Qualitative behavior of two systems of higher‐order difference equations

In this paper, we study the qualitative behavior of following two systems of higher‐order difference equations: and where the parameters α,β,γ,α₁,β₁,γ₁,a,b,c,a₁,b₁,andc₁ and the initial conditions x₀, x_?1, ?, x_?k, y₀, y_?1 ,?, y_?k are positive real numbers. More precisely, we study the equilibrium points, local asymptotic stability, instability, global asymptotic stability of equilibrium points, and rate of convergence of positive solutions that converges to the equilibrium point P₀=(0,0) of these systems. Some numerical examples are given to verify our theoretical results. These examples are experimental verification of our theoretical discussions. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability analysis of a system of second‐order difference equations

In this paper, we study the boundedness character and persistence, existence and uniqueness of the positive equilibrium, local and global behavior, and the rate of convergence of positive solutions of the following system of rational difference equations where the parameters α_i,β_i,a_i,b_i for i∈{1,2} and the initial conditions x_?1,x₀,y_?1,y₀ are positive real numbers. Some numerical example are given to verify our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

The form of solutions and periodicity for some systems of third‐order rational difference equations

In this paper, we deal with the system that has solutions and the periodicity character of the following systems of rational difference equations with order three with initial conditions x_?2,x_?1,x₀,y_?2,y_?1, and y₀ that are arbitrary nonzero real numbers. Some numerical examples will be given to illustrate our results. Copyright © 2015 John Wiley & Sons, Ltd.     

可和系数的二阶差分方程解的渐近性

考虑非线性差分方程△(Pn-1△（yn-1)^σ） qnf(yn)=0,n=1,2,3…其中linn→∞∑s=1^nqs存在且为有限给出了方程（E）具有渐近于非零常数解的必要（充分）条件。     

On a class of systems of rational second-order difference equations solvable in closed form

We show that the following nonlinear system of difference equations where parameters a,b,c,d and initial values x₋₁,x₀,y₋₁,y₀ are real numbers, is solvable in closed form, considerably generalizing some recent results. To do this, we use the method of transformation along with several tricks, transforming the system to some known solvable difference equations, by use of which we obtain some closed-form formulas for general solution to the system. The following five cases are considered separately: (1) c=0; (2) d=0; (3) a=0; (4) b=0; and (5) abcd≠0.     

Qualitative analysis of a fourth order difference equation

In this paper, we will investigate some qualitative behavior of solutions of the following fourth order difference equation $x_{n+1}=ax_{n-1}+\frac{bx_{n-1}}{cx_{n-1}-dx_{n-3}},$ \ $n=0,1,...,$ where the initial conditions $x_{-3,}x_{-2},\ x_{-1}$\ and\ $x_{0}\ $are arbitrary real numbers and the values $a,\ b,\ c\ $and$\;d$ are\ defined as positive real numbers.     

On the solutions of a higher‐order difference equation in terms of generalized Fibonacci sequences

This paper deals with the solutions, stability character, and asymptotic behavior of the difference equation where and the initial values x_?k,x_{?k + 1},…,x₀ are nonzero real numbers, such that their solutions are associated to Horadam numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

Qualitative theory for Volterra difference equations

We investigate l^p boundedness, the topological structure of solutions set and the asymptotic periodicity of Volterra functional difference equations. The theoretical results are complemented with a set of applications.     

Two‐dimensional solvable system of difference equations with periodic coefficients

We show that the following two‐dimensional system of difference equations: where , , , and are periodic sequences, is solvable, considerably extending some results in the literature. In the case when all these four sequences are periodic with period 2 or with period 3, we present closed‐form formulas for the general solutions to the corresponding systems of difference equations. Some comments regarding theoretical and practical solvability of the system, connected to the value of the period of the sequences, are given.     

On a symmetric bilinear system of difference equations

We show in an elegant way how the main result in the recent paper Matsunaga and Suzuki (2018) follows from a known result, and discuss the system appearing therein.     

Dynamical behavior and solution of nonlinear difference equation via Fibonacci sequence

In this paper, we study the behavior of the difference equation $x_{n+1}=ax_{n}+\dfrac{bx_{n}x_{n-1}}{cx_{n-1}+dx_{n-2}},~n=0,1,\ldots,$ where the initial conditions $x_{-2},\ x_{-1},\ x_{0}$ are arbitrary positive real numbers and $a,b,c,d$ are positive constants. Also, we give the solution of some special cases of this equation.     

Solvability and semi‐cycle analysis of a class of nonlinear systems of difference equations

An analysis of semi‐cycles of positive solutions to eight systems of difference equations of the following form where a ∈ [0, + ∞), the sequences p_n, q_n, r_n, s_n are some of the sequences x_n and y_n, with positive initial values x_?j,y_?j, j = 1,2, is conducted in detail, and it is shown that these systems can be solved in closed‐form, which is the main result here. Two methods for showing the solvability are described.     

一类高阶非线性中立型差分方程组非振动解的存在性

研究了一类高阶非线性中立型差分方程组非振动解的存在性.利用Banach空间的压缩映象原理,获得了该方程组存在非振动解的充分条件.     

On the global periodicity of discrete dynamical systems and application to rational difference equations

A new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here. This condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations.     

Dynamics of a three‐dimensional systems of rational difference equations

We investigate in this paper the solutions and the periodicity of the following rational systems of difference equations of three‐dimensional with initial conditions x_?2,x_?1,x₀,y_?2,y_?1,y₀,z_?2,z_?1andz₀ are nonzero real numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

On the periodicity of solutions of max‐type difference equation

This paper is devoted to study the periodic nature of the solution of the following max‐type difference equation: where the initial conditions x_?2,x_?1,x₀ are arbitrary positive real numbers and is a periodic sequence of period two. Copyright © 2014 John Wiley & Sons, Ltd.     

Technical note: A fourth‐order finite difference scheme for a system of a 2D nonlinear elliptic partial differential equations

In this article we present a fourth‐order finite difference scheme, for a system of two‐dimensional, second‐order, nonlinear elliptic partial differential equations with mixed spatial derivative terms, using 13‐point stencils with a uniform mesh size h on a square region R subject to Dirichlet boundary conditions. The scheme of order h⁴ is derived using the local solution of the system on a single stencil. The resulting system of algebraic equations can be solved by iterative methods. The difference scheme can be easily modified to obtain formulae for grid points near the boundary. Computational results are given to demonstrate the performance of the scheme on some problems including Navier‐Stokes equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 43–53, 2001     

More on a hyperbolic‐cotangent class of difference equations

We present a natural method for solving the difference equation where , parameter a, and initial values x_?j, , , are real or complex numbers. As a concrete example, the case k = 1, l = 2 is studied in detail, and several interesting formulas for solutions and objects used in the analysis of the equation are given. A useful remark on solvability of difference equations on complex domains is presented and used here.