Abstract
Using the fixed point method, we prove the Hyers-Ulam stability of a Cauchy-Jensen type additive set-valued functional equation, a Jensen type additive-quadratic set-valued functional equation, a generalized quadratic set-valued functional equation and a Jensen type cubic set-valued functional equation.
Mathematics Subject Classification 2010: 47H10; 54C60; 39B52; 47H04; 91B44.
Similar content being viewed by others
1. Introduction and preliminaries
Set-valued functions in Banach spaces have been developed in the last decades. The pioneering article by Aumann [1] and Debreu [2] were inspired by problems arising in control theory and mathematical economics. We can refer to the articles by Arrow and Debreu [3], McKenzie [4], the monographs by Hindenbrand [5], Aubin and Frankow [6], Castaing and Valadier [7], Klein and Thompson [8] and the survey by Hess [9]. Let Y be a Banach space. We define the following:
2Y : the set of all subsets of Y;
C b (Y): the set of all closed bounded subsets of Y;
C c (Y): the set of all closed convex subsets of Y;
C cb (Y): the set of all closed convex bounded subsets of Y.
On 2Y we consider the addition and the scalar multiplication as follows:
where C, C' ∈ 2Y and λ ∈ ℝ. Further, if C, C' ∈ C c (Y), then we denote by .
It is easy to check that
Furthermore, when C is convex, we obtain (λ + μ)C = λC + μC for all λ, μ ∈ ℝ+.
For a given set C ∈ 2Y, the distance function d(·, C) and the support function s(·, C) are respectively defined by
For every pair C, C' ∈ C b (Y), we define the Hausdorff distance between C and C' by
where B Y is the closed unit ball in Y .
The following proposition reveals some properties of the Hausdorff distance.
Proposition 1.1. For every C, C', K, K' ∈ C cb (Y) and λ > 0, the following properties hold
-
(a)
h(C ⊕ C', K ⊕ K') ≤ h(C, K) + h(C', K');
-
(b)
h(λC, λK) = λh(C, K).
Let (C cb (Y), ⊕, h) be endowed with the Hausdorff distance h. Since Y is a Banach space, (C cb (Y), ⊕, h) is a complete metric semigroup (see [7]). Debreu [2] proved that (C cb (Y), ⊕, h) is isometrically embedded in a Banach space as follows.
Lemma 1.2. [2] Let C(B Y* ) be the Banach space of continuous real-valued functions on B Y* endowed with the uniform norm || · || u . Then the mapping j : (C cb (Y), ⊕, h) → C(B Y* ), given by j(A) = s(·, A), satisfies the following properties:
-
(a)
j(A ⊕ B) = j(A) + j(B);
-
(b)
j(λA) = λj(A);
-
(c)
h(A, B) = ||j(A) - j(B)|| u ;
-
(d)
j(C cb (Y)) is closed in C(B Y * )
for all A, B ∈ C cb (Y ) and all λ ≥ 0.
Let f : Ω → (C cb (Y), h) be a set-valued function from a complete finite measure space (Ω, Σ, ν) into C cb (Y). Then f is Debreu integrable if the composition j ○ f is Bochner integrable (see [10]). In this case, the Debreu integral of f in Ω is the unique element (D) ∫Ω fdν ∈ C cb (Y) such that j((D) ∫Ω fdν) is the Bochner integral of j ○ f. The set of Debreu integrable functions from Ω to C cb (Y) will be denoted by D(Ω, C cb (Y)). Furthermore, on D(Ω, C cb (Y)), we define (f + g)(ω) = f(ω) ⊕ g(ω) for all f, g ∈ D(Ω, C cb (Y)). Then we obtain that ((Ω, C cb (Y)), +) is an abelian semigroup.
Set-valued functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [11–18]).
The stability problem of functional equations was originated from a question of Ulam [19] concerning the stability of group homomorphisms. Hyers [20] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [21] for additive mappings and by Rassias [22] for linear mappings by considering an unbounded Cauchy difference. The article of Rassias [22] has provided a lot of influence in the development of what we call Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by Găvruta [23] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach
The functional equation
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers-Ulam stability of the quadratic functional equation was proved by Skof [24] for mappings f : X → Y, where X is a normed space and Y is a Banach space. Cholewa [25] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [26] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [11, 21–23, 27–61]).
In [52], Najati considered the following functional equation
for all x, y, z ∈ X. It is easy to show that the function f(x) = x satisfies the functional Equation (1.1), which is called a Cauchy-Jensen type additive functional equation and every solution of the Cauchy-Jensen type additive functional equation is said to be a Cauchy-Jensen type additive mapping.
In [57], Park considered the following functional equation
for all x, y ∈ X. It is easy to show that the function f(x) = x+x2 satisfies the functional Equation (1.2), which is called a Jensen type additive-quadratic functional equation and every solution of the Jensen type additive-quadratic functional equation is said to be a Jensen type additive-quadratic mapping.
In [48], Jun and Cho considered the following functional equation
for all x, y, z ∈ X, where r, s are real numbers with r, s ≠ 0. It is easy to show that the function f(x) = x2 satisfies the functional Equation (1.3), which is called a generalized quadratic functional equation and every solution of the generalized quadratic functional equation is said to be a generalized quadratic mapping.
In [62], Kim et al. considered the following functional equation
for all x, y ∈ X. It is easy to show that the function f(x) = x3 satisfies the functional Equation (1.4), which is called a Jensen type cubic functional equation and every solution of the Jensen type cubic functional equation is said to be a Jensen type cubic mapping.
Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies
-
(1)
d(x, y) = 0 if and only if x = y;
-
(2)
d(x, y) = d(y, x) for all x, y ∈ X;
-
(3)
d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
Let (X, d) be a generalized metric space. An operator T : X → X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that d(Tx, Ty) ≤ Ld(x, y) for all x, y ∈ X. If the Lipschitz constant L is less than 1, then the operator T is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Margolis and Diaz.
Theorem 1.3. [31, 63] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either
for all nonnegative integers n or there exists a positive integer n 0 such that
-
(1)
d(Jnx, Jn+1x) < ∞, ∀n ≥ n0;
-
(2)
the sequence {Jnx} converges to a fixed point y* of J;
-
(3)
y* is the unique fixed point of J in the set ;
-
(4)
for all y ∈ Y.
In 1996, Isac and Rassias [47] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [32, 33, 56, 58, 64]).
Using the fixed point method, we prove the Cauchy-Jensen type additive set-valued functional equation, the Jensen type additive-quadratic set-valued functional equation, the generalized quadratic set-valued functional equation and the Jensen type cubic set-valued functional equation.
Throughout this article, let X be a real vector space and Y a Banach space.
2. Stability of the Cauchy-Jensen type additive set-valued functional Equation (1.1)
Using the fixed point method, we prove the Hyers-Ulam stability of the Cauchy-Jensen type additive set-valued functional equation.
Definition 2.1. [52] Let f : X → C cb (Y). The Cauchy-Jensen type additive set-valued functional equation is defined by
for all x, y, z ∈ X. Every solution of the Cauchy-Jensen type additive set-valued functional equation is called an Cauchy-Jensen type additive set-valued mapping.
Theorem 2.2. Let φ : X3 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y, z ∈ X. Suppose that f : X → (C cb (Y), h) is a mapping satisfying
for all x, y, z ∈ X. Then
exists for each x ∈ X and defines a unique Cauchy-Jensen type additive set-valued mapping A : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. Let x = y = z in (2.1). Since f(x) is convex, we get
and so
for all x ∈ X.
Consider
and introduce the generalized metric on X,
where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [41, Theorem 2.4]).
Now we consider the linear mapping J : S → S such that
for all x ∈ X.
Let g, f ∈ S be given such that d(g, f) = ε. Then
for all x ∈ X. Hence
for all x ∈ X. So d(g, f) = ε implies that d(Jg, Jf) ≤ Lε. This means that
for all g, f ∈ S.
It follows from (2.4) that .
By Theorem 1.1, there exists a mapping A : X → Y satisfying the following:
-
(1)
A is a fixed point of J, i.e.,
(2.5)
for all x ∈ X. The mapping A is a unique fixed point of J in the set
This implies that A is a unique mapping satisfying (2.5) such that there exists a μ ∈ (0, ∞) satisfying
for all x ∈ X;
-
(2)
d(Jnf, A) → 0 as n → ∞. This implies the equality
for all x ∈ X;
-
(3)
, which implies the inequality
This implies that the inequality (2.2) holds.
By (2.1),
which tends to zero as n → ∞ for all x, y, z ∈ X. Thus
as desired. □
Corollary 2.3. Let p > 1 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (C cb (Y), h) is a mapping satisfying
for all x, y, z ∈ X. Then
exists for all x ∈ X and defines a unique Cauchy-Jensen type additive set-valued mapping A : X → Y satisfying
and for all x ∈ X.
Proof. The proof follows from Theorem 2.2 by taking
for all x, y, z ∈ X. Then we can choose L = 21-pand we get the desired result. □
Theorem 2.4. Let φ : X3 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y ∈ X. Suppose that f : X → (C cb (Y), h) is a mapping satisfying (2.1).Then there exists a unique Cauchy-Jensen type additive set-valued mapping A : X → (C cb (Y), h) such that
and
for all x ∈ X.
Proof. It follows from (2.3) that
for all x ∈ X.
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 2.5. Let 0 < p < 1 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (C cb (Y), h) is a mapping satisfying (2.6). Then
exists for all x ∈ X and defines a unique Cauchy-Jensen type additive set-valued mapping A : X → Y satisfying
for all x ∈ X.
Proof. The proof follows from Theorem 2.4 by taking
for all x, y, z ∈ X. Then we can choose L = 2p-1 and we get the desired result. □
3. Stability of the Jensen type AQ set-valued functional Equation (1.2)
Using the fixed point method, we prove the Hyers-Ulam stability of the Jensen type additive-quadratic set-valued functional equation.
3.1. An odd case
Theorem 3.1. Let φ : X2 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y ∈ X. Suppose that f : X → (C cb (Y), h) is an odd mapping satisfying
for all x, y ∈ X. Then
exists for all x ∈ X and defines a unique Jensen type additive set-valued mapping A : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. Let y = 0 in (3.1). Since f(x) is convex, we get
for all x ∈ X.
Consider
and introduce the generalized metric on X,
where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [41, Theorem 2.4]).
Now we consider the linear mapping J : S → S such that
for all x ∈ X.
It follows from (3.2) that d(f, Jf) ≤ 1.
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 3.2. Let p > 1 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (C cb (Y), h) is a mapping satisfying
for all x, y ∈ X. Then
exists for all x ∈ X and defines a unique Jensen type additive set-valued mapping A : X → Y satisfying
for all x ∈ X.
Proof. The proof follows from Theorem 3.1 by taking
for all x, y ∈ X. Then we can choose L = 21-pand we get the desired result. □
Theorem 3.3. Let φ : X2 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y ∈ X. Suppose that f : X → (C cb (Y), h) is a mapping satisfying (3.1). Then there exists a unique Jensen type additive set-valued mapping A : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. It follows from (3.2) that
for all x ∈ X.
The rest of the proof is similar to the proofs of Theorems 2.2 and 3.1. □
Corollary 3.4. Let 0 < p < 1 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (C cb (Y), h) is a mapping satisfying (3.3). Then there exists a unique Jensen type additive set-valued mapping A : X → Y satisfying
for all x ∈ X.
Proof. The proof follows from Theorem 3.3 by taking
for all x, y ∈ X. Then we can choose L = 2p-1 and we get the desired result. □
3.2. An even case
Theorem 3.5. Let φ : X2 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y ∈ X. Suppose that f : X → (C cb (Y), h) is an even mapping with f(0) = {0} satisfying (3.1). Then there exists a unique Jensen type quadratic set-valued mapping Q : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. Let y = 0 in (3.1). Since f(x) is convex, we get
for all x ∈ X.
Consider
and introduce the generalized metric on X,
where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [41, Theorem 2.4]).
Now we consider the linear mapping J : S → S such that
for all x ∈ X.
It follows from (3.4) that d(f, Jf) ≤ 1.
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 3.6. Let p > 2 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (C cb (Y), h) is an even mapping with f(0) = {0} satisfying (3.3). Then there exists a unique Jensen type quadratic set-valued mapping Q : X → Y satisfying
for all x ∈ X.
Proof. The proof follows from Theorem 3.5 by taking
for all x, y ∈ X. Then we can choose L = 22-pand we get the desired result. □
Theorem 3.7. Let φ : X2 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y ∈ X. Suppose that f : X → (C cb (Y), h) is an even mapping with f(0) = {0} and satisfying (3.1). Then there exists a unique Jensen type quadratic set-valued mapping Q : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. It follows from (3.4) that
for all x ∈ X.
The rest of the proof is similar to the proofs of Theorems 2.2 and 3.5. □
Corollary 3.8. Let 0 < p < 2 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (C cb (Y), h) is an even mapping with f(0) = {0} and satisfying (3.3). Then there exists a unique Jensen type quadratic set-valued mapping Q : X → Y satisfying
for all x ∈ X.
Proof. The proof follows from Theorem 3.7 by taking
for all x, y ∈ X. Then we can choose L = 2p-2 and we get the desired result. □
4. Stability of the generalized quadratic set-valued functional Equation (1.3)
Using the fixed point method, we define a generalized quadratic set-valued functional equation and prove the Hyers-Ulam stability of the generalized quadratic set-valued functional equation.
Remark 4.1. For convenience, let f(u ± v) = f(u + v) ⊕ f(u - v).
Definition 4.2. Let f : X → C cb (Y). The generalized quadratic set-valued functional equation is defined by
for all x, y, z ∈ X. Every solution of the generalized quadratic set-valued functional equation is called a generalized quadratic set-valued mapping.
Theorem 4.3. Let φ : X3 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y, z ∈ X. Suppose that f : X → (C cb (Y), h) is a mapping with f(0) = {0} and satisfying
for all x, y, z ∈ X, and r, s ≠ 0. Then there exists a unique generalized quadratic set-valued mapping Q : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. Let x = y and z = 0 in (4.1). Since f(x) is convex, we get
and so
for all x ∈ X.
Consider
and introduce the generalized metric on X,
where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [41, Theorem 2.4]).
Now we consider the linear mapping J : S → S such that
for all x ∈ X.
It follows from (4.3) that
The rest of the proof is similar to the proof of Theorem 2.2.
Corollary 4.4. Let p > 2, θ ≥ 0 be real numbers and 0 < |r| < 2. Let X be a real normed space. Suppose that f : X → (C cb (Y), h) is a mapping with f(0) = {0} and satisfying
for all x, y, z ∈ X, and r, s ≠ 0. Then there exists a unique generalized quadratic set-valued mapping Q : X → Y satisfying
for all x ∈ X.
Proof. The proof follows from Theorem 4.3 by taking
for all x, y ∈ X. Then we can choose and we get the desired result. □
Theorem 4.5. Let φ : X3 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y, z ∈ X. Suppose that f : X → (C cb (Y), h) is a mapping with f(0) = {0} and satisfying (4.1). Then there exists a unique generalized quadratic set-valued mapping Q : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. It follows from (4.2) that
for all x ∈ X.
The rest of the proof is similar to the proofs of Theorems 2.2 and 4.3. □
Corollary 4.6. Let 0 < p < 2, θ ≥ 0 be real numbers and |r| > 2. Let X be a real normed space. Suppose that f : X → (C cb (Y), h) is a mapping with f(0) = {0} and satisfying (4.4). Then there exists a unique generalized quadratic set-valued mapping Q : X → Y satisfying
for all x ∈ X.
Proof. The proof follows from Theorem 4.5 by taking
for all x, y ∈ X. Then we can choose and we get the desired result. □
5. Stability of the Jensen type cubic set-valued functional Equation (1.4)
Definition 5.1. Let f : X → C cb (Y). The Jensen type cubic set-valued functional equation is defined by
for all x, y ∈ X. Every solution of the Jensen type cubic set-valued functional equation is called a Jensen type cubic set-valued mapping.
Theorem 5.2. Let φ : X2 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y ∈ X. Suppose that f : X → (C cb (Y), h) is a mapping satisfying
for all x, y ∈ X. Then there exists a unique Jensen type cubic set-valued mapping C : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. Let x = y in (5.1). Since f(x) is convex, we get
and so
for all x ∈ X.
Consider
and introduce the generalized metric on X,
where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [41, Theorem 2.4]).
Now we consider the linear mapping J : S → S such that
for all x ∈ X.
It follows from (5.4) that d(f, Jf) ≤ 4L.
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 5.3. Let p > 3 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (C cb (Y), h) is a mapping satisfying
for all x, y ∈ X Then there exists a unique Jensen type cubic set-valued mapping. C : X → Y satisfying
for all x ∈ X.
Proof. The proof follows from Theorem 5.2 by taking
for all x, y ∈ X. Then we can choose L = 23-pand we get the desired result. □
Theorem 5.4. Let φ : X2 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y ∈ X. Suppose that f : X → (C cb (Y), h) is a mapping satisfying (5.1). Then there exists a unique Jensen type cubic set-valued mapping C : X → (C cb (Y), h) such that
for all x ∈ X.
Proof. It follows from (5.3) that
for all x ∈ X.
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 5.5. Let 0 < p < 3 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (C cb (Y), h) is a mapping satisfying (5.5). Then there exists a unique Jensen type cubic set-valued mapping C : X → Y satisfying
for all × ∈ X.
Proof. The proof follows from Theorem 5.4 by taking
for all x, y ∈ X. Then we can choose L = 2p-3 and we get the desired result. □
6. Conclusions
We have defined the Cauchy-Jensen type additive set-valued functional equation, the Jensen type additive-quadratic set-valued functional equation, the generalized quadratic set-valued functional equation and the Jensen type cubic set-valued functional equation, and we have proved the Hyers-Ulam stability of the Cauchy-Jensen type additive set-valued functional equation, the Jensen type additive-quadratic set-valued functional equation, the generalized quadratic set-valued functional equation and the Jensen type cubic set-valued functional equation by using the fixed point method.
References
Aumann RJ: Integrals of set-valued functions. J Math Anal Appl 1965, 12: 1–12. 10.1016/0022-247X(65)90049-1
Debreu G: Integration of correspondences. Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, Part I, Berkeley, University of California Press, USA 1966.
Arrow KJ, Debreu v: Existence of an equilibrium for a competitive economy. Econometrica 1954, 22: 265–290. 10.2307/1907353
McKenzie LW: On the existence of general equilibrium for a competitive market. Econometrica 1959, 27: 54–71. 10.2307/1907777
Hindenbrand W: Core and Equilibria of a Large Economy. Princeton University Press, Princeton; 1974.
Aubin JP, Frankow H: Set-Valued Analysis. Birkhäuser, Boston; 1990.
Castaing C, Valadier M: Convex Analysis and Measurable Multifunctions, vol. 580. In Lect Notes Math. Springer Berlin; 1977.
Klein E, Thompson A: Theory of Correspondence. Wiley, New York; 1984.
Hess C: Set-valued integration and set-valued probability theory: an overview. In Handbook of Measure Theory, vols. I, II. North-Holland, Amsterdam; 2002.
Cascales T, Rodrigeuz J: Birkhoff integral for multi-valued functions. J Math Anal Appl 2004, 297: 540–560. 10.1016/j.jmaa.2004.03.026
Brzdëk T, Popa D, Xu B: Seletions of set-valued maps satisfying a linear inclusion in a single variable. Nonlinear Anal TMA 2011, 74: 324–330. 10.1016/j.na.2010.08.047
Cardinali T, Nikodem K, Papalini F: Some results on stability and characterization of K -convexity of set-valued functions. Ann Polon Math 1993, 58: 185–192.
Nikodem K: On quadratic set-valued functions. Publ Math Debrecen 1984, 30: 297–301.
Nikodem K: On Jensen's functional equation for set-valued functions. Radovi Mat 1987, 3: 23–33.
Nikodem K: Set-valued solutions of the Pexider functional equation. Funkcialaj Ekvacioj 1988, 31: 227–231.
Nikodem K: K -convex and K -concave set-valued functions. Zeszyty Naukowe Nr 1989., 559: Lodz(Rozprawy Mat. 114)
Piao YJ: The existence and uniqueness of additive selection for ( α, β )-( β, α ) type subadditive set-valued maps. J Northeast Normal Univ 2009, 41: 38–40.
Popa D: Additive selections of ( α, β )-subadditive set-valued maps. Glas Mat Ser III 2001, 36(56):11–16.
Ulam SM: Problems in Modern Mathematics, chapter VI, Science ed. Wiley, New York; 1940.
Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222
Aoki T: On the stability of the linear transformation in Banach spaces. J Math Soc Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1
Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl 1994, 184: 431–436. 10.1006/jmaa.1994.1211
Skof F: Proprietà locali e approssimazione di operatori. Rend Sem Mat Fis Milano 1983, 53: 113–129. 10.1007/BF02924890
Cholewa PW: Remarks on the stability of functional equations. Aequationes Math 1984, 27: 76–86. 10.1007/BF02192660
Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh Math Sem Univ Hamburg 1992, 62: 59–64. 10.1007/BF02941618
Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge Univ Press, Cambridge; 1989.
Azadi Kenary H, Park C: Direct and fixed point methods approach to the generalized Hyers-Ulam stability for a functional equation having monomials as solutions. Iranian J Sci Tech Trans A 2011, A4: 301–307.
Azadi Kenary H, Rezaei H, Talebzadeh S, Park C: Stability for the jensen equation in C* - algebras: a fixed point alternative approach. Adv Diff Eq 2012, 2012: 17. 10.1186/1687-1847-2012-17
Azadi Kenary H, Jang SY, Park C: A fixed point approach to the Hyers-Ulam stability of a functional equation in various normed spaces. Fixed Point Theory Appl 2011, 2011: 67. 10.1186/1687-1812-2011-67
Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. J Inequal Pure Appl Math 2003, 4(1):4.
Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math Ber 2004, 346: 43–52.
Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl 2008, 2008: 1–5. (Article ID 749392)
Cho YJ, Saadati R: Lattice Non-Archimedean Random Stability of ACQ Functional Equation. Adv Diff Equ 2011, 2011: 31. 10.1186/1687-1847-2011-31
Ebadian A, Ghobadipour N, Gordji ME: A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C*- ternary algebras. J Math Phys 2010, 51(1):10. doi:10.1063/1.3496391
Eshaghi Gordji M, Alizadeh Z, Cho YJ, Khodaei H: On approximate C*- ternary m- Homomorphisms: A fixed point approach. Fixed Point Theory Appl 2011., 2011(14): (Article ID FPTA,454093)
Eshaghi Gordji M, Ghaemi MB, Kaboli Gharetapeh S, Shams S, Ebadian A: On the stability of J*- derivations. J Geom Phys 2010, 60(3):454–459. 10.1016/j.geomphys.2009.11.004
Eshaghi Gordji M, Khodaei H: The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces. Discrete Dyn Natl Soc 2010., 15: (Article ID 140767) doi:10.1155/2010/140767
Eshaghi Gordji M, Khodaei H, Rassias JM: Fixed points and stability for quadratic mappings in β- normed left Banach modules on Banach algebras. Results Math 2011. doi:10.1007/s00025–011–0123-z
Eshaghi Gordji M, Najati A: Approximately J* -homomorphisms: a fixed point approach. J Geom Phys 2010, 60: 809–814. 10.1016/j.geomphys.2010.01.012
Gordji ME, Park C, Savadkouhi MB: The stability of a quartic type functional equation with the fixed point alternative. Fixed Point Theory 2010, 11: 265–272.
Eshaghi Gordji M, Azadi Kenary H, Rezaei H, Lee YW, Kim GH: Solution and Hyers-Ulam-Rassias stability of generalized mixed type additive-quadratic functional equations in fuzzy Banach spaces. Abstr Appl Anal 2012., 22: (Article ID 953938) (2012). doi:10.1155/2012/953938
Gordji ME, Savadkouhi MB: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl Math Lett 2010, 23: 1198–1202. 10.1016/j.aml.2010.05.011
Găvruta P, Găvruta L: A new method for the generalized Hyers-Ulam-Rassias stability. Int J Nonlinear Anal Appl 2010, 1(2):11–18.
Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.
Isac G, Rassias ThM: On the Hyers-Ulam stability of ψ -additive mappings. J Approx Theory 1993, 72: 131–137. 10.1006/jath.1993.1010
Isac G, Rassias ThM: Stability of ψ -additive mappings: applications to nonlinear analysis. Int J Math Math Sci 1996, 19: 219–228. 10.1155/S0161171296000324
Jun K, Cho Y: Stability of the generalized Jensen type quadratic functional equations. J. Chungcheong Math. Soc 2007, 20: 515–523.
Kenary HA, Cho YJ: Stability of Mixed Additive-Quadratic Jensen Type Functional Equation in Various Spaces. Comput Math Appl 2011, 61: 2704–2724. 10.1016/j.camwa.2011.03.024
Khodaei H, Rassias ThM: Approximately generalized additive functions in several variables. Int J Nonlinear Anal Appl 2010, 1(1):22–41.
Mohammadi M, Cho YJ, Park C, Vetro P, Saadati R: Random stability of an additive-quadratic-quartic functional equation. J Inequal Appl Vol 2010., 18: (Article ID 754210) (2010). doi:10.1155/2010/724210
Najati A: Stability of homomorphisms on JB* triples associated to a Cauchy-Jensen type functional equation. J Math Inequal 2007, 1(1):83–103.
Najati A, Cho YJ: Generalized Hyers-Ulam stability of the pexiderized cauchy functional equation in non-archimedean spaces. Fixed Point Theory Appl 2011., 11: (Article ID 309026) (2011). doi:10.1155/2011/309026
Najati A, Cho YJ: Generalized Hyers-Ulam stability of the pexiderized cauchy functional equation in non-archimedean spaces. Fixed Point Theory Appl 2011., 11: (Article ID 309026) (2011). doi:10.1155/2011/309026
Najati A, Kang JI, Cho YJ: Local stability of the pexiderized Cauchy and Jensen's equations in fuzzy spaces. J Inequal Appl 2011., 78(2011): doi:10.1186/1029–242X-2011–78
Park C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl 2008., 9: (Article ID 493751) (2008)
Park C: Fuzzy stability of a functional equation associated with inner product spaces. Fuzzy Set Syst 2009, 160: 1632–1642. 10.1016/j.fss.2008.11.027
Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91–96.
Rassias ThM (ed) (Ed): Functional Equations and Inequalities. Kluwer Academic, Dordrecht; 2000.
Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Math Appl 2000, 62: 23–130. 10.1023/A:1006499223572
Saadati R, Cho YJ, Vahidi J: The stability of the quartic functional equation in various spaces. Comput Math Appl 2010, 60: 1994–2002. 10.1016/j.camwa.2010.07.034
Kim H, Ko H, Son J: On the stability of a modified Jensen type cubic mapping. J Chungcheong Math Soc 2008, 21: 129–138.
Diaz J, Margolis B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull Am Math Soc 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0
Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Applications 2007., 2007: 15 (Article ID 50175)
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kenary, H.A., Rezaei, H., Gheisari, Y. et al. On the stability of set-valued functional equations with the fixed point alternative. Fixed Point Theory Appl 2012, 81 (2012). https://doi.org/10.1186/1687-1812-2012-81
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-81