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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.

47H99 Nonlinear operators and their properties
26D10 Inequalities involving derivatives and differential and integral operators
34G20 Nonlinear differential equations in abstract spaces