scholarly journals Existence and Stability of Periodic Solution to Delayed Nonlinear Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Xiang Gu ◽  
Huicheng Wang ◽  
P. J. Y. Wong ◽  
Yonghui Xia
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Lyapunov Functional ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Competition System ◽  
Existence And Stability

The main purpose of this paper is to study the periodicity and global asymptotic stability of a generalized Lotka-Volterra’s competition system with delays. Some sufficient conditions are established for the existence and stability of periodic solution of such nonlinear differential equations. The approaches are based on Mawhin’s coincidence degree theory, matrix spectral theory, and Lyapunov functional.

PERIODIC SOLUTION OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS: VIA TOPOLOGICAL DEGREE THEORY AND MATRIX SPECTRAL THEORY

2012 ◽  
Vol 22 (08) ◽  
pp. 1250196 ◽  
Author(s):  
YONG-HUI XIA
Keyword(s):  
Periodic Solution ◽  
Spectral Theory ◽  
Lyapunov Functional ◽  
Nonlinear Differential Equations ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Topological Degree Theory ◽  
Maximal Eigenvalue ◽  
Existence And Stability ◽  
Better Than

The main purpose of this article is to establish the existence and stability of a periodic solution of nonlinear differential equation connected with a problem from mathematical biology. The existence and stability conditions are given in terms of spectral radius of explicit matrices, which are better than conditions obtained by using classic norms. The approaches are based on Mawhin's coincidence degree theory, matrix spectral theory and Lyapunov functional. It should be noted that the new problem appears due to the introduction of Gilpin–Ayala effect. The standard methods used in the previous literature cannot be used to analyze the asymptotic stability of such systems. To handle this problem, two novel techniques should be employed. One is to rescale the system by [Formula: see text], not [Formula: see text]. The other is to analyze the maximal eigenvalue of matrix. In fact, the analytic technique is nontrivial. It is original and very interesting. Finally, some examples and their simulations show the feasibility of our results.

Dynamics of an N-Species Gilpin–Ayala Impulsive Competition System

2019 ◽  
Vol 20 (7-8) ◽  
pp. 737-746
Author(s):  
Libo Wang ◽  
Guigui Xu
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Almost Periodic Solution ◽  
Asymptotical Stability ◽  
Global Asymptotical Stability ◽  
Almost Periodic ◽  
Competition System

AbstractIn this paper, we consider an N-species Gilpin–Ayala impulsive competition system. By using comparison theorem, Lyapunov functional, and almost periodic functional hull theory of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence, global asymptotical stability, and almost periodic solution of the model. Our results extend some previously known results. The method used in this paper provides a possible method to study the permanence, global asymptotical stability, and almost periodic solution of the models with impulsive perturbations in biological populations.

scholarly journals Application of Mawhin's Coincidence Degree and Matrix Spectral Theory to a Delayed System

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Yong-Hui Xia ◽  
Xiang Gu ◽  
Patricia J. Y. Wong ◽  
Syed Abbas
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Spectral Theory ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Stability Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Existence And Stability ◽  
Delayed System

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model withM-predators andN-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.

Almost Periodic Solution in a Lotka–Volterra Recurrent Neural Networks with Time-Varying Delays

2017 ◽  
Vol 18 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Li Yang ◽  
Zhouhong Li ◽  
Liyan Pang ◽  
Tianwei Zhang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Almost Periodic Solution ◽  
Almost Periodic ◽  
Mawhin’S Continuation Theorem ◽  
Existence And Stability

Abstract:By means of Mawhin’s continuation theorem of coincidence degree theory and Lyapunov function, some simple sufficient conditions are obtained for the existence and stability of a unique positive almost periodic solution of a delayed Lotka–Volterra recurrent neural networks. To a certain extent, the work in this paper corrects the defect of a recent paper. Finally, an example and simulations are given to illustrate the feasibility and effectiveness of the main result.

scholarly journals Existence and Global Attractivity of Positive Periodic Solutions for a Two-Species Competitive System with Stage Structure and Impulse

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Changjin Xu ◽  
Daxue Chen
Keyword(s):  
Periodic Solution ◽  
Lyapunov Functional ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Stage Structure ◽  
Positive Periodic Solution ◽  
Positive Periodic Solutions ◽  
Competitive System

A class of nonautonomous two-species competitive system with stage structure and impulse is considered. By using the continuation theorem of coincidence degree theory, we derive a set of easily verifiable sufficient conditions that guarantee the existence of at least a positive periodic solution, and, by constructing a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are presented. Finally, an illustrative example is given to demonstrate the correctness of the obtained results.

scholarly journals Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks

2005 ◽  
Vol 2005 (3) ◽  
pp. 281-297 ◽  
Author(s):  
Hong Xiang ◽  
Ke-Ming Yan ◽  
Bai-Yan Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Global Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
High Order ◽  
Priori Estimates

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.

EXISTENCE AND GLOBAL EXPONENTIAL STABILITY OF PERIODIC SOLUTION OF A CLASS OF IMPULSIVE NETWORKS WITH INFINITE DELAYS

2007 ◽  
Vol 17 (01) ◽  
pp. 35-42 ◽  
Author(s):  
YONGHUI XIA ◽  
JINDE CAO ◽  
MUREN LIN
Keyword(s):  
Periodic Solution ◽  
Exponential Stability ◽  
Lyapunov Functions ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Mawhin’S Continuation Theorem ◽  
Neuron Networks ◽  
Infinite Delays

Sufficient conditions are obtained for the existence and global exponential stability of a unique periodic solution of a class of impulsive tow-neuron networks with variable and unbounded delays. The approaches are based on Mawhin's continuation theorem of coincidence degree theory and Lyapunov functions.

scholarly journals On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response

2021 ◽  
Vol 73 (4) ◽  
pp. 523-543
Author(s):  
N. N. Pelen
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Analytic Structure ◽  
Predator Prey ◽  
Periodic Environment ◽  
Globally Attractive ◽  
Necessary And Sufficient

UDC 517.9 In this study, the two-dimensional predator-prey system with Beddington–DeAngelis type functional response with impulses is considered in a periodic environment. For this special case, necessary and sufficient conditions are found for the considered system when it has at least one -periodic solution. This result is mainly based on the continuation theorem in the coincidence degree theory and to get the globally attractive -periodic solution of the given system, an inequality is given as the necessary and sufficient condition by using the analytic structure of the system.  

Periodic Solution of a Class Predator-Prey System with Rate Stocking and Time Delay

2013 ◽  
Vol 291-294 ◽  
pp. 2412-2415
Author(s):  
Hui Li ◽  
Yi Fei Wang
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Periodic System ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solution ◽  
Coincidence Degree Theory ◽  
Predator Prey ◽  
Prey System

In this paper, we investigate of a class of predator-prey system with rate stocking and time delay, the existence positive periodic solution by using coincidence degree theory. We obtain the sufficient conditions which guarantee existence of the positive periodic solution of the periodic system. Some new results obtained.

scholarly journals Existence of periodic solution for first order nonlinear neutral delay equations

2001 ◽  
Vol 14 (2) ◽  
pp. 189-194 ◽  
Author(s):  
Genqiang Wang ◽  
Jurang Yan
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Neutral Delay ◽  
First Order ◽  
Neutral Delay Differential Equation ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper by using the coincidence degree theory, sufficient conditions are given for the existence of periodic solutions of the first order nonlinear neutral delay differential equation.

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