#### 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 byfor all with .
Let represent the class of all nonempty, bounded, and closed subsets of . For , we define bywhereNote that is called the Hausdorff -metric induced by the -metric . We recall the following properties from [13].

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 [47]) 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 thatThis implies thatAs . Also is nondecreasing, so and implies . Therefore, (5) impliesand 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 [920] 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 thatfor 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 thatfor all . From (20) and (21), we getfor all . It follows by (22) and property () thatwhich further implies thatfor all . Thus,for all . Since , so letting in (25), we get

This impliesby (). By (), there exists and so that

Suppose that . For this case, let . By definition of the limit, there exists so thatfor all . This implies thatfor all . Thenfor all , where . Now we assume that . Let . From the definition of the limit, there exists such thatfor all , which impliesfor all , where . Hence, in all cases, there exists and such thatfor 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 thatfor all , then there exists such that .

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

Example 2. Let and byIt is easy to see that is a complete -metric space with coefficient . DefineDefine by for all . Now we obtain thatFor , we getTaking for and . Thenalsofor 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 thatfor all . Then there exists such that .

Proof. Define and byThenThus, Theorem 4 can be applied to get such that

Corollary 1. Let be multivalued mapping. Assume that there exists such thatfor 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 inclusionwhere a given set-valued mapping and be such that is given and is unknown function.

Now, for , consider the -metric on defined byfor 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 thatfor all , .(c)There exists so that

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

Proof. Let . Define byfor 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 thatfor holds. This means that for all , there exists such thatfor . For all , it follows from (b) thatIt means that there exists such thatfor all .
Now, we can take the set-valued operator defined byHence, by (a), is lower semicontinuous. It follows that there exists a continuous operator such that for . Then, satisfies thatThat is andfor all . Thus, we obtain thatInterchanging the roles of and , we obtain thatThis implies thatTaking exponential, we haveTaking 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.