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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.

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