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Neumann boundary value problems for impulsive differential inclusions. (English) Zbl 1195.34041
From the text: This paper is concerned with the existence of solutions of boundary value problems (BVP for short) for second order differential inclusions with Neumann boundary. We consider the second order impulsive Neumann BVP, $$x''(t)+k^2x(t)\in F(t,x(t)),\text{ a.e. }t\in J':=[0,1]\setminus \{t_1,\dots,t_m\},$$ $$\Delta x'|_{t=t_k}=I_k(x(t^-_k)),\quad k=1,\dots,m,$$ $$x'(0) = x'(1) = 0,$$ where $F : [0, 1]\times \Bbb R\to{\cal P}(\Bbb R)$ is a compact valued multivalued map, ${\cal P}(\Bbb R)$ is the family of all subsets of $\Bbb R$, $k\in (0,\frac\pi2)$, $0<t_1<t_2<\cdots <t_m<1$, $I_k\in C(\Bbb R,\Bbb R)$ $(k=1,2,\dots,m)$, $\Delta xl_{t=t_k}=x(t^+_k)-x(t^-_k)$, $x(t^+_k)$ and $x(t^-_k)$ represent the right and left limits of $x(t)$ at $t=t_k$ respectively, $k=1,2,\dots,m$. By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values.

34B37Boundary value problems for ODE with impulses
34A60Differential inclusions
47N20Applications of operator theory to differential and integral equations
Full Text: EMIS EuDML