By R. P. Kuzmina (auth.)

In this e-book we think of a Cauchy challenge for a method of standard differential equations with a small parameter. The booklet is split into th ree elements in accordance with 3 ways of related to the small parameter within the procedure. partially 1 we research the quasiregular Cauchy challenge. Th at is, an issue with the singularity integrated in a bounded functionality j , which is dependent upon time and a small parameter. This challenge is a generalization of the regu larly perturbed Cauchy challenge studied by means of Poincare [35]. a few differential equations that are solved by way of the averaging technique might be decreased to a quasiregular Cauchy challenge. for instance, in bankruptcy 2 we give some thought to the van der Pol challenge. partially 2 we research 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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**Sample text**

1. 4. 3. 32) SOLUTION EXPANSIONS OF THE QUASIREGULAR CAUCHY. . 8, n ~ 1. 7, I 31 CE exp(l'i:t) Proof. l)11 < 8. l) with resp ect to u, t. l) . l ), t , E, ! l)) - Fx(O, t, O, ! ) 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.

5. 3 be satisfied. 1) exists, is unique, and satisfies the inequalities for 0 ~ t ~ T , 0 ~ E ~ e* . 6 . 6) be valid. 1) exists, is unique, and satisfie s the in equality fo r t 2 0, 0 ::; E ::; E* . 7) be valid. 1) exists, is unique, and satisfies the inequality T he orem 9. 7. Let, fo r for 0 ::; t ::; T E- x , 0::; E ::; E*. 8. 8) be valid. Th en for any T 2 0, x. 1) exis ts, is un ique, and satisfies the inequalities C*E (e Kt - 1) , II x (t , E) II < Ilx(t , E) - X n(t , E) 11 < C*En+leKt(enKt -1), n > 1 for 0 ::; t::; T - Xln E, 0::; E::; E*.

19 n +1 e X 0205 ... O~+l dO l ... dOn+! En+ l. The brackets with the upper index (~n) denote the partial surn of expansion of the function in brackets into apower series with respect to E . This partial surn includes the powers of E with exponents up to n. rlEq-k n Ski ~ 0, N ~l == ~ ~ kSkl ~ n + 1. k=ll=l }Skl , SOLUTION EXPANSIONS OF THE QUASIREGULAR CAUCHY. . 19) . 8. 27) IIZn(t , e, p)11 0 forn~ O. 19) for 0 ~ OJ ~ 1, j = 1, n + 1. 19) . 6. 28) q=k gg{t. 29) 30 CHAPTER 1 N n-k IT IT {Ct( I<+I)(2k-l) 2: [t 2(1<+1) -r + C} n < k=II =1 q=O < CC I - 2X(I< + I ) < C, - t 2(1<+I)E Cr N n IIII < Sk i, IT IT [Ct( I<+I)(2 k-l) + kl k= I I= 1 < + C.