Approximately Orthogonal Additive Set-valued Mappings

Abstract

Abstract. We investigate the stability of orthogonally additive set-valued functionalequation F ( x + y ) = F ( x ) + F ( y ) ( x ? y )in Hausdorff topology on closed convex subsets of a Banach space. 1. IntroductionA functional equation F is called stable if for any function f satisfying approx-imately to the equation F, there is a true solution of F near to f . In 1940, S. M.Ulam [24] proposed the first stability problem for group homomorphisms. Hyers [9]gave the first significant partial solution to his problem for linear functions. Th. M.Rassias [20] improved Hyers’ theorem by weakening the condition for the Cauchydifference controlled by jjxjj p + jjyjj p , p 2 [0 ; 1). For some recent developments inthis area, we refer the reader to the articles [5, 6, 11, 12, 15, 19] and the referencestherein.In 1985, R¨atz[21] gave a generalization of Birkhoff-James orthogonality [1, 10]in vector spaces. He also investigated some properties of orthogonally additivefunctional equation. This definition motivated some Mathematicians to discussabout the orthogonal stability of functional equations (see e. g. [8, 13, 16, 22]). Onthe other hand, set-valued mappings and their stability have been investigated bysome authors from different point of view [2, 7, 14, 17, 23].In the next section, we prove the stability of set-valued orthogonal additivefunctional equation(1)