then the following hold:(i) converges to as for all ;(ii) , for any ;(iii) is continuous at ;(iv)the series converges for any .

In the following, we recall the notion of generalized metric spaces.

Definition 2 (see [1]). Let be a nonempty set and let satisfy the following conditions for all and all distinct each of which is different from and : Then, the map is called generalized metric. Here, the pair is called a generalized metric space and abbreviated as GMS.

In the above definition, if satisfies only (GMS1) and (GMS2), then it is called semimetric (see, e.g., [15]).

The concepts of convergence, Cauchy sequence, and completeness in a GMS are defined as follows.

Definition 3. (1) A sequence in a GMS is GMS convergent to a limit if and only if as .

(2) A sequence in a GMS is GMS Cauchy if and only if for every there exists positive integer such that for all .

(3) A GMS is called complete if every GMS Cauchy sequence in is GMS convergent.

The following assumption was suggested by Wilson [15] to replace the triangle inequality with the weakened condition.

For each pair of (distinct) points there is a number such that, for every ,

Proposition 4 (see [3]). In a semimetric space, the assumption ( ) is equivalent to the assertion that limits are unique.

Proposition 5 (see [3]). Suppose that is a Cauchy sequence in a GMS with , where . Then for all . In particular, the sequence does not converge to if .

A function is said to be a Meir-Keeler function [16] if, for each , there exists such that for with , we have . Such mapping has been improved and used by several authors [17, 18]. In what follows we recall the notion of weaker Meir-Keeler function.

Definition 6 (see, e.g., [19]). We call a weaker Meir-Keeler function if for each , there exists such that for with , there exists such that .

Let be the class of all nondecreasing function satisfying the following conditions: is a weaker Meir-Keeler function; for all , ; for all , is decreasing; if , then .Let be the class of functions satisfying the following conditions: is continuous; for and .By using the auxiliary functions, defined above, Chen and Sun [19] proved the following theorem.

Theorem 7. Let be a Hausdorff and complete generalized metric space, and let be a function satisfying for all and , . Then has a periodic point in ; that is, there exists such that for some .

Another interesting auxiliary function, -admissible, was defined by Samet et al. [5].

Definition 8 (see [5]). For a nonempty set , let and be mappings. We say that is -admissible if for all .

Example 9. Let and by . Define and Then is -admissible.

Example 10. Let and . Define by and Then is -admissible.

Some interesting examples of such mappings were given in [5].

The notion of an - contractive mapping is defined in the following way.

Definition 11 (see [5]). Let be a metric space and let be a given mapping. We say that is an - contractive mapping if there exist two functions and such that

Clearly, any contractive mapping, that is, a mapping satisfying the Banach contraction, is an - contractive mapping with for all and , .

Very recently, Karapınar [20] gave the analog of the notion of an - contractive mapping, in the context of generalized metric spaces as follows.

Definition 12. Let be a generalized metric space and let be a given mapping. We say that is an - contractive mapping if there exist two functions and such that

Karapınar [20] also stated the following fixed point theorems.

Theorem 13. Let be a complete generalized metric space and let be an - contractive mapping. Suppose that (i) is -admissible;(ii)there exists such that and ;(iii) is continuous.Then there exists a such that .

Theorem 14. Let be a complete generalized metric space and let be an - contractive mapping. Suppose that (i) is -admissible;(ii)there exists such that and ;(iii)if is a sequence in such that for all and as , then for all .Then there exists a such that .

For the uniqueness, Karapınar [20] (see also [21]) added the following additional conditions. For all , we have , where denotes the set of fixed points of . For all , there exists such that and .

Theorem 15. Adding condition ( ) to the hypotheses of Theorem 13 (resp., Theorem 14), one obtains that is the unique fixed point of .

Theorem 16. Adding conditions ( ) and ( ) to the hypotheses of Theorem 13 (resp., Theorem 14), one obtains that is the unique fixed point of .

Corollary 17. Adding condition ( ) to the hypotheses of Theorem 13 (resp., Theorem 14) and assuming that is Hausdorff, one obtains that is the unique fixed point of .

In this paper, we define the notion of weaker generalized - contractive mappings and prove some fixed point results in the setting of generalized metric spaces by using such mappings. We state some examples to illustrate the validity of the main results of this paper.

2. Main Results

In this section, we will state and prove our main results.

We give an extension of the notion of - contractive mappings, in the context of generalized metric space as follows.

Definition 18. Let be a generalized metric space and let be a given mapping. We say that is a ( - - )-contractive mapping of type I if there exist functions , , and such that for all , where

Definition 19. Let be a generalized metric space and let be a given mapping. We say that is a ( - - )-contractive mapping of type II if there exist functions , , and such that for all , where

Next, we introduce the notion of triangular -admissible as follows.

Definition 20. Let and . The mapping is said to be weak triangular -admissible if for all , one has

Now, we state the first fixed point theorem.

Theorem 21. Let be a complete generalized metric space and let be a ( - - )-contractive mapping of type I. Suppose that (i) is weak triangular -admissible;(ii)there exists such that ;(iii) is continuous.Then, has a fixed point ; that is .

Proof. Due to statement (ii) of the theorem, there exists a point such that and . First, we define a sequence in by for all . Notice that if for some , then the proof is completed. Indeed, we have . Thus, for the rest of the proof, we assume that Owing to the fact that is -admissible, we derive that Utilizing the expression above, we find that Since is a weak triangular -admissible mapping, we obtain that Since and , iteratively, we conclude that Taking (10) and (17) into account, we observe that for all , where If , then by (15) and property of the function , inequality (20) turns into Since is decreasing, the inequality above yields a contradiction. Hence, we conclude that and (20) becomes for all . Recursively, we derive that

Owing to the fact that the sequence is decreasing, it converges to some . We will show that . Suppose, on the contrary, that . Taking the definition of weaker Meir-Keeler function into account, there exists such that for with , and there exists such that . Regarding , there exists such that , for all . Hence, we deduce that , which is a contradiction. Thus, , and hence Regarding (10) and (19), we deduce that for all , where

If then inequality (26) turns into for all . By repeating the same argument, inequality (15) implies that Due to the fact that the sequence is decreasing, we conclude that by following the lines at the proof of (25).

If either or , then inequality (26) becomes either or for all . Letting in any of the cases, (31) or (32), together with (25), we have

Let for some with . Without loss of generality, assume that . Thus, . Regarding (15), we consider now where If , then from (34) and (23) we get that If , inequalities (34) and (23) become Due to , inequalities (36) and (37) yield that which is a contradiction. Hence has no periodic point.

In what follows we will prove that the sequence is Cauchy by standard technique. Suppose, on the contrary, that there exists such that for any , there are with satisfying Furthermore, corresponding to , one can choose in a way that it is the smallest integer with . Consequently, we have . Consider Letting , we get that

On the other hand, again by using the quadrilateral inequality, we find Letting , in the inequalities above, we get that On account of (10), we have where Letting , in (44), and regarding definitions of auxiliary functions and (45), we conclude that which yields that . By definition of , we derive that , which is a contradiction. Hence, we conclude that is a Cauchy sequence in . Since is complete, there exists such that

Since is continuous, we obtain from (47) that From (47) and (48) we get immediately that . Taking Proposition 5 into account, we conclude that .

The following result is deduced from the obvious inequality .

Theorem 22. Let be a complete generalized metric space and let be a ( - - )-contractive mapping of type II. Suppose that (i) is a weak triangular -admissible;(ii)there exists such that ;(iii) is continuous.Then there exists a such that .

Theorem 23. Let be a complete generalized metric space and let be a ( - - )-contractive mapping of type I. Suppose that (i) is a weak triangular -admissible and is upper semicontinuous function;(ii)there exists such that ;(iii)if is a sequence in such that for all and as , then for all .Then there exists a such that .

Proof. Following the proof of Theorem 21, we know that the sequence defined by for all converges for some . We will show that . Suppose, on the contrary, that . From (17) and condition (iii), there exists a subsequence of such that for all . By applying the quadrilateral inequality together with (10) and (15), for all , we get that where

Letting in the above equality and regarding that the is an upper semicontinuous mapping, we find that It implies that from ( ) which is a contradiction. Hence, we obtain that is a fixed point of ; that is, .

In the following theorem, we remove the semicontinuity of by weakening the contractive mapping type.

Theorem 24. Let be a complete generalized metric space and be a ( - - )-contractive mapping of type II. Suppose that (i) is a weak triangular -admissible;(ii)there exists such that ;(iii)if is a sequence in such that for all and as , then for all .Then there exists a such that .

Proof. Following the proof of Theorem 21, we know that the sequence defined by for all converges for some . We will show that . Suppose, on the contrary, that . From (17) and condition (iii), there exists a subsequence of such that for all . By applying the quadrilateral inequality together with (10) and (15), for all , we get that where Letting in the above equality and regarding ( ), we find that which is a contradiction. Hence, we obtain that is a fixed point of ; that is, .

Theorem 25. Adding condition ( ) to the hypotheses of Theorem 21 (resp., Theorem 23), one obtains that is the unique fixed point of .

Proof. In what follows we will show that is a unique fixed point of . We will use the reductio ad absurdum. Let be another fixed point of with . It is evident that .

Now, due to (10) and ( ), we have which is a contradiction, where Hence, .

Theorem 26. Adding condition ( ) to the hypotheses of Theorem 22 (resp., Theorem 24), one obtains that is the unique fixed point of .

Proof. The proof is analog of the proof of Theorem 25 which will be concluded by using the reductio ad absurdum. Suppose, on the contrary, that is another fixed point of with . It is evident that .

Now, due to (12) and ( ), we have which is a contradiction, where Hence, .

For the uniqueness, we can also consider the following condition. For all , there exists such that and . Further, , where and for .

Theorem 27. Adding conditions ( ) and ( ) to the hypotheses of Theorem 21 (resp., Theorem 23), one obtains that is the unique fixed point of .

Proof. Suppose that is another fixed point of . From , there exists such that Since is -admissible, from (60), we have Define the sequence in by for all and . From (61), for all , we have where If then by letting in (62) we get that due to the continuity of , and the fact that . If then (62) turns into Iteratively, by using inequality (62), we get that for all . Letting in the above inequality, we obtain Similarly, one can show that Regarding together with (67) and (68), it follows that . Thus we proved that is the unique fixed point of .

Theorem 28. Adding conditions ( ) and ( ) to the hypotheses of Theorem 22 (resp., Theorem 24), one obtains that is the unique fixed point of .

The proof is the analog of the proof of Theorem 27; hence we omit it.

Corollary 29. Adding condition ( ) to the hypotheses of Theorem 21 (resp., Theorems 23, 22, and 24) and assuming that is Hausdorff, one obtains that is the unique fixed point of .

The proof is clear, and hence it is omitted. Indeed, Hausdorffness implies the uniqueness of the limit. Thus, the theorem above yields the conclusions.

3. Consequences

Now, we will show that many existing results in the literature can be deduced easily from Theorems 13 and 14.

Definition 30. Let be a generalized metric space and let be a given mapping. We say that is a ( - - )-contractive mapping of type III if there exist functions , , and such that for all .

Now, we state the first fixed point theorem.

Theorem 31. Let be a complete generalized metric space and let be a ( - - )-contractive mapping of type III. Suppose that (i) is -admissible;(ii)there exists such that and ;(iii) is continuous.Then, has a fixed point ; that is, .

We omit the proof of Theorem 31, since it can be derived easily by following the lines in the proof of Theorem 21, analogously.

Theorem 32. Let be a complete generalized metric space and let be a ( - - )-contractive mapping of type III. Suppose that (i) is -admissible;(ii)there exists such that and ;(iii)if is a sequence in such that for all and as , then for all .Then there exists a such that .

Proof. Following the proof of Theorem 21 (resp., Theorem 31), we know that the sequence defined by for all converges for some . We will show that . Suppose, on the contrary, that . From (17) and condition (iii), there exists a subsequence of such that for all . By applying the quadrilateral inequality together with (10) and (15), for all , we get that

Letting in the above equality and regarding , we find that which is a contradiction. Hence, we obtain that is a fixed point of ; that is, .

Theorem 33. Adding condition ( ) to the hypotheses of Theorem 31 (resp., Theorem 32), one obtains that is the unique fixed point of .

Proof. In what follows we will show that is a unique fixed point of . We will use the reductio ad absurdum. Let be another fixed point of with . It is evident that .

Now, due to (10) and ( ), we have which is a contradiction.

Theorem 34. Adding conditions ( ) and ( ) to the hypotheses of Theorem 31 (resp., Theorem 32), one obtains that is the unique fixed point of .

Corollary 35. Let be a complete generalized metric space and let be a given mapping. Suppose that there exist and such that for all , where Then has a unique fixed point.

Proof. Let be the mapping defined by , for all . Then is a ( - - )-contractive mapping of type I. It is evident that all conditions of Theorem 21 are satisfied. Hence, has a unique fixed point.

Corollary 36. Let be a complete generalized metric space and let be a given mapping. Suppose that there exist and such that for all , where Then has a unique fixed point.

The following Corollary is stronger than the main result of [19]. Notice that we do not need the Hausdorffness condition although it was required in [19].

Corollary 37. Let be a complete generalized metric space and let be a given mapping. Suppose that there exist and such that for all . Then has a unique fixed point.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors' Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

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