The Fuzzy Stability of a Pexiderized Functional Equation

In this paper, Hyers-Ulam-Rassias Stability of the Pexiderized functional equation f(x + y) = g(x) + h(y) is concerned in fuzzy Banach spaces.


Introduction
In 1940 stability problem of a functional equation was initiated by Ulam [14] concerning the stability of group homomorphism.In the next year, Hyers [6] gave answer for Cauchy functional equation in Banach spaces.T. Aoki [15] and Th.M. Rassias [16] generalized Hyers's theorem for additive mappings and linear mappings by considering an unbounded Cauchy difference respectively.Gavruta [10] generalized Rassias theorem by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias's approach.F. Skof [7] generalized Hyers-Ulam stability theorem for the function f : X → Y , where X is a normed linear space and Y is a Banach space.Afterwards, the result of Skof was extended by P. W. Cholewa [11] and S. Czerwik [13].
Fuzzy set theory was initiated by Zadeh [8] and after the introduction of the notion of fuzzy norm on a linear space by Katsaras [2], many authors [12,17], gave various ideas of fuzzy norm.Thereafter various notions of Banach spaces have been generalized in fuzzy Banach spaces.In fact, the notion of the Hyers-Ulam-Rassias stability for various functional equations are being generalized in fuzzy Banach Spaces by several authors [3,5,9,18,19].
In this paper, we investigate the generalized Hyers-Ulam-Rassias stability for the functional equation f (x + y) = g(x) + h(y) in fuzzy Banach spaces.

Preliminaries
In the sequel, we need some definitions which are given bellow.
Example 2.2.Let (X, • ) be a normed linear space, and let a * b = ab or a * b = min {a, b} for all a, b ∈ [0, 1].Let N (x, t) = t t+ x for all x ∈ X and t > 0. Then (X, N, * ) is a fuzzy normed linear space and this fuzzy norm N induced by • is called the standard fuzzy norm.Note 2.1.According to George and Veeramani [1], it can be proved that every fuzzy normed linear space is a metrizable topological space.In fact, also it can be proved that if (X, • ) is a normed linear space, then the topology generated by • coincides with the topology generated by the fuzzy norm N of Example 2.2.As a result, we can say that an ordinary normed linear space is a special case of fuzzy normed linear space.
Remark 2.1.In fuzzy normed linear space (X, N, * ), for all x ∈ X, N (x, •) is non-decreasing with respect to the variable t and lim t→∞ N (x, t) = 1.Definition 2.3.[17] Let (X, N, * ) be a fuzzy normed linear space.A sequence {x n } in X is said to be convergent or converge if there exists an x ∈ X such that lim n→∞ N (x n − x, t) = 1.In this case, x is called the limit of the sequence {x n } and we denote it by N − lim n→∞ x n = x.Definition 2.4.[17] Let (X, N, * ) be a fuzzy normed linear space.A sequence {x n } in X is called Cauchy sequence if for each ε > 0 and t > 0 there exists an n 0 ∈ N such that for all n n 0 and all p > 0, we have N (x n+p − x n , t) > 1 − ε.

Stability of The Functional Equation
Throughout this section, X is assumed to be a real vector space and (Y, N ) is assumed to be a fuzzy Banach space.
Let f, g, h : X → Y be mappings such that uniformly on X 2 .Then there exists a unique mapping A : X → Y such that for all x, y ∈ X and if for some δ > 0, α > 0 for all x, y ∈ X, then Proof.Corresponding to a given > 0 and (3.1), there exists some t 0 > 0 such that (3.5) N (f (x + y) − g(x) − h(y), tφ(x, y)) ≥ 1 − for all x, y ∈ X and t ≥ t 0 .Let The Fuzzy Stability of a Pexiderized Functional Equation for all x, y ∈ X.Now, for all x, y ∈ X and t ≥ t 0 .Define a function for all x ∈ X and t ≥ t 0 .Replacing x and y by −x and 3x respectively in (3.6), we get for all x ∈ X and t ≥ t 0 .Now, (3.9) for all x ∈ X and t ≥ t 0 .Now we show for any positive integer n that (3.10) for all x ∈ X and t ≥ t 0 .
Putting t = t 0 , replacing n by p and x by 3 n x in (3.10), we get p−1 i=0 converges for all x, y ∈ X, for a given δ > 0, there exists [by (3.12), (3.13)] for all x ∈ X, n ≥ n 0 , p > 0. This shows that the sequence The Fuzzy Stability of a Pexiderized Functional Equation some A(x) ∈ Y .So we can define a function A : X → Y by Replacing x and y by 3 n x in (3.5), we get for all x ∈ X, t ≥ t 0 .Since lim n→∞ 3 −n φ(3 n x, 3 n x) = 0, therefore for fixed t > 0 and 0 < < 1, there exists n 0 ∈ N such that (3.16) [by (3.16)].
The first term ≥ 1 − by (3.15) and last term tends to 1 as n → ∞.Thus Again from definition of A we get for all t > 0, x ∈ X.Since lim n→∞ 3 −n φ(3 n x, 3 n y) = 0 for all x, y ∈ X, therefore for fixed t > 0 there exists n 1 ∈ N such that (3.20) Replacing x and y by 3 n+1 x and 3 n x respectively in (3.6) and for t = t 0 , we get From (3.19) and (3.21), we see that first three terms on RHS tend to 1 as n → ∞ and last term ≥ 1 − .Therefore N (2A(2x) − 4A(x), t) = 1 for all t > 0. Thus for all x ∈ X (3.22) A(2x) = 2A(x).
Since lim n→∞ 3 −n φ(3 n x, 3 n y) = 0 for all x, y ∈ X, therefore for fixed t > 0 there exists n 2 ∈ N such that (3.23) for all n ≥ n 2 .Replacing x and y by 3 n x and 3 n y respectively in (3.6) and for t = t 0 , we get for all x, y ∈ X.Now, From (3.19) and (3.24), we see that first three terms on RHS tend to 1 as n → ∞ and last term ≥ 1 − .Therefore N (A(x + y) − A(x) − A(y), t) = 1 for all t > 0 that is, A(x + y) = A(x) + A(y) for all x, y ∈ X.
Let (3.3) hold for some δ > 0, α > 0. Then by similar approach as in the beginning of proof we can deduce from (3.3) that (3.25) The Fuzzy Stability of a Pexiderized Functional Equation Taking limit as n → ∞, we get from (3.14) and (3.25) Because of continuity of N (x, •) and taking limit as t → 0, we get it proves the result (3.4).
To prove the uniqueness of A let us assume that A be another mapping satisfying (3.2) and (3.4).Then for a given > 0, we can find some t 0 > 0 such that converges for all x, y ∈ X, therefore for a fixed c > 0 there exists n 3 ∈ N such that It implies that A(x) = A (x) for all x ∈ X.This proves that A is unique.Replacing x and y by 3 n+1 x and 3 n x respectively in (3.5), we get (3.30) for all x ∈ X, n ≥ 0, t ≥ t 0 .Again, replacing x and y by 3 n x and 3 n+1 x respectively in (3.5), we get (3.31) x, 3 n y) = 0 for all x, y ∈ X, therefore for fixed t > 0, there exists m ∈ N such that (3.32) The Fuzzy Stability of a Pexiderized Functional Equation for all x ∈ X, t > 0, n ≥ m.Let c > 0. Then we can find a positive integer m ≥ m such that This completes the proof of the theorem.
Corollary 3.1.Let a be a fixed real number with 0 ≤ a < 3 and ψ : (a, ∞) → R + be a function such that for all t, s > a Let f, g, h : X → Y be mappings such that for all x, y with x , y > a. Then there exists a unique mapping A : X → Y such that A(x + y) = A(x) + A(y) for all x, y ∈ X and if for some δ > 0, α > 0 for all x ∈ X with x > 2a.
N f (x) − A(x) − f (0), 4δ(3 p + 3) 2 p (3 p − 3) x p ≥ α for all x ∈ X with 0 < x < a.Proof.Define ψ : (0, a) → R + by ψ(t) = t p .Thenψp + 3) 2 p (3 p − 3) x p .4.ConclusionIn this paper the generalized Hyers-Ulam-Rassias stability of the functional equation f (x + y) = g(x) + h(y) has been studied in fuzzy Banach spaces.What could be the general solution of such functional equation and what are the properties of the general solution of this equation, it should be studied in future.Instead of crisp functional equation, if we consider fuzzy functional equation, how can we study the corresponding Hyers-Ulam stability property.