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

By A. M. Samoilenko, Yu V. Teplinskii

This monograph is dedicated to the answer of assorted difficulties within the thought of differential equations within the house "M" of bounded numerical sequences (called countable systems). specifically, the final idea of countable platforms, the idea of oscillating recommendations, and the speculation of countable structures with pulse motion are treated.Main cognizance is given to generalization of the result of various authors, received in recent times for finite-dimensional structures of other equations to the case of platforms from the analysed class.The ebook comprises the subsequent 4 chapters: - basic innovations of the speculation of countless structures of differential equations- Invariant tori- Reducibility of linear platforms- Impulsive systemsThis booklet can be of price and curiosity to somebody operating during this box of differential equations.

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2. Γ (JIT)NJI Ί ;32(n)tti(t)Jl»o-g,|. °° [ g + E ^ n ] n\ (6-41) η and ( — J are the integral and fractional parts of the number —, re- spectively. 3. In what follows, we denote the image of the set D under the mapping A by AD. Let us now study the problem of the existence of periodic solutions of a T system. 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.

Of time t and takes values in the space 371. If a function f(t,x) is periodic in time with period T, then its integral time average is denoted by f(t, x): τ = ^ J f{t,x)dt. ο The meaning of the remaining notation is either obvious or will be explained separately. ,«,.... 4) is defined for any real t and all χ = (χι, X2, • • ·) from a domain D. ) is a point of the space Wl. 4) in the form of a differential equation in the space SOt: dx — = /(*,*). 5), and belonging to the space DJl for all t g ( a , 6).

5) ίο which is absolutely and uniformly convergent on the segment Τ in the strong sense. Section 4 Matrizant of a Linear System 23 Proof. , by choosing XQ in the form of the identity matrix E. As a result, we obtain t XQ = E, X\ = Ε + J P(t) dt, to t t t X2 = E + J P(t) dt + J {P(t) ίο ίο J P(t) dt} dt, ... 4). 1). We denote the length of a segment Τ containing the point t = to by 5. 5) can be majorized by the following convergent number series: _ PQS2 po<53 pQÖk and, therefore, it is absolutely and uniformly convergent.

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