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Existence results for impulsive semilinear damped differential inclusions. (English) Zbl 1046.34017
The paper deals with the existence of mild solutions for first-order impulsive semilinear evolution inclusions \(y'-Ay \in By+ F(t,y)\), \(t\in [0,b]\setminus \{t_1, \dots ,t_m\}\), where \(F :[0,b]\times E \to E\) (\(E\) is a separable Banach space) is a multivalued map, \(A\) is the infinitesimal operator of a semigroup of linear operators and \(B\) is a bounded operator from \(E\) into \(E\). The second-order impulsive semilinear evolution equation \(u''+Ay \in By'+F(t,y)\), where \(B\) and \(F\) be as above and \(A\) is the infinitesimal operator of a cosine family is also considered. The multivalued map \(F\) can be convex- or nonconvex-valued. In the first case, the proof is based on the Bohnenblust/Karlin fixed-point theorem, in the second case on the Covitz/Nadler fixed-point theorem.

34A37 Ordinary differential equations with impulses
34A60 Ordinary differential inclusions
34G20 Nonlinear differential equations in abstract spaces
35R10 Partial functional-differential equations
47H20 Semigroups of nonlinear operators
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