Global asymptotic behavior and boundedness of positive solutions to an odd-order rational difference equation

AbstractIn this note we consider the following high-order rational difference equation xn=1+∏i=1k(1−xn−i)∑i=1kxn−i,n=0,1,…, where k≥3 is odd number, x−k,x−k+1,x−k+2,…,x−1 is positive numbers. We obtain the boundedness of positive solutions for the above equation, and with the perturbation of initial values, we mainly use the transformation method to prove that the positive equilibrium point of this equation is globally asymptotically stable

Antiperiodic Solutions for a Kind of Nonlinear Duffing Equations with a Deviating Argument and Time-Varying Delay

This paper deals with a kind of nonlinear Duffing equation with a deviating argument and time-varying delay. By using differential inequality techniques, some very verifiable criteria on the existence and exponential stability of antiperiodic solutions for the equation are obtained. Our results are new and complementary to previously known results. An example is given to illustrate the feasibility and effectiveness of our main results

Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks

In the present study, we deal with the stability and the onset of Hopf bifurcation of two type delayed BAM neural networks (integer-order case and fractional-order case). By virtue of the characteristic equation of the integer-order delayed BAM neural networks and regarding time delay as critical parameter, a novel delay-independent condition ensuring the stability and the onset of Hopf bifurcation for the involved integer-order delayed BAM neural networks is built. Taking advantage of Laplace transform, stability theory and Hopf bifurcation knowledge of fractional-order differential equations, a novel delay-independent criterion to maintain the stability and the appearance of Hopf bifurcation for the addressed fractional-order BAM neural networks is established. The investigation indicates the important role of time delay in controlling the stability and Hopf bifurcation of the both type delayed BAM neural networks. By adjusting the value of time delay, we can effectively amplify the stability region and postpone the time of onset of Hopf bifurcation for the fractional-order BAM neural networks. Matlab simulation results are clearly presented to sustain the correctness of analytical results. The derived fruits of this study provide an important theoretical basis in regulating networks

Global Asymptotic Stability of a Family of Nonlinear Difference Equations

In this note, we consider global asymptotic stability of the following nonlinear difference equation . . , −1 ∈ (0, ∞), and = max 1≤ ≤V { }. Our result generalizes the corresponding results in the recent literature and simultaneously conforms to a conjecture in the work b

Global Asymptotic Stability of a Family of Nonlinear Difference Equations

In this note, we consider global asymptotic stability of the following nonlinear difference equation xn=(∏i=1v(xn-kiβi+1)+∏i=1v(xn-kiβi-1))/(∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1)),  n=0,1,…, where ki∈ℕ  (i=1,2,…,v),  v≥2, β1∈[-1,1], β2,β3,…,βv∈(-∞,+∞), x-m,x-m+1,…,x-1∈(0,∞), and m=max1≤i≤v{ki}. Our result generalizes the corresponding results in the recent literature and simultaneously conforms to a conjecture in the work by Berenhaut et al. (2007)

Dynamical behavior for a stochastic two-species competitive model

This paper deals with a stochastic two-species competitive model. Some very verifiable criteria on the global stability of the positive equilibrium of the deterministic system are established. An example with its computer simulations is given to illustrate our main theoretical findings

Existence and uniqueness of pseudo almost periodic solutions for Lienard-type systems with delays

This article concerns Lienard-type systems with time-varying delays. By applying the theory of exponential dichotomies, the properties of pseudo almost periodic function, inequality analysis techniques, and contraction mapping principle, new criteria for the existence and uniqueness of pseudo almost periodic solutions are established. An example is given to illustrate the theoretical findings

Existence of Periodic Solutions in a Discrete Predator-Prey System with Beddington-DeAngelis Functional Responses

A discrete predator-prey model with Holling II and Beddington-DeAngelis functional responses is investigated. With the aid of differential equations with piecewise constant arguments, a discrete version of continuous nonautonomous delayed predator-prey model with Beddington-DeAngelis functional responses is proposed. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive solutions of the model are established

On a Conjecture for a Higher-Order Rational Difference Equation

This paper studies the global asymptotic stability for positive solutions to the higher order rational difference equation , where is odd and . Our main result generalizes several others in the recent literature and confirms a conjecture by Berenhaut et al., 2007.</p