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

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 43-53 ON THE STABILITY OF A SUM FORM FUNCTIONAL EQUATION OF MULTIPLICATIVE TYPE Imre Kocsis (Debrecen, Hungary) Abstract. The stability of a so-called sum form functional equation arising in information theory is proved under certain conditions. 1. Introduction A function a is additive, a function M : [0,1] —• R is multilpicative, and a function / : [0,1] —+ R is logarithmic if a(x + y) = a(x) + a(y) for all x,y £ R, M(xy) = M(x)M(y) for all x, y <E]0,1[, M(

0) = 0, Af( 1) = 1, and l(xy) = l{x)+l{y) for all x,y G]0, 1], /(0) = Ü, respectively. We define the following sets of complete probability distributions n rn = {(Pi,...,Pn)e[0,i]

n:5> = i} 2 — 1 and n = {(Pi,---,Pn) €]0,l[

n: J^Pi = 1=1 Through the paper I and A

N shall denote [0,1] or ]0,1[ and T

n or respectively. Let n > 3 and m > 3 be fixed integers, Mi, Mo : I —• R be fixed multiplicative functions and / : / —• R be an unknown function. The functional equation n m n m n m (

10 EE /(?»• ) = E M

i fa

) E /(«;) + E ) E ) i — 1 j = l i — 1 ;=1 1 = 1 j — 1 which holds for all (pi, . ..,p

n) £ A

n and (qi,

. . ., q

m) G A

m plays important role in the characterization of information measures. The general solution of (1.1) is known when M\ or Mo is different from the identity function. The M\(x) — Mo{x

) = x

, x £ I case will be excluded from our investigations, too. In the closed domain case, when the multiplicative functions