Input-output admissibility and exponential trichotomy of difference equations

We propose a new method for the study of the asymptotic behavior of difference equations in infinite-dimensional spaces, providing characterizations for the property of uniform exponential trichotomy. We deduce the structure of the stable, unstable and bounded subspace and prove the uniqueness of the projection families. We introduce a new admissibility concept with respect to a discrete input-output system and prove that this is a necessary and sufficient condition for the existence of uniform exponential trichotomy. Throughout the paper, there is no assumption on the coefficients and the obtained results are applicable to any class of difference equations.     

Asymptotic behaviour of solutions of rational difference systems

In this article, we investigate the asymptotic behaviour of solutions of systems of rational difference equations in arbitrary dimensions. We give conditions for the parameters ensuring that the positive solutions of the considered system are bounded, unbounded, increasing, decreasing, and convergent, respectively.     

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 of a rational difference equation

In this paper, we study the global stability of the difference equation , where the parameters a,a_i(0,∞) for i=0,…,k, x_-k,…, x_-1[0,∞) and x₀(0,∞). We prove that the unique positive equilibrium is globally asymptotically stable if and only if it is locally asymptotically. Also we provide sufficient condition for it to be globally asymptotically stable and our results solve the open problem proposed by Kulenović and Ladas (Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002).     

Global asymptotic stability of a higher order rational difference equation

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

     

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.     

The global attractivity of a higher order rational difference equation

This paper studies global asymptotic stability for positive solutions to the equation

     

On the boundedness of some rational difference equations

We study the boundedness character of solutions of some third order rational difference equations and we confirm some of the conjectures posed in Camouzis et al. [“Progress report on the boundedness character of third-order rational equations”, Journal of Difference Equations and Applications 11 (2005), 1029–1035] and [“On third order rational difference equations, part 6”, Journal of Difference Equations and Applications 11 (2005), 759–777].     

Stability of the Gumowski-Mira equation with period-two coefficient

We investigate the stability of solutions of the Gumowski-Mira equation with a period-two coefficient:

     

On the geometry of a four-parameter rational planar system of difference equations

In this paper, we consider geometric aspects of a rational, planar system of difference equations defined on the open first quadrant and whose behaviour is governed by four independent, non-negative parameters. This system, indexed as (23, 23) in the notation of Ladas (Open problems and conjectures, J. Differential Equ. Appl. 15(3) 2009, pp. 303–323), is one of the 200 systems from Ladas about which little is known. Using geometric techniques, we answer several questions concerning the behaviour of this system.     

Linear barycentric rational collocation method for solving heat conduction equation

The linear barycentric rational collocation method for solving heat conduction equation is presented. The matrix form of discrete heat conduction equation by collocation method is also obtained. With the help of convergence rate of the barycentric interpolation, the convergence rate of linear barycentric rational collocation method for solving heat conduction equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.     

On the solution of a third order linear homogeneous difference equation with variable coefficents

A closed form solution of a third order linear homogeneous difference equation with variable coefficients is presented. From it, the solution for the special case of an equation with constant coefficients is also obtained.     

Average implicit linear difference scheme for generalized Rosenau-Burgers equation

In this paper, a linear three-level average implicit finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-Burgers equation is presented. Existence and uniqueness of numerical solutions are discussed. It is proved that the finite difference scheme is convergent in the order of O(τ² + h²) and stable. Numerical simulations show that the method is efficient.     

The value distribution of meromorphic solutions of some second order nonlinear difference equation

In this paper, we investigate some properties of solutions for some nonlinear difference equation, and obtain some estimates of the exponent of convergence of poles and growth of its transcendental meromorphic solutions $f(z)$ and its difference $\Delta f(z)$. Moreover, we study the existence and forms of rational solutions. We also give some examples to support our theoretical discussion.