*(English)*Zbl 1189.39024

The article deals with the difference equation

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:

If (*) has no prime period-two solutions, then every bounded solution of (*) converges to an equilibrium.

If there exist two equilibrium points ${\overline{x}}_{1},{\overline{x}}_{2}$, $0\le {\overline{x}}_{1}<{\overline{x}}_{2}$, 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.

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, $({x}_{1},{y}_{1})\u2aaf({x}_{2},{y}_{2})\u2aaf({x}_{3},{y}_{3})$, ${E}_{1}$, ${E}_{3}$ are saddle points, ${E}_{2}$ is a local attractor, then the basin of attraction $\mathcal{B}\left({E}_{2}\right)$ is the region between the global stable manifolds ${\mathcal{W}}^{s}\left({E}_{1}\right)$ and ${\mathcal{W}}^{s}\left({E}_{3}\right)$ and the basins of attraction $\mathcal{B}\left({E}_{2}\right)={\mathcal{W}}^{s}\left({E}_{1}\right)$ and $\mathcal{B}\left({E}_{3}\right)={\mathcal{W}}^{s}\left({E}_{3}\right)$ 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:

and some other examples are also presented. Also, some analogues of the main results about equation (*) are formulated for the equation

##### MSC:

39A30 | Stability theory (difference equations) |

39A10 | Additive difference equations |

39A23 | Periodic solutions (difference equations) |