Periodic solutions for nonautonomous differential equations and inclusions in tubes

Grzegorz Gabor

doi:10.1155/s1085337504403042

Abstract

We study the existence of periodic trajectories for nonautonomous differential equations and inclusions remaining in a prescribed compact subset of an extended phase space. These sets of constraints are nonconvex right‐continuous tubes not satisfying the viability tangential condition on the whole boundary. We find sufficient conditions for existence of viable periodic trajectories studying properties of the exit subset of the tube. A new approximation approach for continuous multivalued maps is presented.

Highlights

The present paper is devoted to the existence of solutions to the boundary value problem x(t) ∈ F t,x(t), a.e. in [0,T], (1.1)x(0) = x(T), or, in particular, x(t) = f t,x(t), a.e. in [0,T], (1.2)x(0) = x(T), in a finite-dimensional space, with some additional state constraints, that is, we insist that trajectories do not leave a prescribed closed set W ⊂ [0,T] × Rn, or in other words, that trajectories are viable in W.Following the literature in the subject, we call such trajectories viable periodic
The methods have been mainly based on the degree theory applied to the single-valued or multivalued
Recall that a multivalued map F : R × Rn Rn is a Marchaud map if F has nonempty compact convex values, the map F(·, x) is measurable for every x ∈ Rn, F(t, ·) is upper semicontinuous for almost all t ∈ R, and F has at most the linear growth, which means that “there exists a locally integrable function β : R → [0,∞) such that |F(t, x)| := sup{|y|; y ∈ F(t, x)} ≤ β(t)(1 + |x|) for every (t, x) ∈ R × Rn.”

Summary

Introduction

The present paper is devoted to the existence of solutions to the boundary value problem x(t) ∈ F t,x(t) , a.e. in [0,T], (1.1). We can use terms of tangent cones to the set W. where TW (·) stands for the Bouligand contingent cone and W(t) := {x ∈ Rn | (t, x) ∈ W}, no trajectory leaves W and each T-periodic solution obtained by applying any standard technique is viable, so it is what we look for. The Wazewski retract method has been intensively developed and used, for example, in the Conley index theory It occurs that this exit set W− can be used to find viable T-periodic trajectories for nonautonomous differential equations. Under some regularity assumptions on the tube, we present the new method of studying an existence of viable T-periodic trajectories by a special approximation of the map F by Lipschitz single-valued maps having the same exit set W−

Preliminaries

Differential equations on right-continuous tubes

Differential inclusions