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

52 I. Kocsis satisfies the equation (2.16) G(pq,r) + M(r)G(p,q) = G(p,qr) + M(p)G(q,r), p,q,r e [0,1]. From (2.14) and (2.15) we have that G{p,q) = A

2(p,q) - H(l)M(q). With (2.16) this implies that A

2(p,qr) - A

2(pq,r) + M

1{p)A

2(q,r) = Mi(r)[A

2(p, q) ~ H(l)(pq - Mi(p)g)]. The left hand side is additive in the variable r and the multiplicative function M\ is not the identity function so A

2(p,q) = H(l)(p — Mi(p))q thus (2.14) goes over into (2.17) H{pq) - H(l)pq = M

1(p)(H(q) - H(l)q) + M

x(q){H{p) - H(l)p), where p, q £ [0,1]. Let I : [0,1] — R, /(0) = 0 and II{p)-H(\)p /(p

) ~ M

liv

) ' Then (2.17) shows that / is a logarithmic function and for all p £ [0,1] we have f(p) = a(p) + h(p) = a(p) + H(p) + /i(0) = aip) + M(p)/(p) + H(l)p + h{0). With Bip) = //(l)p-f /i(0),p £ [0,1] we obtain the statement of the theorem. Proof of Theorem 6. Here n = m,Mi / M

2 and, by (2.6), M

1 and M

2 ai'e power functions, that is, Mi(p) = p

a

, M

2(p) = p

1

3

, p £]0,1[ for some 0 < a £ R, 0 < ß £ R. Interchanging P and Q in (1.2) and applying the triangle inequality we have (2.18) j — 1 i — 1 i — 1 j = 1 < 2s. By Lemma 2 we get f(p) = Mp) + cip° + cop? + bip), p £]0, 1[, where A is an additive function, b :]0,1[—

> R is a bounded function, and ci,c

2 £ R. With the definitions o(p) = A(p)-pA{ 1), p£R Bip) = bip) + pA{l) + (ci + c

2)p

a

, p £]0,1[ and C = -c

2 our theorem is proved. Remark. It is clear from the paper that some open problem remains connected with the stability of equation (1.1). For example the case Mi = M

2 or M\ ^ M

2 and n / m. The stability problem is essentially solved in the open domain case.