Quadratic functional equations of Pexider type

Soon-Mo Jung

doi:10.1155/s0161171200004075

Abstract

First, the quadratic functional equation of Pexider type will be solved. By applying this result, we will also solve some functional equations of Pexider type which are closely associated with the quadratic equation.

Highlights

It is easy to see that the quadratic function f (x) = x2 is a solution of each of the following functional equations f (x + y) + f (x − y) = 2f (x) + 2f (y), f (x + y + z) + f (x) + f (y) + f (z) = f (x + y) + f (y + z) + f (z + x), f (x − y − z) + f (x) + f (y) + f (z) = f (x − y) + f (y + z) + f (z − x), f (x + y + z) + f (x − y + z) + f (x + y − z) + f (−x + y + z)
It is natural that each equation is called a quadratic functional equation
It is well known that a function f between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function B such that f (x) = B(x, x) for all x

Summary

Introduction

The general solutions of the quadratic functional equation of Pexider type, SOON-MO JUNG We will find out the general solutions of the functional equation (1.5) which is a “pexiderized” form of the quadratic functional equation (1.1).

Results

Conclusion