On the existence of periodic solutions for nonconvex-valued differential inclusions in 𝐑^{𝐍}

Abstract

Abstract.In this paper we investigate the existence of periodic solutions for differential inclusions with nonconvex-valued orientor field. Using a tangential condition and directionally continuous selectors, we establish the existence of periodic trajectories.

Periodic solutions for nonconvex differential inclusions

“…on T , x(0) = x(b), has a solution. So we recover the result of Hu-Papageorgiou [6] when the "constraint" set K is time independent. Note that in Hu-Papageorgiou [6], K is time varying.…”

“…By translating things if necessary, we can always assume that 0 ∈ int K. Then if by T K (x) we denote the Bouligant tangent cone to K at x ∈ K, we know that int T K (x) = ∅ and x → int T K (x) has open graph (see Aubin-Cellina [1], Proposition 4, p. 221). Then from Proposition 3.5 of Hu-Papageorgiou [6], we have that F (t, x) = F (t, x) ∩ int T K (x) ⊆ F (t, x) ∩ T K (x) (see Papageorgiou [9], Lemma γ) satisfies hypotheses H(F)(i), (ii) and is satisfied with r > 0 such that B r (0) ⊆ K. So according to our theorem here and sinceF (t, x) ⊆ F (t, x), the periodic problem x (t) ∈ F (t, x(t)) a.e. on T , x(0) = x(b), has a solution.…”

“…In this paper our approach is different and is based on degree theoretic arguments. So our hypotheses on the orientor field are different, in some respects weaker, than the ones used by Hu-Papageorgiou [6] (compare the growth hypothesis H(F)(iii) here and in [6]) and also uses a function like the guiding function of Mawhin [8], which gives us a priori bounds and thus makes possible the use of degree theory.…”

“…In a recent paper Hu-Papageorgiou [6] proved the existence of a periodic solution for a nonconvex differential inclusion in R N . To the best of our knowledge, this was the first existence result for nonconvex periodic problems.…”

“…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. In this paper our approach is different and is based on degree theoretic arguments.…”