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On the Hyers-Ulam stability of a differentiable map. (English) Zbl 1025.47041
Let $$B$$ be a Banach space, $$f$$ a differentiable map on $$I= (a,b)$$ into $$B$$ such that $\|f'(t)-\lambda f(t)\|\leq\varepsilon\quad\text{for }t\in I.\tag{1}$ We say that the Ulam-Hyers stability holds for $$f$$, if there exist a $$k\geq 0$$ and a differentiable map $$x: I\to B$$ such that $x'(t)=\lambda x(t)\quad\text{and}\quad\|f(t)- x(t)\|\leq k\varepsilon$ for all $$t\in I$$.
Let $$X$$ be a topological space and $$C(X,\mathbb{C})$$ the Banach space of all complex-valued bounded continuous functions on $$X$$. Let $$A$$ be a uniformly closed linear subspace of $$C(X,\mathbb{C})$$.
In the paper the following results concerning Ulam-Hyers stability are proved:
(1) if $$\text{Re }\lambda\neq 0$$, then Ulam-Hyers stability holds for $$f$$ satisfying (1);
2) if $$\text{Re }\lambda= 0$$, then such a statement need not be true.
As a special case, the author considers an entire function satisfying the condition (1). For such functions Ulam-Hyers stability holds true.

##### MSC:
 47H99 Nonlinear operators and their properties 26D10 Inequalities involving derivatives and differential and integral operators 34G20 Nonlinear differential equations in abstract spaces
##### Keywords:
differential equations; Banach space; Ulam-Hyers stability