Semilinear differential inclusions via weak topologies

AbstractThe paper deals with the multivalued initial value problem x′∈A(t,x)x+F(t,x) for a.a. t∈[a,b], x(a)=x0 in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x and the investigation includes the case when both A and F have a superlinear growth in x. We prove the existence of local and global classical solutions in the Sobolev space W1,p([a,b],E) with 1<p<∞. Introducing a suitably defined Lyapunov-like function, we are able to investigate the topological structure of the solution set. Our main tool is a continuation principle in Frechét spaces and we prove the required pushing condition in two different ways. Some examples complete the discussion