Functional Equations with an Anti-Endomorphism for Functions with Multidimensional Codomains

Abstract Let S be a semigroup, ℍ be the skew field of quaternions, and ψ : S → S be an anti-endomorphism. We determine the general solution of the functional equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>-</m:mo> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>ψ</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>y</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>y</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mo> </m:mo> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> </m:mrow> </m:math> g\left( {xy} \right) - g\left( {x\psi \left( y \right)} \right) = 2g\left( x \right)g\left( y \right),\,\,\,\,\,\,\,\,\,\,x,\,y \in S, where g : S →ℂ is the unknown function. And when S = M is a monoid, we solve the functional equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>ψ</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>y</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>y</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mo> </m:mo> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> </m:mrow> </m:math> g\left( {xy} \right) + g\left( {x\psi \left( y \right)} \right) = 2g\left( x \right)g\left( y \right),\,\,\,\,\,\,\,\,x,\,y \in M, where g : M → ℍ is the unknown function.