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Ulam stabilities of ordinary differential equations in a Banach space. (English) Zbl 1224.34164

Four types of Ulam stability for the following differential equation in a Banach space \(\mathbb B\) are considered: \[ u'(t)=Au(t)+f(t,u(t)),\;t\in I\subset\mathbb R, \] where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup, \(f\in C(I\times\mathbb B,\mathbb B)\).
In the case of a compact interval \(I\), the author proves Ulam-Hyers stability for the considered equation, assuming that \(f\) is Lipschitzian in the second variable. Moreover, in the case \(I=[a,+\infty)\), generalized Ulam-Hyers-Rassias stability with respect to an increasing function \(\varphi\in C(I,\mathbb R_+)\) is established.

MSC:

34D10 Perturbations of ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
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