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First-order periodic impulsive semilinear differential inclusions: existence and structure of solution sets. (English) Zbl 1202.34110

Summary: We present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive semilinear differential inclusions with periodic boundary conditions:

(y ' -Ay)(t)F(t,y(t)),a.e.tJ{t 1 ,,t m }y(t k + )-y(t k - )),k=1,,m,y(0)=y(b)

where J=[0,b] and 0=t 0 <t 1 <<t m <t m+1 =b (m * ) is the infinitesimal generator of a C 0 -semigroup T on a separable Banach space E and F is a multi-valued map. The functions Ik characterize the jump of the solutions at impulse points t k (k=1,,m). We will have to distinguish between the cases when either or neither 1 lies in the resolvent of T(b). Accordingly, the problem is either formulated as a fixed point problem for an integral operator or for a Poincaré translation operator. Our existence results rely on fixed point theory and on a new nonlinear alternative for compact u.s.c. maps respectively. Then, we present some existence results and investigate the topological structure of the solution set. A continuous version of Filippov’s theorem is provided and the continuous dependence of solutions on parameters in both convex and nonconvex cases are examined.

MSC:
34G25Evolution inclusions
34A37Differential equations with impulses
34C25Periodic solutions of ODE
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