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Zbl 1195.34041
Ntouyas, Sotiris K.
Neumann boundary value problems for impulsive differential inclusions.
(English)
[J] Electron. J. Qual. Theory Differ. Equ. 2009, Spec. Iss. I, Paper No. 22, 13 p., electronic only (2009). ISSN 1417-3875/e

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.
MSC 2000:
*34B37 Boundary value problems with impulses
34A60 ODE with multivalued right-hand sides
47N20 Appl. of operator theory to differential and integral equations

Keywords: Neumann boundary value problem; differential inclusions; fixed point

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