On Hilbert Golab-Schinzel type functional equation

Abstract

Let $X$ be a vector space over a field $K$ of real or complex numbers. We will prove the superstability of the following Go{l}c{a}b-Schinzel type equation<br />$$<br />f(x+g(x)y)=f(x)f(y), x,yin X,<br />$$<br />where $f,g:Xrightarrow K$ are unknown functions (satisfying some assumptions). Then we generalize the superstability result for this equation with values in the field of complex numbers to the case of an arbitrary Hilbert space with the Hadamard product. Our result refers to papers by Chudziak and Tabor [J. Math. Anal. Appl. 302 (2005) 196-200], Jabl o'{n}ska [Bull. Aust. Math. Soc. 87 (2013), 10-17] and Rezaei [Math. Ineq. Appl., 17 (2014), 249-258].