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 47 Proof. If — M(x) + c for all x G I then, because of A(x) = M{yfx)

2 + c > c, A is bounded below on I. Thus A(x) = dx, x G R for some d G R. (See Aczél [1].) Therefore M[x) = dx — c, £ G /. Since M is multiplicative we have that c = 0 and de {o,i}. If A(x) = M(x) + B(x) for all x G / we have similarly that A(x) = dx, x G R for some d G R and M(x) = dx — B(x

), a,

1 G /. Thus M is bounded on I that is M(x) = 0 or M(x) = x

a,x G /, for some 0 < a G R. Lemma 4. Let Mi, Mo : / — R, be fíxeci multiplicative functions, Mi ^ M2, A be an additive function and c G R. If M\(x) — Mo{x) — A(x) + c iiolds for all x G I then Mi and Mo are zero or identity functions of 1. Proof. Let ci G /, Mi (a) / M2(a). Then from the equations (1.6) Mi (a?) - Mo(x) = A{x) + c and we get that Mi(a)Mi(z) - Mo{a)Mo(x) = A{ax) + c lf

M 1 0 , Mi (a) , . c(l — Mi (a)) M

2(X) = ——A(ax) - ——— A(X-) + v v " Mi (a) — Mo ( a) v 7 Mi(a) - M

2(a) v y Mi (a) - M

2(a)

' that is, there exist an additive function A* and a constant c* G R such that Mo(x) = + c* for all x G I. Thus, by Lemma 3, Mo is zero or identity function of I. Furthermore, by (1.6), we have the same for Mo. 2. The main results We present two generalizations of Theorem 4. The following theorem says that the functional equation (1.1) is stable on the closed domain. Theorem 5. Let n > 3 and m >3 be fixed integers, 0 < e G R be fixed and Mi, Mo : [0,1] —R be fixed multiplicative functions, Mi or Mo is different from the identity function. If the function f : [0,1] — R satisfies the inequality (1.2) for all (pi,

. . . ,p

n

) G r„ and (q 1,..., q

m

) G T

m then there exists an additive function ci, a logarithmic function I : [0,1] R , a bounded function B : [0,1] —R, and C G R such that f(p) = a(p) + C(M

1(p)-M

2(p))+B(p), p G [0,1] if Mi^Mo, f(p) = ao(p) + Mi(p)l(p) + B(p), p G [0,1] if Mi = M

2.