## Nonlinear boundary value problems for second order differential inclusions.(English)Zbl 1081.34020

Summary: We study two boundary value problems for second-order strongly nonlinear differential inclusions involving a maximal monotone term. The first one is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form $$x\mapsto a(x,x')'$$. In this problem, the maximal monotone term is required to be defined everywhere in the state space $$\mathbb R^N$$. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form $$x\mapsto (a(x)x')'$$. In this case, the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multi-valued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multi-valued right-hand side.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations
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### References:

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