[Retracted] A Solution of Fredholm Integral Inclusions via Suzuki-Type Fuzzy Contractions

Abstract

In this study, we introduce fuzzy weak -contraction and Suzuki-type fuzzy weak -contraction and employ these to prove some fuzzy fixed point results for fuzzy mappings in the setting of metric spaces, which is followed by an example to support our claim. Next, we deduce some corollaries and fixed point results for multivalued mappings from our main result. Finally, as an application of our result, we provide the existence of a solution for a Fredholm integral inclusion.

1. Introduction and Preliminaries

The idea of fuzzy mapping was inspired by the fuzzy set theory given by Zadeh [1]. It was initiated by Heilpern [2] in 1981, defined to be a mapping from an arbitrary set to a subfamily of fuzzy sets in metric linear spaces. He established a fuzzy expansion of Banach contraction principle. It broadens and develops the concept of fuzzy fixed point theory, and several authors worked in this field afterward ([3–7] and references therein).

We describe some related concepts in short in the successive lines.

Here, depicts a metric space. A fuzzy set in is a function with domain and codomain . If is a fuzzy set and , the function value is called grade of membership of in . The collection of all fuzzy set in is denoted by .

Let and . The -level set of , which we denote here by , is defined bywhere denotes closure of set .

Definition 1 (see [2]). A fuzzy subset on is said to be an approximate quantity if and only if its -level set is a compact convex subset of , for each and .

We denote by , the subcollection of approximate quantities. We also denote is nonempty compact convex subset of a metric space . If and , for some , then is an approximation of .

Definition 2 (see [2]). [2] For and , define

Remark 1 (see [2]). is a metric on . Let . Then, is more accurate than , denoted by iff , for each .

Definition 3 (see [2]). Let be a nonempty set and any metric linear space. A mapping is called fuzzy mapping if and only if is a mapping from into (or ), i.e., (or ), for each .

Lemma 1 (see [2]). The following conditions hold for a metric space :(a)If , then (b)(c)If , then For all and .

A fuzzy mapping is a fuzzy subset on with membership function . The function value is the grade of membership of in .

Definition 4 (see [8]). Let and . The fuzzy point of is the fuzzy set given byFor , we have

Definition 5 (see [9]). A fuzzy point in is a fixed fuzzy point of the mapping over if , i.e., or .

Remark 2 (see [2]). If , then is a fixed point of fuzzy mapping .

Also, the generalization of Banach contraction principle has been done in many ways along with providing their applications in different fields. One of them is by generalizing the contraction condition, specially using nonlinear contractions, e.g., Suzuki-type contraction, -contraction, and -contraction [10–16]. One of such generalizations, namely, -weak contraction was done by Alber and Guerre-Delabriere [17] in 1997 to prove fixed point result in the setting of Hilbert space, which was further utilized by Rhoades [18] in metric fixed point theory. Recently, a generalization of the same was furnished by Xue [19]. He used the class of mappings of all continuous nondecreasing functions with and defined generalized -weak contraction as follows:where is a self-mapping on a metric space and . Also, he showed that this contraction condition is more weaker than -weak contraction condition (viz. ). After that, Perveen et al. [20, 21] used his idea and proved results using weaker conditions.

In this study, we utilize the above ideas and define fuzzy weak -contraction and Suzuki-type fuzzy weak -contraction, which we use to prove the existence of fuzzy fixed point supported by an example. Last, we furnish an application of our result to prove the existence of a solution of integral inclusion of Fredholm type.

2. Main Result

First, we define the same class of mappings used in [20, 21].

Let denotes the set of all mappings satisfying the following:(a) is nondecreasing(b) iff and , whenever

We have noticed that [19] used the continuity of . Inspired by [22], we dropped the continuity condition and use a weaker condition, which is given in . In fact, is also weaker than the condition that is lower semicontinuous. Indeed, if is lower semicontinuous, then for a sequence with , we have .

Using the class defined above, we define the following contraction for fuzzy mapping.

Definition 6. Let be a metric space. A fuzzy mapping is(a)a fuzzy weak -contraction mapping if(b)a Suzuki-type fuzzy weak -contraction mapping if the following condition is satisfied: for all , where

Remark 3. If is fuzzy weak -contraction, then is Suzuki-type fuzzy weak -contraction.

Now, we are ready to commence our main theorem.

Theorem 1. Let a complete metric space and be a Suzuki-type fuzzy weak -contraction, such that for every , is closed. Then, there exists , such that is a fuzzy fixed point of , i.e., .

Proof. Let be any arbitrary point. Since , we can choose , such that . If , then we are done. Suppose that . Since , there exists , such thatAgain, if , we are done. Otherwise, we continue this process and obtain a sequence satisfying the following conditions:Thus, we easily obtainwhich implies thatwhich further impliesThus, we see that is a nonincreasing sequence of positive real number bounded below by 0. Hence, converges to a point . We assert that . Suppose it is not so, then taking limit in (11), we obtainwhich is a contradiction. Therefore, we haveNext, we prove that is a Cauchy sequence. Suppose on contrary that it is not so, then there exist two subsequences and of , such that is the smallest positive integer for whichNow, utilizing triangular inequality, we obtainLetting , we obtainAgain, by triangular inequality,which on letting , we obtainNext, by (14) and (17), there exists , such thatThus, for and , by (7), we obtainwhich on letting yields , a contradiction. Thus, is Cauchy in . The completeness of implies that , for some .
Next, we show that . As , there exists , such that for all ,Using the above inequality, we obtain (for all )i.e.,Thus, by (7), we obtainwhich impliesTaking limit , we obtainFurthermore, we prove thatThe above equation holds trivially for . Suppose . Then, for every , there exists , such thatThus, with the help of the above inequality and (27),On taking limit , we obtainSo, (28) holds true for all . Now, if , then we are done. Assume that it is not so, then there exists , such that for every , we can choose , such that for all . For , (28) reduces toTaking , we obtaini.e., , a contradiction. So, , and the proof is completed.

We present the following example to illustrate the utility of our proven result.

Example 1. Let , and is defined byWe define byand a fuzzy mapping byThen, , andWe consider three cases.

Case 1. If , then we haveHence, (7) is satisfied for trivially.

Case 2. If and , then we haveThen,So condition (7) is satisfied.

Case 3. If and , then we haveThus, we getWe see that the assumptions of Theorem 1 are fulfilled in all cases, and hence, has a fuzzy fixed point which is 2.

In view of Remark 3, we deduce the underlying result.

Theorem 2. Let be a complete metric space and a fuzzy weak -contraction, such that for every , is closed. Then, there exists , such that is a fuzzy fixed point of , i.e., .

If the fuzzy mapping is a Suzuki-type fuzzy weak -contraction, then it immediately satisfies the following contraction condition:where and . But the converse need not be true. We justify this claim by showing that the condition is weaker than . For this, we consider the following example.

Let with the metric defined by , ; , and , . Defineand a fuzzy mapping by

So, we get and , and

We observe that the condition is satisfied for all , but for and , the condition is not fulfilled.

Hence, we obtain the following result.

Theorem 3. Let be a complete metric space and a fuzzy mapping satisfying (43). Then, there exists , such that .

Taking , in the above result (viz. (43)), we obtain the next result.

Theorem 4. Let be a complete metric space and a fuzzy mapping satisfying the conditionwhere . Then, there exists , such that .

Remark 4. Let be a fuzzy mapping from to and a closed mapping (where denotes the set of all compact subsets of ). Definefor each . Note that,

In view of above remark, we obtain the fixed point results for multivalued mapping (defined above) from Theorems 1-4.

Theorem 5. Let be a complete metric space and a multivalued closed mapping satisfying and . Then, has a fixed point.

Theorem 6. Let be a complete metric space and a multivalued closed mapping satisfying and . Then, has a fixed point.

Theorem 7. Let be a complete metric space and a multivalued closed mapping satisfying. Then, has a fixed point.

Similarly, we can obtain the results corresponding to Theorem 2.

3. An Application to the Fredholm Integral Inclusion

Consider the following Fredholm integral inclusion:where and ( denotes the class of all nonempty compact and convex subsets of ), and is an unknown function. Consider and take the complete metric space , where

Before proving our claim, we note down the following lemma.

Lemma 2 (see [23, 24]). Let be a metric space and . If there exists , such that(a)For each , there exists , such that (b)For each , there exists , such that Then, .

Theorem 8. Under the conditions given as follows:  for all , the operator is such that is lower semicontinuous on   there exists a continuous function , such that for all and with  The Fredholm integral inclusion (53) has a solution in .

Proof. Define the fuzzy mapping in such a way that It is very obvious that the set of solutions of coincides with the set of fixed points of (53). So, we need to prove that has at least one fixed point.
For this, we consider an arbitrary fixed point and the set-valued operator . Using Michael’s theorem, we obtain a continuous function, such that , for each . Thus, , and so, . Clearly, is closed (hence compact) and convex. So, .
Now, we will check thatLet _α (arbitrary), such that , for . This means for all , there exists , such that . Now, from , we haveThen, there exists , such thatNow, we consider the multivalued operator defined byHence, by , is lower semicontinuous which ensures the existence of a continuous operator , implying thatand hence,So, we get , andAfter interchanging the roles of and and using Lemma 2, we obtain (for each )and by considering (for all ), all the assumptions of Theorem 1 as well as Theorem 2 are satisfied. Hence, the inclusion problem (53) has a solution.

4. Conclusion

In this study, inspired by the work of Suzuki [10] and Xue [19], we define two new contractions, i.e., fuzzy weak -contraction and Suzuki-type fuzzy weak -contraction and use them to prove the existence of fuzzy fixed point and well exemplify them. Also, we provide an application of our proven result to show the existence of solution of Fredholm integral inclusion 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 that they have no conflicts of interest.

Authors’ Contributions

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