scholarly journals Global Attractivity in a Discrete Mutualism Model with Infinite Deviating Arguments

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Xiangdong Xie ◽  
Yalong Xue ◽  
Runxin Wu
Keyword(s):  
Global Attractivity ◽  
Sufficient Conditions ◽  
Deviating Arguments ◽  
Mutualism Model

A set of sufficient conditions is obtained for the global attractivity of the following two-species discrete mutualism model with infinite deviating arguments: x1(k+1)=x1kexp⁡r1(K1+α1∑s=0+∞J2sx2k-s)/(1+∑s=0+∞J2sx2k-s)-x1k and  x2(k+1)=x2kexp⁡r2(K2+α2∑s=0+∞J1sx1k-s)/(1+∑s=0+∞J1sx1k-s)-x2k, where ri,Ki,αi, i=1,2, are all positive constants, ∑j=1+∞Ji(n)=1, and αi>Ki. Our results generalize the main result of Yang et al. (2014).

scholarly journals Permanence in a Discrete Mutualism Model with Infinite Deviating Arguments and Feedback Controls

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Changjin Xu ◽  
Yusen Wu
Keyword(s):  
Sufficient Conditions ◽  
Feedback Controls ◽  
Difference Inequality ◽  
Deviating Arguments ◽  
Inequality Theory ◽  
The Difference ◽  
Mutualism Model

We propose and deal with a discrete mutualism model with infinite deviating arguments and feedback controls. Sufficient conditions which guarantee the permanence of the system are obtained by using the difference inequality theory. The paper ends with brief conclusions.

scholarly journals Global Stability of a Discrete Mutualism Model

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Kun Yang ◽  
Xiangdong Xie ◽  
Fengde Chen
Keyword(s):  
Iterative Method ◽  
Comparison Principle ◽  
Local Stability ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Global Asymptotical Stability ◽  
Interior Equilibrium ◽  
Linear Approximation Method ◽  
Mutualism Model

A discrete mutualism model is studied in this paper. By using the linear approximation method, the local stability of the interior equilibrium of the system is investigated. By using the iterative method and the comparison principle of difference equations, sufficient conditions which ensure the global asymptotical stability of the interior equilibrium of the system are obtained. The conditions which ensure the local stability of the positive equilibrium is enough to ensure the global attractivity are proved.

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

1997 ◽  
Vol 4 (6) ◽  
pp. 557-566
Author(s):  
B. Půža
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Vector Function ◽  
Unique Solvability ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Deviating Arguments ◽  
Measurable Functions

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

The weighted Cauchy problem for linear functional differential equations with strong singularities

2011 ◽  
Vol 18 (3) ◽  
pp. 577-586
Author(s):  
Zaza Sokhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Initial Point ◽  
Higher Order ◽  
Well Posedness ◽  
Deviating Arguments ◽  
Functional Differential

Abstract The sufficient conditions of well-posedness of the weighted Cauchy problem for higher order linear functional differential equations with deviating arguments, whose coefficients have nonintegrable singularities at the initial point, are found.

scholarly journals Permanence and Positive Periodic Solutions of a Discrete Delay Competitive System

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Wenjie Qin ◽  
Zhijun Liu
Keyword(s):  
Periodic Solutions ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Discrete Delay ◽  
Positive Periodic Solutions ◽  
Competitive System ◽  
Discrete Function ◽  
Species Population

A discrete time non-autonomous two-species competitive system with delays is proposed, which involves the influence of many generations on the density of species population. Sufficient conditions for permanence of the system are given. When the system is periodic, by using the continuous theorem of coincidence degree theory and constructing a suitable Lyapunov discrete function, sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions are obtained. As an application, examples and their numerical simulations are presented to illustrate the feasibility of our main results.

scholarly journals Permanence and Global Attractivity of a Discrete Two-Prey One-Predator Model with Infinite Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Zhixiang Yu ◽  
Zhong Li
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Functional ◽  
Global Attractivity ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Predator Model

A discrete two-prey one-predator model with infinite delay is proposed. A set of sufficient conditions which guarantee the permanence of the system is obtained. By constructing a suitable Lyapunov functional, we also obtain sufficient conditions ensuring the global attractivity of the system. An example together with its numerical simulation shows the feasibility of the main results.

Positive solutions to Sturm–Liouville problems with non-local boundary conditions

2014 ◽  
Vol 144 (1) ◽  
pp. 119-138 ◽  
Author(s):  
T. Jankowski
Keyword(s):  
Positive Solutions ◽  
Sufficient Conditions ◽  
Deviating Arguments ◽  
Local Boundary ◽  
Second Order Differential Equations ◽  
Existence Of Positive Solutions ◽  
Delayed Arguments ◽  
Non Local ◽  
Sturm Liouville ◽  
Advanced And Delayed Arguments

In this paper, the existence of at least three non-negative solutions to non-local boundary-value problems for second-order differential equations with deviating arguments α and ϛ is investigated. Sufficient conditions, which guarantee the existence of positive solutions, are obtained using the Avery–Peterson theorem. We discuss our problem for both advanced and delayed arguments. An example is added to illustrate the results.

scholarly journals Global Attractivity of a Periodic DelayedN-Species Model of Facultative Mutualism

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Ahmadjan Muhammadhaji ◽  
Zhidong Teng
Keyword(s):  
Lyapunov Function ◽  
Periodic Solutions ◽  
Function Method ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solutions ◽  
Facultative Mutualism ◽  
Lyapunov Function Method

Two classes of periodicN-species Lotka-Volterra facultative mutualism systems with distributed delays are discussed. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method, some new sufficient conditions on the existence and global attractivity of positive periodic solutions are established.

scholarly journals Oscillation Properties for Systems of Higher-Order Partial Differential Equations with Distributed Deviating Arguments

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Youjun Liu ◽  
Jianwen Zhang ◽  
Jurang Yan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Partial Differential ◽  
Deviating Arguments ◽  
Distributed Deviating Arguments ◽  
Functional Differential ◽  
Oscillation Properties

New sufficient conditions are obtained for oscillation for the solutions of systems of a class of higher-order quasilinear partial functional differential equations with distributed deviating arguments. The obtained results are illustrated by example.

scholarly journals Existence and uniqueness of a periodic solution to certain third order nonlinear delay differential equation with multiple deviating arguments

2013 ◽  
Vol 5 (2) ◽  
pp. 113-1 ◽  
Author(s):  
Adeleke Timothy Ademola
Keyword(s):  
Periodic Solution ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Third Order ◽  
Deviating Arguments ◽  
Multiple Deviating Arguments ◽  
Uniform Ultimate Boundedness ◽  
Lyapunov’S Second Method ◽  
Delay Differential ◽  
Nonlinear Delay Differential Equation

Abstract In this paper, we use Lyapunov’s second method, by constructing a complete Lyapunov functional, sufficient conditions which guarantee existence and uniqueness of a periodic solution, uniform asymptotic stability of the trivial solution and uniform ultimate boundedness of solutions of Eq. (2). New results are obtained and proved, an example is given to illustrate the theoretical analysis in the work and to test the effectiveness of the method employed. The results obtained in this investigation extend many existing and exciting results on nonlinear third order delay differential equations.

Sign in / Sign up

Export Citation Format

Share Document