Abstract
Let M = { 1, 2, . . . , n } and let \({\mathcal {V}=\{\,I \subseteq M: 1 \in I\,\}}\) , where n is an integer greater than 1. Denote \({M{\setminus}{I}}\) by I c for \({I \in \mathcal {V}.}\) We investigate the solution of the following generalized quartic functional equation
in β-Banach modules on a Banach algebra, where \({a_{1},\ldots, a_{n}\in \mathbb{Z}{\setminus}\{0\}}\) with a ℓ ≠ ±1 for all \({\ell \in \{1 , 2, \ldots ,\,n-1\}}\) and a n = 1. Moreover, using the fixed point method, we prove the generalized Hyers–Ulam stability of the above generalized quartic functional equation. Finally, we give an example that the generalized Hyers–Ulam stability does not work.
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Gordji, M.E., Khodaei, H. & Najati, A. Fixed Points and Quartic Functional Equations in β-Banach Modules. Results. Math. 62, 137–155 (2012). https://doi.org/10.1007/s00025-011-0135-8
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DOI: https://doi.org/10.1007/s00025-011-0135-8