On the generalized Hyers-Ulam stability of module left \((m,n)\)-derivations. (English) Zbl 1267.39022
The stability problem of functional equations originates 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 an algebra over the real or complex field \(\mathbb{F}\) and \(M\) be a left \(A\)-module. An additive mapping \(d:A\to M\) is called a module left \((m,n)\)-derivation if
\[
(m+n)d(xy)=2mx.d(y)+2ny.d(x)
\]
holds for all \(x,y\in A\), where \(m\geq 0\), \(n\geq 0\) with \(m+n\neq 0\) are some fixed integers.
The main purpose of the present paper is to prove the following theorem:
Let \(A\) be a normed algebra, \(M\) a Banach left \(A\)-module, and \(F:A^2\to [0,\infty)\) a function such that \(F(2x,y)=\xi F(x,Y)\) and \(F(x,2y)=\chi F(x,y)\) for some scalars \(\xi,\chi\geq 0\), \(\xi\chi<1\). Suppose that \(f:A\to M\) is a map with the properties \[ \|f(x+y)-f(x)f(y)\|\leq F(x,y) \] for all \(x,y\in A\). Then there exists a unique module left \((m,n)\)-derivation \(d:A\to M\) such that \[ \|f(x)-d(x)\|\leq \frac{F(x,x)}{1-\xi\chi} \] for all \(x\in A\).
The main purpose of the present paper is to prove the following theorem:
Let \(A\) be a normed algebra, \(M\) a Banach left \(A\)-module, and \(F:A^2\to [0,\infty)\) a function such that \(F(2x,y)=\xi F(x,Y)\) and \(F(x,2y)=\chi F(x,y)\) for some scalars \(\xi,\chi\geq 0\), \(\xi\chi<1\). Suppose that \(f:A\to M\) is a map with the properties \[ \|f(x+y)-f(x)f(y)\|\leq F(x,y) \] for all \(x,y\in A\). Then there exists a unique module left \((m,n)\)-derivation \(d:A\to M\) such that \[ \|f(x)-d(x)\|\leq \frac{F(x,x)}{1-\xi\chi} \] for all \(x\in A\).
Reviewer: Nasrin Eghbali (Ardabil)
MSC:
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 39B62 | Functional inequalities, including subadditivity, convexity, etc. |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |
| 46L57 | Derivations, dissipations and positive semigroups in \(C^*\)-algebras |
| 16W25 | Derivations, actions of Lie algebras |
Keywords:
generalized Hyers-Ulam stability; normed algebra; Banach left \(A\)-module; module left \((m,n)\)-derivationReferences:
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