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.

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.

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 A third-order multi-point boundary value problem at resonance with one three dimensional kernel space

2014 ◽  
Vol 30 (1) ◽  
pp. 93-100
Author(s):  
XIAOJIE LIN ◽  
◽  
BENSHENG ZHAO ◽  
ZENGJI DU ◽  
◽  
...  
Keyword(s):  
Nonlinear Differential Equations ◽  
Three Dimensional ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Existence Results ◽  
Point Boundary ◽  
Kernel Space ◽  
Third Order ◽  
At Resonance ◽  
Problem At Resonance

This paper deals with a third order nonlinear differential equations with multi-point boundary conditions. By using the coincidence degree theory, we establish some existence results of the problem at resonance under some appropriate conditions. The emphasis here is that the dimension of the linear operator is equal to three. We also give an example to demonstrate our results.

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.  

POSITIVE PERIODIC SOLUTION FOR A GAUSE-TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH DIFFUSION AND TIME DELAY

2008 ◽  
Vol 01 (03) ◽  
pp. 339-354 ◽  
Author(s):  
XIAOQUAN DING ◽  
YUANYUAN WANG
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solution ◽  
Continuation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
System With Time Delay

A two-species Gause-type ratio-dependent predator-prey system with time delay in a two-patch environment is investigated. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solution for the system. As corollaries, some applications are listed. In particular, our results extend and improve some known results.

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