Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains
The following impulsive Cauchy problem is considered: \[ \begin{aligned} &x'(t) \in A(t)x(t) + F(t,x(t))\quad \text{a.e.} \,\, t\in [0,b],\,\,\, t \neq t_{k}, k=1,2,\dots ,m,\\ &x(t_{k}^{+}) = x(t_{k}) + I_{k}(x(t_{k})),\quad k=1,2,\dots ,m, \\ &x(0) = a_0 \in E, \end{aligned} \] 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.
impulsive semilinear evolution differential inclusion
scientific article; zbMATH DE number 5290575
scientific article