On the Hyers-Ulam stability of real continuous function valued differentiable map. (English) Zbl 1002.39039

The authors consider a differentiable map \(f:I \to C(X,\mathbb R)\), where \(I\) is an open real interval and \(X\) a topological space, and assume the following inequality: \[ \|f'(t)-\lambda f(t)\|_{\infty}\leq \varepsilon, \] where \(\varepsilon \geq 0\) and \(\lambda\neq 0\). They prove that there exists an element \(g\) of \(C(X,\mathbb R)\) such that \[ \|f(t)-e^{\lambda t}g\|_{\infty} \leq \frac{3\varepsilon}{|\lambda|} \quad (t \in I). \] An analogous result holds for functions taking values in the space \(C_0(X,\mathbb R)\) of continuous functions vanishing at infinity.
Reviewer: G.L.Forti (Milano)


39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
34G20 Nonlinear differential equations in abstract spaces
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