Global Behavior of a Higher-Order Nonlinear Difference Equation with Many Arbitrary Multivariate Functions

Source abstract

Let $k\ge 0$ and $l\ge 2$ be integers, $c$ a nonnegative number and $f$ an arbitrary multivariate function such that $f(x_1,x_2,x_3,\cdots,x_l)\ge x_1+x_2$ for $x_1,x_2\ge 0$. This work deals with the higher-order nonlinear difference equation $$\begin{equation*} z_{n+1}=\frac {(c+1)z_nz_{n-k}+c[f(z_n,z_{n-k},w_3,\cdots,w_l))-z_n-z_{n-k}]+2c^2}{z_nz_{n-k}+f(z_n,z_{n-k},w_3,\cdots,w_l))+c}, \quad n\ge 0, \end{equation*}$$ where $z_{-k},z_{-k+1},\cdots, z_0$ are positive initial values and $w_i,\ 3\le i\le l,$ arbitrary functions of variables $z_{n-k},z_{n-k+1},\cdots,z_n$. All solutions of this equation are classified into three groups, according to their asymptotic behavior, and a decreasing and increasing characteristic of oscillatory solutions is also explored. Finally, the global asymptotic stability of the positive equilibrium solution $\bar z =c$ is exhibited by establishing a strong negative feedback property.