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

48 I. Kocsis The following theorem states that the functional equation (1.1) is stable on the open domain when n = m > 3 and Mi / Mo. Theorem 6. Let n — m > 3 be a fixed integer, 0 < £ £ H be fixed and Mi, Mo :]0,1[—

»• R be fixed multiplicative functions, Mi / Mo. If the function f :]0,1 [ —> R satisfies the inequality (1.2) for all (pi,...,p

m), • • • I <7m) € then there exists an additive function a, a bounded function B :]0, R, and C G R such that f(p) = a(p) + C(M

1(p) - Mo(p)) + B(p), p 6]0,1[. The proofs of Theorem 5 and Theorem 6 are based on the following arguments. In the closed and open domain case we use Lemma 1 and Lemma 2, respectively. Applying Lemma 1 or Lemma 2 for the function <p(p, Q

) = ]C(/(P9i) - M

1(p)f(q

j) - f(p)Mo(

q

j)) 3=1 with fixed Q — (qi, . .., <?

m) G A

m (1.2) implies that E(/(p?;

) - Afi(p) E ) - /(p) E )) (21) j=l j = l j = l = i4i(p

>Q) + 6i(p,Q) + Li(Q) holds for all p G /, where /ii : R x A

m —+ R is additive in its first variable and 6i : R x A

m —^ R is bounded. In the closed domain case Li(Q) = mf(

0) — /(°) H'j = 1 (íj) particulary. Let P = (pi,

... ,p

m) G A

m,p G J, write pp,- instead of p in (2.1), i = 1 . . .m. and add up the equations we obtained. Thus we get E £(/(PP*?j) - Mx(p) E Mi (pi

) J] /fo) (2.2) 1=1 i=

1 v ' m m m - E AppO E má<h)) = Mp, Q) + E f

ci(ppi. Q) + i = 1 j = 1 Write now P instead of Q in (2.1) to obtain m m m E /(ppo - E fw - f(p) E = ^i(P) p) + &I(P, + 1=1 2 — 1 i — 1