Stability of Bi-Additive Mappings and Bi-Jensen Mappings

Jae-Hyeong Bae,Won-Gil Park

doi:10.3390/sym13071180

Abstract

Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation f(x+y,z+w)=f(x,z)+f(y,w) and the bi-Jensen functional equation 4fx+y2,z+w2=f(x,z)+f(x,w)+f(y,z)+f(y,w).

Highlights

In Cauchy’s equation f ( x + y) = f ( x ) + f (y) we can deal with a class of approximate solutions defined by the functional inequality introduced by Rassias
Jung, and Lee [9] obtained the stability on a bi-Jensen functional equation in Banach spaces
We investigate the generalized Hyers-Ulam stability of (1) in Banach spaces and 2-Banach spaces

Summary

Introduction

In Cauchy’s equation f ( x + y) = f ( x ) + f (y) we can deal with a class of approximate solutions defined by the functional inequality introduced by Rassias. X +y y) = h( x ) + h(y) respectively, the Jensen functional equation 2h 2 = h( x ) + h(y). Jung [4] got the result of the Jensen equation. It was generalized as a functional case by P. Jung, and Lee [9] obtained the stability on a bi-Jensen functional equation in Banach spaces. The authors [10] proved the stability on a Cauchy-Jensen functional equation Banach spaces.

Solution and Stability of a Bi-Additive Functional Equation

Solution and Stability of a Bi-Jensen Functional Equation