Ulam-Hyers Stability for MKC Mappings via Fixed Point Theory

Hassan, Anisa Mukhtar; Karapınar, Erdal; Alsulami, Hamed H.

doi:https://doi.org/10.1155/2016/9623597

Journal of Function Spaces

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Ulam’s Type Stability and Fixed Points Methods 2016

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Research Article | Open Access

Volume 2016 | Article ID 9623597 | https://doi.org/10.1155/2016/9623597

Ulam-Hyers Stability for MKC Mappings via Fixed Point Theory

Anisa Mukhtar Hassan,¹Erdal Karapınar,^2,3and Hamed H. Alsulami³

Academic Editor: Liviu Cădariu

Received07 Jan 2016

Accepted05 Apr 2016

Published25 May 2016

Abstract

We consider some extension of MKC mappings in the framework of complete dislocated metric spaces. Besides the theoretical results, we also consider some illustrative examples. Further, we state and prove that our main results improved the related results in the frame of generalized Ulam-Hyers stability theory.

1. Introduction and Preliminaries

At the beginning of the nineteenth century, Fréchet [1] explicated the concept of “distance” formally by introducing the notion of metric. This conception has been generalized, extended, and improved in several ways, such as fuzzy metric, quasi-metric, partial metric, G-metric, D-metric, b-metric, 2-metric, ultrametric, and modular metric. Among them, the concept of dislocated metric, defined by Hitzler [2], deserves a special interest due to its application potential in several quantitative sciences, in particular, computer science. The concept of dislocated metric spaces was rediscovered by Amini-Harandi [3] as a “metric-like space.”

In this paper, we continue to use dislocated metric space instead of metric-like space.

Definition 1 (see [2]). For a nonempty set , a dislocated metric is a function such that, for all , if , then ; ; . Moreover, the pair is called dislocated metric space (DMS).

The topology of dislocated spaces and basic topological properties (convergence, completeness, etc.) was also considered in [2]. Among them, let us recall the essential notions. Each dislocated metric on a nonempty set has a topology that was generated by the family of open balls

In context of the DMS , a sequence converges to a point if the following limit exists (and is finite):

Moreover, a sequence is said to be Cauchy if the limitexists and is finite. Furthermore, if in (3), then we say that is a 0-Cauchy sequence.

As it is expected, a pair is called complete DMS if, for each Cauchy sequence , there is some such that

Analogously, a pair is called 0-complete DMS if, for each 0-Cauchy sequence , it converges to a point such that in (4).

Let and be DMSs. A mapping is called continuous if then, we have

Remark 2. Each partial metric space is a DMS. Notice also that every metric is necessarily partial metric.

Definition 3 (see [3]). Let be a DMS and let be a subset of . One says that is an open subset of , if for all there exists such that Also, is a closed subset of if is an open subset of .

Lemma 4 (see [4]). For a DMS , one has the following observations:(A)If , then .(B)For a sequence with , one has(C)If , then (D) holds for all , where .(E)Let be a closed subset of and let be a sequence in . If as , then .(F)For a sequence in such that as with , then for all .

Definition 5 (see [5]). Let be a nonempty set and let be a mapping. A self-mapping is called -admissible if the following implications hold:An -admissible is said to be triangular -admissible if

The letter represents the set of positive integers and . Further, the nonnegative real numbers will be denoted by .

1.1. (c)-Comparison Functions

Let be the family of nondecreasing functions satisfying the following conditions: for all , where is the th iterate of . .

It is well known that if is a (c)-comparison function, then for any and is continuous at (see, e.g., [6]).

2. Fixed Point Theorems for -Meir-Keeler Contractions on Dislocated Metric Spaces

2.1. -Meir-Keeler Contractive Mapping

We introduce the following notion which is an improved version of Meir-Keeler contractive mapping.

Definition 6. Let be a DMS. A self-mapping is called -Meir-Keeler contractive (in brief, -MKC) if there exist two functions and satisfying the following condition.
For each , there exists such that

Notice that, for -MKC mapping , we have

In what follows, we will state and prove the first main result of this section.

Theorem 7. Suppose that is a complete DMS and a self-mapping is -MKC. Assume also that(i) is -admissible;(ii)there exists such that ;(iii) is continuous. Then, there exists such that .

Proof. Due to (ii), there exists such that . We define an iterative sequence in asNotice that if there exists some such that , then the proof is completed since .
So, we assume that , for all . Due to Lemma 4 part (C), for , we haveSince is -admissible, again by (ii), we getBy repeating this process, we derive thatRegarding (12) and (15) together with the assumption of the theorem that is an -MKC mapping, for each , we findTaking (13) and the property of into account, the above inequalities yield thatEventually, we conclude that is a nonincreasing and bounded sequence. Hence, there exists such thatIn what follows, we will prove that Suppose, on the contrary, that . Since is an -MKC mapping, for , there exist and a natural number such thatBy taking (15) into account, we get thatwhich is a contradiction since .
Consequently, we haveNow, we will prove that is a Cauchy sequence. That is,First, we observe from (17) thatSince is nondecreasing, by iteration, we conclude thatNow, by using (), (12), (15), and (24), we have the following:Letting in the above inequality, we derive expression (22). That is, the iterative sequence is Cauchy in the DMS .
Since is a complete DMS, then there exists such thatAs is continuous, then we deduce thatSince as and , then, by using Lemma 4, we getFrom (27) and (28), we derive that . By (), we conclude that .

Theorem 8. Suppose that is a complete DMS and a self-mapping is -MKC. Assume also 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. By following the lines in the proof of Theorem 7, we deduce that there exists a Cauchy sequence . Since is complete, it converges to some . On the other hand, from (15) and (iii), we haveBy using () and (29) with the assumption of the theorem that is an ()-MKC mapping, we obtainSince is continuous at , by letting in the inequality above, we find It yields due to ().

Example 9. Let endowed with the dislocated metric for all . Define and by

We can prove easily that is an -MKC mapping and it is -admissible.

Moreover, there exists such that . In fact, for , we have

Now, we show that is continuous. Let That is, Now, we have that is continuous since

So all hypotheses of Theorem 7 are satisfied. Consequently, has a fixed point. Notice that is a fixed point of .

In the following example, a self-mapping is not continuous.

Example 10. Let endowed with the dislocated metric for all . Define and by

Clearly, is not continuous at which shows that Theorem 7 is not applicable in this case.

We will prove that a self-mapping is -MKC. Let be given. Take and suppose that ; we want to show that

Suppose that ; then and so . Hence,

Also, is -admissible. To see this, let ; then, both . Due to the definition of , we have and Thus, we get .

Moreover, there exists such that . Indeed, for , we have

Finally, let be a sequence in such that for all and as . Since for all , by the definition of , we have for all and ; then, .

So, we conclude that all hypotheses of Theorem 8 are fulfilled. Hence, we proved that has a fixed point.

2.2. Generalized--Meir-Keeler Contractive Mapping

Definition 11. Suppose that is a DMS. A self-mapping is said to be generalized--Meir-Keeler contractive (in brief, generalized--MKC) if there exist and such that, for each , there exists such that where

If a self-mapping is a generalized--MKC, then we have

Very recently, Popescu [7] improved the triangular -admissible notion as follows.

Definition 12 (see [7]). Suppose that is a self-mapping and is a function. A self-mapping is called -orbital admissible if A self-mapping is called triangular -orbital admissible if is -orbital admissible and

As it was mentioned in [7], every (triangular) -admissible function is (triangular) -orbital admissible function. The converse is false; see, for example, [7, Example 7].

Lemma 13 (see [7]). Suppose that is a triangular -orbital admissible mapping. Suppose also that there exists such that . Define an iterative sequence by for each . Then, one has for all with .

The following is the first main result of this section.

Theorem 14. Suppose that is a complete DMS, a self-mapping is a generalized--MKC, and the following conditions are fulfilled:(i) is triangular -orbital admissible mapping.(ii)There exists such that .(iii) is continuous. Then, has a fixed point; that is, there exists such that .

Proof. Take such that . As in the proof of Theorem 7, we define an iterative sequence in asWithout loss of generality, we assume that , for all ; then,Indeed, if there exists some such that , then the proof is completed since .
Since is triangular -orbital admissible mapping, by using Lemma 13, we derive thatStep 1. We will prove thatTaking (47) and (48) into account together with the fact that is a generalized--MKC mapping, for each , we derivewhereRegarding , we estimate the last term in the expression as follows:Consequently, we getLet us examine two cases. Let . Since is nondecreasing, from (50), we getwhich is a contradiction. Thus, and again, by (50), we findSo, we deduce that the sequence is nonincreasing and bounded below by zero.
Hence, there exists such thatRecursively, we derive from (55) thatby keeping in mind that is nondecreasing.
On account of (57) and (), we obtainIt is obvious from the triangle inequality thatStep 2. We will prove that is a Cauchy sequence.
Suppose, on the contrary, that there exist and a subsequence of such thatFirst, we will show the existence of such that . Later, we will prove that, for given above, there exists such thatwhich contradicts (60), whereLet . On account of Step , we will choose in a way thatfor all . Due to our construction, clearly, we have . If , then, by using (), we havewhich is a contradiction due to (60). Consequently, there are values of such that and . Indeed, ifthen we obtainwhich contradicts (63).
Moreover, the caseis impossible due to (64).
Hence, we choose the smallest integer with in a way thatSo, necessarily, we have . Hence, we find that and, hence, we get the following approximation:for a natural number satisfying .
Thus, the first three terms of (62) are bounded above by ; that is,Eventually, the last terms of (62) can be estimated from above as follows:Combining estimations (72), (73), (74), and (75), we conclude thatSince is generalized--MKC mapping and since it is triangular -orbital admissible mapping, we haveOn the other hand, by (), we have which is a contradiction with (71).
Thus, claim (60) is false and the constructive sequence is Cauchy; that is,Since is a complete DMS, then there exists such thatOn account of the continuity of the self-mapping , we deduce thatSince as and , then, by using Lemma 4, we getRegarding (81) and (82), we get . Thus, by (), we conclude that .

Theorem 15. Suppose that is a complete DMS, a self-mapping is a generalized--MKC, and the following conditions are fulfilled:(i) is triangular -orbital admissible mapping.(ii)There exists such that .(iii)If is a sequence in such that for all and as , then for all . Then, has a fixed point; that is, there exists such that .

Proof. Define an iterative sequence in as in Theorem 14. Suppose that there is such that ; then, the proof is completed since . So, it is interesting to assume that , for all . Consequently, we have From (48), we find thatFollowing the lines for the proofs of Steps and in Theorem 14, we derive that is a Cauchy sequence andSince is a complete DMS, then there exists such thatWe will prove that . Suppose, on the contrary, that .
Notice from (84), (86), and (iii) thatBy using () and (87) together with the assumption of the theorem that is a generalized-()-MKC mapping, we obtainwhereNotice that as , then we must have .
Regarding (), we have By the above inequality, we haveSuppose that ; then, from (88), we getTaking in (92), we havewhich is a contradiction.
Now, we suppose that ; then, by (88), we find that Letting , this implies that which is again a contradiction.
Finally, we suppose that ; then, again from (88), we have Letting , in the above inequality, we getso we also have a contradiction. Thus, we have and, by (), we have .

2.3. The Uniqueness of the Fixed Point

We propose the following conditions for the uniqueness of the fixed points of the mappings discussed in Sections 2.1 and 2.2. Let denote the set of fixed points of the mapping .

We, first, recollect the following interesting condition for uniqueness of a fixed point of an -MKC mapping. For all , there exists such that and , where we have the following.

Theorem 16. Putting condition () to the statements of Theorem 7 (resp., Theorem 8), one obtains that is the unique fixed point of .

Proof. Let and be two distinct fixed points of . From , there exists such that Due to the fact that is -admissible, we have Inductively, we obtainFrom the above relation and since is an -MKC mapping, we have Iteratively, we getLetting , we obtainSimilarly, we can prove thatUsing (), we have Taking , we find that and so, by (), .

As an alternative uniqueness condition for fixed points of -MKC mapping, we suggest the following hypothesis:(U)For all , then .

Theorem 17. Putting condition (U) to the statements of Theorem 7 (resp., Theorem 8), one finds that is the unique fixed point of .

Proof. Let be two distinct fixed points of . Then, by Lemma 4 part (C), we haveDue to the property of (), we getLet ; then, for any , we find that Regarding (U) and the assumption of the theorem that is an -MKC mapping, we obtainwhich is a contradiction. Thus, .

In what follows, we propose the conditions for the uniqueness of a fixed point of a generalized--MKC mapping:(H1)For all , there exists such that , , and .(H2)Let . If there exists a sequence in such that , , and , then (H3)For any , then .

Theorem 18. Putting conditions (H1), (H2), and (H3) to the statements of Theorem 14 (resp., Theorem 15), one has that is the unique fixed point of .

Proof. Let be two distant fixed points of . From (H1), there exists such thatOwing to the fact that is triangular -orbital admissible and , by (), we haveInductively, we findNow, since and , then, by (), we deduceAgain, by (), since and , we deriveInductively, we getIn an analogous way, we will prove thatDefine an iterative sequence by , for all and
Step 1. We will prove thatBy (117) and the statement of the theorem that is generalized--MKC mapping, we have If , then Now, suppose that ; then, . As is a generalized--MKC mapping, we get whereRegarding (H2) and (), we haveThus, we haveLetting in the inequality above, we obtainwhich is a contradiction. Then,Hence, we getStep 2. We will prove that By (H3) and the assumption of the theorem that is a generalized--MKC mapping, we find thatSuppose, on the contrary, that . Then, from the above inequality, we obtainwhich is a contradiction. Thus, .
In an analogous way of Step , we can complete the proof ofBy (), we haveLetting in the above inequality, we find thatby (), we have .

3. Ulam-Hyers Stability

Definition 19 (see [8]). Let be a complete DMS and let be a self-mapping. The fixed point inclusionis called generalized Ulam-Hyers stable if and only if is increasing and continuous in and , such that, for each and for each solution of the equationthere exists a solution of the fixed point inclusion (135) such that If there exists such that , for each , then the fixed point (135) is said to be Ulam-Hyers stable.

Theorem 20. Let be a complete DMS and let be a self-mapping. Suppose that all the hypotheses of Theorem 18 hold. Additionally, one supposes that (i) and the function , is strictly increasing and onto;(ii)for any solution of (136) one has , where In these conditions, the fixed point inclusion (135) is generalized Ulam-Hyers stable.

Proof. We are in the conditions of Theorem 18; hence, there exists such that . Let and be a solution of (136).
From (ii), we have that , and because is -admissible we will obtain that .
We havewhereWe haveFrom here, we haveIt is clear that if the proof is completed.
Suppose that . Consequently, we have . On account of (138), we derive that From condition (i), we get that and hence Hence, (135) is generalized Ulam-Hyers stable.

Corollary 21. Suppose that is a complete DMS, is a self-mapping, and all the hypotheses of Theorem 16 hold. One supposes also that(i) and the function , is strictly increasing and onto;(ii)for any solution of (136) one has , where . In these conditions, the fixed point inclusion (135) is generalized Ulam-Hyers stable.

The proof is an analog of the proof of Theorem 20.

4. Conclusion

It is possible to list several existing results as a consequence of our main results by choosing both auxiliary functions and in a proper way like in the literature [5, 6, 9–11] and so on.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

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

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Copyright

Copyright © 2016 Anisa Mukhtar Hassan 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.

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