On a new class of two-variable functional equations on semigroups with involutions

Abstract

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 goal of this paper is to find the general non-zero solution f : S 2 → K f\colon S^{2}\to K of the d’Alembert type equation f ⁢ ( x + y , z + w ) + f ⁢ ( x + σ ⁢ ( y ) , z + τ ⁢ ( w ) ) = 2 ⁢ f ⁢ ( x , z ) ⁢ f ⁢ ( y , w ) , x , y , z , w ∈ S , f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z)f(y,w),\quad x,y,z,w\in S, the general non-zero solution f : S 2 → G f\colon S^{2}\to G of the Jensen type equation f ⁢ ( x + y , z + w ) + f ⁢ ( x + σ ⁢ ( y ) , z + τ ⁢ ( w ) ) = 2 ⁢ f ⁢ ( x , z ) , x , y , z , w ∈ S , f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z),\quad x,y,z,w\in S, the general non-zero solution f : S 2 → H f\colon S^{2}\to H of the quadratic type equation f ⁢ ( x + y , z + w ) + f ⁢ ( x + σ ⁢ ( y ) , z + τ ⁢ ( w ) ) = 2 ⁢ f ⁢ ( x , z ) + 2 ⁢ f ⁢ ( y , w ) , x , y , z , w ∈ S , f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z)+2f(y,w),\quad x,y,z,w\in S, where σ , τ : S → S \sigma,\tau\colon S\to S are two involutions.