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**Ivan Andreevich Podlugniy**

Saint-Petersburg State University

Faculty of Mathematics and Mechanics

Dept. of Math.Analysis

student

**Alexandr Alekseevich Florinskiy**

Saint-Petersburg State University

Faculty of Mathematics and Mechanics

Dept. of Math.Analysis

Assoc. Prof.

A nonlinear operator generated by a fixed function of two real variables
is considered. The function is supposed to be smooth, the first argument
is defined on a closed interval,the second one on the real line.
We also assume that this function to be both strictly increasing and bilipshitz
on the second argument.
The operator acts on the space of all infinitely
differentiable real functions defined on the same closed interval as the
first argument of the fixed function, and assings to any such a function
the result of the substitution of its derivative instead of the second argument
in the fixed function of two variables. For any trajectory of
the discrete infinite dimensional dynamical system
(which is chaotic in general case) generated by the operator
we prove the following properties:

- a trajectory of the system is uniformly bounded iff it is pointwise bounded ;

- a trajectory is uniformly convergent with all its derivatives iff
it is pointwise convergent;
- the least pointwise upper bound of the trajectory is also the greatest
lower bound of an other trajectory of the system iff
it is a fixed point of this system.

The last statement gives a serial characteristics of fixed points
of the operator, which is not monotonous.

- fixed point
- infinite dimensional dynamical system
- nonlinear operator
- pointwise bounded trajectory
- the least pointwise upper bound of the trajectory
- uniformly bounded trajectory

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