Special Topics in Differential and Evolution Inclusions

Abstract

In chapters I and II we examined in detail two large classes of evolution inclusions. In this chapter we close the subject of multivalued differential equations, by presenting certain topics that were not yet covered and can not be found in the existing books. So, in section 1 we consider differential inclusions in R N . First we present a unified treatment of the convex and nonconvex problems, which so far have been approached using distinct methods. Using the notions of directionally continuous selectors (see section A-I.5) and the well-known multivalued Filippov regularization of a discontinuous vector field, we are able to transform a nonconvex problem to an equivalent convex one which is solved by standard methods. In the second half of section 1, we go beyond the confines of the existence theory and of the analysis of the topological structure of the solution set of a differential inclusion in R N , by introducing a “metric likelihood” map which is entirely independent of probability theory. This notion allows to characterize the extremal trajectories of a differential inclusion and to give a precise mathematical meaning to a class of estimation, prediction and filtering problems, in a context which is independent of any probability theory (deterministic problems). In section 2 we continue the study of differential inclusions in R N and investigate boundary value problems for second order differential inclusions in R N . The boundary conditions that we consider are nonlinear and include as special cases the Dirichlet (Picard), the Neumann and the periodic problems. In section 3 we investigate evolution inclusions driven by m-accretive operators. We conduct a detailed study analogous to the ones in chapters I and II. Moreover, in the process of our investigation we derive some additional results about semigroups of nonlinear contractions which complement the ones presented in section A-III.8. In section 4 we repeat the same for Volterra integral inclusions in Banach spaces. The Hausdorff measure of noncompactness plays a prominent role in this analysis. In section 5 we examine the reachable set of evolution inclusions and the solution set of differential inclusions in R N with maximal monotone operators. For the reachable set of semilinear evolution inclusions, we determine the infinitesimal generators of the reachable map. For the maximal monotone differential inclusions in R N we consider the solution set as a multifunction of the initial state and determine its differentiability properties and the differential inclusion that the derivative satisfies (variational inclusion). Finally, in section 6 we deal with differential equations with discontinuities. Since such problems need not have solutions, in order to have a reasonable existence theory we pass to a multivalued version of the problem by, roughly speaking, filling in the gaps at the discontinuity points. We consider such problems for ordinary differential equations, elliptic equations and parabolic equations. The ode’s and elliptic equations are solved using a variational approach based on the nonsmooth critical point theory for locally Lipschitz energy functionals, which is outlined in the beginning of section 5. The parabolic problems are analyzed using the method of upper and lower solutions and truncation and penalization techniques. The analytical formalism of evolution triples (see section 1.1) and the theory of operators of monotone-type (see section A-III.6) play a crucial role.