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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.

##### MSC:
 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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