#### Abstract

In this paper, some concepts of -metric spaces are used to study a few fuzzy fixed point theorems. Consequently, corresponding fixed point theorems of multivalued and single-valued mappings are discussed. Moreover, one of our obtained results is applied to establish some conditions for existence of solutions of fuzzy Cauchy problems. It is hoped that the established ideas in this work will awake new research directions in fuzzy fixed point theory and related hybrid models in the framework of -metric spaces.

#### 1. Introduction

One of the challenges in mathematical modeling of practical phenomena relates to the indeterminacy induced by our inability to categorize events with adequate precision. It has been understood that classical mathematics cannot cope effectively with imprecisions. As a result, the concept of fuzzy set was initiated by Zadeh [1] in 1965 as one of the uncertainty approaches to construct mathematical models compatible with real world problems in engineering, life science, economics, medicine, language theory, and so on. The basic ideas of fuzzy set have been extended in different directions. In particular, the notion of fixed point results for fuzzy set-valued mappings and fuzzy contractions was initiated by Heilpern [2] who proved a fixed point theorem parallel to the Banach fixed point theorem (see [3]) in the frame of fuzzy set. Thereafter, several authors have studied and applied fuzzy fixed point results in different settings [4, 5], see, for example [6–14] and the references therein.

Not long ago, Jleli and Samet [9] initiated the concepts of -metric spaces and obtained a generalization of the Banach fixed point theorem. Meanwhile, researchers have picked keen interests in establishing and improving different results in -metric spaces, see, for instance, [15–17].

The aim of this paper is twofold. First, we study some common fuzzy fixed point results in the setting of -complete -metric spaces. Consequently, corresponding fixed point theorems of multivalued and single-valued mappings are derived. Thereafter, one of our obtained results is applied to discuss some solvability conditions of fuzzy initial value problems. As far as we know, in the setting of -metric spaces and fuzzy mappings, the results presented herein are new and fundamental. On this note, it can be improved upon when discussed in other generalized hybrid models within the scope of fuzzy mathematics.

#### 2. Preliminaries

In this section, we record some specific definitions and results that will be useful in what follows hereafter.

*Definition 1 (see [9]). *Let be the set of functions satisfying the following conditions:

() is nondecreasing, i.e., implies

() for every sequence , we have

*Definition 2 (see [9]). *Let be a nonempty set and be a given mapping. Suppose that there exists such that

()

()

() for every , for every , and for every sequence with , we have
Then, is said to be an *-*metric and the pair is called an *-*metric space.

*Example 3 (see [9]). *Let . If we define by
with and , then is an *-*metric space.

*Definition 4 (see [9]). *Let be an *-*metric space and be a sequence in (i)We say that is *-*convergent to if is convergent to with respect to the *-*metric (ii) is said to be *-*Cauchy, if(iii) is *-*complete, if every *-*Cauchy sequence in is *-*convergent to an element in In the sequel, we shall adopt the following notations and definitions in the setting of -metric space. We shall denote an -metric by so that represents an -metric space. Let be the set of all nonempty compact subsets of and . Then
Then, the Hausdorff metric on induced by the metric is defined as

*Definition 5 (see [1, 2]). *Let be an arbitrary nonempty set. Then, a fuzzy set in is a function with domain and values in . If is a fuzzy set in and , then the function values is called the grade of membership of . The *-*level set of , denoted by , is defined as
Here, denotes the closure of the crisp set . Also, the family of fuzzy sets in a metric space shall be denoted by .

*Definition 6. *Let be an *-*metric space. A subset of is called proximal, if for each , there exists such that

Let the set of all nonempty bounded proximal sets in be denoted by and the set of all nonempty closed and bounded subsets of be represented by . Since every compact set is proximal and any proximal set is closed, we have the inclusions:

*Definition 7 (see [2]). *Let be an arbitrary set and a metric space. mapping is called a fuzzy mapping. fuzzy mapping is a fuzzy subset of with membership value

*Definition 8 (see [2, 7]). *Let and be fuzzy mappings from into . A point is called fuzzy fixed point of if The point is known as a common fuzzy fixed point of and if

*Definition 9 (see [18, 19]). *A nondecreasing function is said to be a comparison function, if as for every where denotes the th iterate of .

Denote by the set of all comparison functions.

Lemma 10 (see [18, 19]). *Let . Then, the following properties hold:
*(i)*Each iterate of , for is a comparison function*(ii)* is continuous at 0*(iii)* for all *

#### 3. Main Results

First, we present the following auxiliary result.

Lemma 11. *Let and be nonempty closed and compact subsets of an -metric space . If , then .*

*Proof. *The proof is a direct consequence of the definition of

Theorem 12. *Let be an -complete -metric space and be fuzzy mappings. Assume that for every , there exist such that . Suppose also that the following condition holds:
for all , where . Then, there exists such that .*

*Proof. *Let be arbitrary. By hypothesis, there exists such that . Since is a nonempty compact subset of , there exists such that . Similarly, we can find such that and by compactness of , we can choose such that . For convenience, denote and by, where .

By Lemma 11, we have
Therefore, using (10) together with (9), we have
If , then (11) becomes
which is a contradiction. It follows that . Therefore, we have
By continuous repetition of the above steps, we generate a sequence of elements of with
such that
Consequently, by induction, for all , we have
Let be a given positive number and such that condition is satisfied. By , there exists such that
Let such that . Hence, by (17) and (), we get
Now, for , by (), (16), and (18), we obtain
It follows from () that
This shows that is -Cauchy. Hence, -completeness of implies that there exists such that as . Now, to prove that , assume that . Then by (), we get

Now, we analyze (21) under the following cases:

*Case (i). *If
then (21) becomes
Since as , then by () and the properties of ,
which is a contradiction.

*Case (ii). *If
then,
Hence, by () and the properties of , , a contradiction.

*Case (iii). *If
then,
By condition (), from (28), for , we have
As in (29), we obtain
which is a contradiction.

*Case (iv). *If
then,
By (), from (32), for , and applying for all , we get
which is a contradiction. It follows that . On the same steps, one can show that . Consequently, .

Next, we give an example to support the validity of the hypotheses of Theorem 12.

*Example 17. *Let and be defined by
for all . It can be seen that satisfies (), (), and (); hence, is an -metric space with , , and . Notice that does not satisfy the triangle inequality, since
Moreover, let and consider two fuzzy mappings defined as follows:
(i)If (ii)If Now, define . Clearly, .

Now, for , there exist and such that . Specifically, if , then , and hence
If , then
Similarly, . Therefore, for , by the definition of , we get

Consequently, all the conditions of Theorem 12 are satisfied to find .

By imposing continuity condition on the function , we have the following modification of Theorem 12.

Theorem 18. *Let be an -complete -metric space and be fuzzy mappings. Assume that for every , there exist such that . Suppose also that the following conditions hold:
*(i)

*The function is assumed to be continuous. In addition, suppose satisfies for all*(ii)

*And for all ,, we have*

*Then, there exists such that*

*Proof. *Following the proof of Theorem 12, we obtain that is an -Cauchy sequence in the -complete metric space . Therefore, there exists such that
Now, to prove that , we argue by contradiction. So assume . Then by (), and inequation (41), we get
Taking the limit in (43) as and using (42) together with the continuity of and , we have
which is a contradiction to the condition on . It follows that . On similar steps, we can show that . Consequently, we have

*Example 19. *In line with Example 17, take and ; then, is continuous on and is continuous for all . Notice that the condition for becomes . Therefore, following the remaining constructions of Example 17, one can easily verify that all the hypotheses of Theorem 18 are satisfied to find some such that

#### 4. Consequences

In this section, we apply Theorems 12 and 18 to deduce some fixed point results of multivalued and single-valued mappings in the context of -metric spaces. To this end, recall that a point is called a fixed point of a multivalued (single-valued) mapping on , if .

Corollary 20. *Let be an -complete -metric space and be multivalued mappings. Suppose that the following condition holds:
for all , where . Then, there exists such that .*

*Proof. *Let be any two arbitrary mappings, and consider two fuzzy set-valued maps defined as follows:
Then, for all , we have
Similarly, . Consequently, Theorem 12 can be applied to find such that .

Following the proof of Corollary 20, we can also apply Theorem 18 to establish the following result.

Corollary 21. *Let be an -complete -metric space and be multivalued mappings. Suppose that the following conditions hold:
*(i)

*The function is assumed to be continuous. In addition, suppose satisfies for all*(ii)

*And for all , we have*

*Then, there exists such that .*

Corollary 22. *Let be an -complete -metric space and be single-valued mappings. Suppose that the following condition holds:
for all , where . Then, there exists such that .*

*Proof. *Let for all . Then, define two fuzzy set-valued maps as follows:
Then,
Similarly, . Obviously, , for all . Notice that in this case, . Therefore, Theorem 12 can be applied to find such that and , which further implies that .

Following the proof of Corollary 22, one can also employ Theorem 18 to establish the following result.

Corollary 23. *Let be an -complete -metric space and be single-valued mappings. Suppose that the following conditions hold:
*(i)

*The function is assumed to be continuous. In addition, suppose that satisfies for all and for all*(ii)

*And for all , we have*

*Then, there exists such that .*

*In the following, we apply Corollary 22 to deduce the main result of Jlei and Samet [9].*

Corollary 24 (see [9]). *Let be an -complete -metric space and be a single-valued mapping. If for all , there exists such that
then there exists such that .*

*Proof. *Consider Corollary 22. Define the function by , for all and . Then, by taking , all the hypotheses of Corollary 22 coincide with that of Corollary 24; and so, we can find such that .

*Remark 25. *It is obvious that more consequences of Theorems 12 and 18 can be obtained, but we skip them due to the length of the paper.

#### 5. Application to Fuzzy Initial Value Problems

Fuzzy differential equations (FDEs) and fuzzy integral equations (FIEs) play significant roles in modeling dynamic systems in which uncertainties or vague notions flourish. These concepts have been established in different theoretical directions, and a large number of applications in practical problems have been studied (see, for example, [20–22]). Several techniques for studying FDEs have been presented. The first most popular is using the Hukuhara differentiability (H-differentiability) for fuzzy valued functions (see [20, 23, 24]). On the other hand, the concept of FIEs was initiated by Kaleva [22] and Seikkala [25]. In the study of existence and uniqueness conditions for solutions of FDEs and FIEs, many authors have applied different fixed point theorems. By using the classical Banach fixed point theorem, Subrahmanyam and Sudarsanam [26] proved an existence and uniqueness result for some Volterra integral equations involving fuzzy set-valued mappings. With the help of Shaulder’s fixed point theorem and Arzela-Ascoli’s theorem, Allahviranloo et al. [27] studied the existence and uniqueness conditions of solutions of some nonlinear fuzzy Volterra integral equations. In [28], the authors discussed some existence results for a fuzzy initial value problem (FIVP) by employing some contractive-like mapping techniques. Congxin and Shiji [29] studied a Cauchy problem of fuzzy differential equation on the basis of the definition of H-differentiability for fuzzy set-valued mappings. They obtained the existence and uniqueness theorem for the Cauchy problem under some generalized Lipschitz condition. Similarly, Villamizar-Roa et al. [30] studied the existence and uniqueness of solution of FIVP in the setting of generalized Hukuhara derivatives. For some intricacies involved in the theory of fuzzy differential equations, the interested reader may consult [22, 24, 31].

In this section, using the ideas of fuzzy mappings in an -complete -metric space, we provide some conditions for the existence of solutions of a FIVP. In line with the existence methods, our technique is connected with studying the existence of solutions of the equivalent Volterra integral reformulation of the FIVP.

First, in what follows, we recall a few known results that are needed in the sequel. For most of these basic concepts, we follow [30, 32]. Let denote the family of nonempty compact subsets of . Define addition and multiplication in as usual, that is, for and , we have

The Hausdorff metric in is defined as

It is well known that the couple is a complete metric space. Moreover, the metric satisfies the following properties for all :

In general, , where , and hence is not a linear space (cf. [30]).

*Definition 26. *A fuzzy number in is a function having the following properties:
(i) is normal, that is, there exists such that (ii) is fuzzy convex, that is(iii) is upper semicontinuous, that is, is closed for all (iv) is compactThroughout this section, we shall denote the set of all fuzzy numbers in by . The set denotes the -level set of . It follows from (i) to (iv) that .

The supremum on is defined as
for every , , where is called the diameter of .

We shall call the set of all continuous fuzzy functions defined on . It is verifiable that is an -complete -metric space with respect to the -metric:
The following lemma summarizes some basic properties of the integral of fuzzy functions.

Lemma 27 (see [22]). *Let *