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

50 I. Kocsis The left hand side of (2.3) is aditive in p, while the right hand side can be written in the form Mi(p)Fi(P, Q) + F

2{p, P, Q) + F

3(P, Q), where F

2 is bounded. Applying Lemma 3 with fixed P,Q £ A

r

n we get that Ai(p,P) (2.4) j=

1 X^AfaCp,-)- 1 = P Ai(l,P) E^i)1 i=i i=i futhermore Mi is a bounded multiplicative function or m m m m (2.5) E - E fM = E - E i = l j=l j = 1 i=l If (2.5) holds then, by (1.4), and by Lemma 4 we have that Mi = Mo or (2.6) M i (p) = p

a, M

2(p) = p £ I, 0 < a £R, 0 < ß £ R Proof of Theorem 5. In the case M\ / M

2, by (2.6), we can apply Theorem 4. The case Mi = M

2. If the functions Mi and M

2 are power functions we can apply Theorem 4 again. Suppose now that Mi is not a power function. Fix Q = (f/i,

. . ., q

m) £ r

m for which i ^ 1 (exists such a Q) and let (2.7) A

1(x,Q)-xA

1(l,Q) a{x) = — 7~7~, 7—, X £ R. l-EjLiMifc) ' Then a is additive and a(l) = 0. From (2.4) we get that m (2.8) Ai(p, P) = pA.il, P) + a(p)( 1 - E ^i(pO)» i=i while from (2.1), with p — 1 and P = Q, it follows that m (2.9) Ai(l

f P) = [/(0) - /(!)] E

MiM - m/(°) - *i(l> p)>