A fixed point approach to almost double derivations and Lie \(*\)-double derivations
type: Item
instance of: scholarly article
title: A fixed point approach to almost double derivations and Lie \(*\)-double derivations
zbMATH Open document ID: 1266.39034
author: Ali Ebadian
author: N. Ghobadipour
published in: Results in Mathematics
publication date: 2013-02-08T00:00:00Z
review text: The stability problem of functional equations originated from a question of Ulam in 1940, concerning the stability of group homomorphisms. In 1941, D. H. Hyers gave a first affirmative answer to the question for Banach spaces. Let \(A\) be a subalgebra of an algebra \(B\), \(X\) be a \(B\)-module and \(\sigma:A\to B\) be a linear mapping. A linear mapping \(f :A\to B\) is called \(\sigma\)-derivation if \(f(ab)=f(a)\sigma(b)+\sigma(a)f(b)\) for all \(a,b\in A\). Let \(A\) be an algebra and \(\delta,\varepsilon:A\to A\) be linear mappings. \textit{M. Mirzavaziri} and \textit{E. O. Tehrani} [Bull. Iran. Math. Soc. 35, No. 1, 147--154 (2009; Zbl 1189.46058)] said that a linear mapping \(f:A\to A\) is a \((g,h)\)-double derivation if it satisfies \(f(ab)=f(a)b+g(a)h(b)+h(a)g(b)\). In the present paper, the authors by using the fixed point method prove the Hyers-Ulam stability of double derivations associated with the following mapping: \[ \begin{multlined} \sum_{k=2}^n\left(\sum_{i_1=2}^k\sum_{i_2=i_1+1}^{k+1}\dots \sum_{i_{n-k+1}=i_{n-k}+1}^n\right)\\ \times f\left(\sum_{i=1,i\neq i_1,\dots,i_{n-k+1}}^n x_i -\sum_{r=1}^{n-k+1}x_{i_r}\right)+f\left(\sum_{i=1}^n x_i\right)=2^{n-1}f(x_1)\end{multlined} \] for a fixed positive integer \(n\) with \(n\geq 2\). Also they investigate this problem in a Lie \(C^*\)-algebra.
reviewed by: Nasrin Eghbali
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46L57
zbMATH DE Number: 6135255
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: double derivation
zbMATH Keywords: \(*\)-double derivation
zbMATH Keywords: Lie \(C^*\)-algebra
MaRDI profile type: Publication
full work available at URL: s00025-011-0205-y
OpenAlex ID: W1967631961
cites work: On the stability of the linear transformation in Banach spaces
cites work: Classes of transformations and bordering transformations
cites work: A fixed point theorem of the alternative, for contractions on a generalized complete metric space
cites work: A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C∗-ternary algebras
cites work: Q4910098
cites work: Functional inequalities associated with Cauchy additive functional equation in non-Archimedean spaces
cites work: Stability of generalized derivations on Hilbert \(\mathrm{C}^\ast\)-modules associated with a Pexiderized Cauchy-Jensen type functional equation
cites work: On approximate additive-quartic and quadratic-cubic functional equations in two variables on Abelian groups
cites work: Modal operators on commutative residuated lattices
cites work: Generalized Hyers-Ulam stability of generalized \((N,K)\)-derivations
cites work: GENERALIZED ULAM-HYERS STABILITIES OF QUARTIC DERIVATIONS ON BANACH ALGEBRAS
cites work: On stability of additive mappings
cites work: Q3781518
cites work: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings
cites work: Stability of Isometries
cites work: A perturbation of double derivations on Banach algebras
cites work: On the Stability of the Linear Functional Equation
cites work: Stability of \(\Psi\)-additive mappings: Applications to nonlinear analysis
cites work: Q5397116
cites work: \(\delta\)-double derivations on \(C^*\)-algebras
cites work: Automatic continuity of $\sigma $-derivations on $C^*$-algebras
cites work: Generalized derivations on Lie triple systems
cites work: Hyers-Ulam stability of an \(n\)-Apollonius type quadratic mapping
cites work: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation
cites work: Homomorphisms between Poisson JC*-algebras
cites work: Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach
cites work: Q4470094
cites work: Q4950687
cites work: Q3721188
cites work: On the Stability of the Linear Mapping in Banach Spaces
cites work: On the stability of functional equations in Banach spaces
cites work: On the Behavior of Mappings which do not Satisfy Hyers-Ulam Stability
cites work: Q3259049
DOI: 10.1007/S00025-011-0205-Y
Recommended article: Functions near some \((\alpha _1,\alpha _2)\)-double Jordan derivations in \(p\)-Banach algebras
Recommended article: Stability of homomorphisms and generalized derivations on Banach algebras
Recommended article: Q5405519
Recommended article: A perturbation of double derivations on Banach algebras
Recommended article: STABILITY OF THE JENSEN TYPE FUNCTIONAL EQUATION IN BANACH ALGEBRAS: A FIXED POINT APPROACH
Recommended article: Q5500082
Recommended article: Q3184649
Recommended article: Q5397127
Recommended article: On the generalized Hyers-Ulam stability of module left \((m,n)\)-derivations
Recommended article: Comment on: ``On the stability of quadratic double centralizers on Banach algebras''
description: scientific article; zbMATH DE number 6135255