Alienation of Drygas’ and Cauchy’s Functional Equations

Youssef Aissi,Brahim Fadli,Driss Zeglami

doi:10.2478/amsil-2021-0002

Abstract

Abstract Inspired by the papers [2, 10] we will study, on 2-divisible groups that need not be abelian, the alienation problem between Drygas’ and the exponential Cauchy functional equations, which is expressed by the equation f ( x + y ) + g ( x + y ) g ( x - y ) = f ( x ) f ( y ) + 2 g ( x ) + g ( y ) + g ( - y ) . f\left( {x + y} \right) + g\left( {x + y} \right)g\left( {x - y} \right) = f\left( x \right)f\left( y \right) + 2g\left( x \right) + g\left( y \right) + g\left( { - y} \right). We also consider an analogous problem for Drygas’ and the additive Cauchy functional equations as well as for Drygas’ and the logarithmic Cauchy functional equations. Interesting consequences of these results are presented.

Highlights

The alienation and strong alienation problems are introduced by Dhombres ([3]), who gave the following definitions
For more details concerning the alienation phenomenon in the theory of functional equations we refer to the survey article [12], which was authored by Ger and Sablik
The aim of the present paper is to study the alienation phenomenon between Drygas’ and Cauchy’s functional equations

Summary

Introduction

The alienation and strong alienation problems are introduced by Dhombres ([3]), who gave the following definitions. For more details concerning the alienation phenomenon in the theory of functional equations we refer to the survey article [12], which was authored by Ger and Sablik. The aim of the present paper is to study the alienation phenomenon between Drygas’ and Cauchy’s functional equations. We will examine, on 2-divisible abelian groups, the alienation phenomenon between the equation (1.2) and the Jensen-additive functional equation, that is x+y. We will study the alienation problem, on a ring, of Drygas’ and the logarithmic Cauchy functional equations, which is expressed as follows (1.8) f (xy)+g(x+y)+g(x−y) = f (x)+f (y)+2g(x)+g(y)+g(−y), x, y ∈ X.

Drygas’ and the exponential Cauchy functional equations

Drygas’ and the additive Cauchy functional equations

Drygas’ and the logarithmic Cauchy functional equations