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 house "M" of bounded numerical sequences (called countable systems). particularly, the overall thought of countable platforms, the idea of oscillating strategies, and the speculation of countable structures with pulse motion are treated.Main awareness is given to generalization of the result of various authors, bought in recent times for finite-dimensional platforms of alternative equations to the case of structures from the analysed class.The ebook includes the next 4 chapters: - basic ideas of the speculation of endless structures of differential equations- Invariant tori- Reducibility of linear platforms- Impulsive systemsThis booklet should be of worth and curiosity to someone operating during this box of differential equations.

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Example text

3 2 ( f - l ) a i W ' (T Vβ ai(t)\ + 3 J Further, for qm, we obtain Ύ< m T qm*i{t) τη > — — j r — 2( t t l ( t ) . 1. 2. 6) and conditions A and B. 22) uniformly in (t,r,x0) ( MT\ G (-00,00) χ (-00,00) χ f D Moreover, thefiinctionXoo(t, r, isfies the equation Xo) — J . is periodic in t and τ with period Τ and sat- t = X0 + J ( / ( i , x(t, τ, x0)) - f(t, x(t, T, z 0 ) ) ) dt. 23) Proof. In what follows, we denote the set of points from D contained in D /M T\ together with their I — J -neighborhoods by Df.

2. 7) τ > 0. ). 8) Proof. 1). 3), we obtain _ r ΩΚΨΜΜΜΜΨ))), R< o, \ & τ ( Μ φ ) ) [ 0 ( φ τ ( φ ) ) ) ~ E], r > 0, Γ ^τ+Λψ)0{φτ+Μ), ~ { ί11+,(φ)[0(φτ+ί(φ)) -Ε), = Gt(r + t, φ). 3. ο(Τ) v)ll ^ e x p { —7111 }, where the constants Κ = const > 0 and 7 = const > 0 are independent of φ, then ||Gt(0,y>)|| < Kexp{-j\t\}. 3, \\Οο(τ,φζ(φ))\\

Dt Denote \P\\o = s u p ^ m a x | p i i ( i ) | , i j=i T Ω<0 = Κ(ί)]~=1. Thus, we get ||P||o < Po, ||Ω*0||ο < eP0<5, where δ is the length of the segment T , and dÜlto < p0exp{p05}, dt ΙΙ(Ω^) 1 Ho < e x p { p 0 5 } = φ 0 . 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).

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