Periodic solutions of functional differential inclusions and fixed points of σ-selectionable correspondences

Delay Differential Inclusions with Constraints

“…In case the constraint set K is time independent, we prove that the solution set is an TxVset and a periodic solution exists if the system is periodic. This way we generalize an earlier result of Haddad-Lasry [7]. When K is timedependent, since the graph of K is not convex in general, we can only establish some existence results and pose as an open problem the question of whether the solution set is an Rs-set.…”

“…The following fixed point result is due to Haddad-Lasry [7]: Lemma 3 [7]. Let H ç X be nonempty, convex, and compact and assume that T is a a-selectionable multifunction from H into X such that T(x) ç H for all x £ H. Then T(-) has a fixed point in H (i.e., there exists x £ H such that x £ r(x)).…”

“…It is clear that for each fixed zz > 1 , by choosing 6 > 0 sufficiently small, we can get measurable A" ç T with X(An) < ^ such that (7) \\gl"'e(t,¥)-g2n'£(t,¥)\\<Y for all (t,ip)£ (T\A") xJT. Set D" = U£l" ^ , e = 7, and define C*(i, ip) = Gn(t, y/) + Bl +XDn(t)Bßi0, where ß = m + 1 with m = sup{||g"2' "(t, y/)\\ ■ (t, y/)£Tx&}.…”

“…License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use [7], who assume that the orientor field F(t, y/) is jointly u.s.c.…”

Periodic solutions for nonconvex differential inclusions

“…All earlier works had assumed that the orientor field (multivalued vector field) is convex valued. We refer to the works of Haddad-Lasry [4], Aubin-Cellina [1], Macki-Nistri-Zecca [7] and Plaskacz [11]. The approach of Hu-Papageorgiou [6] was based on directionally continuous selectors for the orientor field and on a Nagumo type tangential condition.…”