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

44 I. Kocsis are power functions, the general solution was given by L. Losonczi and Gy. Maksa in [8]. Theorem 1. Let n > 3 and m > 3 be fixed integers, a, ß £ R, a / 1 or ß / 1, Mi(p) = p

Q, M

2(p) — p

ß, p E [0, 1], 0

a = 0

0 = 0. The general solution of equation (1.1) is f(p) = «i(p) + C(p° - /), p £ [0,1] if a^ß f(p) = a

2(p)+p

Ql(p), pe[

0,1] if <x = ßt 1 where a\ and a

2 are additive functions, fli(l) = «2(1) = 0, / is a logarithmic function, and c E R. In the open domain case the general solution of (1.1) was given by B. R. Ebanks, R Kannappan, P. K. Sahoo, and W. Sander in [2]: Theorem 2. Let n > 3 and m > 3 be ßxed integers, :]0,1 [ —R be fixed multiplicative functions, M

1 or Mo is different from the identity function. The general solution of equation (1.1) is f(p)=a

1(p) + C(M

1(p)-M

2(p)), p E]0,1[ if M

x±M

2 f(p)=a

2(p) + M

1(p)l(p)-b, p E]0, 1[ if Mi = M

2 where a

x and a

2 are additive functions, ai(l) = 0, / is a logarithmic function, c E R, and b = 0.3(1) = 0, if Mi = M

2 i {0,1}, 6 _ 03(1

}f Mi = M2 = q nm b = + m - 1), if Ml = M

2 = 1. nm Applying the methods used in the proof of Theorem 1 in Losonczi-Maksa [8] with arbitrary multiplicative functions (which are not both identity functions) instead of power functions we have the following generalization of Theorem 1. Theorem 3. Let n > 3 and m >3 be ßxed integers, Mi, M

2 : [0,1] —> R be fíxed multiplicative functions, Mi or M

2 is different from the identity function. Then the general solution of equation (1.1) is f(p) = a

1(p)+C(M

1(p)-M

2(p)), p G [0,1] if Mi / M

2 f(p) = a

2(p)+Mi(p)l(p), P E [0,1] if Mi = M

2 where ai and a

2 are additive functions, a 1 (1) = 0.2(1) = 0, / is a logarithmic function and c £ R. For the problem of the stability of functional equations in Hyers-Ulam sense we refer to the survey paper of Hyers and Rassias [4]. By the stability problem