On the Stability of a Parametric Additive Functional Equation in Quasi-Banach Spaces

and Applied Analysis 3 2. Stability of Functional Equation 1.4 in Quasi-Banach Spaces For simplicity, we use the following abbreviation for a given mapping f : X → Y : Df x1, x2, . . . , xm m ∑ i 1 f ⎛ ⎝mxi m ∑ j 1,j / i xj ⎞ ⎠ f ( m ∑


Introduction
It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation.
The first stability problem was raised by Ulam 1 during his talk at the University of Wisconsin in 1940. The stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? If the answer is affirmative, we would say that the equation is stable.
In 1941, Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces. Let f : E → E be a mapping between Banach spaces such that for all x, y ∈ E, and for some δ > 0. Then there exists a unique additive mapping T : E → E such that 3 ψ t < t; t > 1.
These stability results can be applied in stochastic analysis 10 , financial, and actuarial mathematics, as well as in psychology and sociology.
In 1987 Gajda and Ger 11 showed that one can get analogous stability results for subadditive multifunctions. In 1978 Gruber 12 remarked that Ulam's problem is of particular interest in probability theory and in the case of functional equations of different types. We refer the readers to 2, 5-32 and references therein for more detailed results on the stability problems of various functional equations.
We recall some basic facts concerning quasi-Banach space. A quasi-norm is a realvalued function on X satisfying the following. 2 λ · x |λ| · x for all λ ∈ R and all x ∈ X.
3 There is a constant K ≥ 1 such that x y ≤ K x y for all x, y ∈ X. The pair X, · is called a quasi-normed space if · is a quasi-norm on X. A quasi-Banach space is a complete quasi-normed space. A quasi-norm · is called a p-norm 0 < p ≤ 1 if for all x, y ∈ X. In this case, a quasi-Banach space is called a p-Banach space. Given a pnorm, the formula d x, y : x − y p gives us a translation invariant metric on X. By the Aoki-Rolewicz theorem see 33 , each quasi-norm is equivalent to some p-norm. Since it is much easier to work with p-norms, henceforth we restrict our attention mainly to p-norms. In this paper, we consider the generalized Hyers-Ulam stability of the following functional equation: for a fixed positive integer m with m ≥ 2 in quasi-Banach spaces. Throughout this paper, assume that X is a quasi-normed space with quasi-norm · X and that Y is a p-Banach space with p-norm · Y . Abstract and Applied Analysis 3

Stability of Functional Equation 1.4 in Quasi-Banach Spaces
For simplicity, we use the following abbreviation for a given mapping f : X → Y : We start our work with the following theorem which can be regard as a general solution of functional equation 1.4 . Putting Putting So, 2.6 turns to the following: From 2.5 and 2.8 , we have Replacing x by y and y by x in 2.8 and comparing it with 2.9 , we get

2.10
Letting x y in 2.5 , 2.8 , and 2.10 , respectively, we obtain Replacing f 2x and f 2y by their equivalents by using 2.12 in 2.10 , we get Replacing y by −x in 2.13 , we get Similarly, replacing y by y − x in 2.13 , we obtain Replacing y by −y and x by −x in 2.15 and 2.16 , respectively, we obtain

2.17
Abstract and Applied Analysis for all x ∈ X, and for all x ∈ X and the mapping T : X → Y satisfies the inequality: for all x ∈ X.
Proof. Putting for all x ∈ X. The right-hand side tends to zero as j → ∞, hence T x T x for all x ∈ X. This show the uniqueness of T . Corollary 2.3. Let θ, r j 1 ≤ j ≤ m be nonnegative real numbers such that 0 < r j < 1. Suppose that a mapping f : X → Y with f 0 0 satisfies the inequality: for all x j ∈ X 1 ≤ j ≤ m . Then there exists a unique additive mapping T : X → Y such that

2.33
for all x ∈ X.
Proof. This is a simple consequence of Theorem 2.2.
The following corollary is Hyers-Ulam-type stability for the functional equation 1.4 .

Corollary 2.4.
Let θ be nonnegative real number. Suppose that a mapping f : X → Y with f 0 0 satisfies the inequality: for all x j ∈ X 1 ≤ j ≤ m . Then there exists a unique additive mapping T : X → Y such that

2.35
for all x ∈ X. for all x ∈ X and the mapping T : X → Y satisfies the inequality: for all x ∈ X.
Proof. Putting for all x ∈ X and all nonnegative n and r with n ≥ r.