On extensions of the generalized quadratic functions from “large” subsets of semigroups

Abstract

Let \((G,+)\) be a commutative semigroup, \(\tau \) be an endomorphism of G and involution, D be a nonempty subset of G, and \((H,+)\) be an abelian group, uniquely divisible by 2. Motivated by the extension problem of J. Aczel and the stability problem of S.M. Ulam, we show that if the set D is “sufficiently large”, then each function \(g{:} D\rightarrow H\) such that \(g(x+y)+g(x+\tau (y))=2g(x)+2g(y)\) for \(x,y\in D\) with \(x+y,x+\tau (y)\in D\) can be extended to a unique solution \(f{:} G\rightarrow H\) of the functional equation \(f(x+y)+f(x+\tau (y))=2f(x)+2f(y)\).