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Zbl 1147.34045
Cardinali, Tiziana; Rubbioni, Paola
Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 1, A, 73-84 (2008). ISSN 0362-546X

The following impulsive Cauchy problem is considered: \aligned &x'(t) \in A(t)x(t) + F(t,x(t))\quad \text{a.e.} \,\, t\in [0,b],\,\,\, t \neq t_{k}, k=1,2,\dots ,m,\\ &x(t_{k}^{+}) = x(t_{k}) + I_{k}(x(t_{k})),\quad k=1,2,\dots ,m, \\ &x(0) = a_0 \in E, \endaligned where $\{A(t)\}_{t\in [0,b]}$ is a family of linear operators (not necessarily bounded) in a Banach space $E$ generating an evolution operator, $F$ is a multifunction of Carathéodory type, $0<t_0<t_1<\ldots<t_m<t_{m+1}=b,$ $I_k:E\to E, k=1,\ldots,m,$ are impulse functions and $x(t^+)=\lim_{s\to t^+}x(s).$ First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on noncompact domains. An example illustrating the abstract results is also presented.
[Sotiris K. Ntouyas (Ioannina)]
MSC 2000:
*34G25 Evolution inclusions
34A37 Differential equations with impulses

Keywords: impulsive semilinear evolution differential inclusion; evolution system; mild solution; measure of non-compactness; upper semicontinuous multifunction

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