Global behavior of the difference equation $x_{n+1}=\frac{ax_{n-3} }{b+ cx_{n-1}x_{n-3}}$

Global Behavior of the Difference Equation

The main objective of this paper is to study the qualitative behavior for a class of nonlinear rational difference equation. We study the local stability, periodicity, Oscillation, boundedness, and the global stability for the positive solutions of equation. Examples illustrate the importance of the results Keywords— Difference equation, stability, oscillation, boundedness, globale stability an...

On the nature of solutions of the difference equation $mathbf{x_{n+1}=x_{n}x_{n-3}-1}$

We investigate the long-term behavior of solutions of the difference equation[ x_{n+1}=x_{n}x_{n-3}-1 ,, n=0 ,, 1 ,, ldots ,, ]noindent where the initial conditions $x_{-3} ,, x_{-2} ,, x_{-1} ,, x_{0}$ are real numbers.  In particular, we look at the periodicity and asymptotic periodicity of solutions, as well as the existence of unbounded solutions.

Global Behavior of a Higher-order Rational Difference Equation

We investigate in this paper the global behavior of the following difference equation: xn+1 = (Pk(xn i0 ,xn i1 , . . . ,xn i2k ) + b)/(Qk(xn i0 ,xn i1 , . . . ,xn i2k ) + b), n = 0,1, . . ., under appropriate assumptions, where b [0, ), k 1, i0, i1, . . . , i2k 0,1, . . . with i0 < i1 < < i2k, the initial conditions xi 2k ,xi 2k+1, . . . ,x0 (0, ). We prove that unique equilibrium x = 1 of that...

Global behavior of a higher order difference equation

The Periodic Character of a Rational Difference Equation 3

with y−s, y−s+1, . . . , y−1 ∈ (0,∞) and k, m ∈ {1, 2, 3, 4, . . .}, where s = k+m. We prove that if gcd(2k,m) = 1, then solutions to this equation are asymptotically 2k-periodic. This generalizes the corresponding result for m = 3 and k = 1, proved in K. S. Berenhaut et al., Periodic Solutions of the Rational Difference Equation yn = yn−3+yn−4 yn−1 ,Journal of Difference Equations and Applicat...