Orthogonal stability of the Cauchy functional equation on balls in normed linear spaces

Abstract

We study the stability of some functional equations postulated for orthogonal vectors in a ball centered at the origin. The maps considered are defined on a finite-dimensional normed linear space with Birkhoff-James orthogonality and take their values in a real sequentially complete linear topological space. The main results establish the stability of the corresponding conditional Cauchy functional equation on a half-ball and in uniformly convex spaces on a whole ball. The methods used in the first part of the paper are similar to those from [10]. Since, however, now in a general structure, some additional problems arise, we need several new tools.