A Variant of D’alembert’s Matrix Functional Equation

The aim of this paper is to characterize the solutions Φ : G → M₂(ℂ) of the following matrix functional equations Φ(xy)+Φ(σ(y)x)2=Φ(x)Φ(y), x,y,∈G,{{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, and Φ(xy)−Φ(σ(y)x)2=Φ(x)Φ(y), x,y,∈G,{{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, where G is a group that need not be abelian, and σ : G → G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.