# Countable Systems of Differential Equations by A. M. Samoilenko, Yu V. Teplinskii

By A. M. Samoilenko, Yu V. Teplinskii

This monograph is dedicated to the answer of assorted difficulties within the idea of differential equations within the area "M" of bounded numerical sequences (called countable systems). particularly, the final idea of countable platforms, the speculation of oscillating recommendations, and the idea of countable platforms with pulse motion are treated.Main realization is given to generalization of the result of a variety of authors, bought in recent times for finite-dimensional structures of other equations to the case of platforms from the analysed class.The publication comprises the next 4 chapters: - common innovations of the speculation of countless structures of differential equations- Invariant tori- Reducibility of linear platforms- Impulsive systemsThis booklet may be of price and curiosity to someone operating during this box of differential equations.

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**Sample text**

4. Let D be a closed bounded domain in the space ÜTt and let AQ and A be continuous mappings from D into VJl such that \\AX — < ε· If, in this case, AQ is a topological mapping, then AD contains AQD — ε : AQD - ε C AD. Proof. Since AQ is a topological mapping, the continuous mapping AAQ 1 defined on AQD maps the set AQD onto AD. 4 to the case where AQ = Ε (E is the identity mapping). Thus, let AQ = E. It is necessary to prove that D - ε C AD. We take XQ e D - ε. Then the ball Τε(χ0): \\X - rco|| < ε belongs to D: ΤΕ(Χ0) C D.

1. 1) be such that the following conditions are satisfied in the domain H: (i) ||/(x)|| < β = const < oo; (ii) | | / ( x ' ) — f{x")|| < OL\\X' — x"\\, where a = const < oo. 3. 1): 1. 1) has two solutions χ = x(t) and χ = x\(t), solutions either have no common points or coincide. then these 2. 1) is such that there exist ίχ φ t2 for which x{t\) = x(t2), then this solution either corresponds to the equilibrium state or is periodic. , [Pon]). 2. 1) determines a dynamical system in the space 97Ϊ. 1) such that φ(χ, 0) = χ, χ e 9Ji.

Which, in view of the theorem on the compactness of a set in the space C°°, implies that the set of points {ufc} is compact as a set in the space C°°. By virtue of the Schauder theorem, we conclude that the operator A* has a fixed point in C*: A*x* = 2*. 43), this yields Ax* = XQ. 46) means that XQ is the image of a point x* under the mapping A. 46) holds for any point xo £ D — ε, and the point x* belongs to the ball Te{xo) and, hence, to the set D. 46) means that the entire set D — ε is the image of the set D under the mapping A.