A Purely Fixed Point Approach to the Ulam-Hyers Stability and Hyperstability of a General Functional Equation

Abstract

In this paper, using a purely fixed point approach, we produce a new proof of the Ulam-Hyers stability and hyperstability of the general functional equation:

$$\displaystyle \sum _{i=1}^m A_i f(\sum _{j=1}^n a_{ij} x_j) + A = 0,\qquad (x_1, x_2, \dots , x_n) \in X^n, $$

considered in Bahyrycz and Olko (Aequationes Math 89:1461, 2015. https://doi.org/10.1007/s00010-014-0317-z), and in Bahyrycz and Olko (Aequationes Math 90:527, 2016. https://doi.org/10.1007/s00010-016-0418-y). Here m and n are positive integers, f is a mapping from a vector space X into a Banach space (Y, ∥ ∥), A ∈ Y  and, for every i ∈{1, 2, …, m} and j ∈{1, …, n}, A _i and a _ij are scalars.