BRZDĘK’S FIXED POINT METHOD FOR THE GENERALISED HYPERSTABILITY OF BI-JENSEN FUNCTIONAL EQUATION IN (2, β)-BANACH SPACES

Abstract

In this article, we introduce the notions of (2, β)-Banach spaces and we will reformulate the fixed point theorem [11, Theorem 1] in this space. Using this fixed point theorem, we prove the generalized hyperstability results of the bi-Jensen functional equation 4f((x + z)/2,(y + w)/2)= f(x, y) + f(x,w) + f(z, y) + f(y,w)
in (2, β)-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. The method we use here can be applied to various similar equations in many variables.