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.     

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.     

Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.     

Solvability of eight classes of nonlinear systems of difference equations

We have studied recently solvability and semi‐cycles of 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, in detail. This paper is devoted to the study of the other eight systems of the form. We show that these systems are also solvable in closed form and describe semi‐cycles of their solutions complementing our previous results on such systems of difference equations.     

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.     

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.     

EXISTENCE OF PERIODIC SOLUTIONS OF SYSTEM OF DIFFERENCE EQUATIONS

This paper studies the nonautonomous nonlinear system of difference equationsΔx(n)=A(n)x(n)+f(n,x(n)),n∈Z,(*) where x(n)∈R~N,A(n)=(a_(ij)(n))N×N is an N×N matrix,with a-(ij)∈C(R,R) for i,j= 1,2,3,...,N,and f=(f_1,f_2,...,f_N)~T∈C(R×R~N,R~N),satisfying A(t+ω)=A(t),f(t+ω,z)=f(t,z) for any t∈R,(t,z)∈R×R~N andωis a positive integer.Sufficient conditions for the existence ofω-periodic solutions to equations (*) are obtained.     

Solution form of a higher‐order system of difference equations and dynamical behavior of its special case

The solution form of the system of nonlinear difference equations where the coefficients a ,b ,α ,β and the initial values x  _? i ,y  _? i ,i ∈{0,1,…,k } are non‐zero real numbers, is obtained. Furthermore, the behavior of solutions of the aforementioned system when p  = 1 is examined. Copyright © 2016 John Wiley & Sons, Ltd.     

正负系数中立型差分方程的全局吸引性

A neutral difference equation with positive and negative coefficients△(xn-cnxn-k) pnxn-l-qnxn-r=0,n=0,1,2.…，is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.     

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 the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

Solving the linear difference equation with periodic coefficients via Fibonacci sequences

ABSTRACT

The aim of this paper is to establish explicit solutions of homogeneous linear difference equations with periodic coefficients. For this purpose, we get around the problem by converting each equation of this class to an equivalent linear difference equation with constant coefficients. Second, we provide some expressions of the solutions via the combinatorial and the Binet formulas of weighted generalized Fibonacci sequences. Finally, some numerical examples and applications are proposed.     

正负系数中立型差分方程的正解的存在性

In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions in the literature     

Asymptotical stability of partial difference equations with variable coefficients

The dynamical systems considered have scalar state, are multivariate, linear, time-discrete, and time-variable and are described by an initial value problem for a class of evolutionary partial difference equations. The time set is the nonnegative part of the integer lattice in several dimensions. Parts of the asymptotical stability set in the parameter space spanned by the time-variable coefficients are explicitly found. To assess the quality of the sufficient stability criteria, a comparison with the exact stability set is made in an example.