Stability of a fractional difference equation of high order (original) (raw)

Global Asymptotic Stability for Linear Fractional Difference Equation

and the initial conditions , ∈ {− ,. .. , 0} are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

Stability and asymptotic properties of a linear fractional difference equation

Advances in Difference Equations, 2012

This paper discusses qualitative properties of the two-term linear fractional difference equation where α, λ ∈ R, 0 < α < 1, λ = 1 and 0 ∇ α is the αth order Riemann-Liouville difference operator. For this purpose, we show that this fractional equation is the Volterra equation of convolution type. This enables us to analyse its qualitative properties by use of tools standardly employed in the qualitative investigation of Volterra difference equations. As the main result, we derive a sharp condition for the asymptotic stability of the studied equation and, moreover, give a precise asymptotic description of its solutions.

Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation

The Scientific World Journal, 2013

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the formxn+1=xn-12/(axn2+bxnxn-1+cxn-12),n=0,1,2,…,where the parameters a, b, and c are positive numbers and the initial conditionsx-1andx0are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

Oscillation of solutions for a class of nonlinear fractional difference equations

Journal of Nonlinear Sciences and Applications

In this paper, we investigate the oscillation of the following nonlinear fractional difference equations, ∆ (a (t) [∆ (r (t) (∆ α x (t)) γ 1)] γ 2) + q (t) f t−1+α s=t 0 (t − s − 1) (−α) x (s) = 0, where t ∈ N t 0 +1−α , γ 1 and γ 2 are the quotient of two odd positive number, and ∆ α denotes the Riemann-Liouville fractional difference operator of order 0 < α ≤ 1.

Fractional Order Difference Equations J

2014

A difference equation is a relation between the differences of a function at one or more general values of the independent variable. These equations usually describe the evolution of certain phenomena over the course of time. The present paper deals with the existence and uniqueness of solutions of fractional difference equations.

Oscillatory Behavior of Fractional Difference Equations

In this paper, we study oscillatory behavior of the fractional difference equations of the following form 0 0 1 ( ) 1 ( ( ) ( ( ))) ( ) ( 1) ( ) 0, , t t s t ptg xt qt f t s xs t N α α α α − + − + − = ⎛ ⎞ Δ Δ + −− = ∈ ⎜ ⎟ ⎝ ⎠ ∑ where ∆α denotes the Riemann-Liouville difference operator of order α, 0 < α ≤ 1. We establish some oscillation criteria for the equation using Riccati transformation technique and Hardy inequality. Examples are provided to illustrate our main results.