New Results about Quadratic Functional Equation on Semigroups

Let S be a semigroup, let ( H, +) be a uniquely 2-divisible, abelian group and let φ, ψ be two endomorphisms of S that need not be involutive. In this paper, we express the solutions f : S → H of the following quadratic functional equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mi>φ</m:mi> <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:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ψ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>y</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mn>2</m:mn> <m:mi>f</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:mi>x</m:mi> <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> f( {xφ ( y )} ) + f( {ψ ( y )x} ) = 2f( x ) + 2f( y ), \;\;\;\;x,y ∈ S, in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. Some applications of this result are presented.