A family of functional equations related to the monomial functions and its stability

TL;DR

This paper investigates a family of complex functional equations involving additive, quadratic, cubic, and quartic components in vector and Banach spaces, establishing their solutions as monomial functions and analyzing their stability.

Contribution

It introduces a general family of mixed functional equations and proves that solutions are monomial functions, also exploring their Hyers-Ulam stability in Banach spaces.

Findings

Solutions are monomial functions of specific degrees.

The family of equations exhibits Hyers-Ulam stability.

Generalized stability results in Banach spaces.

Abstract

Our aim of this paper is to study a family of functional equation in vector and Banach spaces with difference operators, where this family of functional equation is a general mixed additive-quadratic-cubic-quartic functional equations. We show that every function satisfies the our functional equation is a monomial function with a certain degree. Furthermore, we deal with the generalized Hyers-Ulam stability of this family of functional equations in Banach space.