ON EXTENSIONS OF THE GENERALISED JENSEN FUNCTIONS ON SEMIGROUPS

Abstract

Assume that $(G,+)$ is a commutative semigroup, $\unicode[STIX]{x1D70F}$ is an endomorphism of $G$ and an involution, $D$ is a nonempty subset of $G$ and $(H,+)$ is an abelian group uniquely divisible by two. We prove that if $D$ is ‘sufficiently large’, then each function $g:D\rightarrow H$ satisfying $g(x+y)+g(x+\unicode[STIX]{x1D70F}(y))=2g(x)$ for $x,y\in D$ with $x+y,x+\unicode[STIX]{x1D70F}(y)\in D$ can be extended to a unique solution $f:G\rightarrow H$ of the generalised Jensen functional equation $f(x+y)+f(x+\unicode[STIX]{x1D70F}(y))=2f(x)$.