Fixed Points and Stability of the Cauchy Functional Equation in C*-Algebras

The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki 3 for additive mappings and by Th. M. Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias 4 has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa 5 by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias’ approach. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem see 6–19 . J. M. Rassias 20, 21 following the spirit of the innovative approach of Th. M. Rassias 4 for the unbounded Cauchy difference proved a similar stability theorem in which he replaced the factor ‖x‖p ‖y‖p by ‖x‖p · ‖y‖q for p, q ∈ R with p q / 1 see also 22 for a number of other new results . We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → 0,∞ is called a generalized metric on X if d satisfies