Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
Kocsis, I., On the stability of a sum form functional equation of multiplicative type
On the stability of a sum form functional equation of multiplicative type 45 for equation (1.1) we mean the following: Let n > 3 and m > 3 be fixed integers, Mi, M2 : I —> R be fixed multiplicative functions, and 0 < £ G R be fixed. Prove or disprove that the functions f : I R satisfying the funtional inequality (1.2) E E v ) - E Mi (pi ] E ^ ) - E ) E u> («*) i=1 j — 1 i=1 j — 1 Z = 1 J = 1 < £ for all (pi , . . ., p n) G and (c/i , . . . , G A m are the sum of a solution of (1.1) and a bounded function. The stability of equation (1.1) on closed domain, when the multiplicative functions are power functions was proved in Kocsis-Maksa [6]. Theorem 4. Let n > 3 and m > 3 he fixed integers, e, a, ß G R, £ > 0, a ^ 1 or ß ^ 1. If the function f : [0,1] —+ R satisfies the inequality (1.2) for all (pi , . . ., p n) G r n and (q 1, . . ., f/ m) G r m then there exists an additive function a, a logarithmic function I : [0. 1] — R . a bounded function B : [0,1] —R. and C G R such that a(l) = U and f(p) =a(p) + C(p a-q ß) + B(p), p G [0, 1] if cxfß, f(p) = a(p)+p al(p) + B(p), pG [0,1] if <* = /?# 1. In this paper we deal with the stability of (1.1) 011 closed domain and on open domain when the functions M 1 and M2 are arbitrary multiplicative functions (Mi or M2 is different from the identity function). We notice that the condition n = m or (n / rn) is essential in the open domain case when zero probabilities are excluded, while it is not essential in the closed domain case. The basic tool for the proof of the stability theorems is the stability of the sum form functional eqation n (1.3) E^') = 0, i = 1 where n > 3 is a fixed integer, <p : I R is an unknown function and (1.3) holds for all (p 1, ... ,p n) G A n. The general solution of equation (1.3) in the closed domain case was given by L. Losonczi and Gy. Maksa in [8] and in the open domain case by L. Losonczi in [7]. In both cases the general solution of (1.3) is (1.4) v>(p) = a(p)-— , pel, where a is an additive function.
