The Construction of a Pfaff System with Arbitrary Piecewise Continuous Characteristic Power-Law Functions

Transcription

1 Differentia Equations, Vo. 41, No. 2, 2005, pp Transated from Differentsia nye Uravneniya, Vo. 41, No. 2, 2005, pp Origina Russian Text Copyright c 2005 by Izobov, Krupchik. ORDINARY DIFFERENTIAL EQUATIONS The Construction of a Pfaff System with Arbitrary Piecewise Continuous Characteristic Power-Law Functions N.A.IzobovandE.N.Krupchik Institute of Mathematics, Nationa Academy of Sciences, Minsk, Bearus Bearus State University, Minsk, Bearus Received March 24, 2004 Consider the inear Pfaff system x/ t i = A i tx, x R n, t =t 1,t 2 R 2 >1, i =1, 2, n N, 1 n with bounded continuousy differentiabe matrix functions A i t satisfying the compete integrabiity condition [1, pp ; 2, pp A 1 t/ t 2 + A 1 ta 2 t = A 2 t/ t 1 + A 2 ta 1 t, t R 2 >1. Let p = p[x R 2 and λ = λ[x R 2 be the ower characteristic vector [3 and the characteristic vector [4, respectivey, of a nontrivia soution x : R 2 >1 Rn \{0} of system 1 n, and et P x = {p[x} and Λ x = {λ[x} be the ower characteristic set and the characteristic set of the same soution; they are bounded and cosed and can be represented [3, 4, respectivey, by monotone decreasing concave and convex curves p 2 = ϕ p 1 :[α 1,α 2 [β 1,β 2 andλ 2 = f λ 1 :[a 1,a 2 [b 1,b 2 on the pane R 2. We introduce anaogs of the Demidovich characteristic degree [5 of a function of a singe variabe, namey, the ower characteristic degree [6 d = d x p R 2 and the upper characteristic degree [7 d = d x λ R 2 of the soution x 0, given by the conditions n xt p, t d, n t n x p, d im =0, t n t n x p, d + εe i < 0, ε >0, i =1, 2, n xt λ, t d, n t n x λ, d im =0, t n t n x λ, d εei > 0, ε >0, i =1, 2, where e i =2 i, i 1 and n t n t 1, n t 2 R+ 2, as we as the ower degree set D xp ={d x p} and the upper degree set D x λ = { dx λ }. By [6, if for an interior point p =p 1,ϕp 1 P x, p 1 α 1,α 2, and for an exterior point λ =λ 1,fλ 1 Λ x, λ 1 a 1,a 2, there exist finite imits c x p 1 2n x p 1,ϕp 1, 0 and c x λ 1 2 n x λ 1,fλ 1, 0, respectivey, then the individua interior ower degree set D x p and upper degree set D x λ ofthesoutionx 0 are the ines d 1 +d 2 = c x p 1 and d 1 + d 2 = c x λ 1, respectivey, on the pane R 2. It was shown in [8 that the eft boundary ower degree set D x p, p =α 1,β 2, the right boundary ower degree set D x p, p = α 2,β 1, the eft boundary upper degree set D x λ, λ =a 1,b 2, and the right boundary upper degree set D x λ, λ =a 2,b 1, of the soution x 0 of system 1 n do not necessariy coincide with any ine on the two-dimensiona pane and even need not contain a rectiinear segment /05/ c 2005 Peiades Pubishing, Inc.

2 THE CONSTRUCTION OF A PFAFF SYSTEM WITH ARBITRARY PIECEWISE Therefore, it is natura to introduce the ower characteristic degree function c x p 1 and the upper characteristic degree function c x λ 1 ofthesoutionx 0 of system 1 n defined on the intervas α 1,α 2 anda 1,a 2, respectivey. The investigation of these functions was initiated in [6; more precisey, competey integrabe Pfaff systems 1 n with infinitey differentiabe bounded coefficients such that the characteristic degree functions c x p 1 and c x λ 1 of nontrivia soutions of these systems coincide with given functions were constructed for some piecewise constant functions and some continuous functions. The next step in the description of the functions c x p 1 and c x λ 1 is to impement an arbitrary given piecewise continuous function by characteristic degree functions of some nontrivia soution of some system 1 n. By [9, p. 106 of the Russian transation, a piecewise continuous function on an interva is defined as a function continuous everywhere on that interva except for finitey many jump discontinuities. Theorem 1. For an arbitrary positive integer n and for an arbitrary piecewise continuous function g :[α 1,α 2 R, there exists a competey integrabe Pfaff system 1 n with infinitey differentiabe bounded coefficients such that, for each nontrivia soution x : R>1 2 R n \{0} of this system, the domain of the curve P x coincides with the interva [α 1,α 2 and c x p 1 =gp 1, p 1 α 1,α 2. 2 Proof. Note that it suffices to prove the desired assertion for n = 1. Indeed, having constructed a one-dimensiona competey integrabe Pfaff equation x/ t i = a i tx, x R, t R>1, 2 i =1, 2, 1 1 with infinitey differentiabe bounded coefficients such that the ower characteristic degree function c x1 p 1 [x 1 of a nontrivia soution x 1 : R>1 2 R\{0} coincides with the function g p 1 [x 1 for a p 1 [x 1 α 1,α 2, one can choose the desired n-dimensiona system 1 n intheformofthe diagona system with coefficient matrix A i t =diag[a i t,...,a i t, i =1, 2, of order n. Then Xt =x 1 te n,wheree n is the identity matrix of order n, is the principa soution matrix of system 1 n, and each nontrivia soution x : R>1 2 R n \{0} of this system can be represented in the form xt =Xtc = x 1 tc with some c R n \{0}. Therefore, the ower characteristic vectors p[x =p [x 1 and hence the ower characteristic degree functions c x p 1 [x = c x1 p 1 [x 1 of this soution and the soution x 1 of Eq. 1 1 coincide. First, consider the case of an interva [α 1,α 2 on the ine R such that α 1 <α 2 < CONSTRUCTION OF THE SOLUTION We construct a soution x of the Pfaff equation 1 1 intheform n xt =nφt+νtnψt, t R 2 >1. 3 The function φ is defined so as to ensure that the domain of the curve P φ of its ower characteristic set coincides with the interva [α 1,α 2 and a points p[φ of the ower characteristic set P φ are reaized in different directions θ t t 2 /t 1 depending on p[φ. We define the function ν on the basis of the function g θ t so as to ensure that the soution x satisfies reation 2, n x is an infinitey differentiabe function, and its derivatives n xt/ t i, i =1, 2, are bounded. We use the infinitey differentiabe standard function [10, p. 54 of the Russian transation { { [ exp τ τ e 01 τ; τ 1,τ 2 = 1 2 exp τ 2 τ 2} for τ τ 1,τ 2 [1 + sgn τ 2 1 τ 1 + τ 2 /2 for τ τ 1,τ 2, <τ 1 <τ 2 < +, and define infinitey differentiabe functions φ and ψ by the reations n φt = 2 t 1 t t 1 t 2 + t 2 /α 1 e01 θt / 2α 2 1 ;1/2, [ t 1 t 2 + α 2 t 1 1 e01 θt /α 2 2 ;1/2, 1 4, n ψt =e 01 θt /α 2 2;1/2, 1 [ 1 e 01 θt / 2α1 2 ;1/2, 1 n t1, t R> DIFFERENTIAL EQUATIONS Vo. 41 No

3 186 IZOBOV, KRUPCHIK Suppose that, on the interva α 1,α 2, the function g has k points α 1 < 1 < 2 < < k <α 2 of discontinuity. For convenience, we set 0 = α 1, k+1 = α 2,andτ j = j j 1, j =1,...,k+1. On the basis of the function g, we introduce the new functions g j :[ j 1, j R, j =1,...,k+1, given by the reations g j = g, j 1, j, g j j 1 =g j 1 +0, g j j =g j 0. Obviousy, each function g j is continuous on the cosed interva [ j 1, j, j =1,...,k+1. By the Weierstrass approximation theorem, for each function g j, there exists a sequence {P j } N of agebraic poynomias uniformy converging on [ j 1, j to the function g j. By virtue of the Weierstrass theorem on continuous functions on an interva, the foowing notation is we defined: µ = max max P j < +, j=1,...,k+1 [ j 1, j ϱ = max dp j /d < +. max j=1,...,k+1 [ j 1, j We use some vaue δ 0 > max j=0,...,k+1 g j α and introduce the numbers γ =δ 1 +1+ϱ 2 + µ α 2 1 2, δ = γ e 2, N. We spit the quadrant R>1 2 that is, the domain of the soution x to be constructed into disjoint basic strips Π = { } t R 2 >1 : δ t 1 + t 2 ζt γ +1, transition strips Π = { t R 2 >1 : γ } <ζt <δ, N, and the triange T = {t R>1 2 : ζt γ 1 }. Let us now proceed to the construction of the function ν. We first define auxiiary functions ν, N, as foows: ν t =P j [ θ t, θ t j τ j / 2 2, j 1 + τ j / 2 2, 6 1 j =1,...,k+1, ν t =g j, ν t =P j j =1,...,k, θ t θ t [ + [ j + τ j+1/ 4 2, j τ j / 4 2, g j P j θ t [1 e 01 2 j + / θ t τ j ;1/2, 1, / θ t j τ j 4 2 /, j τ j 2 [ ν t =g j + P j+1 θ t g j [1 e 01 2 j + / θ t τ j+1 ; 1, 1/2 θ t j + τ j+1 2, j =1,...,k+1,, / 2 2 /, j + τ j+1 4 2, j =0,...,k, DIFFERENTIAL EQUATIONS Vo. 41 No

4 THE CONSTRUCTION OF A PFAFF SYSTEM WITH ARBITRARY PIECEWISE ν t =g α 1, / θ t α 1 + τ 1 4 2, 65 ν t =g α 2, / θ t α 2 τ k Reations define the function ν in the entire quadrant R 2 >1. We set the desired function ν in the basic strips Π to be equa to the function ν ; i.e., νt =ν t, t Π, N. 7 1 In the transition strips, we pass from the function ν 1 to the function ν : νt =ν 1 t+[ν t ν 1 t e 01 n ζt; n γ, n δ, t Π, =2, 3, Finay, we set νt =ν 1 te 01 n ζt; n γ 1, n δ 1, t T Π CONSTRUCTION OF THE LOWER CHARACTERISTIC SET OF THE SOLUTION x 2.1. Construction of the Lower Characteristic Set of the Function φ Let us show that the set P { } p R 2 : p 1 p 2 =1, α 1 p 1 α 2 is the ower characteristic set of the function φ. Obviousy, the inequaities 2 t 1 t 2 + t 2 /α 1 2t 2 /α 1 + t 2 /α 1 = 2 1 t 2 /α 1 0, 2 t 1 t 2 + α 2 t 1 2α 2 t 1 + α 2 t 1 = α t 1 0 are vaid for α 2 2/2 θ t 2α 2 1. We take an arbitrary vector p P and set R φ p, t n φt p, t; then we estimate the quantity R φ p, t frombeow.forα 2 2 /2 θ t 2α 2 1, we obtain the inequaity R φ p, t 2 t 1 t 2 p 1 t 1 t 2 /p 1 = t 1 p θ t /p Reation 4 with θ t 2α 2 1 impies the estimate R φ p, t =t 2 /α 1 p 1 t 1 t 2 /p 1 1/α 1 1/p 1 t 2 p 1 t 1 p 1 t 1 > Finay, by using 4, we obtain the inequaity R φ p, t =α 2 t 1 p 1 t 1 t 2 /p 1 α 2 p 1 t for θ t α 2 2/2. The estimates impy the inequaity φ p im R φ p, t/ t 0. t The reation φ p = 0 and the second condition φ p + εe i < 0, ε>0, i =1, 2, in the definition [3 of the ower characteristic vector are estabished in the direction t 2 = p 2 1 t 1, t 1 +. We have thereby justified the incusion P P φ. On the other hand, for an arbitrary ower characteristic vector p =p 1,p 2 P φ, in the directions t 2 = e, t 1 +, andt 1 = e, t 2 +, weobtain the inequaities p 1 α 2 and p 2 1/α 1, respectivey. Since the set P φ can be represented [3 by a stricty monotone decreasing curve, we find that it necessariy coincides with P. DIFFERENTIAL EQUATIONS Vo. 41 No

5 188 IZOBOV, KRUPCHIK 2.2. PROOF OF THE COINCIDENCE OF THE LOWER CHARACTERISTIC SET OF THE SOLUTION x WITH THE SET P Let us show that the ower characteristic set P x of the soution x coincides with the ower characteristic set P φ of the function φ. To this end, we prove the existence of the imit im t t 1 νtnψt =0. 9 By setting ΠL = Π Π Π +1, N, and by taking some N, weestimatethe function ν in the strip ΠL. Note first that since θ t beongs to the interva [ / j τ j 2 2 /, j 1 + τ j 2 2, we have the inequaities j 1 < j 1 + τ j /2 θ t j τ j /2 < j, which impy that P j θt µ.sinceζt µ α2 1 inthestripπl, it foows from 6 1that ν t = P j θ t µ 4 t 1, [ / θ t j τ j 2 2 /, j 1 + τ j 2 2, 10 1 j =1,...,k+1, t Π. By using the inequaity ζt max j=0,...,k+1 g j α 2 1, which is vaid in each strip ΠL, N, fromreation6 2, we obtain the estimates ν t = g j 4 t 1, θ t [ j + τ j+1/ 4 2 /, j τ j 4 2, 10 2 j =1,...,k, t ΠL. By virtue of 6 5 and6 6, we have the inequaities ν t = g α 1 4 / t 1, 2α 2 1 θ t α 1 + τ 1 4 2, t ΠL, 103 ν t = g α 2 4 / t 1, θ t α 2 τ k+1 4 2, t ΠL. 104 Now et θ t / j τ j 4 2 /, j τ j 2 2 with some j {1,...,k+1} and t ΠL. If g j P j θt 0, then from 63, we obtain the estimates ν t P j θ t ν t P j P j θ t + g j P j But if g j P j θt < 0, then we have the inequaities ν t P j θ t 4 t 1, ν t P j θ t + g j P j θ t 4 t 1, θ t g j 4 t 1. θ t g j 4 t 1. DIFFERENTIAL EQUATIONS Vo. 41 No

6 THE CONSTRUCTION OF A PFAFF SYSTEM WITH ARBITRARY PIECEWISE We have thereby proved the estimate ν t 4 t 1, θ t j =1,...,k+1, In a simiar way, one can show that ν t 4 t 1, θ t j + τ j+1 j =0,...,k, / j τ j 4 2 /, j τ j 2 2, t ΠL. t ΠL. / 2 2 /, j + τ j+1 4 2, Therefore, for each N from the estimates , we obtain the inequaity ν t 4 t 1, t ΠL, θ t 2α By virtue of definitions of the function νt and inequaity 11, we have the estimate νt 4 t 1, t R 2 >1, θ t 2α By using reation 5 and the estimate 12, we obtain the inequaity νtnψt 4 t 1 n t 1, t R 2 >1, 13 which impies 9. We have thereby shown that the ower characteristic set P x of the soution x of Eq. 1 1 coincideswithp. 3. PROOF OF RELATION 2 FOR THE SOLUTION x We take an arbitrary interior point p =p 1,ϕp 1, α 1 <p 1 <α 2, of the ower characteristic set P x. Let the imit n x p, 0 be reaized aong a sequence {tm} [6 for which there exists a finite imit im θ tm = θ>0. There are ony two possibe cases: θ p 2 1 and θ = p2 1. Let θ p 2 1. Since im p1 + 2 θ tm = p θ > 0, without oss of generaity, one can assume that the sequence {tm} satisfies the inequaity p θ tm p / θ 2 > 0, m N. It foows from that R φ p, t Mt 1 m, { M =min p } θ /2p 1 ; p 1 ; α 2 p 1 > 0. Then, by using 3, we obtain the estimate n x p, 0 im Mt 1 m 4 t1 mnt 1 m / n tm =+. Now we suppose that θ = p 2 1. Without oss of generaity, we assume that a eements of the sequence tm reaizing the ower imit n x p, 0 beong to the strips Π m Π m with distinct indices m > 1, m+1 > m + as m +. The foowing two cases are possibe: i either p 1 does not coincide with any point of discontinuity of the function g, i.e., p 1 j0 1, j0, j 0 {1,...,k+1}, or ii p 1 coincides with some point of discontinuity j0, j 0 {1,...,k}, ofthe function g. DIFFERENTIAL EQUATIONS Vo. 41 No

7 190 IZOBOV, KRUPCHIK By virtue of the incusion im θ tm = p 2 1 j0 τ j0 /2 2 0, j0 1 + τ j0 /2 2 0 with some 0 N, in the first case without oss of generaity, we assume that the sequence {tm} satisfies the inequaities j0 τ j0 / < θtm < j0 1 + τ j0 / for a m N. Again, without oss of generaity, we assume that m 1 0 for a m N. Consequenty, / j0 τ j0 2 2 / m 1 <θtm < j0 1 + τ j0 2 2 m 1, m N. 14 Without oss of generaity, one can restrict considerations to the foowing two possibiities: 1 tm Π m for a m N; 2 tm Π m for a m N as we. In the first possibe case, by using 7 1, 6 1, and inequaity 14, we obtain νtm = P j0 m θ tm, and reations 5 and 14 impy that n φtm = n t 1 m. Since the poynomias P j0 m uniformy converge to the function g j0 as m on the interva [ j0 1, j0, j0 1 < θ tm < j0 for a m N and since g j0 is a continuous function on the interva [ j0 1, j0, it foows from that n x p, 0 im P j0 m / θ tm n t 1 m n tm = im P j0 m θ tm / 2=g p 1 / 2. But if the second possibiity takes pace, then, without oss of generaity, we consider ony two cases: either ν m tm ν m 1tm for a m N, orν m tm >ν m 1tm for a m N. In the first case, from 7 2, we obtain the estimate νtm ν m tm, whence it foows that n x p, 0 g p 1 / 2. In the second case, from 7 2 and 14, we have νtm ν m 1tm = P j0 m 1 θ tm. Therefore, we have again obtained the estimates n x p, 0 im P j0 m 1 / θ tm n t 1 m n tm = g p 1 / 2. By virtue of the inequaities j0 τ j0 /2 m 2 <θ τ m < j0 1 + τ j0 /2 m 2, m 0, m N, aong some sequence {τm}, τm Πm, θ τ m = p 2 1, m 0, m N, reations7 1 and6 1 impy that ντm = ν m τm = P m j0 θτ m, and consequenty, im R φ p, τm + P j0 m = im P j0 m / θ τ m n τ 1 m n τm θ tm / 2=g p1 / 2. We have thereby proved the desired inequaity 2 for a points p 1 of continuity of the function g. Now et p 1 coincide with some point of discontinuity j0, j 0 {1,...,k}, of the function g. Without oss of generaity, we consider the foowing three possibe cases: 1 j0 + τ j0+1/ 4 2 m θtm j0 τ j0 / 4 2 m for a m N; DIFFERENTIAL EQUATIONS Vo. 41 No

8 THE CONSTRUCTION OF A PFAFF SYSTEM WITH ARBITRARY PIECEWISE α 2 1 >θ tm > j0 τ j0 / 4 2 m for a m N; 3 α 2 2 <θ tm < j0 + τ j0+1/ 4 2 m for a m N. In the first case, we have ν m tm = g j0, m N, and the inequaities / j0 + τ j / m 1 θtm j0 τ j0 4 2 m 1, m N, and 6 2 impythat ν m 1tm = g j0, m N. It foows from 7 1 and7 2 thatνtm = g j0, m N. Therefore, we obtain the estimates n x p, 0 im g j 0 nt 1 m/ n tm = g j0 / 2. If the second possibiity takes pace, then θtm > j0 + τ j0 /4 m, m N. Since tm Π m Π m, it foows from the definition of the strips that ζtm m α 2 1, m N, whence t 1 m m.byusing8 1, we obtain the estimates R φ p, tm t 1 m j0 + 2 / θ tm j0 t 1 mτ 2 j 0 /16 m j0 t 1 mτ 2 j 0 /16 j0. Consequenty, by virtue of 13, we have n x p, 0 im / t 1 mτ 2 j 0 /16 j0 4 t1 mnt 1 m n tm =+. For the third possibiity, we have θtm < j0 + τ j0+1/ 4 m, m N. Since tm Π m Π m, we again have the inequaity t 1 m m. By using inequaity 8 1, we obtain the estimates R φ p, tm t 1 m j0 + 2 / θ tm j0 t 1 mτ 2 j / m j0 t 1 mτ 2 j 0+1/16 j0, whence it foows that n x p, 0 im t 1 mτ 2 j /16 / 0+1 j 0 4 t1 mnt 1 m n tm =+. In addition, the inequaity ντm = g j0 is vaid aong the sequence {τm}, τm Πm, θ τ m = 2 j 0, m N; consequenty, im R φp, τm + g j0 nτ 1 m/ n τm = g j0 / 2. This competes the proof of reation 2 for points j, j =1,...,k, of discontinuity of the function g as we. DIFFERENTIAL EQUATIONS Vo. 41 No

9 192 IZOBOV, KRUPCHIK 4. CONSTRUCTION OF THE EQUATION. BOUNDEDNESS OF THE COEFFICIENTS The function x>0given by 3 is a soution of Eq. 1 1 with the coefficients a i t = n xt/ t i, t R>1, 2 i =1, 2, satisfying the condition of the compete integrabiity by virtue of the infinite differentiabiity of n x in R>1 2. By using the inequaity 0 de 01 τ; τ 1,τ 2 dτ { [ 2exp 2τ 2 τ 1 2 if τ 2 τ 1 1/2 4 if τ 2 τ 1 2, from the emma in [11, we show that these coefficients are bounded. First, et us show that the derivatives n φt/ t i, i =1, 2, are bounded. If α 2 2 /2 θ t 2α 2 1, then we have the estimates n φt/ t 1 3 θ t α 2 + e 8 n φt/ t 2 3 t θ 2 t /α 1 /α 21 +2e8 2θ 3/2 t α 2 θ t /α 2 2 σ 1, 2 / θ t θ t /α 1 α e 8 2 θ t α 2 /α 2 2 σ 1 2θ 3/2 θ 1 t 1/α 1 + e 8 with some constant σ 1 > 0. The fact that the same derivatives are bounded for θ t <α 2 2 /2and θ t > 2α 2 1 is obvious. Let us now proceed to proving the boundedness of partia derivatives of the product νtnψt. By 5, we have 15 νtnψt =0, t R 2 >1, θ t 0,α 2 2 /2 [ 2α 2 1, +, 16 which impies that it suffices to prove the boundedness of the derivatives νtnψt/ t i, i =1, 2, for θ t α 2 2 /2, 2α2 1. By using 12 and 5, we obtain νt n ψt/ t 1 4 t 1 2e 8 2α 2 1/α n t 1 /t 1 +1/t 1 σ2, t R 2 >1, θ t α 2 2 /2, 1 2α2, νt n ψt/ t 2 4 t 1 2e 8 / t 1 α2 2 + e 8 / t 1 α1 2 n t1 σ 2, t R 2 >1, θ t α 2 2 /2, 1 2α2, with some constant σ 2 > 0. We take some N and estimate the derivatives of the function ν in the strip ΠL. Let [ / θ t j τ j 2 2 /, j 1 + τ j 2 2 with some j {1,...,k+1}, t ΠL. Then from 6 1 and from the inequaity ζt ϱ α2 1, t ΠL, we obtain ν t/ t 1 = dp j /d = θt /2t 1 ϱ α 1 /2t 1 α 1 / 2 t 1, 171 θ / t ν t/ t 2 ϱ 2 θ t t 1 1/ 2α 2 t But if either θ t [ j + τ j+1/ 4 2 /, j τ j 4 2, j {1,...,k}, or θ t / α 1 + τ / or θt α 2 τ k+1 4 2,then ν t/ t i =0, i =1, DIFFERENTIAL EQUATIONS Vo. 41 No

10 THE CONSTRUCTION OF A PFAFF SYSTEM WITH ARBITRARY PIECEWISE / If θ t j τ j 4 2 /, j τ j 2 2, j {1,...,k+1}, t ΠL, then, by anaogy with 17 1 and17 2, from 6 3 and the inequaities t 1 µ 4 2 and t 1 g j 4 2, we obtain the estimates ν t/ t 1 α 1 / t 1 +2e 8 g j + µ θ t /t 1 τ j α 1 / / t 1 4e 8 4 α 1 t 3 1 τ j, ν t/ t 2 1/ α 2 t1 +2e 8 g j + µ / θt t 1 τ j 1/ / α 2 t1 4e 8 4 α 2 t 3 1 τ j / Finay, if θ t j + τ j /, j + τ j+1 4 2, j {0,...,k}, t ΠL, then, in a simiar way, it foows from 6 4 that ν t/ t 1 α 1 / 2 t 1 +2e 8 µ + g j θ t /t 1 τ j+1 α 1 / 2 / t 1 4e 8 4 α 1 t 3 1 τ j+1, ν t/ t 2 1/ 2α 2 t1 +2e 8 µ + g j / θt t 1 τ j+1 1/ / 2α 2 t1 4e 8 4 α 2 t 3 1 τ j The estimates impy the inequaity ν t/ t i σ 3 / t 1, t ΠL, i =1, 2, 18 with some constant σ 3 > 0. From definition 7 1 of the function νt in the basic strips Π and from 18, we obtain the estimates νt/ t i σ 3 / t 1, t Π, N, i =1, By using 7 2, 7 3, 18, 11, and the second inequaity in 15, we obtain the estimates νt/ t i σ 3 / t t 1 /ζt σ 3 / t 1 +8/ 4 t 3 1, t Π, N, θ t 2α 2 1, i =1, Therefore, it foows from inequaities 19 1, 19 2, the reation νt =0,t T, the estimate n ψt n t 1, t R>1, 2 and from 16 that the products νt/ t i nψt, t R>1, 2 i =1, 2, are bounded. We have thereby shown that the coefficients of system 1 1 are bounded. The proof of Theorem 1 for α 1 <α 2 < 0 is compete. But if α 2 0, then on the basis of the function g, we introduce a new function g 1 :[α 1 α 2 1, 1 R by setting g 1 = g + α Then, just as above, for the function g 1, we construct an infinitey differentiabe function x 1 > 0 so as to ensure that this function has bounded derivatives n x 1 t/ t i, i =1, 2, the domain of the curve of its ower characteristic set P x1 coincides with the interva [α 1 α 2 1, 1, and the function itsef satisfies the reation c x1 p 1 [x 1 = g 1 p 1 [x 1 for a p 1 [x 1 α 1 α 2 1, 1. The ower characteristic vectors p 1 [x,ϕp 1 [x P x and p 1 [x 1,ϕ 1 p 1 [x 1 P x1 of the functions xt =x 1 texp[α 2 +1t 1 andx 1 t, respectivey, are reated by the conditions p 1 [x =p 1 [x 1 +α 2 +1 [α 1,α 2 andϕp 1 [x = ϕ 1 p 1 [x α 2 1, and the ower characteristic degree functions are reated by the formuas c x p 1 [x = c x1 p 1 [x 1 = g 1 p 1 [x 1 = g p 1 [x. DIFFERENTIAL EQUATIONS Vo. 41 No

11 194 IZOBOV, KRUPCHIK Obviousy, the function x is a soution of a competey integrabe Pfaff equation 1 1 with bounded infinitey differentiabe coefficients, which competes the proof of Theorem 1. Theorem 2. For an arbitrary positive integer n and for an arbitrary piecewise continuous function g :[a 1,a 2 R, there exists a competey integrabe Pfaff system 1 n with infinitey differentiabe bounded coefficients such that, for each nontrivia soution x : R>1 2 R n \{0} of this system, the domain of the curve Λ x coincides with the interva [a 1,a 2 and c x λ 1 =gλ 1 for a λ 1 a 1,a 2. Proof. We use the function g to introduce a new piecewise continuous function g 1 :[ a 2, a 1 R by setting g 1 = g. Foowing Theorem 1, we construct an infinitey differentiabe function x 1 > 0 such that the function has bounded derivatives n x 1 t/ t i, i =1, 2, the domain of the curve P x1 of this function coincides with the interva [ α 2, α 1, and c x1 p 1 [x 1 = g 1 p 1 [x 1 for a p 1 [x 1 α 2, α 1. We use the function x 1 to define a soution x = x 1 1 of the competey integrabe Pfaff equation 1 1 with bounded infinitey differentiabe coefficients a i t = n xt/ t i, t R>1, 2 i =1, 2. Note that the characteristic vector λ[x ofthesoutionx is equa to the ower characteristic vector p [x 1 of the function x 1 with the opposite sign, i.e., λ[x = p [x 1. Therefore, c x λ 1 [x = 2 n x λ[x, 0 = 2n x1 p [x 1, 0 = c x1 p 1 [x 1 = g 1 p 1 [x 1 = g p 1 [x 1 = g λ 1 [x for a λ 1 [x a 1,a 2, which competes the proof of Theorem 2. REFERENCES 1. Gaishun, I.V., Vpone razreshimye mnogomernye differentsia nye uravneniya Competey Integrabe Mutidimensiona Differentia Equations, Minsk, Gaishun, I.V., Lineinye uravneniya v ponykh proizvodnykh Linear Equations in Tota Derivatives, Minsk, Izobov, N.A., Differents. Uravn., 1997, vo. 33, no. 12, pp Grudo, E.I., Differents. Uravn., 1976, vo. 12, no. 12, pp Demidovich, B.P., Mat. Sb., 1965, vo. 66, no. 3, pp Krupchik, E.N., Differents. Uravn., 1999, vo. 35, no. 7, pp Lasyi, P.G., A Remark on the Theory of Characteristic Vector Degrees of Soutions of Pfaff Linear Systems, Preprint IM AS BSSR, Minsk, 1982, no Izobov, N.A. and Krupchik, E.N., Differents. Uravn., 2001, vo. 37, no. 5, pp Korn, G. and Korn, T., Mathematica Handbook for Scientists and Engineers, New York: McGraw-Hi, Transated under the tite Spravochnik po matematike dya nauchnykh rabotnikov i inzhenerov, Moscow: Nauka, Gebaum, B. and Omsted, J., Counterexampes in Anaysis, San Francisco, Transated under the tite Kontrprimery v anaize, Moscow: Mir, Izobov, N.A. and Krupchik, E.N., Differents. Uravn., 2003, vo. 39, no. 3, pp DIFFERENTIAL EQUATIONS Vo. 41 No