差分方程X_n+1=ax_n+bX²_n/(cx_n+d_n-1)的解性质的研究

Studying of Property of the Solution of the Nonlinear Difference EquationX_n+1=ax_n+bX²n/(cx_n+d_n-1)

摘要:

文献[1]中研究了差分方程X_n+1=ax_n+bX²_n/(cx_n+d_n-1)的解性质的研究，(a,b,c,d是正实数)解的有界性、平衡点的局部及全局吸引性，并给出了其几类特殊情形下的解的表达式。本文首先指出文献[1]中的第2、3段的讨论方法是不正确的。然后本文中我们对平衡点的性质进行了讨论，分为两种情况进行研究，分别为 (1-a )(c+d)≠ b及(1-a )(c+d)= b 。在第一种情况中可以得到平衡解在适当的条件下是全局吸引的，在第二种情况下得到任意实数都是平衡解且是不渐进稳定的。在每种情况中举了一些数值例子，并用相应的软件画出其图像，从而进一步证明了结论的正确性。

In paper [1], some properties are investigated about difference equation X_n+1=ax_n+bX²n/(cx_n+d_n-1)

Abstract: Studying of Property of the Solution of the Nonlinear Difference EquationX_n+1=ax_n+bX²n/(cx_n+d_n-1)，where a>0,b>0,c>0,d>0, and the initial conditions x1,x 0∈（0，+∞）such as the boundedness of the solution and the properties of the equilibrium point. In this paper we will prove that some methods used in paper [1] are wrong at first. Then we continue to study some properties of the difference equation under dif-ferent circumstances. We get some interesting conclusions. At last, we use some numerical examples to verify the conclusions that we get in this paper.