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 51 where p £ [0,1], P G r m. Equations (2.8) and (2.9) imply that A 1(p tP) = a(p) (l-f]M 1(p i)) < 2i o> ( v ;r 7 . v + P Í [/( 0 ) - /(1)] E Mi (pi ) - m/( 0) - bi(l, P)j After some calculations we have that m Hp, P) - p[f( 1) + (777. - l)/(0) + 6i(l, P)]) £ Mi( q j) 3= 1 m (2.11) = (bi(p, Q) - p[f( 1) + (m - l)/(0) + 6i(l, Q)}) £ Mi( P i) i-i m m + X] (P9j > P) - E MM, Q) + PM*. Q) - P% j= 1 j=i Since the right hand side of (2.11) is bounded in Q , while Yl'j'=i is not, we have (2.12) 6i(p,P)=p[6i(l,P) + /(l) + (m-l)/(0)] > pG[0,l], P G T m. By (2.11), it follows from (2.1) that m (h{pqj) - Mi(p)h( q j)~ h(p)M 1(q j) + h(Q)Mi( q j) j= 1 - p[h( 0) - A(l)]Afi( ff i) - /7(0) - [/7(1) - h(0)] Pq j) = 0, where h(p) - f(p) — a(p), p G [0,1]. Applying Lemma 1 we get that 1 3 Ä(P9) - Mi(p)h(q) - Mi(q)h(p) + h(0)Mi(q) - p[h(0) - h(l)]Mi(q) - h ( 0 ) - pq[h( 1) - MO)] + M!(p)Ä(0) = A 2(p, q) p,q G [0,1], where Ao : [0,1] x [0,1] —R is additive in its second variable. Define the function II on [0,1] by H(p) = h(p) — /?.(0). Thus (2.13) can be written in the form (2.14) II(pq) - Mi(p)H(q) - Mi(q)h(p) + H(\)pM{q) = A 2(p, q). A calculation shows that the function G : [0, l] 2 —» R defined by (2.15) G{p, q) = H(p, q) - M 1{p)h{q) - Mi(q)H(p)
