ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES

Abstract

Abstract. In this paper, we solve the additive ‰ -functional equations f ( x + y + z ) i f ( x ) i f ( y ) i f ( z )= ‰ ‡2 f ‡ x + y + z 2· i f ( x ) i f ( y ) i f ( z )·(0.1) ;where ‰ is a flxed number with ‰ 6= 1 ; 2, and f ( x + y + z ) i f ( x ) i f ( y ) i f ( z )= ‰ ‡2 f ‡ x + y 2+ z · i f ( x ) i f ( y ) i 2 f ( z )·(0.2) ;where ‰ is a flxed number with ‰ 6= 1.Using the direct method, we prove the Hyers-Ulam stability of the additive ‰ -functional equations (0.1) and (0.2) in Banach spaces. 1. Introduction The stability problem of functional equations originated from a question of Ulam[5] concerning the stability of group homomorphisms.The functional equation f ( x + y ) = f ( x )+ f ( y ) is called the Cauchy equation . Inparticular, every solution of the Cauchy equation is said to be an additive mapping .Hyers [3] gave a flrst a–rmative partial answer to the question of Ulam for Banachspaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and byRassias [4] for linear mappings by considering an unbounded Cauchy difierence. Ageneralization of the Rassias theorem was obtained by G‚avruta [2] by replacing theunbounded Cauchy difierence by a general control function in the spirit of Rassias’approach.In Section 2, we solve the additive functional equation (0.1) and prove the Hyers-Ulam stability of the additive functional equation (0.1) in Banach spaces.