A set-valued map (D Omegarightarrow 2^{X}) is said to be a closed (compact) random set if (D omega)) is closed (compact) for P-a.s.s
Let x denote a diffusion process defined on a closed compact manifold.
Denote P w ( f ) k c ( X ) = { A ⊂ X : nonempty, weakly (closed) compact and convex }.
Therefore, (N_{2}) is a completely continuous multivalued map, u.s.c. with convex closed, compact values.
As (R(B)) is closed (compact perturbation of semi-Fredholm operator), we deduce that B is left invertible.
Therefore, the multivalued (R_{|B}:Brightarrow2^{B} ) is a closed compact map with non-empty convex compact values, and hence (u.s.c).s.c
Let (P_{mathrm{bd},mathrm{cl}}(X)), (P_{mathrm{cp},mathrm{cv}}(X)), (P_{mathrm{bd},mathrm{cl},mathrm{cv}}(X)), and (P_{mathrm{cd}}(X)) denote, respectively, the family of all nonempty bounded-closed, compact-convex, bounded-closed-convex and compact-acyclic (see [9]) subset of X.
Let (mathcal{P}(H)) denote the class of all nonempty subsets of H. Let (mathcal{P}_{bd,cl}(H) ), (mathcal{P}_{cp,cv}(H) ), (mathcal{P}_{bd,cl,cv}(H) ) and (mathcal{P}_{cd}(H) ) denote respectively the families of all nonempty bounded-closed, compact-convex, bounded-closed-convex and compact-acyclic (see [30]) subsets of H.
Suppose further that there exist a nonempty closed and compact resp., weakly closed and weakly compact subset of and a point such that and (2.2).
Let be a nonempty -bounded subset of Hausdorff locally convex (resp., complete) space and let and be self-maps of Suppose that is -starshaped, and -closed (resp., -weakly closed), is compact (resp., is weakly compact), is continuous on (resp., is demiclosed at ), and are Banach operator pairs and satisfy (3.4) for all then.
Theorem 4.5 Let ( E, d ) be a complete CAT ( 0 ) space, C ⊆ E be a closed locally compact convex set, K be a nonempty compact subset of C, and G, H : C → 2 E be two upper semicontinuous set-valued mappings with nonempty compact values.
