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 [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 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 [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 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 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 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.