arrow
Volume 9, Issue 4
Global Behavior of a Higher-Order Nonlinear Difference Equation with Many Arbitrary Multivariate Functions

Wen-Xiu Ma

East Asian J. Appl. Math., 9 (2019), pp. 643-650.

Published online: 2019-10

Export citation
  • 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. 

  • AMS Subject Headings

39A11, 39A30, 39A10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mawx@cas.usf.edu (Wen-Xiu Ma)

  • BibTex
  • RIS
  • TXT
@Article{EAJAM-9-643, author = {Ma , Wen-Xiu}, title = {Global Behavior of a Higher-Order Nonlinear Difference Equation with Many Arbitrary Multivariate Functions}, journal = {East Asian Journal on Applied Mathematics}, year = {2019}, volume = {9}, number = {4}, pages = {643--650}, 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. 

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.140219.070519}, url = {http://global-sci.org/intro/article_detail/eajam/13324.html} }
TY - JOUR T1 - Global Behavior of a Higher-Order Nonlinear Difference Equation with Many Arbitrary Multivariate Functions AU - Ma , Wen-Xiu JO - East Asian Journal on Applied Mathematics VL - 4 SP - 643 EP - 650 PY - 2019 DA - 2019/10 SN - 9 DO - http://doi.org/10.4208/eajam.140219.070519 UR - https://global-sci.org/intro/article_detail/eajam/13324.html KW - Difference equation, positive equilibrium, oscillatory solution, strong negative feedback, global asymptotic stability. AB -

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. 

Wen-Xiu Ma. (2019). Global Behavior of a Higher-Order Nonlinear Difference Equation with Many Arbitrary Multivariate Functions. East Asian Journal on Applied Mathematics. 9 (4). 643-650. doi:10.4208/eajam.140219.070519
Copy to clipboard
The citation has been copied to your clipboard