Fixed Point Approach to the Stability of the Quadratic Functional Equation

Abstract

In the present paper, we apply a fixed point theorem to prove the Hyers–Ulam–Rassias stability of the quadratic functional equation $$f(kx+ y)+f\bigl(kx+\sigma(y)\bigr)=2k^{2}f(x)+2f(y),\quad x,y\in E_{1} $$ from a normed space E 1 into a complete β-normed space E 2, where σ:E 1⟶E 1 is an involution and k is a fixed positive integer larger than 2. Furthermore, we investigate the Hyers–Ulam–Rassias stability for the functional equation in question on restricted domains.The concept of Hyers–Ulam–Rassias stability originated essentially with the Th.M. Rassias’ stability theorem that appeared in his paper “On the stability of linear mapping in Banach spaces” (Proc. Am. Math. Soc. 72:297–300, ).