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Nonlinear second-order multivalued boundary value problems. (English) Zbl 1052.34022
The authors study the following nonlinear multivalued boundary value problem $\varphi(x'(t))'\in A(x(t))+ F(t,x(t))\quad \text{ a.e. on}\,\,\, t\in[0,T],$
$(\varphi(x'(0)),-\varphi(x'(T)))\in\xi(x(0),x(T)),$ where $$\varphi:{\mathbb R}^{N}\to {\mathbb R}^{N}$$ is the function defined by $$\varphi (\xi):= \| \xi\| ^{p-2}\xi$$, $$p \geq 2$$, $$A:D(A) \subseteq \mathbb{R}^N\to 2^{{\mathbb R}^{N}}$$ is a maximal monotone map, $$F:[0,T]\times {\mathbb R}^{N}\to 2^{{\mathbb R}^{N}}$$ is a multivalued vector field and $$\xi:{\mathbb R}^{N}\times {\mathbb R}^{N}\to 2^{{\mathbb R}^{N}\times {\mathbb R}^{N}}$$ is a maximal monotone map describing the boundary conditions. Using notions and techniques from nonlinear operator theory and multivalued analysis, some existence theorems for both convex and nonconvex problems are obtained under the hypothesis that $$F$$ satisfies the well-known Hartman condition on a priori bound. Their framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities, and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Finally, they present some special cases for illustrating the generality and unifying character of their results.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34A60 Ordinary differential inclusions
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