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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 \(F : [0, 1]\times \mathbb R\to{\mathcal P}(\mathbb R)\) is a compact valued multivalued map, \({\mathcal P}(\mathbb R)\) is the family of all subsets of \(\mathbb R\), \(k\in (0,\frac\pi2)\), \(0<t_1<t_2<\cdots <t_m<1\), \(I_k\in C(\mathbb R,\mathbb R)\) \((k=1,2,\dots,m)\), \(\Delta xl_{t=t_k}=x(t^+_k)-x(t^-_k)\), \(x(t^+_k)\) 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:

34B37 Boundary value problems with impulses for ordinary differential equations
34A60 Ordinary differential inclusions
47N20 Applications of operator theory to differential and integral equations
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