Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations

For nonnegative real numbers $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$ such that $B+C>0$ and $\alpha+\beta+\gamma >0$, the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,... %, \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} has a unique positive equilibrium. A proof is given here for the following statements: \medskip \noindent Theorem 1. {\it For every choice of positive parameters $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.} \medskip \noindent Theorem 2. {\it For every choice of positive parameters $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}= \displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in (0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.}Comment: 23 page

Research Article Boundedness and Global Attractivity of a Higher-Order Nonlinear Difference Equation

We investigate the local stability, prime period-two solutions, boundedness, invariant intervals, and global attractivity of all positive solutions of the following difference equation: y n 1 r py n y n−k / qy n y n−k , n ∈ N 0 , where the parameters p, q, r ∈ 0, ∞ , k ∈ {1, 2, 3, . . .} and the initial conditions y −k , . . . , y 0 ∈ 0, ∞ . We show that the unique positive equilibrium of this equation is a global attractor under certain conditions

The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations

In this paper, the circular system of Riccati type complex difference equations of the form $u_{n+1}^{(j)} = \frac{a_ju_n^{(j-1)}+b_j}{c_ju_n^{(j-1)}+d_j}, \; n = 0, 1, 2, \cdots, \; j = 1, 2, \cdots, k,$ where $u_n^{(0)}: = u_n^{(k)}$ for all $n$, is investigated. First, the forbidden set of the equation is given. Then the solvability of the system is examined and the expression of the solutions, given in terms of their initial values. Next, the asymptotic behaviour of the solutions is studied. Finally, in case of negative Riccati real numbers $R_j: = \frac{a_jd_j-b_jc_j}{[a_j+d_j]^2}, \; j\in\overline{1, k},$ it is shown that there exists a unique positive fixed point which attracts all solutions starting from positive states.</p