Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras

Abstract

The goal of the present paper is to investigate some new stability results by applying the alternative fixed point of generalized quadratic functional equation $$\begin{array}{ll}f\left(\sum\limits_{i=1}^{n}a_ix_i\right)+{\sum\limits_{i=1}^{n-1}}{\sum\limits_{j=i+1}^{n}}\left[f(a_ix_i+a_jx_j)+2f(a_ix_i-a_jx_j)\right]\\ \qquad \quad = (3n-2){\sum\limits_{i=1}^{n}}a^2_{i}f(x_{i})\end{array}$$ in β–Banach modules on Banach algebras, where $${a_{1},\dots,a_{n}\in \mathbb{Z}{\setminus}\{0\}}$$ and some $${\ell\in\{1 , 2 ,\dots, n-1\},}$$ a l ≠ ±1 and a n = 1, where n is a positive integer greater or at least equal to two.