The Generalized Hyers-Ulam-Rassias Stability of a Quadratic Functional Equation

International Journal of Research and Scientific Innovation (IJRSI) | Volume V, Issue VII, July 2018 | ISSN 2321–2705

The Generalized Hyers-Ulam-Rassias Stability of a Quadratic Functional Equation

Rajkumar.V

Hindusthan Institute of Technology, Coimbatore, Tamil Nadu, India

Abstract: – I give the general solution of the functional equation

f(x+y+a)+f(x-y+a)=2f(x+a)+2f(y)

and investigate its generalized Hyers-Ulam-Rassias-Stability.

Keywords: Quadratic functional equation; Hyers-Ulam-Rassias-Stability.

I. INTRODUCTION

In 1940, S.M.Ulam[11] posed a problem on the stability of the linear functional equation before the mathematics club of the university of Wisconsin. The problem can be stated as follows.

Let X and Y be Banach spaces.For every ε>0, does there exist δ>0 such that for a function f:X→Y satisfying a δ-linear condition, ‖f(x+y)-f(x)-f(y) ‖≤δ, for all x,y∈X,there exists a function T:X→Y such that ‖f(x)-T(x)‖≤ε for all x∈X ?

This problem was answered by D.H.Hyers [6] and was generalized by Th.M.Rassias [10]. Since then, the stability of various functional equations, including linear functional equations [5,4,7,8] and quadratic functional equations [1,2,3,9] , was thoroughly studied.

The classical quadratic functional equation:

f(x+y)+f(x-y)=2f(x)+2f(y)

admits a solution in the form f(x)=cx^2 on the set of real numbers and every solution of this equation is said to be a quadratic function [3]. J.M.Rassias [9] derived the stability of the generalized version of the above quadratic functional equation:

Q(ax+by)+Q(bx-ay)=(a^2+b^2)[Q(x)+Q(y)] ,

which covers a wide range of quadratic functional equations in two variables.

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