Hyperstability of some functional equation on restricted domain‎: direct and fixed point methods

The study of stability problems of functional equations was motivated by a question of S.M. Ulam asked in 1940. The first result giving answer to this question is due to D.H. Hyers. Subsequently, his result was extended and generalized in several ways. We prove some hyperstability results for the equation
g(ax)(cx)=Ag(x)(y)
on restricted domain. Namely, we show, under some weak natural assumptions, that functions satisfying the above equation approximately (in some sense) must be actually solutions to it.