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

46 I. Kocsis The stability problem for equation (1.3) was solved by Gy. Maksa in [10] on closed domain and by I. Kocsis in [5] on open domain. Lemma 1. (Maksa [10]) Let n > 3 be a fixed, integer and 0 < £ £ R be fixed. If the function ip : [0, 1] —* R satisfies the inequality (1.5) 2 — 1 for all (pi

,..., pn) £ r„, then there exist an additive function A and a bounded function B : [0,1] — R such that 73(0) = 0, \B(p)\ < ISe, and V>(p)-<p(0)=A(p) + B(p), p £ [0,1]. Lemma 2. (Kocsis [5]) Let n > 3 be a fixed integer and 0 < e £ R be fixed. If the function ip :]0,1[—

> R satisfies (1.5) for all (pi

,. .., p

n) £ , then there exist an additive function A and a bounded function B : [0, 1] —» R such that \B(p)\ < 220s, and <p(p) = A(p)-^ + B(p), p £ ]0,1[

. In what follows the following two lemmata will also be needed. Lemma 3. Let A is an additive function, M : / —• R is a multiplicative function B : I —+ R is a bounded function, and c £ R. If A(x) = M{x) + c for all x £ I then A(x) = dx, x £ R for some d £ R and M(x) = 0 or M{x

) - x, x £ I. If A{x) = M(x) + B(x) for all x £ I then A{x) = dx, x £ R for some d £ R and M(x) = 0 or M(x) = x

a, x £ I for some 0 < a £ R.