Periodic solutions of dissipative systems revisited
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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. ´ Copyright © 2006 J. Andres and L. Gorniewicz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction 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 Carath´eodory 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 →∞
(1.2)
holds, for all solutions x(·) of (1.1). Theorem 1.1 (classical). Assume the uniqueness of solutions of (1.1). If system (1.1) is dissipative, then it admits a τ-periodic solution x(·) ∈ AC([0,τ], Rn ) (with |x(t)| < D, for all t ∈ R). The standard proof of Theorem 1.1 (see, e.g., [30, pages 172-173]) is based on the application of Browder’s fixed point theorem [7], jointly with the fact that, in the case of Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 65195, Pages 1–12 DOI 10.1155/FPTA/2006/65195
2
Dissipative systems
uniqueness, time periodic dissipative systems are uniformly dissipative, that is, ∀D1 > 0
∃ t > 0 : t0 ∈ R, x0 < D1 , t ≥ t0 + t =⇒ x(t) < D2 ,
(1.3)
where D2 > 0 is a common constant, for all D1 > 0, and x(·) = x(·,t0 ,x0 ) is a solution of (1.1) such that x(t0 ) = x(t0 ,t0 ,x0 ) = x0 ∈ Rn , and that their solutions are uniformly bounded (see [26]). 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. also [27]). So far, many extensions of Theorem 1.1 were obtained especially for abstract dissipative processes or in infinite dimensions (see, e.g., [1, 2, 4, 6, 8, 10, 14, 19–22, 30]). The aim of this paper is first to reprove Theorem 1.1 in an extremely simple way by means of asymptotic fixed point theorems and to demonstrate that a very recent theorem of this type in [28] is only a very particular case of much older results, for example, in [11–13, 24, 25] (cf. also [2, 18]). Furthermore, 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 asymptot
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