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The notion of eventual stability has been recently discussed. We extend this notion to impulsive systems of differential equations. Our technique depends on Liapunov's direct method. (C) 2002 Elsevier Science Inc. All rights reser...
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The notion of eventual stability has been recently discussed. We extend this notion to impulsive systems of differential equations. Our technique depends on Liapunov's direct method. (C) 2002 Elsevier Science Inc. All rights reserved. [References: 7]
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The notion of Lipschitz stability of impulsive systems of differential equations (DEs) was introduced. In this paper, we will extend the notion of eventual stability to impulsive systems of DEs and extend the notion of Lipschitz s...
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The notion of Lipschitz stability of impulsive systems of differential equations (DEs) was introduced. In this paper, we will extend the notion of eventual stability to impulsive systems of DEs and extend the notion of Lipschitz stability of impulsive systems of DEs to a new type of stability called eventual Lipschitz stability. Some criteria and results are given. Our technique depends on Liapunov's direct method and comparison principle. (C) 2002 Elsevier Science Inc. All rights reserved. [References: 6]
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This paper discusses the uniform eventual Lipschitz stability and uniform eventual asymptotic stability of impulsive differential system. Some criteria of such stability for impulsive differential system are given. Compared with t...
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This paper discusses the uniform eventual Lipschitz stability and uniform eventual asymptotic stability of impulsive differential system. Some criteria of such stability for impulsive differential system are given. Compared with the existing result, the condition is less conservative. Our technique depends on Lyapunov's direct method and comparison principle. (C) 2003 Elsevier Inc. All rights reserved.
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In this paper we first give a criterion on stability of equilibrium solutions for autonomous systems with constraints. Then we discuss the relationship between asymptotic behaviors of an asymptotically autonomous system with const...
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In this paper we first give a criterion on stability of equilibrium solutions for autonomous systems with constraints. Then we discuss the relationship between asymptotic behaviors of an asymptotically autonomous system with constraint and its limit system. Finally as an example, we revisit an extreme ideology model proposed in the literature and give a more detailed description on the dynamics of the system.
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The notion of phi(0)-stability recently was introduced. In this paper, we will extend this notion to the so-called eventual phi(0)-stability for impulsive systems of differential equations under more relax conditions. Our techniqu...
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The notion of phi(0)-stability recently was introduced. In this paper, we will extend this notion to the so-called eventual phi(0)-stability for impulsive systems of differential equations under more relax conditions. Our technique depends on Lyapunov's direct method. (C) 2003 Published by Elsevier Inc.
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In this paper we develop a unified approach to study the eventual smoothness and exponential stabilization of global weak solutions of two different chemotaxis systems. One is a Keller-Segel system with consumption of chemo-attrac...
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In this paper we develop a unified approach to study the eventual smoothness and exponential stabilization of global weak solutions of two different chemotaxis systems. One is a Keller-Segel system with consumption of chemo-attractants recently studied in [Y. Tao and M. Winkler, J. Differential Equations, 252 (2012), pp. 2520-2543] and the other is a chemo-repulsion system studied in [T. Cieslak, P. Laurencot, and C. Morales-Rodrigo, Banach Center Publ., 81 (2008), pp. 105-117]. For both systems in dimension three, we prove the existence of weak solutions that become regular after certain time T > 0 and obtain the exponential convergence rate toward spatially homogeneous steady states. Our method relies on the stability of constant steady states of these chemotaxis systems in the corresponding scaling-invariant spaces. For the first system, we improve the results in Tao and Winkler and [M. Winkler, Trans. Amer. Math. Soc., 369 (2017), pp. 3067-3125] in a sense that the convexity assumption on the domain is removed and, moreover, exponential stabilization with an optimal convergence rate is obtained for the first time, while for the second system, our result is completely new. In addition, we provide an alternative proof for the chemo-repulsion system via an energy method by deriving delicate higher-order estimates.
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A nonlinear dynamical system is called eventually competitive (or cooperative) provided that it preserves a partial order in backward (or forward) time only after some reasonable initial transient. We present in this paper the Non...
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A nonlinear dynamical system is called eventually competitive (or cooperative) provided that it preserves a partial order in backward (or forward) time only after some reasonable initial transient. We present in this paper the Non-oscillation Principle for eventually competitive or cooperative systems, by which the non-ordering of (both omega- and alpha-) limit sets is obtained for such systems; and moreover, we established the Poincare-Bendixson Theorem and structural stability for three-dimensional eventually competitive and cooperative systems.
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In terms of the spectral properties of the associated operator matrix, we obtain a new criterion to judge the uniform exponential stability of the solutions to abstract Volterra equations. In particular, we study in detail the cas...
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In terms of the spectral properties of the associated operator matrix, we obtain a new criterion to judge the uniform exponential stability of the solutions to abstract Volterra equations. In particular, we study in detail the case where the kernel function a(t) takes the form α e ?βt (β > 0, α ≠ 0). Moreover, we give examples to illustrate our results.
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