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An existence theorem for inclusions of the type $$\Psi$$ (u)(t)$$\in F(t,\Phi (u)(t))$$ and an application to a multi-valued boundary value problem. (English) Zbl 0687.47044
This paper contains a general existence result on inclusions of the type $$\Psi (u)(t)\in F(t,\Phi (u)(t)),$$ where each of $$\Psi$$ and $$\Phi$$ is an operator from a set V into an $$L^ p$$ space of vector-valued functions. An application of such result yields the following
Theorem. Let $$F: [a,b]\times {\mathbb{R}}^ n\times {\mathbb{R}}^ n\to 2^{{\mathbb{R}}^ n}$$ be a multifunction, with non-empty closed convex values, satisfying the following conditions:
(1) for almost every $$t\in [a,b]$$, the multifunction $$(x,y)\to F(t,x,y)$$ has closed graph;
(2) the set $$\{(x,y)\in {\mathbb{R}}^ n\times {\mathbb{R}}^ n:$$ the multifunction $$t\to F(t,x,y)$$ is measurable} is dense in $${\mathbb{R}}^ n\times {\mathbb{R}}^ n;$$
(3) there exists $$p\in [1,+\infty[$$ and $$r\in]0,+\infty[$$ such that $(\int^{b}_{a}(\sup_{\| x\|,\| y\| \leq cr}dist(0,F(t,x,y)))^ p dt)^{1\quad /p}\leq r,$
where $$c=\max \{(b-a)^{1-1/p},(b-a)^{2-1/p}\}.$$
Under such hypotheses, there exists $$u\in W^{2,p}([a,b],{\mathbb{R}}^ n)$$ such that $u''(t)\in F(t,u(t),u'(t))\quad a.e.\quad in\quad [a,b],\quad u(a)=u(b)=0.$
Reviewer: B.Ricceri

##### MSC:
 47J05 Equations involving nonlinear operators (general)
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