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On first order impulsive differential inclusions with periodic boundary conditions. (English) Zbl 1017.34008

Consider the impulsive periodic multivalued problem of the form \[ y'\in F(t,y),\quad t\in J= [0,T],\quad t\neq t_k,\quad k= 1,\dots, m,\tag{1} \]
\[ \Delta y|_{t= t_k}= I_k(y(t^-_k)),\quad k= 1,\dots, m,\tag{2} \]
\[ y(0)= y(T),\tag{3} \] where \(F: J\times \mathbb{R}\to 2^{\mathbb{R}}\) is a compact and convex-valued multivalued map, \(0= t_0< t_1<\cdots< t_m< t_{m+1}= T\), \(I_k\in C(\mathbb{R},\mathbb{R})\), \(k= 1,2,\dots, m\), \(\Delta y |_{t= t_k}= y(t^+_k)- y(t^-_k)\).
The existence of lower and upper solutions to (1)–(3) is supposed. The authors find sufficient conditions for the existence of solutions to (1)–(3). The proof is based on a fixed-point theorem for multivalued condensing operators.

MSC:

34A37 Ordinary differential equations with impulses
34B37 Boundary value problems with impulses for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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