Quadratic--quartic functional equations in RN--spaces

TL;DR

This paper investigates the general solutions and stability of a specific quadratic-quartic functional equation within random normed spaces, expanding understanding of such equations under various t-norms.

Contribution

It provides the first comprehensive solution and stability analysis for this functional equation in the context of random normed spaces with arbitrary t-norms.

Findings

Derived the general solution for the functional equation.

Established stability results under arbitrary t-norms.

Extended the theory of functional equations in random normed spaces.

Abstract

In this paper, we obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary $t$-norms $$f(2x+y)+f(2x-y)=4[f(x+y)+f(x-y)]+2[f(2x)-4f(x)]-6f(y).$$