Nonlinear Difference Equation Xn Research Articles

Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci sequence

In this paper, we solve in closed-form the following third-order system of nonlinear difference equations xn+1 = ynyn−1xn−1p xn(anyn−2q + bnynyn−1),yn+1 = xnxn−1yn−1q yn(cnxn−2p + dnxnxn−1),p,q ∈ ℕ,n ∈ ℕ0 where the initial values x−i,y−i,i = 0,1,2 and the parameters (an)n∈ℕ0,(bn)n∈ℕ0,(cn)n∈ℕ0, (dn)n∈ℕ0 are non-zero real numbers. The form of the solutions of the one dimensional case of our system and a more general system defined by one to one functions are also presented.

On the Solutions of a System of Third-Order Rational Difference Equations

The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn-2), yn+1=xnxn-2/(±yn-1±xn-2), n=0,1,…, is clarified in which the initial conditions x-2, x-1, x0, y-2, y-1, y0 are considered as arbitrary positive real numbers. To exemplify the theoretical discussion, some numerical examples are presented.

差分方程x<sub>n+1</sub> =x<sub>n</sub><sup style="margin-left:-8px;">2</sup>+ x<sub>n-1</sub><sup style="margin-left:-8px;">2</sup>解的渐近性质

本文讨论了非线性差分方程xn+1 =xn2+ xn-12 的解的渐近性质，给出了零解的收敛域的一个子域以及得到了初始值x0,x1 在满足一定的条件下其解发散到无穷大的结论。 This paper considers a nonlinear difference equation xn+1 =xn2+ xn-12 with the initial values .The sufficient conditions under which the solutions converge to zero or diverge to infinity have been obtained.

On the dynamics of the nonlinear rational difference equation xn+1=Axn+Bxn−k+Cxn−l+bxn−k/dxn−k−exn−l

Abstract In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation x n + 1 = Ax n + Bx n - k + Cx n - l + bx n - k dx n - k - ex n - l , n = 0 , 1 , 2 , … where the coefficients A , B , C , b , d , e ∈ ( 0 , ∞ ) , while k and l are positive integers. The initial conditions x - l , … , x - k , … , x - 1 , x 0 are arbitrary positive real numbers such that k l . Some numerical examples will be given to illustrate our results.

On the Dynamics of the Solutions of the Rational Recursive Sequences

In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation xn+1 = Axn +Bxn−k + Cxn−l +Dxn−σ + bxnxn−kxn−l dxn−k − exn−l , n = 0, 1, 2, ..... where the coefficients A,B,C,D, b, d, e ∈ (0,∞), while k, l and σ are positive integers. The initial conditions x−σ,..., x−l,..., x−k, ..., x−1, x0 are arbitrary positive real numbers such that k < l < σ. Some numerical examples will be given to illustrate our results.

Dynamical Behavior of a System of Second-Order Nonlinear Difference Equations

This paper is concerned with local stability, oscillatory character of positive solutions to the system of the two nonlinear difference equationsxn+1=A+xn-1p/ynpandyn+1=A+yn-1p/xnp,n=0,1,…, whereA∈(0,∞),p∈[1,∞),xi∈(0,∞), andyi∈(0,∞),i=-1,0.

On the dynamics of the nonlinear difference equation xn+1=α+βxn−1+xn−1/xn

On the dynamics of the nonlinear difference equation xn+1=α+βxn−1+xn−1/xn

Global Attractivity of a Rational Difference Equation

In this paper, we investigate the global attractivity of positive solutions of the nonlinear difference equation xn+1 = ax

BEHAVIOR OF SOLUTIONS TO A FUZZY NONLINEAR DIFFERENCE EQUATION

In this paper, we study the existence, asymptotic behavior of the positive solutions of a fuzzy nonlinear difference equation xn+1 = Axn + xn 1 B + xn 1 ; n = 0; 1; ; where (xn) is a sequence of positive fuzzy number, A;B are positive fuzzy numbers and the initial conditions x 1;x0 are positive fuzzy numbers.

Dynamic behavior of a nonlinear rational difference equation and generalization

This paper is concerned about the dynamic behavior for the following high order nonlinear difference equation xn =( xn-k + xn-m + xn-l)/(xn-kxn-m + xn-mxn-l +1) with the initial data {x−l,x−l+1, ... ,x−1 }∈ R l and 1 ≤ k ≤ m ≤ l. The convergence of solution to this equation is investigated by introducing a new sequence, which extends and includes corresponding results obtained in the references (Li in J Math Anal Appl 312:103-111, 2005; Berenhaut et al. Appl. Math. Lett. 20:54-58, 2007; Papaschinopoulos and Schinas J Math Anal Appl 294:614-620, 2004) to a large extent. In addition, some propositions for generalized equations are reported.

Global asymptotic stability of equilibrium point for a family of rational difference equations

This paper is concerned with the following nonlinear difference equations xn+1=∑i=1lAsixn−siB+C∏j=1kxn−tj,n=0,1,…, where the initial data x−m,x−m+1,…,x−1,x0∈R+, m=max{s1,…,sl,t1,…,tk}, s1,…,sl,t1,…,tk are non-negative integers, and Asi,B,C are arbitrary positive real numbers. We give sufficient conditions under which the unique equilibrium x̄=0 of this equation is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references Cinar (2004) [6], Yang et al. (2005) [7] and Berenhaut et al. (2007) [8].

Dynamics of a Higher‐Order Nonlinear Difference Equation

We consider the higher‐order nonlinear difference equation xn+1 = (α + xn)/(A + Bxn + xn−k), n = 0, 1, …, where parameters are positive real numbers and initial conditions x−k, …, x0 are nonnegative real numbers, k ≥ 2. We investigate the periodic character, the invariant intervals, and the global asymptotic stability of all positive solutions of the abovementioned equation. We show that the unique equilibrium of the equation is globally asymptotically stable under certain conditions.

Existence of Monotone Solutions of a Difference Equation

We consider the nonlinear difference equationxn+1=f(xn−k,xn−k+1,…,xn),n=0,1,…,wherek∈{1,2,…}and the initial valuesx−k,x−k+1,…,x0∈(0,+∞). We give sufficient conditions under which this equation has monotone positive solutions which converge to the equilibrium, extending and including in this way some results of the literature.

Global Attractivity of the Equilibrium of a Nonlinear Difference Equation

The authors consider the nonlinear difference equation xn+1=αxn+xn-kf(xn-k),n=0,1,(0.1) where α∈(0,1), k ∈ 0,1, and f ∈ C1[[0,∞), [0,∞)] with f′(x) < 0.They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.

On the recursive sequence xn+1 = −1/xn + A/xn−1

We investigate the periodic character of solutions of the nonlinear difference equation xn+1 = −1/xn + A/xn−1. We give sufficient conditions under which every positive solution of this equation converges to a period two solution. This confirms a conjecture in the work of DeVault et al. (2000).

On the system of two nonlinear difference equations xn+1 = A + xn−1/yn, yn+1 = A + yn−1/xn

We study the oscillatory behavior, the periodicity and the asymptotic behavior of the positive solutions of the system of two nonlinear difference equations xn+1 = A + xn−1/yn and yn+1 = A + yn−1/xn, where A is a positive constant, and n = 0, 1, ….

On a System of Two Nonlinear Difference Equations

In this paper we study the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equationsxn+1=A+yn/xn−p,yn+1=A+xn/yn−q,n=0,1,…,p,qare positive integers.

Piecewise polynomial galerkin approximation to invariant densities of one‐dimensional difference equations

AbstractProposed herein is a piecewise polynomial Galerkin approximation to invariant densities of 1‐dimensional nonlinear difference equations xn+1 = τ(xn) (n = 0,1…), in which map τ is an endmorphism of an interval I. The piecewise polynomial is a linear combination of N × K polynomial bases of degree K‐1; N is the number of partitioned subintervals of I. The approximate solution for K = 1 is shown to be identical to the Ulam‐Li solution. Forthermore, the sufficient condition for the L1 convergence of the approximate solution (K = 2, 3) for large N is given. Numerical experiments demonstrate that the approximate solution for K = 3 produces a significant improvement of the Ulam‐Li solution.