Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias Stabilities of an Additive Functional Equation in Several Variables

In 1940, Ulam [13] proposed the Ulam stability problem of additive mappings. In the next year, Hyers [5] considered the case of approximately additive mappings f : E→ E′, where E and E′ are Banach spaces and f satisfies inequality ‖ f (x+ y)− f (x)− f (y)‖ ≤ ε for all x, y ∈ E. It was shown that the limit L(x) = limn→∞ 2−n f (2nx) exists for all x ∈ E and that L is the unique additive mapping satisfying ‖ f (x)−L(x)‖ ≤ ε. In 1978, Rassias [14] generalized the result to an approximation involving a sum of powers of norms. In 1982–1989, Rassias [8–11] treated the Ulam-Gavruta-Rassias stability on linear and nonlinear mappings and generalized Hyers result to the following theorem.


Introduction
In 1940, Ulam [13] proposed the Ulam stability problem of additive mappings.In the next year, Hyers [5] considered the case of approximately additive mappings f : E → E , where E and E are Banach spaces and f satisfies inequality f (x + y) − f (x) − f (y) ≤ ε for all x, y ∈ E. It was shown that the limit L(x) = lim n→∞ 2 −n f (2 n x) exists for all x ∈ E and that L is the unique additive mapping satisfying f (x) − L(x) ≤ ε.In 1978, Rassias [14] generalized the result to an approximation involving a sum of powers of norms.In 1982-1989, Rassias [8][9][10][11] treated the Ulam-Gavruta-Rassias stability on linear and nonlinear mappings and generalized Hyers result to the following theorem.
Theorem 1.1 (J.M. Rassias).Let f : E → E be a mapping, where E is a real-normed space and E is a Banach space.Assume that there exist θ > 0 such that for all x, y ∈ E, where r = p + q = 1.Then there exists a unique additive mapping L : for all x ∈ E.
However, the case r = 1 in the above inequality is singular.A counterexample has been given by Gȃvruta [2].The above-mentioned stability involving a product of different powers of norms is called Ulam-Gavruta-Rassias stability by Bouikhalene and Elqorachi [1], Ravi and ArunKumar [12], and Nakmahachalasint [6].In recent years, some other authors [3,4,7] have investigated the stability of additive mapping in various forms.
In this paper, we propose an n-dimensional additive functional equation and investigate its Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities.

The functional equation and the solution
Theorem 2.1.Let n > 1 be an integer and let X, Y be real vector spaces.A mapping f : X → Y satisfies the functional equation if and only if f satisfies the Cauchy functional equation Proof.We first suppose that a mapping f : X → Y satisfies (2.2).By the additivity of the Cauchy functional equation, we have for all x 1 ,x 2 ,...,x n ∈ X.Hence, f satisfies (2.1).Now suppose that a mapping f : X → Y satisfies (2.1).Putting (2.4) which simplifies to f (x + y) = f (x) + f (y) as desired.

Hyers-Ulam-Rassias stability
The following theorem treats the Hyers-Ulam-Rassias stability of (2.1).Theorem 3.1.Let n > 1 be an integer, let X be a real vector space, and let Y be a Banach space.Given real numbers δ,θ ≥ 0 and p ∈ (0,1) for all x 1 ,x 2 ,...,x n ∈ X, then there exists a unique additive mapping L : X → Y that satisfies (2.1) and the inequality The mapping L is given by which simplifies to We first consider the case where 0 < p < 1. Rewrite the above inequality (3.6) as For every positive integer m, (3.8) Substituting x with x,2x,2 2 x,...,2 m−1 x in (3.7), the above inequality becomes 1) . (3.9) Consider the sequence {2 −m f (2 m x)}.For all positive integers k < l, we have 1) . (3.10) The right-hand side of the above inequality approaches 0 as k → ∞.Therefore, L(x) = lim m→∞ 2 −m f (2 m x) is well defined.Taking the limit of (3.9) as m → ∞, we have To show that L satisfies (2.1), replace each x i in (3.1) with 2 m x i .This results in (3.12) Dividing the above inequality by 2 m and taking the limit as m → ∞, we obtain which verifies that L indeed satisfies (2.1).
Paisan Nakmahachalasint 5 To prove the uniqueness of L, suppose there is a mapping L : X → Y such that L satisfies (2.1) and (3.2).The additivity of L and L is asserted by Theorem 2.1; hence, (3.14) Thus, L(x) = L (x) for all x ∈ X.
For the case p > 1, δ = 0 and (3.7) must be replaced by The rest of the proof can be done in the same fashion as that of the case 0 < p < 1.
Theorem 4.1.Let n > 1 be an integer, let X be a real vector space, and let Y be a Banach space.Given real numbers δ,θ ≥ 0 and p ∈ (0,1) for all x 1 ,x 2 ,...,x n ∈ X, then there exists a unique additive mapping L : X → Y that satisfies (2.1) and the inequality The mapping L is given by (3.3).
Proof.We make the same substitution as in the proof of Theorem 3.1 and obtain instead of (3.5) the following inequality: 3) The rest of the proof, apart from a multiplicative factor of 2 appears before θ, can be carried over from that of Theorem 3.1.
It should be remarked that in the case where n = 2, functional equation (2.1) reduces to the Cauchy functional equation, and the Ulam-Gavruta-Rassias stability of this problem has been treated by J. M. Rassias, and the result has been restated in Theorem 1.1.