Strongly nonlinear multivalued boundary value problems
The authors study the periodic boundary value problem \[ \begin{aligned} &(a(x(t))\| x'(t)\| ^{p-2}x'(t))' \in A(x(t)) + F(t,x(t)) \quad \text{a.e. on } [0,T],\\ &x(0) = x(T), x'(0)=x'(T); \end{aligned} \] where \(A : \mathbb R^N \to \mathbb R^N\) is a maximal monotone operator, \(a : \mathbb R^N \to [c,d]\) is continuous, and \(F\) is a multivalued map with compact values satisfying a Hartman-type condition: ``\(\exists M> 0\) such that \(\langle x, y \rangle \geq 0\) \(\forall y \in F(t,x)\), for a.e. \(t \in [0,1]\) and all \(x \in \mathbb R^N\) with \(\| x\| = M\).'' The existence of a solution is established, respectively, for \(x \mapsto F(t,x)\) u.s.c. with convex values, and for \(x \mapsto F(t,x)\) l.s.c. The proofs rely on the theory of monotone and generalized pseudo-monotone operators, the theory of multivalued maps, and on the Fryskowski-Bressan-Colombo-selection theorem.
scientific article; zbMATH DE number 1880005
scientific article