Random stability of a functional equation associated with inner product: a fixed point approach

Abstract

Th.M. Rassias [Bull. Sci. Math. 108 (1984), 95{99] proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed positive integer l2l12l∑ 2li= 1xi2+∑2li=1xi ? 12l∑2lj=1xj2=∑ 2li=1?xi? 2holds for all x1; ; x2l ? V. For the above equality, we can define the following functionalequation 2lf(12l∑2li=1xi)+∑2li=1fxi ? 12l∑2lj=1xj =∑2li=1f(xi); (1) whose solution is realized as the sum of an additive mapping and a quadratic mapping. Using fixed point method, we prove the Hyers-Ulam stability of the functional equation (1) in random Banach spaces.