Hyers-Ulam-Rassias Stability of Some Additive Fuzzy Set-Valued Functional Equations with the Fixed Point Alternative

. Let Y be a real separable Banach space and let ( K 𝐶 (𝑌),𝑑 ∞ ) be the subspace of all normal fuzzy convex and upper semicontinuous fuzzy sets of Y equipped with the supremum metric 𝑑 ∞ . In this paper, we introduce several types of additive fuzzy set-valued functional equations in ( K 𝐶 (𝑌),𝑑 ∞ ) . Using the fixed point technique, we discuss the Hyers-Ulam-Rassias stability of three types additive fuzzy set-valued functional equations, that is, the generalized Cauchy type, the Jensen type, and the Cauchy-Jensen type additive fuzzy set-valued functional equations. Our results can be regarded as important extensions of stability results corresponding to single-valued functional equations and set-valued functional equations, respectively.


Introduction
In 1940, Ulam [1] proposed the following question concerning the stability of group homomorphisms.
Let  1 be a group and let  2 be a metric group with the metric (⋅, ⋅).Given  > 0, does there exist a  > 0 such that if a function ℎ :  1 →  2 satisfies the inequality (ℎ(), ℎ()ℎ()) <  for all ,  ∈  1 , then there is a homomorphism  :  1 →  2 with (ℎ(), ()) <  for all  ∈  1 ?Afterwards, Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces.Later, the result of Hyers was generalized by Aoki [3] (for some historical comments regarding the work of Aoki, see [4]) for additive mappings and by Rassias [5] for linear mappings in which the Cauchy difference is allowed to be unbounded.However, the paper of Rassias [5] has provided a lot of influence in the development of what we call Hyer-Ulam stability or Hyers-Ulam-Rassias stability of functional equations.Hereafter, a generalization of Rassias's Theorem obtained by G ȃ vrut ¸a [6] by replacing the Cauchy difference with a more general majorant function in the spirit of Rassias' approach.Until now, the stability problems for different types of functional equations in various spaces have been extensively studied.For more detail, the reader can refer to [7].Among the studies of these problems, it is worth mentioning that Radu [8] discarded the direct method which was frequently used and proposed a novel method to establish the stability of Cauchy functional equation via fixed point technique.Recently, Ciepliński [9] summarized some applications of several types of fixed point theorems to the Hyers-Ulam stability of functional equations.As of now, this method has been successfully used in the study of stability problems of many types of functional equations in abstract spaces.
As defined in [7], let  1 and  2 be two appropriate spaces, where  2 is equipped with the metric .For some ,  ∈ N (N denotes the set of all natural numbers), the following functions for all  ∈  1 , then we say that the functional equation has the Hyers-Ulam-Rassias stability or the above functional equation is stable in the sense of Hyers-Ulam-Rassias.In particularly, if the functions  and Φ are replaced by two constants,  and  ( > 0), respectively, then we say that the functional equation has the Hyers-Ulam stability or it is stable in the sense of Hyers-Ulam.
In 2008, Mirmostafaee and Moslehian [10] initiated the study of stability problems of functional equations in fuzzy setting.Specifically, they considered the stability of the Cauchy functional equation in a fuzzy normed space.In the same year, they together with Mirzavaziri [11] proved the stability of the Jensen functional equation in the same space.Since then, the fuzzy stability problems of various types of functional equations have been extensively investigated by different authors [12,13].At the same time, the fixed point method has been widely used to prove the fuzzy stability of several types of functional equations [14,15].
In summary, one can see that the (fuzzy) stability for a single-valued functional equation is whether, for a given mapping satisfying almost a functional equation (which means that the mapping is close to a solution of the functional equation), there exists an exact solution of the functional equation which can be used to approximate the given mapping.Typically, a metric associated with the corresponding space is chosen to characterize the functional inequality.In 2009, Nikodem and Popa [16] considered the general solution of set-valued maps satisfying linear inclusion relation, which can be regarded as a generalization of the additive single-valued functional equation.By means of the inclusion relation, Lu and Park [17] first investigated the stability of two types of additive set-valued functional equations.In the following, Park et al. [18] further studied the stability problems of the quadratic, cubic, and quartic set-valued functional equations in a similar way.However, it should be pointed out that, in their studies, the inclusion relation is applied to characterize the set-valued functional inequality rather than an appropriate metric.Recently, similar to the method that is used to deal with the single-valued functional equations, Kenary et al. [19] proved the stability of several types of set-valued functional equations via the fixed point approach, in which the Hausdorff metric is adopted to characterize the set-valued functional inequality.
The purpose of this paper is to extend the set-valued functional equations to fuzzy set-valued functional equations and establish some stability results of several important fuzzy set-valued functional equations, including the generalized Cauchy, Jensen, and Cauchy-Jensen type fuzzy set-valued functional equations.Notice that the supremum metric, as a generalization of the Hausdorff metric, is applied to characterize the fuzzy set-valued functional inequality.More importantly, the corresponding single-valued and set-valued functional equations acted as special cases will be included in our results.

Preliminaries
In what follows, we begin with some related concepts and fundamental results, which are mainly derived from [20][21][22][23].Let R, R + , and R  denote the set of all real numbers, the set of all nonnegative real numbers, and the -dimensional Euclidean space, respectively.
Let  be a real separable Banach space with the norm ‖ ⋅ ‖  .We denote by K() and K  () the set of all nonempty compact subsets of  and the set of all nonempty compact convex subsets of , respectively.
Let  and  be two nonempty subsets of  and let  ∈ R. The (Minkowski) addition and scalar multiplication can be defined by Notice that the sets K() and K  () are closed under the operations of addition and scalar multiplication.In fact, these two operations induce a linear structure on K() and K  () with zero element {0}, respectively.It should be noted that this linear structure is just a cone rather than a vector space because, in general,  + (−1) ̸ = {0}.Moreover, for all ,  ∈ R, it follows that In particular, if  is convex and  ≥ 0, then ( + ) =  + .
Furthermore, we can define the Hausdorff separation of  from  by where  1 denotes the closed unit ball in ; that is,  1 = { ∈  | ‖‖  ≤ 1}.Meantime, the Hausdorff separation of  from  can also be defined in a similar way.
Abstract and Applied Analysis 3 Based on these two types of separations, the Hausdorff distance between nonempty subsets  and  is defined by In general, if ,  ∈ K() or K  (), then   (, ) = ||  (, ) for all  ∈ R. In addition, according to some of the properties of Hausdorff distance, if we restrict our attention to the nonemtpy closed subsets C() of , then it can be verified that (C(),   ) is a metric space.In fact, it follows from [20] that (C(),   ) is a complete metric space.Clearly, K() and K  () are closed subsets of C().Hence, (K(),   ) and (K  (),   ) are also complete metric spaces.
In 1991, Inoue [22] introduced the concept of Banach space valued fuzzy sets in order to extend the usual fuzzy sets defined on R or R  .In other words, the base space of a fuzzy set is replaced by a more general Banach space.
For a given real separable Banach space , a fuzzy set defined on  is a mapping  :  → [0,1].Denote by F() the set of all fuzzy sets defined on .Let F  () denote the class of fuzzy sets  :  → [0, 1] with the following properties: Notice that the conditions (ii) and (iv) imply that [] 0 is also compact.Moreover, we use the notation F  () to denote the subspace of F() whose members also satisfy A linear structure can be defined in F() in a similar way to fuzzy sets in R or R  by for , V ∈ F() and  ∈ R, where  0 () = 0 if  ̸ = 0 and  0 (0) = 1.Then F() is closed under these operations and level setwise for each  ∈ [0, 1] and  ∈ R. Similar to the closeness of K  (), it is easy to know that F  () is also closed under these operations.Based on the statement mentioned above, we can easily obtain the following lemma.
Remark 2. The Lemma 1 shows that F  () is just a cone defined on  rather than a vector space.
As a generalization of the Hausdorff metric   in K(), we will define the supremum metric  ∞ in F  ().For , V ∈ F  (), the supremum metric is defined by Remark 3. Every ordinary crisp subset  of  can be identified with the fuzzy set on  by its characteristic function , and vice versa.
In view of the property of the Hausdorff metric, it is easy to see that  ∞ (, V) =  ∞ (, V) for any  ≥ 0. Restricting attention to the set F  (), we can prove that (F  (),  ∞ ) is a complete metric space by the method analogous to that used in [21] (see Proposition 7.2.3).
Finally, we quote a fundamental result in fixed point theory.

Stability of the Generalized Cauchy Type Additive Fuzzy Set-Valued Functional Equation
In this section, we will establish the Hyers-Ulam-Rassias stability of the generalized Cauchy type additive fuzzy setvalued functional equation by employing the fixed point method.
Definition 5. Let  be a cone with the vertex 0 and let  :  → F  () be a fuzzy set-valued mapping.The generalized Cauchy type additive fuzzy set-valued functional equation is defined by for all ,  ∈  and for some ,  > 0 with  +  ̸ = 1.
Especially, if  =  = 1, then ( 15) is called the standard Cauchy type additive fuzzy set-valued functional equation.Every solution of ( 15) is called a generalized Cauchy type additive fuzzy set-valued mapping.Example 6.Let  = R + and  = R. Suppose that  : R + → F  (R) is a triangular fuzzy set-valued mapping, that is, for every  ∈ R + , () is a triangular fuzzy number in R, which is defined by where  and  are two nonnegative real numbers.By the definition of -level set, we can obtain that for every  ∈ R + .Then, for every  ∈ [0, 1], it is easy to verify that for all ,  ∈ R + and ,  > 0 with  +  ̸ = 1.That is,  is a solution of (15) in R + .

Remark 7. A triangular fuzzy number
Remark 8.More generally, if  = R + , by Lemma 1, it is easy to see that () =  0 is a solution of (15) for any  ∈ R + and any fixed  0 ∈ F  ().
for all  ∈ .
(i)  is a fixed point of : that is, for all  ∈ .The mapping  is the unique fixed point of  in the set which implies that  is the unique mapping satisfying (30) such that there exists a  ∈ (0, 1) satisfying for all  ∈ .
Remark 18. Theorem 16 and Corollary 17 can be viewed as a direct extension of the stability results of the single-valued Jensen functional equation obtained by C ȃ dariu and Radu [27] and Jung [28], respectively.

Stability of the Cauchy-Jensen Type Additive Fuzzy Set-Valued Functional Equation
As a combination of the Cauchy and Jensen functional equations, in this section, we will prove that the Hyers-Ulam-Rassias stability of the Cauchy-Jensen type additive fuzzy setvalued functional equation in a similar way as shown before.
where, as usual, inf 0 = ∞.It can easily be verified that (, ) is a complete generalized metric space (see [25], Theorem 2.4).We now define the linear mapping  :  →  by for all  ∈ .Moreover, we can infer from (65) and (66) that The rest of the proof is similar to the proof of Theorem 9.
Remark 22.In Theorem 21, if the fuzzy set-valued mapping  degenerates into a set-valued mapping, then the supremum metric  ∞ will reduce to the Hausdorff metric   .Thus, this theorem is obviously an extension of Theorems 2.2 and 2.4 in [19].