Second Order Nonlinear Evolution Inclusions Existence and Relaxation Results

This is the first part of a work on second order nonlinear,nonmonotone evolution inclusionsdefined in the framework of an evolution triple of spaces and with a multivalued nonlinearity dependingon both x(t)and (t).In this first part we prove existence and relaxation theorems.We consider thecase of an usc,convex valued nonlinearity and we show that for this problem the solution set is nonemptyand compact in C~1(T,H).Also we examine the lsc,nonconvex case and again we prove the existenceof solutions.In addition we establish the existence of extremal solutions and by strengthening ourhypotheses,we show that the extremal solutions are dense in C~1(T,H)to the solutions of the originalconvex problem(strong relaxation).An example of a nonlinear hyperbolic optimal control problem isalso discussed.