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

Authors: Yonghong Shen; Yaoyao Lan; Wei Chen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

LetY 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.


---

## Body

## 1. Introduction

In 1940, Ulam [1] proposed the following question concerning the stability of group homomorphisms.LetG
1 be a group and let G
2 be a metric group with the metric d
(
·
,
·
). Given ϵ
>
0, does there exist a δ
>
0 such that if a function h
:
G
1
→
G
2 satisfies the inequality d
(
h
(
x
y
)
,
h
(
x
)
h
(
y
)
)
<
δ for all x, y
∈
G
1, then there is a homomorphism H
:
G
1
→
G
2 with d
(
h
(
x
)
,
H
(
x
)
)
<
ϵ for all x
∈
G
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 Găvruţa [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 E
1 and E
2 be two appropriate spaces, where E
2 is equipped with the metric d. For some p, q
∈
ℕ (ℕ denotes the set of all natural numbers), the following functions
(1)
g
i
:
E
1
q
⟶
E
1
(
i
∈
1,2
,
…
,
p
)
,
G
:
E
2
p
×
E
1
q
⟶
E
2are given. Assume that φ, Φ
:
E
1
q
→
[
0
,
∞
) are functions satisfying some given conditions. If for every function f
:
E
1
→
E
2 satisfying the inequality
(2)
d
(
G
(
f
(
g
1
(
x
1
,
x
2
,
…
,
x
q
)
)
,
f
(
g
2
(
x
1
,
x
2
,
…
,
x
q
)
)
,
…
,
f
(
g
p
(
x
1
,
x
2
,
…
,
x
q
)
)
,
x
1
,
x
2
,
…
,
x
p
)
,
0
)
≤
φ
(
x
1
,
x
2
,
…
,
x
q
)for allx
1, x
2, …, x
q
∈
E
1
q, there exists a function F
:
E
1
→
E
2 such that
(3)
G
(
F
(
g
1
(
x
1
,
x
2
,
…
,
x
q
)
)
,
F
(
g
2
(
x
1
,
x
2
,
…
,
x
q
)
)
,
…
,
F
(
g
p
(
x
1
,
x
2
,
…
,
x
q
)
)
,
x
1
,
x
2
,
…
,
x
p
)
=
0for allx
1, x
2, …, x
q
∈
E
1
q and
(4)
d
(
f
(
x
)
,
F
(
x
)
)
≤
Φ
(
x
,
x
,
…
,
x
)for allx
∈
E
1, then we say that the functional equation
(5)
G
(
f
(
g
1
(
x
1
,
x
2
,
…
,
x
q
)
)
,
f
(
g
2
(
x
1
,
x
2
,
…
,
x
q
)
)
,
…
,
f
(
g
p
(
x
1
,
x
2
,
…
,
x
q
)
)
,
x
1
,
x
2
,
…
,
x
p
)
=
0has the Hyers-Ulam-Rassias stability or the above functional equation is stable in the sense of Hyers-Ulam-Rassias. In particularly, if the functionsφ and Φ are replaced by two constants, δ and K
δ (K
>
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.


## 2. Preliminaries

In what follows, we begin with some related concepts and fundamental results, which are mainly derived from [20–23]. Let ℝ, ℝ
+, and ℝ
n denote the set of all real numbers, the set of all nonnegative real numbers, and the n-dimensional Euclidean space, respectively.LetY be a real separable Banach space with the norm ∥
·
∥
Y. We denote by 𝒦
(
Y
) and 𝒦
C
(
Y
) the set of all nonempty compact subsets of Y and the set of all nonempty compact convex subsets of Y, respectively.LetA and B be two nonempty subsets of Y and let λ
∈
ℝ. The (Minkowski) addition and scalar multiplication can be defined by
(6)
A
+
B
=
{
a
+
b
∣
a
∈
A
,
b
∈
B
}
,
λ
A
=
{
λ
a
∣
a
∈
A
}
.Notice that the sets𝒦
(
Y
) and 𝒦
C
(
Y
) are closed under the operations of addition and scalar multiplication. In fact, these two operations induce a linear structure on 𝒦
(
Y
) and 𝒦
C
(
Y
) 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, A
+
(
-
1
)
A
≠
{
0
}. Moreover, for all λ, μ
∈
ℝ, it follows that
(7)
λ
(
A
+
B
)
=
λ
A
+
λ
B
,
(
λ
+
μ
)
A
⊆
λ
A
+
μ
A
.In particular, ifA is convex and λ
μ
≥
0, then (
λ
+
μ
)
A
=
λ
A
+
μ
A.Furthermore, we can define the Hausdorff separation ofB from A by
(8)
d
H
*
(
B
,
A
)
=
inf
⁡
⁡
{
ϵ
>
0
∣
B
⊆
A
+
ϵ
S
1
¯
}
,whereS
1
¯ denotes the closed unit ball in Y; that is, S
1
¯
=
{
y
∈
Y
∣
∥
y
∥
Y
≤
1
}. Meantime, the Hausdorff separation of A from B can also be defined in a similar way.Based on these two types of separations, the Hausdorff distance between nonempty subsetsA and B is defined by
(9)
d
H
(
A
,
B
)
=
max
⁡
⁡
{
d
H
*
(
A
,
B
)
,
d
H
*
(
B
,
A
)
}
.In general, ifA, B
∈
𝒦
(
Y
) or 𝒦
C
(
Y
), then d
H
(
λ
A
,
λ
B
)
=
|
λ
|
d
H
(
A
,
B
) for all λ
∈
ℝ. In addition, according to some of the properties of Hausdorff distance, if we restrict our attention to the nonemtpy closed subsets 𝒞
(
Y
) of Y, then it can be verified that (
𝒞
(
Y
)
,
d
H
) is a metric space. In fact, it follows from [20] that (
𝒞
(
Y
)
,
d
H
) is a complete metric space. Clearly, 𝒦
(
Y
) and 𝒦
C
(
Y
) are closed subsets of 𝒞
(
Y
). Hence, (
𝒦
(
Y
)
,
d
H
) and (
𝒦
C
(
Y
)
,
d
H
) 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
n. In other words, the base space of a fuzzy set is replaced by a more general Banach space.For a given real separable Banach spaceY, a fuzzy set defined on Y is a mapping u
:
Y
→
[
0,1
]. Denote by ℱ
(
Y
) the set of all fuzzy sets defined on Y. Let ℱ
K
(
Y
) denote the class of fuzzy sets u
:
Y
→
[
0,1
] with the following properties:(i)
u is normal, that is, [
u
]
1
=
{
y
∈
Y
∣
u
(
y
)
≥
1
} is nonempty;
(ii)
u is upper semicontinuous;
(iii)
[
u
]
α
=
{
y
∈
Y
∣
u
(
y
)
≥
α
}is compact for each α
∈
(
0,1
];
(iv)
[
u
]
0
=
⋃
α
∈
(
0,1
]
‍
[
u
]
α
¯ is a bound subset of Y.Notice that the conditions (ii) and (iv) imply that[
u
]
0 is also compact. Moreover, we use the notation ℱ
K
C
(
Y
) to denote the subspace of ℱ
(
Y
) whose members also satisfy(v)
u is fuzzy convex; that is, [
u
]
αis convex for each α
∈
(
0,1
].A linear structure can be defined inℱ
(
Y
) in a similar way to fuzzy sets in ℝ or ℝ
n by
(10)
(
u
⊕
v
)
(
y
)
=
sup
⁡
x
+
z
=
y
min
⁡
⁡
{
u
(
x
)
,
v
(
z
)
}
,
(
γ
u
)
(
y
)
=
{
u
(
y
γ
)
,
if
γ
≠
0
,
I
0
(
y
)
,
if
γ
=
0
,
for u
,
v
∈
ℱ
(
Y
) and γ
∈
ℝ, where I
0
(
y
)
=
0 if y
≠
0 and I
0
(
0
)
=
1. Then ℱ
(
Y
) is closed under these operations and level setwise
(11)
[
u
⊕
v
]
α
=
[
u
]
α
+
[
v
]
α
,
[
λ
u
]
α
=
λ
[
u
]
αfor eachα
∈
[
0,1
] and λ
∈
ℝ. Similar to the closeness of 𝒦
C
(
Y
), it is easy to know that ℱ
K
C
(
Y
) is also closed under these operations. Based on the statement mentioned above, we can easily obtain the following lemma.Lemma 1.
For anyu
,
v
∈
ℱ
K
C
(
Y
) and λ, μ
∈
ℝ, the following equalities hold:(i)
λ
(
u
⊕
v
)
=
λ
u
⊕
λ
v;
(ii)
λ
(
μ
u
)
=
(
λ
μ
)
u;
(iii)
(
λ
+
μ
)
u
=
λ
u
⊕
μ
u for any λ
,
μ
≥
0.Remark 2.
The Lemma1 shows that ℱ
K
C
(
Y
) is just a cone defined on Y rather than a vector space.As a generalization of the Hausdorff metricd
H in 𝒦
(
Y
), we will define the supremum metric d
∞ in ℱ
C
(
Y
). For u, v
∈
ℱ
C
(
Y
), the supremum metric is defined by
(12)
d
∞
(
u
,
v
)
=
sup
⁡
α
∈
(
0,1
]
d
H
(
[
u
]
α
,
[
v
]
α
)
.Remark 3.
Every ordinary crisp subsetA of Y can be identified with the fuzzy set on Y by its characteristic function χ
A
:
Y
→
{
0,1
}, that is, with χ
A
(
y
)
=
1 if y
∈
A and χ
A
(
y
)
=
0 if y
∉
A. Therefore, if A
∈
𝒦
(
Y
)
(
or
A
∈
𝒦
C
(
Y
)
), then χ
A
∈
ℱ
C
(
Y
) (or ℱ
K
C
(
Y
)), and vice versa.From Remark3, for any A, B
∈
𝒦
(
Y
) (or ℱ
K
C
(
Y
)), it follows that
(13)
d
∞
(
χ
A
,
χ
B
)
=
sup
⁡
α
∈
(
0,1
]
d
H
(
[
χ
A
]
α
,
[
χ
B
]
α
)
=
d
H
(
A
,
B
)
.In particular, ifA and B degenerate into two singleton sets {
a
} and {
b
}, then we can infer from equality (13) that d
∞
(
χ
{
a
}
,
χ
{
b
}
)
=
d
(
a
,
b
), where d denotes the usual metric between a and b.In view of the property of the Hausdorff metric, it is easy to see thatd
∞
(
λ
u
,
λ
v
)
=
λ
d
∞
(
u
,
v
) for any λ
≥
0. Restricting attention to the set ℱ
K
C
(
Y
), we can prove that (
ℱ
K
C
(
Y
)
,
d
∞
) 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.Theorem 4 (Diaz and Margolis [24]).
Let(
X
,
d
) be a complete generalized metric space, that is, one for which d may assume infinite values. Suppose that J
:
X
→
X is a strictly contractive mapping with Lipschitz constant L
<
1. Then, for each given element x
∈
X, either
(14)
d
(
J
n
x
,
J
n
+
1
x
)
=
∞
for alln
≥
0 or there exists an n
0
∈
ℕ such that(i)
d
(
J
n
x
,
J
n
+
1
x
)
<
∞ for all n
≥
n
0;
(ii)
the sequence{
J
n
x
} converges to a fixed point y
* of J;
(iii)
y
* is the unique fixed point of J in the set Y
=
{
y
∈
X
∣
d
(
J
n
0
x
,
y
)
<
∞
};
(iv)
d
(
y
,
y
*
)
≤
(
1
/
(
1
-
L
)
)
d
(
y
,
J
y
) for all y
∈
Y.


## 3. 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 set-valued functional equation by employing the fixed point method.Definition 5.
LetX be a cone with the vertex 0 and let f
:
X
→
ℱ
K
C
(
Y
) be a fuzzy set-valued mapping. The generalized Cauchy type additive fuzzy set-valued functional equation is defined by
(15)
f
(
ξ
x
+
η
y
)
=
ξ
f
(
x
)
⊕
η
f
(
y
)
for allx
,
y
∈
X and for some ξ
,
η
>
0 with ξ
+
η
≠
1.Especially, ifξ
=
η
=
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.
LetX
=
ℝ
+ and Y
=
ℝ. Suppose that f
:
ℝ
+
→
ℱ
K
C
(
ℝ
) is a triangular fuzzy set-valued mapping, that is, for every t
∈
ℝ
+, f
(
t
) is a triangular fuzzy number in ℝ, which is defined by
(16)
f
(
t
)
=
(
t
-
a
t
,
t
,
t
+
b
t
)
,
t
∈
ℝ
+
,
wherea and b are two nonnegative real numbers. By the definition of α-level set, we can obtain that
(17)
[
f
(
t
)
]
α
=
[
t
-
a
t
(
1
-
α
)
,
t
+
b
t
(
1
-
α
)
]
for everyt
∈
ℝ
+. Then, for every α
∈
[
0,1
], it is easy to verify that
(18)
[
f
(
ξ
x
+
η
y
)
]
α
=
ξ
[
f
(
x
)
]
α
+
η
[
f
(
y
)
]
α
for allx
,
y
∈
ℝ
+ and ξ, η
>
0 with ξ
+
η
≠
1. That is, f is a solution of (15) in ℝ
+.Remark 7.
A triangular fuzzy numberu
∈
ℱ
K
C
(
ℝ
) is characterized by an ordered triple (
x
l
,
x
c
,
x
r
)
∈
ℝ
3 with x
l
≤
x
c
≤
x
r such that the support set [
u
]
0
=
[
x
l
,
x
r
] and 1-level set [
u
]
1
=
{
x
c
}.Remark 8.
More generally, ifX
=
ℝ
+, by Lemma 1, it is easy to see that f
(
t
)
=
t
u
0 is a solution of (15) for any t
∈
ℝ
+ and any fixed u
0
∈
ℱ
K
C
(
Y
).Theorem 9.
Letφ
:
X
×
X
→
[
0
,
+
∞
) be a function such that there exists a positive constant L
<
1 satisfying
(19)
φ
(
x
,
y
)
≤
L
ξ
+
η
φ
(
(
ξ
+
η
)
x
,
(
ξ
+
η
)
y
)
for allx, y
∈
X and for some ξ, η
>
0 with ξ
+
η
≠
1. Suppose that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(20)
d
∞
(
f
(
ξ
x
+
η
y
)
,
ξ
f
(
x
)
⊕
η
f
(
y
)
)
≤
φ
(
x
,
y
)
for allx, y
∈
X. Then
(21)
A
(
x
)
=
lim
⁡
n
→
∞
(
ξ
+
η
)
n
f
(
x
(
ξ
+
η
)
n
)
exists for eachx
∈
X and defines a unique generalized Cauchy type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(22)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
L
(
ξ
+
η
)
(
1
-
L
)
φ
(
x
,
x
)
for allx
∈
X.Proof.
Replacingx
=
y in (20), by Lemma 1, we get
(23)
d
∞
(
f
(
(
ξ
+
η
)
x
)
,
(
ξ
+
η
)
f
(
x
)
)
≤
φ
(
x
,
x
)
for allx
∈
X. Thus, we can obtain
(24)
d
∞
(
f
(
x
)
,
(
ξ
+
η
)
f
(
x
ξ
+
η
)
)
≤
φ
(
x
ξ
+
η
,
x
ξ
+
η
)
≤
L
ξ
+
η
φ
(
x
,
x
)
for allx
∈
X.
Consider the setE
=
{
g
∣
g
:
X
→
ℱ
K
C
(
Y
)
,
g
(
0
)
=
I
0
} and introduce the generalized metric D on E, which is defined by
(25)
D
(
g
,
h
)
=
inf
⁡
⁡
{
μ
∈
(
0
,
∞
)
∣
d
∞
(
g
(
x
)
,
h
(
x
)
)
≤
μ
φ
(
x
,
x
)
,
∀
x
∈
X
}
,
where, as usual,inf
⁡
∅
=
∞. In can easily be verified that (
E
,
D
) is a complete generalized metric space (see [25], Theorem 2.4).
Now, we consider the linear mappingJ
:
E
→
E such that
(26)
J
g
(
x
)
=
(
ξ
+
η
)
g
(
x
ξ
+
η
)
for allx
∈
X.
Letg, h be given such that D
(
g
,
h
)
≤
ϵ. Then
(27)
d
∞
(
g
(
x
)
,
h
(
x
)
)
≤
ϵ
φ
(
x
,
x
)
for allx
∈
X. Hence, we have
(28)
d
∞
(
J
g
(
x
)
,
J
h
(
x
)
)
=
d
∞
(
(
ξ
+
η
)
g
(
x
ξ
+
η
)
,
(
ξ
+
η
)
h
(
x
ξ
+
η
)
)
=
(
ξ
+
η
)
d
∞
(
g
(
x
ξ
+
η
)
,
h
(
x
ξ
+
η
)
)
≤
ϵ
(
ξ
+
η
)
φ
(
x
ξ
+
η
,
x
ξ
+
η
)
≤
L
ξ
+
η
ϵ
(
ξ
+
η
)
φ
(
x
,
x
)
=
L
ϵ
φ
(
x
,
x
)
for allx
∈
X. So D
(
g
,
h
)
≤
ϵ implies that D
(
J
g
,
J
h
)
≤
L
ϵ. This means that
(29)
D
(
J
g
,
J
h
)
≤
L
D
(
g
,
h
)
for allg, h
∈
E. Evidently, J is a strictly contractive self-mapping on E with the Lipschitz constant L
<
1.
Moreover, it follows from (24) that D
(
f
,
J
f
)
≤
L
/
(
ξ
+
η
). According to Theorem 4, there exists a mapping A
:
X
→
ℱ
c
(
Y
) satisfying the following.(i)
A is a fixed point of J: that is,
(30)
1
ξ
+
η
A
(
x
)
=
A
(
x
ξ
+
η
)
for allx
∈
X. The mapping A is the unique fixed point of J in the set
(31)
M
=
{
g
∈
E
∣
D
(
f
,
g
)
<
∞
}
,
which implies thatA is the unique mapping satisfying (30) such that there exists a r
∈
(
0,1
) satisfying
(32)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
r
φ
(
x
,
x
)
for allx
∈
X.(ii)
D
(
J
n
f
,
A
)
→
0 as n
→
∞. This implies the equality
(33)
lim
⁡
n
→
∞
(
ξ
+
η
)
n
f
(
x
(
α
+
β
)
n
)
=
A
(
x
)
for allx
∈
X.(iii)
D
(
f
,
A
)
≤
(
1
/
(
1
-
L
)
)
D
(
f
,
J
f
), which implies the inequality
(34)
D
(
f
,
A
)
≤
1
1
-
L
·
L
ξ
+
η
=
L
(
ξ
+
η
)
(
1
-
L
)
.
This implies that the inequality (22) holds.
By (20), we can obtain that
(35)
d
∞
(
(
ξ
f
(
x
(
ξ
+
η
)
n
)
⊕
η
f
(
y
(
ξ
+
η
)
n
)
)
(
ξ
+
η
)
n
f
(
ξ
x
(
ξ
+
η
)
n
+
η
x
(
ξ
+
η
)
n
)
,
(
ξ
+
η
)
n
(
ξ
f
(
x
(
ξ
+
η
)
n
)
⊕
η
f
(
y
(
ξ
+
η
)
n
)
)
)
≤
(
ξ
+
η
)
n
φ
(
x
(
ξ
+
η
)
n
,
y
(
ξ
+
η
)
n
)
≤
(
ξ
+
η
)
n
·
L
n
(
ξ
+
η
)
n
φ
(
x
,
x
)
=
L
n
φ
(
x
,
y
)
,
which tends to zero asn
→
∞ for all x, y
∈
X. Thus,
(36)
A
(
ξ
x
+
η
y
)
=
ξ
A
(
x
)
⊕
η
A
(
y
)
for allx, y
∈
X and therefore the mapping A
:
X
→
ℱ
K
C
(
Y
) is a generalized Cauchy type additive fuzzy set-valued mapping as desired.Corollary 10.
LetX be a cone with the vertex 0 contained in a real normed space and let p, θ be positive real numbers with p
>
1 (resp. 0
<
p
<
1). Suppose that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(37)
d
∞
(
f
(
ξ
x
+
η
y
)
,
ξ
f
(
x
)
⊕
η
f
(
y
)
)
≤
θ
(
∥
x
∥
p
+
∥
y
∥
p
)
for allx, y
∈
X and for some ξ, η
>
0 with ξ
+
η
>
1 (resp. <
1). Then
(38)
A
(
x
)
=
lim
⁡
n
→
∞
(
ξ
+
η
)
n
f
(
x
(
ξ
+
η
)
n
)
exists for eachx
∈
X and defines a unique generalized Cauchy type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(39)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
2
θ
∥
x
∥
p
(
ξ
+
η
)
p
-
(
ξ
+
η
)
for allx
∈
X.Proof.
In Theorem9, let φ
(
x
,
y
)
=
θ
(
∥
x
∥
p
+
∥
x
∥
p
). Then we can choose L
=
(
ξ
+
η
)
1
-
p and we get the desired result.Corollary 11.
LetX be a cone with the vertex 0 contained in a real normed space and let p, θ be positive real numbers with p
>
1
/
2 (resp. p
<
1
/
2). Suppose that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(40)
d
∞
(
f
(
ξ
x
+
η
y
)
,
ξ
f
(
x
)
⊕
η
f
(
y
)
)
≤
θ
∥
x
∥
p
∥
y
∥
p
for allx, y
∈
X and for some ξ, η
>
0 with ξ
+
η
>
1 (resp. <
1). Then
(41)
A
(
x
)
=
lim
⁡
n
→
∞
(
ξ
+
η
)
n
f
(
x
(
ξ
+
η
)
n
)
exists for eachx
∈
X and defines a unique generalized Cauchy type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(42)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
θ
∥
x
∥
2
p
(
ξ
+
η
)
2
p
-
(
ξ
+
η
)
for allx
∈
X.Proof.
In Theorem9, let φ
(
x
,
y
)
=
θ
∥
x
∥
p
∥
y
∥
p. Then we can choose L
=
(
ξ
+
η
)
1
-
2
p and we get the desired result.Corollary 12.
LetX be a cone with the vertex 0 contained in a real normed space and let p, q, θ be positive real numbers with p
+
q
>
1 (resp. p
+
q
<
1). Suppose that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(43)
d
∞
(
f
(
ξ
x
+
η
y
)
,
ξ
f
(
x
)
⊕
η
f
(
y
)
)
≤
θ
(
∥
x
∥
p
∥
y
∥
q
+
∥
x
∥
p
+
q
+
∥
y
∥
p
+
q
)
for allx, y
∈
X and for some ξ, η
>
0 with ξ
+
η
>
1 (resp. <
1). Then
(44)
A
(
x
)
=
lim
⁡
n
→
∞
(
ξ
+
η
)
n
f
(
x
(
ξ
+
η
)
n
)
exists for eachx
∈
X and defines a unique generalized Cauchy type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(45)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
3
θ
∥
x
∥
p
+
q
(
ξ
+
η
)
p
+
q
-
(
ξ
+
η
)
for allx
∈
X.Proof.
In Theorem9, let φ
(
x
,
y
)
=
θ
(
∥
x
∥
p
∥
y
∥
q
+
∥
x
∥
p
+
q
+
∥
y
∥
p
+
q
). Then we can choose L
=
(
ξ
+
η
)
1
-
(
p
+
q
) and we get the desired result.Remark 13.
According to the equality (13), all results obtained in this section still hold true, even if the fuzzy set-valued mapping f
:
X
→
ℱ
K
C
(
Y
) degenerates into a set-valued mapping f
:
X
→
𝒦
C
(
Y
) or a single-valued mapping f
:
X
→
Y. Therefore, these results can be regarded as an important extension of the stability results of the classical additive Cauchy type functional equations and the additive Cauchy type set-valued functional equations. In essence, Theorem 9 extends some main results obtained by Gajda [26], Hyers [2], and Rassias [5].


## 4. Stability of the Jensen Type Additive Fuzzy Set-Valued Functional Equation

In this section, we will prove the Hyers-Ulam-Rassias stability of the Jensen type additive fuzzy set-valued functional equation by using the same method as employed in the previous section.Definition 14.
LetX be a cone with the vertex 0 and let f
:
X
→
ℱ
K
C
(
Y
) be a fuzzy set-valued mapping. The Jensen type additive fuzzy set-valued functional equation is defined by
(46)
2
f
(
x
+
y
2
)
=
f
(
x
)
⊕
f
(
y
)
for allx, y
∈
X. Every solution of (46) is called a Jensen type additive fuzzy set-valued mapping.Remark 15.
Obviously, it can be checked that the triangular fuzzy set-valued mappingf as defined in Example 6 is also a solution of (46) in ℝ
+. Similarly, f
(
t
)
=
t
u
0 is a solution of (46) for any t
∈
ℝ
+ and any fixed u
0
∈
ℱ
K
C
(
Y
).Theorem 16.
Letj
∈
{
-
1,1
} be fixed and let φ
:
X
×
X
→
[
0
,
+
∞
) be a function such that there exists a positive constant L
<
1 satisfying
(47)
φ
(
x
,
0
)
≤
2
j
L
φ
(
2
-
j
x
,
0
)
for allx
∈
X. Moreover, assume that φ satisfies
(48)
lim
⁡
n
→
∞
2
-
j
n
φ
(
2
j
n
x
,
2
j
n
y
)
=
0
for allx, y
∈
X. If a mapping f
:
X
→
ℱ
K
C
(
Y
) satisfies f
(
0
)
=
I
0 and the inequality
(49)
d
∞
(
2
f
(
x
+
y
2
)
,
f
(
x
)
⊕
f
(
y
)
)
≤
φ
(
x
,
y
)
for allx, y
∈
X, then
(50)
A
(
x
)
=
lim
⁡
n
→
∞
2
-
j
n
f
(
2
j
n
x
)
exists for eachx
∈
X and defines a unique Jensen type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(51)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
{
1
1
-
L
φ
(
x
,
0
)
,
i
f
j
=
-
1
,
L
1
-
L
φ
(
x
,
0
)
,
i
f
j
=
1
for allx
∈
X.Proof.
Lettingy
=
0 in (49). Since u
⊕
I
0
=
u for any u
∈
ℱ
K
C
(
Y
), we get
(52)
d
∞
(
2
f
(
x
2
)
,
f
(
x
)
)
≤
φ
(
x
,
0
)
for allx
∈
X. Furthermore, it follows from (47) that
(53)
d
∞
(
f
(
x
)
,
1
2
f
(
2
x
)
)
≤
1
2
φ
(
2
x
,
0
)
≤
L
φ
(
x
,
0
)
for allx
∈
X.
Consider the setE
=
{
g
∣
g
:
X
→
ℱ
K
C
(
Y
)
,
g
(
0
)
=
I
0
} and introduce the generalized metric D on E, which is defined by
(54)
D
(
g
,
h
)
=
inf
⁡
⁡
{
μ
∈
(
0
,
∞
)
∣
d
∞
(
g
(
x
)
,
h
(
x
)
)
≤
μ
φ
(
x
,
0
)
,
∀
x
∈
X
}
,
where, as usual,inf
⁡
∅
=
∞. In can easily be verified that (
E
,
D
) is a complete generalized metric space (see [25], Theorem 2.4).
We now define the linear mappingJ
:
E
→
E by
(55)
J
g
(
x
)
=
2
-
j
g
(
2
j
x
)
for allx
∈
X.
Moreover, we can infer from (52) and (53) that
(56)
D
(
f
,
J
f
)
≤
{
1
,
if
j
=
-
1
,
L
,
if
j
=
1
.
The rest of the proof is similar to the proof of Theorem9.Corollary 17.
Letj
∈
{
-
1,1
} be fixed and let p
,
θ be positive real numbers with p
≠
1, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(57)
d
∞
(
2
f
(
x
+
y
2
)
,
f
(
x
)
⊕
f
(
y
)
)
≤
θ
(
∥
x
∥
p
+
∥
y
∥
p
)
for allx, y
∈
X. Then
(58)
A
(
x
)
=
lim
⁡
n
→
∞
2
-
j
n
f
(
2
j
n
x
)
exists for eachx
∈
X and defines a unique Jensen type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(59)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
{
θ
∥
x
∥
p
1
-
2
1
-
p
,
i
f
j
=
-
1
,
p
>
1
,
θ
∥
x
∥
p
2
1
-
p
-
1
,
i
f
j
=
1
,
0
<
p
<
1
for allx
∈
X.Proof.
In Theorem16, let φ
(
x
,
y
)
=
θ
(
∥
x
∥
p
+
∥
y
∥
p
). Then we can choose L
=
2
j
(
p
-
1
) and hence we can obtain the desired result.Remark 18.
Theorem16 and Corollary 17 can be viewed as a direct extension of the stability results of the single-valued Jensen functional equation obtained by Cădariu and Radu [27] and Jung [28], respectively.


## 5. 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 set-valued functional equation in a similar way as shown before.Definition 19.
LetX be a cone with the vertex 0 and let f
:
X
→
ℱ
K
C
(
Y
) be a fuzzy set-valued mapping. The Cauchy-Jensen type additive fuzzy set-valued functional equation is defined by(60)
f
(
x
+
y
2
+
z
)
⊕
f
(
x
+
z
2
+
y
)
⊕
f
(
y
+
z
2
+
x
)
=
2
(
f
(
x
)
⊕
f
(
y
)
⊕
f
(
z
)
)
for allx, y, z
∈
X. Every solution of (60) is called a Cauchy-Jensen type additive fuzzy set-valued mapping.Remark 20.
It is easy to see that the Example6 is also suitable for (60). Similarly, f
(
t
)
=
t
u
0 is a solution of (60) for any t
∈
ℝ
+ and any fixed u
0
∈
ℱ
K
C
(
Y
).Theorem 21.
Letj
∈
{
-
1,1
} be fixed and let ϕ
:
X
3
→
[
0
,
+
∞
) be a function such that there exists a positive constant L
<
1 satisfying
(61)
φ
(
x
,
y
,
z
)
≤
2
j
L
φ
(
2
-
j
x
,
2
-
j
y
,
2
-
j
z
)
for allx, y, z
∈
X. Assume that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(62)
d
∞
(
f
(
x
+
y
2
+
z
)
⊕
f
(
x
+
z
2
+
y
)
⊕
f
(
y
+
z
2
+
x
)
,
2
(
f
(
x
)
⊕
f
(
y
)
⊕
f
(
z
)
)
(
x
+
y
2
+
z
)
)
≤
ϕ
(
x
,
y
,
z
)
for allx, y, z
∈
X. Then
(63)
A
(
x
)
=
lim
⁡
n
→
∞
2
-
j
n
f
(
2
j
n
x
)
exists for eachx
∈
X and defines a unique Cauchy-Jensen type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that(64)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
{
L
6
(
1
-
L
)
φ
(
x
,
x
,
x
)
,
i
f
j
=
-
1
,
1
6
(
1
-
L
)
φ
(
x
,
x
,
x
)
,
i
f
j
=
1
for allx
∈
X.Proof.
Lettingx
=
y
=
z in (62). By Lemma 1, we can get
(65)
d
∞
(
1
2
f
(
2
x
)
,
f
(
x
)
)
≤
1
6
ϕ
(
x
,
x
,
x
)
for allx
∈
X. Furthermore, it follows from (61) that
(66)
d
∞
(
f
(
x
)
,
2
f
(
x
2
)
)
≤
1
3
ϕ
(
x
2
,
x
2
,
x
2
)
≤
L
6
ϕ
(
x
,
x
,
x
)
for allx
∈
X.
Consider the setE
=
{
g
∣
g
:
X
→
ℱ
K
C
(
Y
)
,
g
(
0
)
=
I
0
} and introduce the generalized metric D on E, which is defined by
(67)
D
(
g
,
h
)
=
inf
⁡
⁡
{
μ
∈
(
0
,
∞
)
∣
d
∞
(
g
(
x
)
,
h
(
x
)
)
≤
μ
φ
(
x
,
x
,
x
)
,
∀
x
∈
X
}
,
where, as usual,inf
⁡
∅
=
∞. It can easily be verified that (
E
,
D
) is a complete generalized metric space (see [25], Theorem 2.4).
We now define the linear mappingJ
:
E
→
E by
(68)
J
g
(
x
)
=
2
-
j
g
(
2
j
x
)
for allx
∈
X.
Moreover, we can infer from (65) and (66) that
(69)
D
(
f
,
J
f
)
≤
{
L
6
,
if
j
=
-
1
,
1
6
,
if
j
=
1
.
The rest of the proof is similar to the proof of Theorem9.Remark 22.
In Theorem21, if the fuzzy set-valued mapping f degenerates into a set-valued mapping, then the supremum metric d
∞ will reduce to the Hausdorff metric d
H. Thus, this theorem is obviously an extension of Theorems 2.2 and 2.4 in [19].Corollary 23.
Letj
∈
{
-
1,1
} be fixed and let p
,
θ be positive real numbers with p
≠
1 and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(70)
d
∞
(
f
(
x
+
y
2
+
z
)
⊕
f
(
x
+
z
2
+
y
)
⊕
f
(
y
+
z
2
+
x
)
,
2
(
f
(
x
)
⊕
f
(
y
)
⊕
f
(
z
)
)
(
x
+
y
2
+
z
)
)
≤
θ
(
∥
x
∥
p
+
∥
y
∥
p
+
∥
z
∥
p
)
for allx
,
y
,
z
∈
X. Then
(71)
A
(
x
)
=
lim
⁡
n
→
∞
2
-
j
n
f
(
2
j
n
x
)
exists for eachx
∈
X and defines a unique Cauchy-Jensen type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(72)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
{
θ
∥
x
∥
p
2
p
-
2
,
i
f
j
=
-
1
,
p
>
1
,
θ
∥
x
∥
p
2
-
2
p
,
i
f
j
=
1
,
0
<
p
<
1
for allx
∈
X.Proof.
In Theorem21, let φ
(
x
,
y
,
z
)
=
θ
(
∥
x
∥
p
+
∥
y
∥
p
+
∥
z
∥
p
). Then we can choose L
=
2
j
(
p
-
1
) and hence we can obtain the desired result.Corollary 24.
Letj
∈
{
-
1,1
} be fixed and let p
,
θ be positive real numbers with p
≠
1
/
3 and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(73)
d
∞
(
f
(
x
+
y
2
+
z
)
⊕
f
(
x
+
z
2
+
y
)
⊕
f
(
y
+
z
2
+
x
)
,
(
x
+
y
2
+
z
)
2
(
f
(
x
)
⊕
f
(
y
)
⊕
f
(
z
)
)
)
≤
θ
∥
x
∥
p
∥
y
∥
p
∥
z
∥
p
for allx, y, z
∈
X. Then
(74)
A
(
x
)
=
lim
⁡
n
→
∞
2
-
j
n
f
(
2
j
n
x
)
exists for eachx
∈
X and defines a unique Cauchy-Jensen type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(75)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
{
θ
∥
x
∥
3
p
3
(
2
3
p
-
2
)
,
i
f
j
=
-
1
,
p
>
1
3
,
θ
∥
x
∥
3
p
3
(
2
-
2
3
p
)
,
i
f
j
=
1
,
0
<
p
<
1
3
for allx
∈
X.Proof.
In Theorem21, let φ
(
x
,
y
,
z
)
=
θ
∥
x
∥
p
∥
y
∥
p
∥
z
∥
p. Then we can choose L
=
2
j
(
3
p
-
1
) and hence we can obtain the desired result.Corollary 25.
Letj
∈
{
-
1,1
} be fixed and let p, q, t, θ be positive real numbers with p
+
q
+
t
≠
1 and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f
:
X
→
ℱ
K
C
(
Y
) is a mapping satisfying
(76)
d
∞
(
f
(
x
+
y
2
+
z
)
⊕
f
(
x
+
z
2
+
y
)
⊕
f
(
y
+
z
2
+
x
)
,
(
x
+
y
2
+
z
)
2
(
f
(
x
)
⊕
f
(
y
)
⊕
f
(
z
)
)
)
≤
θ
(
∥
x
∥
p
∥
y
∥
q
∥
z
∥
t
+
∥
x
∥
p
+
q
+
t
+
∥
y
∥
p
+
q
+
t
+
∥
z
∥
p
+
q
+
t
)
for allx, y, z
∈
X. Then
(77)
A
(
x
)
=
lim
⁡
n
→
∞
2
-
j
n
f
(
2
j
n
x
)
exists for eachx
∈
X and defines a unique Cauchy-Jensen type additive fuzzy set-valued mapping A
:
X
→
ℱ
K
C
(
Y
) such that
(78)
d
∞
(
f
(
x
)
,
A
(
x
)
)
≤
{
4
θ
∥
x
∥
3
p
3
(
2
p
+
q
+
t
-
2
)
,
i
f
j
=
-
1
,
p
+
q
+
t
>
1
,
4
θ
∥
x
∥
p
+
q
+
t
3
(
2
-
2
p
+
q
+
t
)
,
i
f
j
=
1
,
0
<
p
+
q
+
t
<
1
for allx
∈
X.Proof.
In Theorem21, let φ
(
x
,
y
,
z
)
=
θ
(
∥
x
∥
p
∥
y
∥
q
∥
z
∥
t
+
∥
x
∥
p
+
q
+
t
+
∥
y
∥
p
+
q
+
t
+
∥
z
∥
p
+
q
+
t
). Then we can choose L
=
2
j
[
(
p
+
q
+
t
)
-
1
] and hence we can obtain the desired result.


## 6. Concluding Remark

In this paper, we proved the Hyers-Ulam-Rassias stability of several types of additive fuzzy set-valued functional equations, including Cauchy, Jensen, and Cauchy-Jensen type fuzzy set-valued functional equations. Our results generalized certain important results obtained by other authors for these equations when they are a single-valued or a set-valued one. Obviously, this paper provided us a novel idea to discuss the stability of functional equations from a more unified perspective. Certainly, further work will focus on the stability of other types of functional equations by using this idea.



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*Source: 139175-2014-03-05.xml*