Journal of Mathematics
Volume 2021 (2021), Article ID 9945842, 7 pages
https://doi.org/10.1155/2021/9945842

Research Article

Common -Fuzzy Fixed Point Results with Applications to Volterra Integral Inclusions

Abdullah Eqal Al-Mazrooei and Jamshaid Ahmad

Department of Mathematics, University of Jeddah, P.O.Box 80327, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Jamshaid Ahmad; jamshaid_jasim@yahoo.com

Received 16 March 2021; Accepted 28 April 2021; Published 12 May 2021

Academic Editor: Ljubisa Kocinac

Copyright © 2021 Abdullah Eqal Al-Mazrooei and Jamshaid Ahmad. 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.

The purpose of this paper is to establish some common -fuzzy fixed point theorems for a pair fuzzy mappings and obtain some results of literature for multivalued mappings. For it, we define the notion of generalized -contractions in the context of -metric spaces. As applications, we investigate the solutions of Volterra integral inclusions by our established results.

1. Introduction and Preliminaries

Among all the impressive and inspiring generalizations of metric spaces, -metric space has an integral place. Czerwik [1] in 1993 extended the notion of metric space by introducing the conception of -metric space in this way.

Definition 1. Let . A mapping : is called -metric if it satisfies these assertions:

for all , where .
Then, is called a -metric space. A standard example of -metric space which is not metric space is the following:
and defined by

for all with .
Let represent the class of all nonempty, bounded, and closed subsets of . For , we define by

where

Note that is called the Hausdorff -metric induced by the -metric . We recall the following properties from [1–3].

Lemma 1. (see [2]). Let be a -metric space. For any and any , we have the following:

(i) for any (ii)(iii) for any (iv)(v)(vi)(vii).

Later on, many authors (see [4–7]) worked in this way. Recently Jleli and Samet [8] gave the notion of -contractions and proved a contemporary result for such contractions in generalized metric spaces. Afterwards, Hancer et al. [9] revised the foregoing definitions by including a broad condition (). Inspired by Jleli and Samet [8] and Hancer et al. [9], Alamri et al. [10] initiated the above notions in the context of -metric spaces and introduced a more general condition () along with above axioms.

Definition 2. (see [10]). We represent by the family of all mappings satisfying these properties:

 ()  () for , if and only if  () there exists and such that  () for all with  () for all such that , and some , then , for all

They supported this condition by the following nontrivial example.

Example 1. (see [10]). Let be given by . Clearly, satisfies ()-(). Here we show only (). Assume that, for all and some , we have , which implies that

This implies that

As . Also is nondecreasing, so and implies . Therefore, (5) implies

and hence () holds.
On the other side, Kumam et al. [11] utilized the concept of -metric space and obtained common - fuzzy fixed points for fuzzy mappings under generalized rational contractions. For more details in the direction of fixed point results for fuzzy mappings, we refer [9–20] to the readers.
We need the following lemma of Czerwik [2].

Lemma 2. (see [11]). Let be a -metric space and , then

In this paper, we obtain common -fuzzy fixed point results for a pair of fuzzy mappings and establish some theorems to generalize some results from the literature. We solve the Volterra integral inclusions as application of our established results.

2. Main Results

In this way, we state our main result.

Theorem 1. Let be a complete -metric space with coefficient such that is continuous. Assume that : and for each , there exist such that . If there exist and such that

for all with , then there exists such that .

Proof.  

Let , then by assumption there exists such that . Let . For this , there exists such that . By Lemma 2, () and (8), we have

Thus,

From (), we know that

Thus from (10), we get

Then, from (12), there exists (obviously, ) such that

For this , there exists such that . By Lemma 2, (), and (8), we have

Thus,

From (), we know that

Thus from (15), we get

Then, from (17), there exists (obviously, ) such that

So, continuing in the same way, we construct in such that

for all . From (20) and (21), we get

for all . It follows by (22) and property () that

which further implies that

for all . Thus,

for all . Since , so letting in (25), we get

This implies

by (). By (), there exists and so that

Suppose that . For this case, let . By definition of the limit, there exists so that

for all . This implies that

for all . Then

for all , where . Now we assume that . Let . From the definition of the limit, there exists such that

for all , which implies

for all , where . Hence, in all cases, there exists and such that

for all . Hence by (25) and (34), we obtain

Taking the limit , we get

Thus which implies that is convergent. Thus is a Cauchy sequence in . Since is a complete -metric space, so there exists a such that

Now, we prove that . We suppose on the contrary that , then there exist and of such that , for all . Now, using (8) with and , we obtain

As , so by () so we obtain

Letting , we have

Hence, . Likewise, one can straightforwardly prove that . Thus, .

Note: From now to onwards, we consider as continuous functional and as complete -metric space.

The following corollary follows from Theorem 1 by considering for .

Theorem 2. Let , and for each , such that . If such that

for all , then there exists such that .

Theorem 3. Let , and for each , there exist such that . If there exists such that

for all , then there exists such that .

Example 2. Let and by

It is easy to see that is a complete -metric space with coefficient . Define

Define by for all . Now we obtain that

For , we get

Taking for and . Then

also

for all . As a result, all assertions of Theorem 6 hold and there exists such that is an -fuzzy fixed point of .

3. Set-Valued Results

Theorem 4. Let . Suppose that such that

for all . Then there exists such that .

Proof. Define and by

Then

Thus, Theorem 4 can be applied to get such that

Corollary 1. Let be multivalued mapping. Assume that there exists such that

for all . Then there exists such that .

Remark 1. If , then -metric spaces turns into complete metric space and we obtain some new results for fuzzy mappings as well as multivalued mappings in metric spaces.

4. Applications

Consider the Volterra integral inclusion

where a given set-valued mapping and be such that is given and is unknown function.

Now, for , consider the -metric on defined by

for all . Then, is a complete -metric space.

We will assume the following:

(a)For each , the mapping is such that is lower semicontinuous in (b)There exists which is continuous such that

 for all , .(c)There exists so that

Theorem 5. Under the assumptions (a)–(c), the integral inclusion (54) has a solution in .

Proof. Let . Define by

for all . Let be arbitrary, then there exists . For , it follows from Michael’s selection theorem that there exists such that for each . It follows that . Hence, . It is a simple matter to show that is closed, and so details are excluded (see also [17]). Moreover, since is continuous on , and is continuous, their ranges are bounded. This means that is bounded. Thus, . ■
Let , then there exists such that . Let be arbitrary such that

for holds. This means that for all , there exists such that

for . For all , it follows from (b) that

It means that there exists such that

for all .
Now, we can take the set-valued operator defined by

Hence, by (a), is lower semicontinuous. It follows that there exists a continuous operator such that for . Then, satisfies that

That is and

for all . Thus, we obtain that

Interchanging the roles of and , we obtain that

This implies that

Taking exponential, we have

Taking the function defined by for , we get that the condition (8) is satisfied. Using the result 6, we conclude that (54) has a solution.

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.

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