A NOTE ON STABILITY OF THE POPOVICIU FUNCTIONAL EQUATION ON RESTRICTED DOMAIN

Abstract

We prove the Hyers-Ulam stability, on restricted domain, of a functional equation of Jensen type, introduced by T. Popoviciu (1965). The conviction that the well known problem of stability of functional equations (at the moment named the Hyers-Ulam stability) was started by a paper of D.H. Hyers [13], solving a problem of S. Ulam, appears to be commonly accepted, though, the first known result of this kind is due to Gy. Polya and G. Szego [25, Teil I, Aufgabe 99] (cf. [9, p. 125]). Generalizations and extensions of that problem has been proposed in [2, 3, 4, 12, 14, 17, 18, 19]; next some of them have been rediscovered by Th.M. Rassias [26, 27] (see also [10]), who has thus given a strong impulse for the further development of this field of research. For more details, discussions and surveys on the results obtained so far we refer to [1, 6, 7, 11, 15, 20, 21, 22]. Usually the problem has been considered for functions with values in Banach spaces. However, originally it had been stated for functions with values in metric spaces. For instance, in [30] one can find the following problem: Given a group G1, a metric group G2 with metric d and a positive number e, find a positive number δ such that, for every f : G1 → G2 satisfying: d(f(xy), f(x)f(y)) ≤ δ for all x, y ∈ G1, there exists a homomorphism Mathematics Subject Classification (2000): 39B82.