Asymptotic behavior analysis on multivalued evolution inclusion with projection in Hilbert space

Let be a real Hilbert space, a closed convex subset of , and continuous convex and bounded from below on . We study the asymptotic behaviour of the non-autonomous differential inclusion with project operator as follows where is an absolutely continuous decreasing control function with strictly positive value and . On the one hand, when , each solution of PS with initial point (not necessarily in ) converges to an element of in the weak topology. On the other hand, when and , the weak and strong convergence of PS to the minimizer of over can be guaranteed under some conditions. Strong convergence can be obtained when or is strongly convex on . Another two sufficient conditions are given to ensure the weak convergence. Moreover, asymptotic analysis for the special case that is also considered. Finally, we present some discussion on the global existence of the solutions of PS.