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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)

##### MSC:
 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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