Stability of Functional Inequalities with Cauchy-Jensen Additive Mappings

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms. We are given a group G and a metric group G′ with metric ρ(·,·). Given > 0, does there exist a δ > 0 such that if f :G→G′ satisfies ρ( f (xy), f (x) f (y)) < δ for all x, y ∈G, then a homomorphism h :G→G′ exists with ρ( f (x),h(x)) < for all x ∈G? In 1941, Hyers [2] considered the case of approximately additive mappings f : E→ E′, where E and E′ are Banach spaces and f satisfies Hyers’ inequality


Introduction
In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems.Among these was the following question concerning the stability of homomorphisms.
We are given a group G and a metric group G with metric ρ(•,•).Given > 0, does there exist a δ > 0 such that if f : G → G satisfies ρ( f (xy), f (x) f (y)) < δ for all x, y ∈ G, then a homomorphism h : G → G exists with ρ( f (x),h(x)) < for all x ∈ G?
In 1941, Hyers [2] considered the case of approximately additive mappings f : E → E , where E and E are Banach spaces and f satisfies Hyers' inequality for all x, y ∈ E. It was shown that the limit L(x) = lim n→∞ ( f (2 n x)/2 n ) exists for all x ∈ E and that L : E → E is the unique additive mapping satisfying In 1978, Rassias [3] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded.
Let f : E → E be a mapping from a normed vector space E into a Banach space E subject to the inequality f (x + y) − f (x) − f (y) ≤ x p + y p (1.3) for all x, y ∈ E, where and p are constants with > 0 and p < 1.
Then, the limit L(x) = lim n→∞ ( f (2 n x)/2 n ) exists for all x ∈ E and L : E → E is the unique additive mapping which satisfies for all x ∈ E. If p < 0, then inequality (1.3) holds for x, y = 0 and (1.4) for x = 0.In 1991, Gajda [4], following the same approach as in Rassias [3], gave an affirmative solution to this question for p > 1.It was shown by Gajda [4] as well as by Rassias and Šemrl [5] that one cannot prove a Rassias-type theorem when p = 1.Inequality (1.3) that was introduced for the first time by Rassias [3] provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept.This new concept of stability is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations (cf. the books of Czerwik [6], Hyers et al. [7]).
Gȃvrut ¸a [8] provided a further generalization of Rassias' theorem.During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [9][10][11][12][13][14]).
Gilányi [15] and Rätz [16] showed that if f satisfies the functional inequality then f satisfies the Jordan-von Neumann functional equation Gilányi [17] and Fechner [18] proved the generalized Hyers-Ulam stability of the functional inequality (1.3).Now, we consider the following functional inequalities: which are associated with Jordan-von Neumann-type Cauchy-Jensen additive functional equations.
The purpose of this paper is to prove that if f satisfies one of the inequalities (1.7) and (1.8) which satisfies certain conditions, then we can find a Cauchy-Jensen additive mapping near f , and thus we prove the generalized Hyers-Ulam stability of the functional inequalities (1.7) and (1.8).

Stability of functional inequality (1.7)
We prove the generalized Hyers-Ulam stability of a functional inequality (1.7) associated with a Jordan-von Neumann-type 3-variable Cauchy-Jensen additive functional equation.Throughout this paper, let G be a normed vector space and Y a Banach space.
for all x, y,z ∈ G.Then, f is Cauchy-Jensen additive.
Theorem 2.2.Assume that a mapping f : G → Y satisfies the inequality and that the map φ : for all x, y,z ∈ G.Then, there exists a unique Cauchy-Jensen additive mapping A : for all x ∈ G.
Next, we claim that the mapping A : G → Y is Cauchy-Jensen additive.In fact, it follows easily from (2.3) and condition of φ that (2.10) Thus, the mapping A : G → Y is Cauchy-Jensen additive by Lemma 2.1.Now, let T : G → Y be another Cauchy-Jensen additive mapping satisfying (2.5).Then we obtain Y.-S.Cho and H.-M. Kim 5 which tends to zero as n → ∞.So, we can conclude that A(x) = T(x) for all x ∈ G.This proves the uniqueness of A. Hence, the mapping A : G → Y is a unique Cauchy-Jensen additive mapping satisfying (2.5).
Theorem 2.3.Assume that a mapping f : G → Y satisfies inequality (2.3) and that the map for all x, y,z ∈ G.
Then, there exists a unique Cauchy-Jensen additive mapping A : for all x ∈ G.
Proof.We get by (2.8) for all nonnegative integers m and l with m > l and all x ∈ G.It means that a sequence Moreover, letting l = 0 and passing the limit m → ∞ in (2.14), we get (2.13).
The remaining proof goes through by the similar argument to Theorem 2.2.
Theorem 2.4.Assume that a mapping f : G → Y satisfies inequality (2.3) and that the map for all x, y,z ∈ G.If there exists a number L with 0 ≤ L < 1 such that the mapping then there exists a unique Cauchy-Jensen additive mapping A : for all x ∈ G.
Proof.We get by (2.8) for all x ∈ G. Hence, we get for all nonnegative integers m and l with m > l and all x ∈ G.It means that a sequence {3 n f (x/3 n )} is a Cauchy sequence for all x ∈ G. Since Y is complete, the sequence {3 n f (x/ 3 n )} converges.So, one can define a mapping A : G → Y by A(x) := lim n→∞ 3 n f (x/3 n ) for all x ∈ G.Moreover, letting l = 0 and passing the limit m → ∞ in (2.19), we get (2.17).
The remaining proof goes through by the similar argument to Theorem 2.2.
Corollary 2.5.Assume that there exist nonnegative numbers θ and a real p > 1 such that a mapping f : G → Y satisfies the inequality for all x, y,z ∈ G.
Then, there exists a unique Cauchy-Jensen additive mapping A : for all x ∈ G.
Proof.We get by (2.8) for all nonnegative integers m and l with m > l and all x ∈ G.It means that a sequence ) for all x ∈ G.Moreover, letting l = 0 and passing the limit m → ∞ in (2.25), we get (2.24).
The remaining proof goes through by the similar argument to Theorem 2.3.
Corollary 2.7.Assume that there exist nonnegative numbers θ, δ, and a real p < 1 such that a mapping f : G → Y satisfies the inequality for all x, y,z ∈ G.
Then, there exists a unique Cauchy-Jensen additive mapping A : for all x ∈ G.

Stability of functional inequality (1.8)
We prove the generalized Hyers-Ulam stability of a functional inequality (1.8) associated with a Jordan-von Neumann-type 3-variable Cauchy-Jensen additive functional equation.
Theorem 3.1.Assume that a mapping f : G → Y satisfies the inequality and that the map φ : Then, there exists a unique Cauchy-Jensen additive mapping A : for all x ∈ G.
Proof.Letting x, y,z := 0 in (3.1), we get f (0) ≤ (1/2)φ(0,0,0).And by setting x := 2x, y := 0, and z := −x in (3.1), we get for all x ∈ G. Also by letting y := −x and z := 0 or by letting y := x and z := −x in (3.1), we get for all x ∈ G. Hence, we get by (3.4) and (3.5) Y.-S.Cho and H.-M. Kim 9 for all nonnegative integers m and l with m > l and all x ∈ G.It means that a sequence x) for all x ∈ G.Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.3).The remaining proof is similar to that of Theorem 2.3.
Theorem 3.2.Assume that a mapping f : G → Y satisfies inequality (3.1) and that the map Then, there exists a unique Cauchy-Jensen additive mapping A : for all x ∈ G.
The rest of proof is similar to that of Theorem 2.2.
Proof.Let g(x) := ( f (x) − f (−x))/2.Then, we get by (3.4) for all x ∈ G. Hence, we get by (3.11) x,0,2 j x + φ 2 j+1 x,0,−2 j x + 6 f (0) (3.12) for all nonnegative integers m and l with m > l and all x ∈ G.It means that a sequence {(1/2 n )g(2 n x)} is a Cauchy sequence for all x ∈ G. So, one can define a mapping Moreover, letting l = 0 and passing the limit m → ∞ in (3.12), we get (3.10).Next, we claim that the mapping L : G → Y is a Cauchy-Jensen additive mapping.Note that Y.-S.Cho and H.-M. Kim 11 and so we obtain by (3.1) and (3.4), which tends to zero as n → ∞ for all x ∈ G. Hence, we see that L is additive.The remaining proof is similar to the corresponding part of Theorem 2.3.
Remark 3.4.Assume that a mapping f : G → X satisfies inequality (3.1) and that the map ) lim n→∞ 2 n φ(x/2 n , y/2 n ,z/2 n ) = 0 for all x, y,z ∈ G.Then, there exists a unique Cauchy-Jensen additive mapping L : for all x ∈ G. (3.17) for all nonnegative integers m and l with m > l and all x ∈ G.It means that the sequence {2 n g(x/2 n )} is a Cauchy sequence for all x ∈ G. So, one can define a mapping L : G → Y by L(x) := lim n→∞ 2 n g(x/2 n ) = lim n→∞ 2 n [( f (x/2 n ) − f (−x/2 n ))/2] for all x ∈ G.Moreover, letting l = 0 and passing the limit m → ∞ in (3.17), we get (3.15).
Next, we claim that the mapping L : G → Y is a Cauchy-Jensen additive mapping.Note that L(−x) = −L(x) because g(−x) = −g(x).So, we obtain by (3.1)The remaining proof is similar to that of Theorem 2.2.