On some properties of the solution set map to Volterra integral inclusion (Q2403363)

The author continues the study of the following Volterra integral inclusion: \[ x(t)\in h(t) + \int\limits_0^t k(t,s) F(s,x(s)) ds, \;\;t \in I=[0,T], \tag{1} \] where \(E\) is a Banach space, \(h \in C(I,E)\), \(k(t,s) \in L(E)\), \(F: I \times E \to conv(E)\), \(L(E)\) is the space of bounded linear endomorphism's of \(E\) and \(conv(E)\) is the set of all nonempty closed convex subsets in \(E\). A function \(x \in C(I,E)\) is a solution of (1) if for almost every \(t \in I\) the relation \( x(t)\in h(t) + \int\limits_0^t k(t,s) W(s) ds \) holds for some \(w \in L^p(I,E), p \geq 1\) and \(w \in F(t,x(t))\). Under some natural conditions concerning \(F\) and \(k(t,s)\), the following interesting results are proved: -- for \(p \in [1,\infty)\) the solution set \(S_F^p (h)\) of the inclusion (1) is an \(R_\delta\) set in the space \(C(I,E)\) if \(E\) is a reflexive space ; -- the solution map \(S_F^p (h) : C(I,E) \to P(C(I,E))\) for each \(h \in C(I,E)\) and \(p \in [1,\infty)\) has a continuous single valued selection in the case when \(E\) is a separable space; -- the set \(\bigcup\limits_{h \in M} S_F^p (h)\) is connected if the set \(M \subset C(I,E)\) is connected too; -- if \(E\) is reflexive for \(p \in (1,\infty)\) and the set \(M\), \(M \subset C(I,E)\) is compact and convex, then the image of the set \(M\) trough the solution map \(S_F^p(h)\) is compact and acyclic. In addition, since the solutions of inclusion (1) are understood in the sense of the Aumann integral, the author studies the properties of the selection set map \(Sel_F^p (h) : C(I,E) \to L^p(I,E)\). In the case when \(E\) is separable it is proved that the set \(Sel_F^p (h)\) is an absolute retract for each \(h \in C(I,E)\). Moreover, \(Sel_F^p (h)\) has a continuous single valued selection.