ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Three Properties of a Discrete Dynamical System in the Space of Infinitely Differentiable Functions

### Author(s):

Ivan Andreevich Podlugniy

Saint-Petersburg State University
Faculty of Mathematics and Mechanics
Dept. of Math.Analysis
student

sherman97@mail.ru

Alexandr Alekseevich Florinskiy

Saint-Petersburg State University
Faculty of Mathematics and Mechanics
Dept. of Math.Analysis
Assoc. Prof.

florinskiy.a@gmail.com

### Abstract:

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.

### Keywords

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

### References:

1. R. P. Boas «When is a function analytic? » Math. Intellegencer 11, №4, 1989, 34-37
2. B. M. Makarov, M. G. Golusina, A. A. Lodkin, A. N. Podkorytov. Problemes d’analise reele. Paris, Cassini, 2010, 593 P
3. Ilyin Y. A. “General problems of integration of differential inequalities in explicit form” , Vestnik SPbSU. Mathematics. Mechanics. Astronomy. 2017, vol. 4(62), issue 4. p. 597-607
4. Karl-Gosurin, Grosse Erdman “Universial Families and Hypercyclic operators” Bulletin of the AMS, vol. 36 №3 345-381
5. G. R. MacLane. Sequences of derivatives and normal families, J. Analise Math. (1952/53), MR 14:741d