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

Ricceri, Ornella Naselli; Ricceri, Biagio

Appl. Anal. 38, No. 4, 259-270 (1990).

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)

Keywords:

multi-valued boundary value problem; operator inclusions; \(L^ p\)-space of vector-valued functions; multifunction, with non-empty closed convex values

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