A Fixed Point Approach to the Stability of an n-Dimensional Mixed-Type Additive and Quadratic Functional Equation

and Applied Analysis 3 Moreover, they also investigated the Hyers-Ulam-Rassias stability of 1.3 by using the direct method see 18 . Indeed, they tried to approximate the even and odd parts of each solution of a perturbed inequality by the even and odd parts of an “exact” solution of 1.3 , respectively. In Theorems 3.1 and 3.3 of this paper, we will apply the fixed point method and prove the Hyers-Ulam-Rassias stability of the n-dimensional mixed-type additive and quadratic functional equation. The advantage of this paper, in comparison with 18 , is to approximate each solution of a perturbed inequality by an “exact” solution of 1.3 , and we obtain sharper estimations in consequence of this advantage. Throughout this paper, let V be a real or complex vector space, Y a Banach space, and n an integer larger than 1. 2. Preliminaries Let X be a nonempty set. A function d : X2 → 0,∞ is called a generalized metric on X if and only if d satisfies the following: M1 d x, y 0 if and only if x y; M2 d x, y d y, x for all x, y ∈ X; M3 d x, z ≤ d x, y d y, z for all x, y, z ∈ X. We remark that the only difference between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to 19 . Theorem 2.1. Let X, d be a complete generalized metric space. Assume that Λ : X → X is a strict contraction with the Lipschitz constant L < 1. If there exists a nonnegative integer n0 such that d Λ0 1x,Λn0x < ∞ for some x ∈ X, then the following statements are true. i The sequence {Λnx} converges to a fixed point x∗ of Λ. ii x∗ is the unique fixed point of Λ in X∗ {y ∈ X | d Λ0x, y < ∞}. iii If y ∈ X∗, then d ( y, x∗ ) ≤ 1 1 − L ( Λy, y ) . 2.1 In 1991, Baker applied the fixed point method to prove the Hyers-Ulam stability of a nonlinear functional equation see 20 . Thereafter, Radu noticed that many theorems concerning the Hyers-Ulam stability of various functional equations follow from the fixed point alternative Theorem 2.1 . Indeed, he applied the fixed point method to prove the existence of a solution of the inequality 1.1 and investigated the Hyers-Ulam stability of the additive Cauchy equation see 21 and also 22–26 . For a somewhat different fixed point approach to stability of functional equations, see 27, 28 . 4 Abstract and Applied Analysis 3. Hyers-Ulam-Rassias Stability Let V be a real or complex vector space and let Y be a Banach space. For a given function f : V → Y , we use the following abbreviation: Df x1, x2, . . . , xn : 2f ⎛ ⎝ n ∑