## Basins of attraction of equilibrium points of monotone difference equations.(English)Zbl 1189.39024

The article deals with the difference equation
$x_{n+1} = f(x_n,x_{n-1}), \qquad n = 0,1,\dots,\tag{*}$
with $$f \in C[I \times I,I]$$ ($$I$$ is an interval in $${\mathbb R}$$); it is assumed that $$f$$ is increasing in both variables. More precisely, the equilibria and their basins of attraction are studied. The main results are the following: 7mm
(1)
If (*) has no prime period-two solutions, then every bounded solution of (*) converges to an equilibrium.
(2)
If there exist two equilibrium points $$\overline{x}_1, \overline{x}_2$$, $$0 \leq \overline{x}_1 < \overline{x}_2$$, then the box $$(\overline{x}_1,\overline{x}_2)^2$$ is a part of the basin of attraction of an equilibrium, $$\overline{x}_1$$ if $$(x - \overline{x}_1)(f(x,x) - x)< 0$$ for $$x \in (\overline{x}_1,\overline{x}_2)$$ is satisfied and $$\overline{x}_2$$ if $$(x - \overline{x}_2)(f(x,x) - x) < 0$$ for $$x \in (\overline{x}_1,\overline{x}_2)$$ is satisfied.
(3)
If (*) has no minimal period-two solutions and $$E_1(x_1,y_1)$$, $$E_2(x_2,y_2)$$, $$E_3(x_3,y_3)$$ are three consecutive equilibrium points, $$(x_1,y_1) \preceq (x_2,y_2) \preceq(x_3,y_3)$$, $$E_1$$, $$E_3$$ are saddle points, $$E_2$$ is a local attractor, then the basin of attraction $${\mathcal B}(E_2)$$ is the region between the global stable manifolds $${\mathcal W}^s(E_1)$$ and $${\mathcal W}^s(E_3)$$ and the basins of attraction $${\mathcal B}(E_2) = {\mathcal W}^s(E_1)$$ and $${\mathcal B}(E_3) = {\mathcal W}^s(E_3)$$ are exactly the global stable manifolds of $$E_1$$ and $$E_2$$. [The latter statement is vague and puzzles: the equilibrium of (*) are real numbers of $$I$$; $$E_1$$, $$E_2$$, $$E_3$$ are not elements of $$I$$.]
The following examples with the corresponding nice illustrations are considered:
\begin{aligned} x_{n+1} &= \tfrac12(x_n + x_{n-1} + \sin x_{n-1}),\\ x_{n+1} &= \tfrac12(x_n + x_{n-1} + \sin x_n + \sin x_{n-1}),\\ x_{n+1} &= x_n^3 + x_{n-1}^3;\end{aligned}
and some other examples are also presented. Also, some analogues of the main results about equation (*) are formulated for the equation
$x_{n+1} = f(x_n,x_{n-1},\dots,x_{n-k+1}), \qquad n = 0,1,\dots$

### MSC:

 39A30 Stability theory for difference equations 39A10 Additive difference equations 39A23 Periodic solutions of difference equations