A new type of functional equations on semigroups with involutions

TL;DR

This paper characterizes the general solutions of various functional equations of d'Alembert, Jensen, and quadratic types on semigroups with involutions, extending classical equations to more general algebraic structures.

Contribution

It provides explicit solutions for these functional equations on semigroups with involutions, generalizing known results to broader algebraic contexts.

Findings

Solutions for d'Alembert type equations on semigroups with involutions.

Solutions for Jensen type equations on semigroups with involutions.

Solutions for quadratic type equations on semigroups with involutions.

Abstract

Let $S$ be a commutative semigroup, $K$ a quadratically closed commutative field of characteristic different from $2$, $G$ a $2$-cancellative abelian group and $H$ an abelian group uniquely divisible by $2$. The aim of this paper is to determine the general solution $f:S^2\to K$ of the d'Alembert type equation: $$ f(x+y,z+w)+f(x+\sigma(y),z+\tau(w)) =2f(x,z)f(y,w),\quad\quad (x,y,z,w\in S) $$ the general solution $f:S^2\to G$ of the Jensen type equation: $$ f(x+y,z+w)+f(x+\sigma(y),z+\tau(w)) =2f(x,z),\quad\quad (x,y,z,w\in S) $$ the general solution $f:S^2\to H$ of the quadratic type equation quation: $$ f(x+y,z+w)+f(x+\sigma(y),z+\tau(w)) =2f(x,z)+2f(y,w),\quad\quad (x,y,z,w\in S) $$ where $\sigma,\tau: S\to S$ are two involutions.