Almost additive-quadratic-cubic mappings in modular spaces

Mohammad Maghsoudi,Abasalt Bodaghi,Abolfazl Niazi Motlagh,Majid Karami

doi:10.33044/revuma.v60n2a05

Abstract

We introduce and obtain the general solution of a class of generalized mixed additive, quadratic and cubic functional equations. We investigate the stability of such modified functional equations in the modular space Xρ by applying the ∆2-condition and the Fatou property (in some results) on the modular function ρ. Furthermore, a counterexample for the even case (quadratic mapping) is presented.

Highlights

In 1940, Ulam raised the first stability problem before a Mathematical Colloquium at the University of Wisconsin, where he presented a list of unsolved problems that were published years later in the book [38]
An answer to the problem was partially given by Hyers [12] for linear functional equations in the setting of Banach spaces
The stability results of additive functional equations in modular spaces equipped with the Fatou property and ∆2condition were investigated by Sadeghi [35], who used Khamsi’s fixed point theorem

Summary

Introduction

In 1940, Ulam raised the first stability problem before a Mathematical Colloquium at the University of Wisconsin, where he presented a list of unsolved problems that were published years later in the book [38]. Additive-quadratic-cubic functional equation; Modular spaces; Hyers– Ulam stability. They determined the general solution and proved the Hyers–Ulam stability problem for the functional equation (1.3) related to cubic Jordan

Results

Conclusion