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

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