By A. M. Samoilenko, Yu V. Teplinskii

This monograph is dedicated to the answer of assorted difficulties within the conception of differential equations within the area "M" of bounded numerical sequences (called countable systems). specifically, the overall conception of countable platforms, the speculation of oscillating strategies, and the speculation of countable platforms with pulse motion are treated.Main consciousness is given to generalization of the result of quite a few authors, got lately for finite-dimensional platforms of alternative equations to the case of structures from the analysed class.The e-book includes the next 4 chapters: - basic options of the idea of endless platforms of differential equations- Invariant tori- Reducibility of linear platforms- Impulsive systemsThis e-book could be of price and curiosity to an individual operating during this box of differential equations.

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**Extra resources for Countable Systems of Differential Equations**

**Example text**

11) can be majorized by the convergent number series i= 1 m a * Iu s i (t)Iz° and JZ m£x i= 1 > s i ' 26 General Concepts of the Theory of Infinite Systems Chapter 1 respectively, for any s = 1 , 2 , 3 , . . , are uniformly convergent on this segment. Therefore, the equality dt dt is true for a function z(t) satisfying conditions (a) and (b). 10) to an equation of the form ( p m i n t ) + Q I ^ - = p(mi0z(t)) Since ( P ( t ) Q , l 0 ) z ( t ) = Ρ(ί)(Ωΐ0ζ(ί)), + ζ®. we arrive at the equation rt7 or τ z{t) = J Q-\t0,T)f(T)dr + c, to where c is an arbitrary constant vector such that ||c|| < oo.

We assume that 0(φ) is an infinite matrix whose elements Cij(φ), i,j 1 , 2 , . . , are continuous in φ and satisfy the inequality oo sup i sup I j=l φ Cij ( φ ) I < oo. 5) = Invariant Tori 52 Chapter 2 The set of matrices satisfying the last estimate is denoted by Ο φ . 1). As follows from its definition, the Green function Gq(t, ψ) is 27r-periodic with respect to ψί, i = 1, m, provided that α(φ), Ρ(φ), and 0(φ) are also periodic and, moreover, G o ( 0 , v ? ) - G o ( + 0 ,φ) = Ε. 2. 7) τ > 0.

2. 8) that passes through the point χ = xo for t = τ is T-periodic, provided that the indicated value of Δ is unique. 1. 9) Oim(s)ds, m = 0,1,2,... Then there exist positive where πι 2. and ( ^ constants ) are, respectively, the integral and fractional parts of the number —, such that am+i(t) < Tmqm2(t - r) ( l - for all tfrom the segment [r, τ + Τ]. Proof. 12) for all t G [τ, τ + Τ]. 11) is satisfied. 12) that, for t G [τ, τ + Τ], β4(ί)<^θ2(ί) = ^αι(ί), Τ2 0Cm{t) < — a m _ 2 ( i ) , rn > 3. 13) imply the estimate <*m(t)< 2 , m - M e i (t), rn = 2 , 3 , .