Abstract and Applied Analysis

Volume 2014 (2014), Article ID 379358, 15 pages

http://dx.doi.org/10.1155/2014/379358

Research Article

Fixed Point Results for Various -Admissible Contractive Mappings on Metric-Like Spaces

Saleh A. Al-Mezel,¹ Chi-Ming Chen,² Erdal Karapınar,1,3 and Vladimir Rakočević⁴

¹Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah 21491, Saudi Arabia

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

³Department of Mathematics, Atılım University, İncek, 06836 Ankara, Turkey

⁴Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia

Received 18 February 2014; Accepted 11 April 2014; Published 11 May 2014

Academic Editor: Ngai-Ching Wong

Copyright © 2014 Saleh A. Al-Mezel 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 establish some fixed point theorems for -admissible mappings in the context of metric-like space via various auxiliary functions. In particular, we prove the existence of a fixed point of the generalized Meir-Keeler type -contractive self-mapping defined on a metric-like space . The given results generalize, improve, and unify several fixed point theorems for the generalized cyclic contractive mappings that have appeared recently in the literature.

1. Introduction and Preliminaries

Nonlinear functional analysis is one of the most dynamic research fields in mathematics. In particular, fixed point theory that has a wide application potential to several quantitative sciences has attracted a number of authors. In the recent decades, several new abstract spaces and new contractive type mappings have been considered to develop the fixed point theory and to increase application potential to existing open problems. Among them, Samet et al. [1] proved very interesting fixed point theorem by introducing the --contractive self-mapping in the setting of complete metric space . In this notion, is a -distance function (see, e.g., [2–5]) and self-mapping is -admissible. The notion of mapping --contractive mappings has charmed a number of authors (see, e.g., [1, 6–14]).

In this paper, we combine some of the notions to get more general results in the research field of fixed point theory. In particular, we investigate the existence of a fixed point of -admissible mapping in the context of metric-like space via implicit functions.

Throughout this paper, by , we denote the set of all nonnegative numbers, while is the set of all natural numbers. In 1994, Matthews [15] introduced the following notion of partial metric spaces.

Definition 1 (see [15]). A partial metric on a nonempty set is a function such that for all

() if and only if ;();();().

A partial metric space is a pair such that is a nonempty set and is a partial metric on .

Remark 2. It is clear that if , then, from and , . But if , may not be .

Later, fixed point theory has developed rapidly on partial metric spaces; see [16–23]. Further, in 2012, Amini-Harandi [24] introduced the concept of a metric-like space.

Definition 3 (see [24]). A function , where is a nonempty set, is said to be metric-like on if the following conditions are satisfied, for all :

()if , then ;();().

Then the pair is called a metric-like space.

Remark 4 (see [24]). (1) A metric-like on satisfies all of the conditions of a metric except that may be positive for .

(2) Every partial metric space is a metric-like space. But the converse is not true.

Each metric-like on generates a topology on whose base is the family of open -balls , where for all and . We recall some definitions on a metric-like space as follows.

Definition 5 (see [24]). Let be a metric-like space. Then

(1)a sequence in a metric-like space converges to if and only if ;(2)a sequence in a metric-like space is called a -Cauchy sequence if and only if exists (and is finite);(3)a metric-like space is said to be complete if every -Cauchy sequence in converges, with respect to , to a point such that

(4)a mapping is continuous, if the following limits exist (finite) and

Definition 6 (see [24]). Let be a metric-like space and be a subset of . Then is a -open subset of if, for all , there exists such that . Also, is a -closed subset of if is a -open subset of .

Further, Karapınar and Salimi [25] proved the following crucial properties in the setting of metric-like space .

Lemma 7 (see [25]). Let be a metric-like space. Then

(A)if , ;(B)if is a sequence such that ,

(C)if , ;(D) holds for all , where .

Lemma 8. Let be a metric-like space and be a sequence in such that as and . Then for all .

We recall the notion of cyclic map which was introduced by Kirk et al. [26]. A mapping is called cyclic if and . Kirk et al. [26] proved the analog of the Banach contraction mapping principle via cyclic mappings.

Theorem 9 (see [26]). Let and be two nonempty closed subsets of a complete metric space , and suppose satisfies the following:

(i) is a cyclic map,(ii) for all , , and .

Then is nonempty and has a unique fixed point in .

Furthermore, Kirk et al. [26] also introduced the following notion of the cyclic representation.

Definition 10 (see [26]). Let be a nonempty set, , and an operator. Then is called a cyclic representation of with respect to if

(1), , are nonempty subsets of ;(2), , and .

By using the notion in the definition above, Kirk et al. [26] proved the following theorem.

Theorem 11 (see [26]). Let be a complete metric space, , be closed nonempty subsets of , and . Suppose that satisfies the following condition:

where is upper semicontinuous from the right and for . Then has a fixed point .

In 2012, Karapınar et al. [22] investigated the existence and uniqueness of a fixed point for cyclic generalized --contractive type mappings in the context of partial metric space. Very recently, Karapınar and Salimi [25] improved the results in [22] by introducing the notion of cyclic generalized --contractive mapping . In this paper [25], the authors proved fixed theorems for such a mapping in the setting of a metric-like space with a cyclic representation of with respect to .

Definition 12 (see [25]). Let be a metric-like space, be -closed nonempty subsets of , and . One says that is called a generalized cyclic --contractive mapping if

(1) is a cyclic representation of with respect to ;(2)One considers

for all and , , where is nondecreasing and continuous and is lower semicontinuous.

Theorem 13 (see [25]). Let be a metric-like space, be -closed nonempty subsets of , and . If is a generalized cyclic --contractive mapping, then has a fixed point .

In this study, we also discuss the notion of -admissible mappings. The following definition was introduced in [1].

Definition 14 (see [1]). For a nonempty set , let and be mappings. One says that is -admissible, if, for all , one has

Recall that Samet et al. [1] introduced the following concepts.

Definition 15 (see [1]). Let be a metric space and let be a given mapping. One says that is an - contractive mapping if there exist two functions and a certain such that

It is evident that a mapping satisfying the Banach contraction is a - contractive mapping equipped with for all and , .

The notion of transitivity of mapping was introduced in [13, 14] as follows.

Definition 16 (see [13, 14]). Let . One says that is -transitive (on ) if

for all .

In particular, one says that is transitive if it is 1-transitive; that is,

As consequences of Definition 16, one obtains the following remarks.

Remark 17 (see [13, 14]). (1) Any function is 0-transitive.

(2) If is -transitive, then it is -transitive for all .

(3) If is transitive, then it is -transitive for all .

(4) If is -transitive, then it is not necessarily transitive for all .

In this paper, we investigate the existence and uniqueness of a fixed point of several -admissible mappings in the context of metric-like space. In particular, we establish fixed point theorem for the generalized cyclic Meir-Keeler type --contractive mappings, the generalized --contractive mappings, and the generalized weaker Meir-Keeler type --contractive mappings. Our results generalize or improve many recent fixed point theorems for the generalized cyclic contractive mappings in the literature.

2. Fixed Point Theorem via the -Admissible Meir-Keeler Type Mappings

In this section, first of all, we will introduce the notion of the generalized Meir-Keeler type -contractive mappings. Later, we investigate the existence and uniqueness of such mappings in the context of metric-like spaces. We start with recalling the notion of the Meir-Keeler type mappings.

A function is said to be a Meir-Keeler type mapping (see [27]), if, for each , there exists such that, for with , we have .

Let be the class of all function satisfying the following conditions:

() is an increasing and continuous function in each coordinate;()for , , , and ;() if and only if .

We will introduce the notion of the generalized Meir-Keeler type -contractive mappings in metric-like spaces as follows.

Definition 18. Let be a metric-like space and let . One says that is called a generalized Meir-Keeler type -contractive mapping if for each there exists such that

for all and .

Remark 19. Note that if is a generalized Meir-Keeler type -contractive mapping, then we have, for all and ,

In what follows, we state the main fixed point theorem for a generalized Meir-Keeler type -contractive mapping in the setting of complete metric-like space.

Theorem 20. Let be a complete metric-like space and let be a generalized Meir-Keeler type -contractive mapping where is transitive. Suppose that

(i) is -admissible;(ii)there exists such that ;(iii) is continuous.

Then there exists such that .

Proof. Our proof consists of four steps. In the first step, we prove that , for all . Due to assumption (ii) of the theorem, there exists such that . We will construct an iterative sequence in as follows:

If we have , for some , then the proof is completed. Indeed, is a fixed point of . Hence, throughout the proof, we presume that

Since is -admissible, we have

By elementary calculations, we derive that

In the second step, we will prove that . Notice that we have for all by (13) and Lemma 7(C). Since is a generalized Meir-Keeler type -contractive mapping, by taking and in (11), we have

We assert that is decreasing; that is, for all . Suppose, on the contrary, that for some . By taking and in (11) and (16), we have

which is a contradiction. So the is decreasing, and it must converge to some ; that is,

By condition , inequality (16) becomes

We next claim that . If not, we assume that . By taking limit as in (19), we have

which is a contradiction. Hence, we have .

In the third step, we will prove that is a -Cauchy sequence. We will use the method of reductio ad absurdum. Suppose, on the contrary, that is not a -Cauchy sequence. Hence, there exists and subsequences and of with satisfying

Since is transitive, from (15), we have and hence . Consider the following:

Letting , we obtain that

Also we have

We get

Letting in the inequality above, we find that

Analogously, we derive that

Further, we have

Letting in the above inequality, we get that

Notice also that

Letting in the above inequality and taking the property into account, we get that

which is a contradiction. Thus, is a -Cauchy sequence.

In the fourth and last step, we will prove that has a fixed point . Owing to the fact that is complete, there exists such that ; equivalently,

Since is continuous, we obtain from (32) that

Due to Lemma 8, we also have

Combining (32)–(34) and Lemma 7(A), we get immediately that is a fixed point of ; that is, .

In the next theorem the continuity of is not required.

Theorem 21. Let be a complete metric-like space and let be a generalized Meir-Keeler type -contractive mapping, mapping, where is transitive. Suppose that

(i) is -admissible;(ii)there exists such that ;(iii)if is a sequence in such that for all and as , then for all .

Then there exists such that .

Proof. Following the proof of Theorem 20, we know that the sequence defined by , for all , converges to where . It is enough to show that is the fixed point of . Suppose, on the contrary, that . From (15) and condition (iii), there exists a subsequence of such that for all . Applying (11), for all , we get that

Letting in the above equality and taking (34) into account, we get that

By we get that

which is a contradiction. Thus we get , and, by Lemma 7(A), we have .

For the uniqueness, we need an additional condition.

(U) For all , we have , where denotes the set of fixed points of .

In what follows we will show that is a unique fixed point of .

Theorem 22. Adding condition () to the hypotheses of Theorem 20 (resp., Theorem 21), one obtains that is the unique fixed point of .

Proof. We will use the reductio ad absurdum. Let be another fixed point of with and hence . By hypothesis ,

Due to inequality (11) we have

Taking property into account, we get that

which is a contradiction. Hence, . It follows from Lemma 7(A) that . Thus we proved that is the unique fixed point of .

3. Fixed Point Theorem via Auxiliary Functions

In the section, we will discuss the notion of generalized --contractive mappings and prove fixed point theorems for these mappings in complete metric-like spaces. We denote by the class of functions satisfying the following conditions:

() is continuous and nondecreasing;() for all and .

Let be the class of all function satisfying the following conditions:

(ϕ ₁) is an increasing and continuous function in each coordinate;(ϕ ₂)for , , , and , where ;(ϕ ₃) if and only if .

We use the following notations to specify the collection of the given functions:

We now state the new notion of generalized --contractive mappings in metric-like spaces is as follows.

Definition 23. Let be a metric-like space and let . One says that is called a generalized --contractive mapping if is -admissible and satisfies the following inequality:

for all , where , , , and .

One now states the main fixed point of this section as follows.

Theorem 24. Let be a complete metric-like space and let be a --contractive mapping where is transitive. Suppose that

(i) is -admissible;(ii)there exists such that ;(iii) is continuous.

Then there exists such that .

Proof. As in the proof of Theorem 20, we construct an iterative sequence in as follows:

If we have , for some , then the proof is completed. Indeed, is a fixed point of . Hence, from now on, we assume that

Moreover, due to Lemmas 7(C) and (D), we have

Again, as in the proof of Theorem 20, since is -admissible, we deduce that

Owing to the fact that is a generalized --contraction, by taking and in (42), we have

As a first step, we prove that

For this goal, we show that is decreasing; that is, for all . Suppose, on the contrary, that for some . By substituting and in (42) and (47), we have

Regarding the condition for all and by using inequality (49), we derive that , which contradicts to (45). Hence, we deduce that

From the arguments above, we also have, for each ,

It follows from (50) that the sequence is monotone decreasing. Hence, it should be convergent to some ; that is,

Letting in (51) and by using the continuities of and and the lower semicontinuity of , we have

which implies that .

As in the proof of Theorem 20, we will use the same techniques, method of reductio ad absurdum, to prove that is a -Cauchy sequence. Suppose, on the contrary, that is not a -Cauchy sequence. Hence, there exists and subsequences and of with satisfying

By repeating the related lines in the proof of Theorem 20, we find the following limits:

By assumption of the theorem, we have

Letting in (56), we find that

which implies that . This is a contradiction. Therefore, the sequence is a -Cauchy sequence.

As a last step, we will prove that has a fixed point . Owing to the fact that is complete, there exists such that , equivalently,

Since is continuous, we obtain from (58) that

Due to Lemma 8, we also have

On account of (58)–(60) and Lemma 7(A), we derive that is a fixed point of ; that is, .

Theorem 25. Let be a complete metric-like space and let be a --contractive mapping where is transitive. Suppose that

(i) is -admissible;(ii)there exists such that ;(iii)if is a sequence in such that for all and as , then for all .

Then there exists such that .

Proof. Following the proof of Theorem 24, we know that the sequence defined by , for all , converges to where . It is enough to show that is the fixed point of . Suppose, on the contrary, that . From (46) and condition (iii), there exists a subsequence of such that for all . Applying (42), for all , we get that

Letting in the above equality and taking (60) into account, we get that

By we get that

which is a contradiction. Thus we get , and, by Lemma 7(A), we have .

In the next theorem we will show that is a unique fixed point of .

Theorem 26. Adding condition () to the hypotheses of Theorem 24 (resp., Theorem 25), one obtains that is the unique fixed point of .

Proof. We will use the reductio ad absurdum. Let be another fixed point of with and hence . By hypothesis ,

Due to inequality (42) we have

Taking property into account, we get that

which is a contradiction. Hence, . By Lemma 7(A) we get that . Thus we proved that is the unique fixed point of .

4. Fixed Point Theorems via the Weaker Meir-Keeler Function

In the section, we will investigate the existence and uniqueness of a fixed point of certain mappings by using the Meir-Keeler function. Now, we recall the notion of the weaker Meir-Keeler function .

Definition 27 (see [28]). One calls a weaker Meir-Keeler function if, for each , there exists such that, for with , there exists such that .

One denotes by the class of nondecreasing functions satisfying the following conditions:

(μ ₁) is a weaker Meir-Keeler function;(μ ₂) for and ;(μ ₃)for all , is decreasing;(μ ₄)if , then .

And one denotes by the class of functions satisfying the following conditions:

 () is continuous; () for and .

We state the notion of the generalized weaker Meir-Keeler type --contractive mappings in metric-like spaces as follows.

Definition 28. Let be a metric-like space, and let . One says that is called a generalized weaker Meir-Keeler type --contractive mapping if is -admissible and satisfies

for all , where , , and

The main result of this section is the following.

Theorem 29. Let be a complete metric-like space and let be a generalized weaker Meir-Keeler type --contractive mapping where is transitive. Suppose that

(i) is -admissible;(ii)there exists such that ;(iii) is continuous.

Then there exists such that .

Proof. Following the lines in the proof of Theorem 20, we construct an iterative sequence in as follows:

If we have , for some , then the proof is completed. Indeed, is a fixed point of . Hence, from now on, we assume that

Moreover, due to Lemmas 7(C) and (D), we have

Again, by following the lines in the proof of Theorem 20, we get that

We divide the proof into three steps.

Step 1. We will prove that . Since is a generalized weaker Meir-Keeler type --contractive mapping, by taking and in (67), we have

where

If , then, by (73) and the properties of the functions and , we have

Since is decreasing, the inequality above yields a contradiction. Thus, we conclude that and inequality (71) becomes

for all . Recursively, we conclude that

for all .

Since is decreasing, it must converge to some . We claim that . On the contrary, assume that . Then, by the definition of the weaker Meir-Keeler function , there exists such that, for with , there exists such that . Since , there exists such that , for all . Thus, we conclude that . So we get a contradiction. Therefore ; that is,

Step 2. We prove that is a -Cauchy sequence.

We will use the method of reductio ad absurdum, as in the proof of Theorem 20. Suppose, on the contrary, that is not a -Cauchy sequence. Hence, there exists and subsequences and of with satisfying

By repeating the related lines in the proof of Theorem 20, we find the following limits:

By the assumption of the theorem, we have

where

Case 1. If , letting , then (81) becomes

which yields that , and so we conclude that . Therefore, we get a contradiction.

Case 2. If or , letting , then (81) turns into

which yields that . It is a contradiction.

Case 3. If , letting , then (81) becomes

which yields that , and hence . So, we get a contradiction.

Following the arguments above, we show also that is a -Cauchy sequence.

Step 3. In this step, we prove that has a fixed point . Since is complete, there exists such that ; equivalently,

Since is continuous, we obtain from (86) that

Due to Lemma 8, we also have

On account of (58)–(88) and Lemma 7(A), we derive that is a fixed point of ; that is, .

Theorem 30. Let be a complete metric-like space and let be a generalized weaker Meir-Keeler type --contractive mapping where is transitive. Suppose that

(i) is -admissible;(ii)there exists such that ;(iii)if is a sequence in such that for all and as , then for all .

Then there exists such that .

Proof. Following the proof of Theorem 29, we know that the sequence defined by , for all , converges to where . It is enough to show that is the fixed point of . Suppose, on the contrary, that . From (71) and condition (iii), there exists a subsequence of such that for all . Applying (67), for all , we get that

where

Letting in equality (89) and taking (88) into account, we get that

Since , for all , we conclude that ; that is, .

In what follows we will show that is a unique fixed point of .

Theorem 31. Adding condition () to the hypotheses of Theorem 29 (resp., Theorem 30), one obtains that is the unique fixed point of .

Proof. We will use the reductio ad absurdum. Let be another fixed point of with and hence . By hypothesis ,

Due to inequality (67), we have

where

Hence, we have

since , for all , which is a contradiction. Thus we proved that is the unique fixed point of .

5. Consequences

In this section, we will demonstrate that several existing fixed point results in the literature can be deduced easily from our main results: Theorem 22, Theorem 26, and Theorem 31.

5.1. Standard Fixed Point Theorems

If we substitute for all in Theorem 22, we derive immediately the following fixed point theorem.

Theorem 32. Let be a complete metric-like space and let be a mapping. Suppose that for each there exists such that

for all and . Then there exists a unique fixed point such that .

If we take for all in Theorem 26, we get the following fixed point theorem.

Theorem 33. Let be a metric-like space and let be self-mapping. Suppose that satisfies the following inequality:

for all , where , , , and . Then there exists a unique fixed point such that .

If we take for all in Theorem 31, we get the following fixed point theorem.

Theorem 34. Let be a complete metric-like space and let be mapping. Suppose that satisfies

for all , where , , and

Then there exists a unique fixed point such that .

5.2. Fixed Point Theorems on Metric Spaces Endowed with a Partial Order

In the last decade, the investigation of the existence of fixed point on metric spaces endowed with partial orders has been appreciated by several authors. The initial results in this direction were reported by Turinici [29], Ran and Reurings in [30]. Now, we consider the partially ordered versions of our theorems. For this purpose, we need to recall some concepts.

Definition 35. Let be a partially ordered set and let be a given mapping. One says that is nondecreasing with respect to if

Definition 36. Let be a partially ordered set. A sequence is said to be nondecreasing with respect to if for all .

Definition 37. Let be a partially ordered set and let be a metric on . One says that is regular if, for every nondecreasing sequence such that as , there exists a subsequence of such that for all .

Theorem 38. Let be a complete metric-like space and let be a mapping. Suppose that for each there exists such that

for all and . Then there exists a unique fixed point such that .

We have the following result.

Corollary 39. Let be a partially ordered set and let be a metric-like mapping on such that is complete metric-like space. Let be a nondecreasing mapping with respect to . Suppose that for each there exists such that

for all with and . Suppose also that the following conditions hold:

(i)there exists such that ;(ii) is continuous or is regular.

Then has a fixed point. Moreover, if for all there exists such that and , one has uniqueness of the fixed point.

Proof. Define the mapping by

Clearly, for each , there exists such that

for all and . From condition (i), we have . Moreover, for all , from the monotone property of , we have

Thus is -admissible. Now, if is continuous, the existence of a fixed point follows from Theorem 20. Suppose now that is regular. Let be a sequence in such that for all and as . From the regularity hypothesis, there exists a subsequence of such that for all . This implies from the definition of that for all . In this case, the existence of a fixed point follows from Theorem 21. To show the uniqueness, let . By the hypothesis, there exists such that and , which implies from the definition of that and . Thus we deduce the uniqueness of the fixed point by Theorem 22.

By using the same argument in the proof of Corollary 39, we can conclude the following two corollaries. We omit the proofs of these corollaries to avoid the repetition.

Corollary 40. Let be a partially ordered set and let be a metric-like mapping on such that is complete metric-like space. Let be a nondecreasing mapping with respect to . Suppose that satisfies the following inequality:

for all with , where , , , and .

Suppose also that the following conditions hold:

(i)there exists such that ;(ii) is continuous or is regular.

Then has a fixed point. Moreover, if for all there exists such that and , one has uniqueness of the fixed point.

Corollary 41. Let be a partially ordered set and be a metric-like mapping on such that is complete metric-like space. Let be a nondecreasing mapping with respect to . Suppose that satisfies

for all with , where , , and

Suppose also that the following conditions hold:

(i)there exists such that ;(ii) is continuous or is regular.

Then has a fixed point. Moreover, if for all there exists such that and , one has uniqueness of the fixed point.

5.3. Fixed Point Theorems for Cyclic Contractive Mappings

In this subsection, we consider the cyclic contraction and related fixed point as a consequence of our main results. Notice that this trend was initiated by Kirk et al. [31]. Following this paper [31], a number of fixed point theorems for cyclic contractive mappings have been reported (see, e.g., [32–37]).

We have the following result.

Corollary 42. Let be nonempty closed subsets of a complete metric-like space and let be a given mapping such that

(I) and ,

where . Suppose that for each there exists such that

for all and . Then has a unique fixed point that belongs to .

Proof. Since and are closed subsets of the complete metric space , then is complete. Define the mapping by

From (111) and the definition of , for each , there exists such that

for all and . Thus satisfies the contractive condition (104).

Let such that . If , from (I), , which implies that . If , from (I), , which implies that . Thus, in all cases, we have . This implies that is -admissible.

Also, from (I), for any , we have , which implies that .

Now, let be a sequence in such that for all and as . This implies from the definition of that

Since is a closed set with respect to the Euclidean metric, we get that

which implies that . Thus we get immediately from the definition of that for all .

Let be distinct fixed points of from (I); this implies that . So, for any , we have and . Thus condition is satisfied.

Now, all the hypotheses of Theorem 22 are satisfied; we deduce that has a unique fixed point that belongs to (from (I)).

As in the previous section, we can conclude the following two corollaries by using the same argument in the proof of Corollary 42. We omit the proofs of the following corollaries to avoid the repetition.

Corollary 43. Let be nonempty closed subsets of a complete metric-like space and let be a given mapping such that

(I) and , where . Suppose that satisfies the following inequality:

for all , where , , , and . Then has a unique fixed point that belongs to .

Corollary 44. Let be nonempty closed subsets of a complete metric-like space and let be a given mapping such that

(I) and , where . Suppose that satisfies

for all , where , , and

Then has a unique fixed point that belongs to .

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

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

Acknowledgments

This research was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors give thanks to anonymous referees for their remarkable comments, suggestion, and ideas that help to improve this paper.

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