Global asymptotic stability of a higher order rational difference equation

In this note, we consider the following rational difference equation:

     

Global attractivity of a higher-order nonlinear difference equation

The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation

     

Global stability of a rational symmetric difference equation

Motivated by ideas from papers: C. Cinar, I. Yalçinkaya, S. Stević, A note on global asymptotic stability of a family of rational equations, Rostock. Math. Kolloq. 59 (2004), 41–49, and S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl. 316 (2006), 60–68, here we confirm a conjecture on a rational symmetric difference equation from the paper: K. Berenhaut, J. Foley, S. Stević, The global attractivity of the rational difference equation , Appl. Math. Lett. 20 (2007), 54–58.     

Global stability for a class of difference equation

Abstract. In this paper the global stability of difference equation     

Global asymptotic stability of a family of difference equations

We consider the family of difference equations of the form

     

Global behavior of a higher order nonlinear difference equation

In this paper, we consider a higher order difference equation of the form

     

Stationary patterns and stability in a tumor-immune interaction model with immunotherapy

This paper examines a diffusive tumor-immune system with immunotherapy under homogeneous Neumann boundary conditions. We first investigate the large-time behavior of nonnegative equilibria and then explore the persistence of solutions to the time-dependent system. In particular, we present the sufficient conditions for tumor-free states. We also determine whether nonconstant positive steady-state solutions (i.e., a stationary pattern) exist for this coupled reaction-diffusion system when the parameter of the immunotherapy effect is small. The results indicate that this stationary pattern is driven by diffusion effects. For this study, we employ the comparison principle for parabolic systems and the Leray-Schauder degree.     

Global behavior for a fourth-order rational difference equation

In this paper, we use a method different from the known literature to investigate the global behavior of the following fourth-order rational difference equation:

     

Global stability of some classes of higher-order nonlinear difference equations

We give elegant and short proofs of some recent results on global stability of some classes of higher-order nonlinear difference equations.     

Global stability of a generalized epidemic model

The competitive exclusion principle is one of the most interesting and important phenomena in both theoretical epidemiology and biology. We show that the equilibrium in which only the strain with the maximum basic reproductive number exists is globally asymptotically stable by using an average Lyapunov function theorem and some dynamical system theory. This result is anticipated by H.J. Bremermann and H.R. Thieme (1989) [6] where they showed that the equilibrium is locally stable — the global result has not been established previously.     

Dynamics of a higher-order rational difference equation

We consider a higher order rational difference equation. Firstly, we skillfully give a sufficient and necessary condition for the existence and uniqueness of the initial value problem. And then we investigate the local stability, asymptotic behavior, periodicity and oscillation of solutions for the difference equation. Finally, we give some numerical simulations to illustrate our results.     

一类二阶非线性差分方程的全局吸引性

考虑二阶非线性差分方程xn＋1=a＋bxn/A＋xn-1,n=0,1,2,….证明了当条件a,b,A∈（0,∞）成立时方程的唯一正平衡点x^-=（b-A＋√（（b-A）2＋4a））～（1/2））/2是方程的所有正解的一个全局吸引子,所得推论证明了由Kocic和Ladas提出的一个猜想是正确的.     

Global asymptotical stability in a rational difference equation

In this paper we prove a global attractivity result for the unique positive equilibrium point of a difference equation, which improves and generalizes some known ones in the existing literature. Especially, our results completely solve an open problem and some conjectures proposed in [1, 2, 3, 4].     

Qualitative properties for a fourth-order rational difference equation

In this paper, we use a method different from the known literature to investigate the qualitative properties of the following fourth-order rational difference equation:

     

A stability criterion for a neutral difference equation with delay

A stability criterion for a neutral difference equation with delay is established which extends and improves a result of Ladas et al. [1]. Our derivation is based on Lyapunov's direct method for stability, and avoids the approach employed by Ladas et al. who had considered asymptotic behaviors of oscillatory and nonoscillatory solutions.     

Global stability and asymptotics of some classes of rational difference equations

In this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable. Using a Berg's result, we also find asymptotics of some solutions of these equations.     

Global stability in difference equations satisfying the generalized Yorke condition

We present several conditions sufficient for global stability of the zero solution of nonautonomous difference equation xn+1=qxn+fn(xn,…,xn−k), n∈Z, when the nonlinearities fn satisfy a sort of negative feedback condition. Moreover, for every k∈N, we indicate qk such that one of our stability conditions is sharp if q∈(0,qk]. We apply our results to discrete versions of Nicholson's blowflies equation, the Mackey-Glass equations, and the Wazewska and Lasota equation.     

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form

     

高阶非线性差分方程的全局吸引性

研究了一类高阶非线性差分方程所有正解的周期性,不变区间及全局吸引性.证明了方程的正平衡点是在一个依赖于参数的盆里的全局吸引子.