By R. P. Kuzmina (auth.)

ISBN-10: 9048155002

ISBN-13: 9789048155002

In this publication we reflect on a Cauchy challenge for a approach of standard differential equations with a small parameter. The booklet is split into th ree elements in line with 3 ways of regarding the small parameter within the method. partially 1 we learn the quasiregular Cauchy challenge. Th at is, an issue with the singularity incorporated in a bounded functionality j , which depends upon time and a small parameter. This challenge is a generalization of the regu larly perturbed Cauchy challenge studied through Poincare [35]. a few differential equations that are solved by way of the averaging strategy should be lowered to a quasiregular Cauchy challenge. as an instance, in bankruptcy 2 we think of the van der Pol challenge. partly 2 we examine the Tikhonov challenge. this is often, a Cauchy challenge for a process of standard differential equations the place the coefficients by way of the derivatives are integer levels of a small parameter.

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**Example text**

1) in the domain of definition. 4 . t ~ t*(c), 0 §4. 1. 1. 8. Proof. 1 that z(O)(t,/1) = 0. 2 ). 1 for k = 0. 3) for j = 0, k - 1. 6) for z(k)(t, /1). 4) F(k) (t, /1) for k ~ 2. 2). 5) A(t, /1) = Fx(O, t, 0, f(t, /1) ). 2) . 2) . 6. 7) SOLUTION EXPANSIONS OF THE QUASIREGULAR CAUCHY. . 4) F(k)(t,f-l) is a linear combination of the function F derivatives. 4. 8) 1=1 j=l with constant coefficients. Here N Slj ~ 0, == L El k-l Ljslj < k. j. = 1 holds. The rest of the numbers Slj vanish. ) I. -l) +C < Ct(K+l)(2k-3) + C.

E} _ BUI + UI dB l dB (u - u). ( t ,ll) Her e ] 32 CHAPT ER 1 Ilxll < < II Zn (t, e, J-L) II + B Ilull + (1 - II Zn(t,c, J-L)1I + 8 B) Wil li < 8. 33) . 31) . 4. 30)). 32) . 5. 32) . 10 for any valu es of e, J-L (0 ~ e ~ C2, 0 < J-L ~ €') . 10). Co nside r t he functions a, b, c. 20) it follows t hat q a = a(t ,e, J-L ) = J II U (q, s, J-L )II· G(O, s,e, J-L ) ds O~q9 0 max SOLUTION EXPANSIONS OF THE QUASIREGULAR CAUCHY. . 33 q < max f II U (q, s , /1)11 · go(s) ds En+1. 8, n = O. 6, + C ] .

31) . 4. 30)). 32) . 5. 32) . 10 for any valu es of e, J-L (0 ~ e ~ C2, 0 < J-L ~ €') . 10). Co nside r t he functions a, b, c. 20) it follows t hat q a = a(t ,e, J-L ) = J II U (q, s, J-L )II· G(O, s,e, J-L ) ds O~q9 0 max SOLUTION EXPANSIONS OF THE QUASIREGULAR CAUCHY. . 33 q < max f II U (q, s , /1)11 · go(s) ds En+1. 8, n = O. 6, + C ] . 8, n = O. 8. 6, { Ct K+! 8 . 38) > 0, SOLUTION EXPANSIONS OF THE QUASIREGULAR CAUCHY. . 2al(t,E) 35 < 8' [Pl (t , E) + V rl (t , E) ] , o < t ::; t*(E), 0::;E::;E2, o < fL ::; E.