Ulam stability of functional equations in 2-Banach spaces via the fixed point method
type: Item
instance of: scholarly article
title: Ulam stability of functional equations in 2-Banach spaces via the fixed point method
zbMATH Open document ID: 1473.39047
author: Krzysztof Ciepliński
published in: Journal of Fixed Point Theory and Applications
publication date: 2021-08-12T00:00:00Z
review text: Let \(X\) be a linear space and \(Y\) be a \(2\)-Banach space over the field \(\mathbb{F}=\mathbb{R}\) or \(\mathbb{C}\). Let \(a_{11}, a_{12},\dots, a_{n1}, a_{n2}\in \mathbb{F}, a_{1,j_1,\dots,j_n}, \dots, a_{n,j_1,\dots,j_n}\in\mathbb{F}\) for \(j_1,\dots, j_n\in\{-1, 1\}\) and \(A_{i_1\dots,i_n}\in \mathbb{F}\) for \(i_1,\dots, i_n\in \{1, 2\}\) be given scalars. Set \(A := \sum_ {i_1,\dots, i_n\in\{1,2\}} A_{i_1,\dots,i_n}\) and assume that \(|A|>1\). The authors prove the Ulam stability of the following functional equations \[f(a_{11}x_{11} + a_{12}x_{12}, \dots, a_{n1}x_{n1} + a_{n2}x_{n2}) =\sum_ {i_1,\dots, i_n\in\{1,2\}} A_{i_1,\dots,i_n}f(x_{1i_1},\dots, x_{ni_n}) \] and \[ \begin{multlined} \sum_ {j_1,\dots, j_n\in\{-1,1\}}f( a_{1, j_1},\dots,j_n(x_{11}+j_1x_{12}),\dots, a_{n, j_1},\dots, j_n(x_{n1}+j_nx_{n2}))\\ =\sum_ {i_1,\dots, i_n\in\{1,2\}} A_{i_1,\dots,i_n}f(x_{1i_1},\dots, x_{ni_n}) \end{multlined} \] by using the fixed point method.
reviewed by: Maryam Amyari
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 39B72
zbMATH DE Number: 7381141
zbMATH Keywords: Ulam stability
zbMATH Keywords: fixed point method
zbMATH Keywords: Banach space
zbMATH Keywords: Cauchy equation
zbMATH Keywords: Jordan-von Neumann equation
zbMATH Keywords: linear functional equation
MaRDI profile type: Publication
full work available at URL: s11784-021-00869-x
OpenAlex ID: W3163227330
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DOI: 10.1007/S11784-021-00869-X
description: scientific article; zbMATH DE number 7381141