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.


34B15 Nonlinear boundary value problems for ordinary differential equations
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