Two classes of piecewise-linear difference equations with eventual periodicity six

In this paper, we will study two classes of difference equations which are piecewise-linear and of similar forms. We will show that all non-trivial solutions of both equations are eventually periodic with prime period six.     

On global periodicity of difference equations

In this research, we consider a difference equation of order k?2 of the following form:

     

Periodically forced nonlinear systems of difference equations

The existance of nontrivial (x=0( periodic solutions of a general class of periodic nonlinear difference equations is proved using bifurcation theory methods. Specifically, the existance of a global continuum of nontrivial periodicsolutions that bifurcates from the trivial solution (x=0) is proved. Conditions are given under which the nontrivial solutions are positive. A prerrequisite Fredholm and adjoint operator theory for linear periodic systems is developed. An application to application dynamics is made.     

On the Behavior of Solutions of x n+1 = p+(x n−1/xn )

Our goal in this article is to complete the study of the behavior of solutions of the equation in the title when the parameter p is positive and the initial conditions are arbitrary positive numbers. Our main focus is the case 0 < p < 1. We will show that in this case, all solutions which do not monotonically converge to the equilibrium have a subsequence which converges to p and a subsequence which diverges to infinity. For the sake of completeness, we will also present the results (which were previously known) with alternative proofs for the case p = 1 and the case p > 1.     

Periodic solutions of the Lyness max equation

We investigate the periodic nature of solutions of a “max-type” difference equation sometimes referred to as the “Lyness max” equation. The equation we consider is x_n+1=max{x_n,A}/x_n−1, n=0,1,…, where A is a positive real parameter and the initial conditions are arbitrary positive numbers. We also present related results for a similar equation sometimes referred to as the “period 7 max” equation.     

Existence of multi-valued solutions with asymptotic behavior of Hessian equations

In this paper, we use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of Hessian equations.     

Asymptotic behavior of solutions for a class of retarded difference equations

Consider the retarded difference equationx _n −x _n−1 =F(−f(x _n )+g(x _n−k )), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞. Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan     

Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations

     

Existence of periodic and subharmonic solutions for second order functional difference equations

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.     

Asymptotic behavior of nonlinear difference systems

In this paper, we consider two-dimensional nonlinear difference systems of the form

We classify their solutions according to asymptotic behavior and give some necessary and sufficient conditions for the existence of solutions of such classes.     

Trichotomy of a system of two difference equations

In this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations

     

On two classes of autonomous third-order nonlinear ordinary differential equations

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.     

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when pmax{m,α}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t→∞, as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N=1, the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.     

Periodic solutions to a difference equation with maximum

An open problem posed by G. Ladas is to investigate the difference equation

where are any nonnegative real numbers with 0$">. We prove that there exists a positive integer such that every positive solution of this equation is eventually periodic of period .

     

Asymptotic properties of solutions of nonlinear difference equations

Using discrete inequalities and Schauder's fixed point theorem we study the problem of asymptotic equilibrium for difference equations.     

Global behavior of a three-dimensional linear fractional system of difference equations

We investigate the global asymptotic behavior of solutions of the system of difference equations

where the parameters a, b, c, d, e, and f are in (0,∞) and the initial conditions x₀, y₀, and z₀ are arbitrary non-negative numbers. We obtain some global attractivity results for the positive equilibrium of this system for different values of the parameters.     

Asymptotic solutions for mixed-type equations with a small deviation

An asymptotic matrix solution is formulated for a class of mixed-type linear vector equations with a single variable deviation which is small at infinity. This matrix solution describes the asymptotic behavior of all exponentially bounded solutions. A sufficient condition is obtained for there to be no other solutions.