Functional equation originating from sum of higher powers of arithmetic progression using difference operator is stable in Banach space: direct and fixed point methods

Abstract

In this paper, the authors has proved the solution of a new type of functional equation$$f\left(\sum_{j=1}^k j^p x_j\right)=\sum_{j=1}^k\left(j^p f\left(x_j\right)\right), \quad k, p \geq 1$$which is originating from sum of higher powers of an arithmetic progression. Its generalized Ulam - Hyers stability in Banach space using direct and fixed point methods are investigated. An application of this functional equation is also studied.