On Fuzzy Fixed Points and an Application to Ordinary Fuzzy Differential Equations

Ameer, Eskandar; Aydi, Hassen; Arshad, Muhammad

doi:https://doi.org/10.1155/2020/8835751

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Inhomogeneous Nonlinear Partial Differential Problems: Existence and Non-Existence of Solutions

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Volume 2020 | Article ID 8835751 | https://doi.org/10.1155/2020/8835751

On Fuzzy Fixed Points and an Application to Ordinary Fuzzy Differential Equations

Eskandar Ameer,^1,2Hassen Aydi ,^3,4and Muhammad Arshad¹

Academic Editor: Mohamed Boussairi Jleli

Received25 Sept 2020

Revised15 Oct 2020

Accepted27 Oct 2020

Published16 Nov 2020

Abstract

The aim of this paper is to obtain the common fuzz fixed points of -fuzzy mappings satisfying generalized almost -contraction in complete metric spaces. Our results are extensions and improvements of the several well-known recent and classical results in literature. We give an example for supporting these results. As an application, we apply our obtained results to study the existence of a solution for a second order nonlinear boundary value problem.

1. Introduction and Preliminaries

The fixed point result of Banach [1] is an interesting tool that ensures the existence and uniqueness of a fixed point of self-mappings defined on metric spaces. Later, various extensions and generalizations of Banach’s theorem appeared by defining a variety of contractive type conditions for self and non-self-mappings on different spaces (see [2–27]).

In [17, 18], Berinde discussed many contraction type mappings. He initiated the notion of almost contractions.

Definition 1. A self-mapping on a metric space (in short MS) is named as an almost contraction, if there are and so that for all , This almost contraction has been generalized as follows.

Definition 2. A self-mapping on a MS is called a generalized almost contraction, if there are and so that for all , Wardowski [28] presented the notion of -contractions and established a related fixed point theorem.

Definition 3 (see [28]). A self-mapping on a MS said to be an -contraction if there are and such that where F is the set of functions such that
(F1) For all , ,
(F2) For each sequence of positive numbers, (F3) There is so that ,
Later on, Altun et al. [22] modified Definition 3 by adding the condition (F4):
(F4) , for all with
Denote by the set of functions verifying the conditions -.

Theorem 4 (see [28]). Let be an -contraction on a complete MS. Then, admits a unique fixed point , and for each , the sequence is convergent to .

Definition 5 (see [4]). A self-mapping on a MS is said to be a -contraction, if there are and so that where is the set of functions so that
() is nondecreasing,
() For each sequence , () There are and such that
() is continuous
The following result was established by Jleli and Samet [4].

Theorem 6 (see [4]). Each -contraction mapping on a complete MS possesses a unique fixed point.
As in [22], Hancer et al. [12] took the following:
() , for all with .
Denote by the set of functions so that , and hold.
As in [9], we denote by the set of functions so that
() is nondecreasing,
()
() is continuous.

Theorem 7 (see [9]). Let be a self-mapping on a MS . We have the equivalence of the following: (i) is a -contraction with ,(ii) is a -contraction with .Very recently, Liu et al. [9] introduced the notion of ()-type contractions and established new fixed point theorems for such kind of mappings in complete MS.

Definition 8. A self-mapping on a MS is said to be a -type contraction, if there are a comparison function and such that for all , where and is the set of functions so that
() is nondecreasing,
() For each sequence , () is continuous
As in [2], a function is named as a comparison function if (1) is increasing, that is, ,(2) for all , where stands for the nth iterate of .Denote by the set of comparison functions.

Lemma 9 (see [9]). Let be a nondecreasing and continuous function with , and be a sequence in . Then,

Example 10 (see [2]). Consider the following comparison functions(1) with ,

Example 11 (see [9]). For all , consider Here, .

On the other hand, using the notion of a fuzzy set, Heilpern [29] initiated the family of fuzzy mappings, which generalized the concept of set-valued mappings, and presented a fixed point result for fuzzy contraction mappings in the context of metric linear spaces. Mention that the theorem of Heilpern [29] is considered as a fuzzy extension of the BCP. Later on, several researchers worked on fixed point results involving fuzzy mappings, see ([30–46]).

Let be the set family of bounded and closed subsets of a MS . For and , set

Given as

is a metric on . It is named as the Pompeiu-Hausdorff metric induced by . A fuzzy set in is a function with domain and values in . Denote by the set fuzzy sets in . Let be a fuzzy set and , then the function values is named as the grade of membership of in . The -level set of is denoted by and is defined by

Here, denotes the closure of the set . Let be the collection of all fuzzy sets in . For , means for each . The fuzzy set is denoted by unless and until it is stated, where is the characteristic function of the crisp set . If there is so that , then consider

If for each , then take

We write instead of . A fuzzy set in a metric linear space is said to be an approximate quantity if and only if is compact and convex in for each and . The set of approximate quantities in is denoted by . Let be an arbitrary set and be a MS. A mapping is called fuzzy mapping (in short, FM) if is a mapping from into . A fuzzy mapping is a fuzzy subset on with the membership function . The function is the grade of membership of in .

Definition 12. Let be FM from into . An element in is said to be an -fuzzy fixed point of if there is so that . is named as a common -fuzzy fixed point of and if there is so that . is called a common fixed point of fuzzy mappings.

In this manuscript, we ensure the existence of some common -fuzzy fixed points for fuzzy mappings for almost -contractions in the class of complete MS. Our theorems generalize some known results in literature.

Now, we need the following.

Lemma 13 (see [16]). Let be a MS and , then for every ,

Lemma 14 (see [30]). Let be a metric linear space and be a fuzzy mapping and . Then, there is so that .

Lemma 15 (see [47]). Let be a metric space, and be FM such that is a nonempty compact set for each . Then, if and only if for each .

2. Main Results

From now on, is assumed to be a complete MS.

Theorem 16. Let be FM and for each , and there exist , such that and are nonempty, closed, and bounded subsets of . Assume that there exist , , and such that for all implies where If is continuous, then there exists some such that

Proof. Let . By hypotheses, there exists such that is a nonempty, closed, and bounded subset of . For convenience, we denote by . Let , then there exists such that is a nonempty, closed, and bounded subset of . Since is nondecreasing, we have used (19) and Lemma 13where By , we have Thus, there exists such that Then from (22), we have where If then from (27), we have which is a contradiction. Thus, By (27), we get that Next, there exists such that is a nonempty, closed, and bounded subset of . By Lemma 13, using (19) and the fact that is nondecreasing, we have where From , we have Thus, there exists such that Then from (32), we have where If then from (37), we have which is a contradiction. Thus, By (37), we get that By continuing this process, we construct a sequence in such that and Thus, from (42) and (43), we have This implies that Letting , we get It yields that This together with and Lemma 9 gives that Now, we will prove that is a Cauchy sequence. Arguing by contradiction, we assume that there are and sequences and of integers so that for all with Thus, Letting in we get Again, Taking in (51) and (52), we get From (48) and (51), we can choose an integer so that by (19), we get where Letting in the above inequality, since and are continuous and by using (48), (49), (51), and (54), we get It is a contradiction, so is Cauchy. Since is complete, converges to i.e., . We claim that . Assume that (that is, ) then there are and a subsequence of so that , for all Since , for all so by we have where Letting and using the continuity of and , we have which is a contradiction. Hence, and Similarly, one can easily prove that . Therefore,

Example 17. Let endowed with the metric . Consider as and . Here, and are continuous. For , given bysuch that For with we have where Hence, all the conditions of Theorem 16 are satisfied, and .

Corollary 18. Let be FM from into , and for each , there exist , such that and are nonempty, closed, and bounded subsets of . Assume that there are and such that for all where is defined by (20). If is continuous, then there is such that

Proof. Set in Theorem 16.

Corollary 19. Let be a FM from into , and for each , there exist such that is nonempty, closed, and bounded subsets of . Assume that there are , , and such that for all where If is continuous, then there is so that

Proof. Take in Theorem 16.
Now, we consider multivalued mappings.

Theorem 20. Let be multivalued mappings. Assume that there are , , and such that for all where If is continuous, then there is such that

Proof. Consider a mapping and a pair of fuzzy mappings defined by Then, Thus, by Theorem 16, there is so that

Theorem 21. Let be FM. Assume that there are , , and such that for all where If is continuous, then there is so that and

Proof. Let ; then by Lemma 14, there is so that . Similarly, we can find such that . It follows that for each , and are nonempty, closed, and bounded subsets of . As , by the definition of a -metric for fuzzy sets, we have Since is nondecreasing, we have where for all Since , we have It yields that Similarly, It yields that for all , where By Theorem 16, we obtain such that , that is, and

Corollary 22. Let be a complete MS, and be FM. Assume that there exist and such that for all where is defined by (80). If is continuous, then there is so that and

Proof. Take in Theorem 21.
We denote by (for details, see [43, 44]) the setvalued mapping induced by a FM i.e.,

Corollary 23. Let be FM such that for all , and are nonempty, closed, and bounded subsets of . Assume that there exist , , and such that for all where If is continuous, then there is so that and for all

Proof. By Theorem 16, there is so that . Then by Lemma 15, we have

Corollary 24. Let be FM such that for all , and are nonempty, closed, and bounded subsets of . Assume that there are , , and so that for all where If is continuous, then there is such that and for all

Proof. Take in Corollary 23.

3. Some Consequences

Corollary 25. Let be FM, and for each , there exist , such that and are nonempty, closed, and bounded subsets of . Assume that there are and such that for all where and are defined by (20) and (21), respectively. Then, there is such that

Proof. Take and in Theorem 16.

Corollary 26. Let be FM, and for each , there are , , such that and are nonempty, closed, and bounded subsets of . Assume that there are and such that for all implies where is defined by (20). Then, there is such that

Proof. Take , , and in Theorem 16.

Corollary 27. Let be FM, and for each , there exist , , such that and are nonempty, closed, and bounded subsets of . Assume that there are and such that for all implies where is defined by (20). Then, there is so that

Proof. Take , , and in Theorem 16.

Corollary 28. Let be FM, and for each , there exist , , such that and nonempty, closed, and bounded subsets of . Assume that there is such that for all implies where and are defined by (20) and (21), respectively, and is such that for each Then, there is such that

Proof. It follows from Theorem 16 by taking and

4. Application to an Ordinary Fuzzy Differential Equation

In this section, we apply our obtained results to study the existence of a solution for the second order nonlinear boundary value problem: where is a continuous function. This problem is equivalent to the integral equation ([45, 46, 48]): where Green’s function is given by and satisfies , , and . Let us recall some properties of . Particularly,

Our investigation is based on the existence of a common fixed point for a pair of integral operators given as follows: where , , and .

Theorem 29. Suppose that (a) are increasing in their second and third variables,(b)There is such that for all , we have (c)There are such that for all , we have for all , with (d)For and , we have (e)if is comparable, then every and every are comparable.Then, the pair of nonlinear integral equations has a common solution in .

Proof. Consider with the metric

Note that is a complete linear MS. Let be two integral operators defined by (108) and (109). Clearly, and are well defined since and are continuous functions. Now, is a solution of (113) and (114) if and only if is a common fixed point of and . By hypothesis (a), is increasing. Next, for all with , by hypothesis (c), we have successively

From (116) and (117), we easily obtain

It implies that

Therefore,

Let be defined by respectively.

Thus, we have

Therefore, by Corollary 28, and have a common fixed point , that is, is a common solution of (113) and (114). As an immediate consequence of Theorem 29, in the case of , we find that the integral equation (105) has a solution in , and hence, the second order nonlinear boundary value problem (104) has a solution.

5. Conclusion

In the present work, we introduced a new concept of fuzzy mappings in complete metric spaces. Also, we derived the existence of -fuzzy common fixed points for two fuzzy mappings under generalized almost -contractions in complete metric spaces. We also gave an illustrative example to support our main results. We further showed some relations between multivalued mappings and fuzzy mappings, which can be utilized to ensure the existence of a common fixed point for multivalued mappings. Finally, we applied our main results to provide a solution for a second order nonlinear boundary value problem.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally in writing this article. All authors read and approved the final manuscript.

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