A generalization of d’alembert, jensen and quadratic functional equations on semigroups

Abstract EN

In this paper we shall find the general solution g : S1× S2 ×…× Sn→ F of a generalization of d’Alembert equation g (x1 + y1, x2 + y2, …, xn + yn ) +g( x1 y1, x2 + σ2 y2, …, xn + σn yn ) = 2g(x1, x2, …, xn) g(y1, y2,…, yn) ( (x1 , x2,…, xn) (y1, y2 ,…, yn) I S1 × S2 ×…× Sn ) the general solution h : S1× S2 ×…× Sn ×T1× T2 ×…× Tm → G of generalizations of Jensen equation h(x1 + y1, x2 + y2,…, xn + yn, t1, t2,.., tm )+ h(x1 + σ1 y1, x2 + σ2 y2,…, xn + σn yn , t1, t2, .., tm) = 2 h(x1, x2,…, xn, t1, t2,.., tm) (x1, x2,…, xn, t1, t2,..., tm), (y1 ,y2,…,yn, t1, t2,.., tm) I S1× S2 ×…× Sn × T1× T2 ×…× Tm) ) and h( (x1 + y1, t1), ( x2 + y2, t2),…,( xn + yn, tn ) ) + h( (x1 + σ1y1, t1), ( x2 + σ2y2 , t2,) xn + σnyn, tn ) ) = 2 h( (x1 , t1), ( x2 , t2), …, xn, tn ) (x1 , x2,…, xn), (y1 y2,…, yn,), (t1 ,t2,…, tn) I S1× S2 ×…× Sn, and the general solution k : S1 × S2 ×…× Sn→ F of a generalization of the quadratic equation k( x1 + y1, x2 + y2 , …, xn + yn ) + k( x1 + σ1 y1, x2 + σ2 y2, …, xn + n yn ) 2= k(x1, x2, …, xn)+ 2k(y1, y2,…, yn) ( (x1, x2,…, xn), (y1, y2, …, yn) I S1× S2 ×…× Sn), where S1,S2,…,Sn,T1, T2,…, and Tm are commutative semi groups ,F is a quadratic ally closed field of characteristic different from 2, G is a 2-cancelative Aeolian group, H is an Aeolian group uniquely divisible by 2 and for each i = 1, 2,…n, σi is an endomorphism of Si with σi (σi xi) = xi.