The Stability of a Quadratic Functional Equation with the Fixed Point Alternative

and Applied Analysis 3 Theorem 1.3 see 26–28 . Let X, d be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d ( Jx, J 1x ) ∞ 1.7 for all nonnegative integers n or there exists a positive integer n0 such that 1 d Jx, J 1x < ∞, for all n ≥ n0; 2 the sequence {Jnx} converges to a fixed point y∗ of J ; 3 y∗ is the unique fixed point of J in the set Y {y ∈ X | d J0x, y < ∞}; 4 d y, y∗ ≤ 1/ 1 − L d y, Jy for all y ∈ Y . Lee et al. 29 proved that a mapping f : X → Y satisfies f ( 2x y ) f ( 2x − y) 8f x 2f(y) 1.8 for all x, y ∈ X if and only if the mapping f : X → Y satisfies f ( x y ) f ( x − y) 2f x 2f(y) 1.9 for all x, y ∈ X. Using the fixed point method, Park 14 proved the generalized Hyers-Ulam stability of the quadratic functional equation f ( 2x y ) 4f x f ( y ) f ( x y ) − f(x − y) 1.10 in Banach spaces. In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation 1.8 in Banach spaces. Throughout this paper, assume that X is a normed vector space with norm || · || and that Y is a Banach space with norm ‖ · ‖. 2. Fixed Points and Generalized Hyers-Ulam Stability of a Quadratic Functional Equation For a given mapping f : X → Y , we define Cf ( x, y ) : f ( 2x y ) f ( 2x − y) − 8f x − 2f(y) 2.1 for all x, y ∈ X. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation Cf x, y 0. 4 Abstract and Applied Analysis Theorem 2.1. Let f : X → Y be a mapping for which there exists a function φ : X2 → 0,∞ with f 0 0 such that ∥ ∥Df ( x, y )∥ ∥ ≤ φ(x, y) 2.2 for all x, y ∈ X. If there exists an L < 1 such that φ x, y ≤ 4Lφ x/2, y/2 for all x, y ∈ X, then there exists a unique quadratic mapping Q : X → Y satisfying 1.8 and ∥ ∥f x −Q x ∥ ≤ 1 8 − 8L x, 0 2.3 for all x ∈ X. Proof. Consider the set S : { g : X −→ Y}, 2.4 and introduce the generalized metric on S: d ( g, h ) inf { K ∈ R : ∥ ∥g x − h x ∥ ≤ Kφ x, 0 , ∀x ∈ X}. 2.5 It is easy to show that S, d is complete. Now we consider the linear mapping J : S → S such that