The article deals with the difference equation
with ( is an interval in ); it is assumed that 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 , , then the box is a part of the basin of attraction of an equilibrium, if for is satisfied and if for is satisfied.
If (*) has no minimal period-two solutions and , , are three consecutive equilibrium points, , , are saddle points, is a local attractor, then the basin of attraction is the region between the global stable manifolds and and the basins of attraction and are exactly the global stable manifolds of and . [The latter statement is vague and puzzles: the equilibrium of (*) are real numbers of ; , , are not elements of .]
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