Periodic Time Dependent Gross-Substitute Systems

Nakajima, Fumio

doi:10.1137/0136032

Cited by 19 publications

(14 citation statements)

References 2 publications

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“…As we will explain at the end of Section 2, our main results in the present paper are much more general than those earlier results, so that they have a considerably wider range of applications. Furthermore, the proofs of our results are exceedingly simple and elementary, as we will see in Section 3. Let us also mention the related results in [21,15], which deal with systems of ODE's of the cooperative type and establish convergence results that are similar to ours. Cooperative ODE systems are classical examples of finite dimensional orderpreserving systems, but since [21,15] do not assume irreducibility of the systems, the strong comparison principle (F2 ) defined in Section 2 does not necessarily hold for their systems.…”

supporting

confidence: 70%

Convergence and structure theorems for order-preserving dynamical systems with mass conservation

Ogiwara

Hilhorst

Matano³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which 0 is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczyński (1987 and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.1. Introduction. In this paper we study order-preserving dynamical systems having a certain kind of "mass conservation property" and establish a general theory on the convergence of bounded orbits and the existence of fixed points. We will then apply our general theory to a number of problems including reaction-diffusion equations possibly with nonlocal terms and some delay-differential equations. Our theory implies the existence of multiple stationary and/or time-periodic solutions as an immediate corollary.Our present study was originally motivated by the analysis of mathematical models for transportation in molecular motor systems and those for chemical reversible

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supporting

confidence: 70%

Convergence and structure theorems for order-preserving dynamical systems with mass conservation

Ogiwara

Hilhorst

Matano³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…The third type convergence criterion is that the monotone systems possess a first integral or invariant function with positive gradient. Many model systems have this property, see, e.g., [21], [29], [30], [40], [46] for ordinary di¤erential equations; [45] for partial di¤erential equations; [4], [5], [25], [26], [51], [52] for functional di¤erential equations. These authors showed that every forward orbit of such a system is convergent to an equilibrium or a fixed point, and that the set of all fixed points is a totally ordered curve.…”

Section: Introductionmentioning

confidence: 99%

Convergence in monotone and uniformly stable skew-product semiflows with applications

Jiang¹,

Zhao²

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

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The 1-covering property of omega limit sets is established for monotone and uniformly stable skew-product semiflows with the componentwise separating property of bounded and ordered full orbits. Then these results are applied to study the asymptotic almost periodicity of solutions to almost periodic reaction-di¤usion equations and di¤eren-tial systems with time delays. The earlier convergence results for autonomous and periodic monotone systems are generalized to the almost periodic case without the strong monotonicity assumption.

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“…The convergence of positive bounded solutions in time independent and periodic as well as some special almost periodic cooperative systems with a first integral has been discussed in several papers. For example, Nakajima [9] proved the asymptotic periodicity of bounded solutions for periodic and cooperative gross-substitute systems with a linear first integral, and this convergence result was further generalized to the almost periodic case by Sell and Nakajima [13]. Mierczynski [8] proved the convergence of bounded solutions for autonomous and strictly cooperative systems with a first integral, and Jiang [6] obtained the same result in the case where these systems are cooperative.…”

Section: Introductionmentioning

confidence: 97%

Convergence in almost periodic cooperative systems with a first integral

Shen

Zhao

2004

Proc. Amer. Math. Soc.

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Abstract. This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skew-product semiflows we establish the asymptotic almost periodicity of bounded solutions to such systems, which extends the existing convergence results for time independent and periodic cooperative systems with a first integral and proves a conjecture of

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Periodic Time Dependent Gross-Substitute Systems

Cited by 19 publications

References 2 publications

Convergence and structure theorems for order-preserving dynamical systems with mass conservation

Convergence and structure theorems for order-preserving dynamical systems with mass conservation

Convergence in monotone and uniformly stable skew-product semiflows with applications

Convergence in almost periodic cooperative systems with a first integral

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