Periodic solutions of dissipative systems revisited

Jan Andres,Lech Górniewicz

doi:10.1155/fpta/2006/65195

Jan Andres, Lech Górniewicz

Open Access

https://doi.org/10.1155/fpta/2006/65195

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Abstract

We reprove in an extremely simple way the classical theorem that time periodic dissipative systems imply the existence of harmonic periodic solutions, in the case of uniqueness. We will also show that, in the lack of uniqueness, the existence of harmonics is implied by uniform dissipativity. The localization of starting points and multiplicity of periodic solutions will be established, under suitable additional assumptions, as well. The arguments are based on the application of various asymptotic fixed point theorems of the Lefschetz and Nielsen type.

Highlights

Consider the system x = F(t,x), F(t,x) ≡ F(t + τ,x), τ > 0, (1.1)where F : [0, τ] × Rn → Rn is a Caratheodory function
Let us note that the same idea of the proof was already present in [9], but since that time Browder’s theorem was not at our disposal, only subharmonic solutions were deduced by means of the Brouwer fixed point theorem
We will obtain more precise information about localization of the starting point of the implied τ-periodic solution of (1.1) by means of the asymptotic relative Lefschetz theorem [17], and discuss possible multiplicity results by means of the asymptotic relative Nielsen theorem [5]

Summary

Introduction

Where F : [0, τ] × Rn → Rn is a Caratheodory function. We say that system (1.1) is dissipative (in the sense of Levinson [23]) if there exists a common constant D > 0 such that limsup x(t) < D t→∞. If system (1.1) is dissipative, it admits a τ-periodic solution x(·) ∈ AC([0, τ], Rn) (with |x(t)| < D, for all t ∈ R). Let us note that the same idea of the proof was already present in [9], but since that time Browder’s theorem was not at our disposal, only subharmonic (i.e., kτ-periodic; k ∈ N) solutions were deduced by means of the Brouwer fixed point theorem (cf [27]). We will obtain more precise information about localization of the starting point of the implied τ-periodic solution of (1.1) by means of the asymptotic relative Lefschetz theorem [17], and discuss possible multiplicity results by means of the asymptotic relative Nielsen theorem [5]. We will generalize Theorem 1.1, jointly with the relative and multiplicity results, in the lack of uniqueness

Asymptotic fixed point theorems

Some further information

Lack of uniqueness

Concluding remarks