The Existence of Periodic Solutions to Nonautonomous Differential Inclusions

Abstract

ABSTRACT. For an m-dimensional differential inclusion of the form ie A(t)x(t) + F[t,x(t)],with A and F T-periodic in t, we prove the existence of a nonconstant periodic solution. Our hypotheses require m to be odd, and require F to have different growth behavior for |i| small and |i| large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin.We prove the existence of a nonconstant T-periodic solution to a time-dependent differential inclusion of the formWe assume the following: In the following, the maps A(-) and F(-,x) will be considered extended from the interval [0, T] to the real line R by T-periodicity. (L~)m is the Banach space of functions in L1^,T],Rm), extended to R by T-periodicity. In the same way ÍACt)™ and (Gr)m are the Banach spaces of T-periodic functions which are absolutely continuous and continuous endowed with the usual norms ||x||>ic and HxHoo respectively.

Periodic solutions for semi-linear evolution inclusions

“…All earlier works had assumed that the orientor field is convex valued. We refer to the works of Haddad-Lasry [14], Aubin-Cellina [1], Macki-Nistri-Zecca [21] and Plaskacz [24]. For the nonconvex case, we refer to works of Hu-Papageorgiou [18], Hu-Kandilakis-Papageorgiou [16], De Blasi-Górniewicz-Pianigiani [4] and Benchohra-Nieto-Ouahabi [3].…”

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

“…All earlier results on (1) assumed that the orientor field is convex-valued. We refer to the works of Haddad-Lasry [6] (Theorem B-II-1), Aubin-Cellina [1] (Theorem 4, p. 237), Macki-Nistri-Zecca [10] and Plaskacz [14] (Theorem 4.5).…”

“…Bressan [3]) and a Nagumo type tangential condition, analogous to the one employed in [1], [6] and [14], we are able to establish the existence of solutions for (1). The approach of Macki-Nistri-Zecca [10] was based on degree-theoretic techniques.…”

Periodic solutions for nonlinear differential inclusions with multivalued perturbations

“…Nonlinear differential equations arise in many fields of sciences ( such as physics, mechanics, and material science), and have been studied by many authors in the last decades (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14] and the references therein). Differential inclusions governed by subdifferential, which was first studied by Brezis in [1], play an important role in the theory of the nonlinear evolution equations (see [1,2,3,4,5,6,7]).…”