On The Solutions of The Equation f(xf(y)k+yf(x)‘)=f(x)f(y)

Abstract

The functional equation (1) f(xf(y)k + yf(x)1) « f(x)f(y), where k and 1 are positive integers and the unknown function f maps 1R. into itself, has appeared in connection with determining some subsemigroups of the group Lg (of. [2]). Putting k » 0 and 1 = 1 we get the Gołąb-Sohinzel equation as a particular case of (1) which has been studied by many authors including N. Brillouet who in [l] has also dealt with continuous solutions of equation f(xf(y) +yf(x)) «cxf(x)f(y). Our results presented here generalize those from [4j and [1] (in the oase ot= 1). They are also more general than it was announoed by M. Sablik at the 21st Symposium on Functional Bquations (of. [33) (Fragment tekstu).