Evolution Equations

In this chapter, we focus on dynamic (evolution) equations. In Sect. 6.1, we deal with Cauchy, periodic and antiperiodic systems defined in the framework of an evolution triple. We also study second order evolution inclusions and second order periodic systems involving maximal monotone operators which need not be defined on all \(\mathbb {R}^{N}\) (differential variational inequalities). In addition we consider Lienard systems. In Sect. 6.2, we examine evolution inclusions involving nonlocal conditions. We show the nonemptiness (and compactness) of the solution set (the compactness for the convex problem). In Sect. 6.3, we show the topological triviality of the solution set. Finally in Sect. 6.4, using the notions of G and of PG convergences, we prove continuous dependence results for the solution multifunction. The main tools in this chapter, are the Lebesgue-Bochner spaces, multivalued analysis and the theory of operators of monotone-type.