On a nonautonomous difference equation with bounded coefficient

In this paper we investigate the boundedness, the persistence and the attractivity of the positive solutions of the nonautonomous difference equation

     

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

     

The existence of continuous invariants for linear difference equations

We study real continuous invariants for systems of linear difference equations. We shall prove a conjecture by Ladas about the existence of such invariants. In fact, necessary and sufficient conditions on existence of such invariants will be established. The invariants will be constructed when they exist.     

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

Boundedness,persistence and extinction of a stochastic non-autonomous logistic system with time delays

This paper investigates the stochastic non-autonomous logistic system with time delays. Under two simple assumptions on the environmental noise, it is shown that the stochastic system has a unique global positive solution, and this positive solution is asymptotically bounded. The conditions for extinction, weak persistence of solutions are also obtained by the exponential martingale inequality. Finally, a numerical example is provided to illustrate our results.     

A note on thetheoretical existence of invariants: Part II

We propose a framework in which to study invariants of difference equations with the ultimate goal of establishing the (theoretical) existence of effective invariants. The paper is divided into two parts.Part I began with several examples which illustrated some of the difficulties to be overcome and ended with a discussion of the connection between invariants and decompositions of the phase space.In PartII, which is more topological in nature, we now pursue the theme of invariant decompositions     

Boundedness of extremal solutions in dimension 4

In this paper we establish the boundedness of the extremal solution u^∗$u^{\ast }$ in dimension N=4$N=4$ of the semilinear elliptic equation −Δu=λf(u)$-\Delta u=\lambda f\left(u\right)$, in a general smooth bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, with Dirichlet data u_|∂Ω=0$u{|}_{\partial \Omega }=0$, where f$f$ is a C¹$C^1$ positive, nondecreasing and convex function in [0,∞)$[0,\infty )$ such that f(s)/s→∞$f\left(s\right)/s\to \infty$ as s→∞$s\to \infty$.     

Stability of abstract nonlinear nonautonomous differential-delay equations with unbounded history-responsive operators

We consider a class of nonautonomous functional-differential equations in a Banach space with unbounded nonlinear history-responsive operators, which have the local Lipshitz property. Conditions for the boundedness of solutions, Lyapunov stability, absolute stability and input-output one are established. Our approach is based on a combined usage of properties of sectorial operators and spectral properties of commuting operators.     

A Note on the Difference Equation x n +1 = ∑ i =0 k α i x n − i p i

In this note we improve Theorem 2 in Ref. [3] , about the difference equation x n +1 = ~ i =0 k f i x n m i p i , n =0,1,2,..., where k is a positive integer, f i , p i ] (0, X ) for i =0,..., k , and the initial conditions x m k , x m k +1 ,..., x 0 are arbitrary positive numbers.     

Oscillations of the system of linear nonautonomous difference equations

We consider the linear nonautonomous system of difference equations x_n+1−x_n+P(n)x_n−k=0, n=0,1,2,… , where k∈Z, P(n)∈R^rxr. We obtain sufficient conditions for the system to be oscillatory. The conditions based on the eigenvalues of the matrix coefficients of the system.     

Pullback attractors in nonautonomous difference equations

Nonautonomous difference equations are formulated as cocycles which generalize semigroups corresponding to autonomous difference equations. Pullback attractors are the appropriate generalization of autonomous attractors to cocycles. The existence of a pullback attractor follows when the difference equation cocycle has a pullback absorbing set. Results from the literature are outlined, including the construction of a Lyapunov function characterizing pullback attraction, and illustrated with several examples.     

Sufficient conditions for the global stability of nonautonomous higher order difference equations

We present some explicit sufficient conditions for the global stability of the zero solution in nonautonomous higher order difference equations. The linear case is discussed in detail. We illustrate our main results with some examples. In particular, the stability properties of the equilibrium in a nonlinear model in macroeconomics is addressed.     

On a class of third-order nonlinear difference equations

This paper studies the boundedness character of the positive solutions of the difference equation