Multivalued periodic Lienard systems (Q2314830)

The following differential inclusion is considered \begin{align*} &a(u'(t))'+ \frac{d}{dt}\nabla G(u(t))\in A(u(t))+F(t,u(t)) \quad \text{for a.a.}\quad t\in [0,b],\\ &u(0)=u(b), \; u'(0)=u'(b), \end{align*} which is called multivalued Lienard system. Here \(a:\mathbb{R}^ N\to \mathbb{R}^N\) is a monotone homeomorphism, \(A:\mathbb{R}^N\to 2^{\mathbb{R}^N}\) is a maximal monotone map (not necessarily \(A(u)\neq \emptyset\) everywhere). \(G\in C^2(\mathbb{R}^N, \mathbb{R})\) and \(F\) is a multivalued perturbation. The authors prove the existence of solutions under appropriate sets of assumptions on \(F\), both in convex-valued case, and the nonconvex one. Among other interesting things they use a condition by \textit{P. Hartman} [Trans. Am. Math. Soc. 96, 493--509 (1960; Zbl 0098.06101)] in place of global growth condition on \(F(t,\cdot)\). Properties of monotonicity, Bressan-Colombo selection theorem, a multivalued extension of Leray-Schauder alternative are the tools for the proofs. Several examples of \(a\) are given, including \(a(y)=|y|^{p-2}y, 1 < p <\infty\), which corresponds to the vector \(p\)-Laplacian. Then they consider the existence of extremal solutions (the set \(F(t,x)\) is replaced by the set of extreme points \(ext F(t,x)\) in the inclusion) under slightly modified assumptions. A strong relaxation theorem for a class of parametrized Lienard systems is also proved.