Карпатські математичні публікації T2 N2

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i

i

ii

.2, 2

2010

i

.. i i -

iL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

.., .., .. i i

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

.I. i i -

i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

.., I..

i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

i I.. i - i

ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

.I., .., .. i i -

i i i . . . . . . . . . . . . . . 48

.I., .., .. i i

. . . . . 55

.., I.. i 74

.. i i i ii . . . . . . . . . . . . 83

I.. i ii i-

i . . . . . . . . . . . . . . . . . . . . . 101

. . . . . . . . . . . . . . . 111

.I., .I. i -

i i . . . . . . . . . . . . . . . . . 116

.. iii i i i . . . . . . . . . . . . . . 123

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Carpathian

mathematical

publications

Scientific journal

V.2, 2

2010

Contents

Balan V.A. Construction of radially bounded antiproximinal sets inL1 . . . . . . . . 4

Voloshyn H.A., Maslyuchenko V.K., Maslyuchenko O.V. On approximation of the se-

parately and jointly continuous functions . . . . . . . . . . . . . . . . . . . . . . . 10

Dmytryshyn M.I. The spaces of exponential type vectors of complex degrees of positiveoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Zagorodnyuk A.V., Chernega I.V. Spectra of algebras of symmetric and subsymmetric

analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Zarichnyi I.M. Characterization of the macro-Cantor set up to coarse equivalence . . . 39

Kopach M.I., Obshta A.F., Shuvar B.A. An application of analogues of two-sided

Kurpels methods to ordinary differential equation . . . . . . . . . . . . . . . . . . 48

Kopytko B.I., Mylyo O.Ya., Tsapovska Zh.Ya. A parabolic conjugation problem with

general boundary condition and a conjugation condition of Wentzel type . . . . . . 55

Malytska H.P., Burtnyak I.V. Method of parametris for the ultraparabolic systems . . 74

Nykyforchyn O.R. Atomized measures and semiconvex compacta . . . . . . . . . . . . 83

Savka I.Ya. Nonlocal problem with dependent coefficients in conditions for the second-

order equation in time variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Savchenko A. Extensions of partial fuzzy metrics . . . . . . . . . . . . . . . . . . . . . 111

Sobkovich R.I., Kazmerchuk A.I. Solvability of n-point problems with parameter for

system of differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Sharyn S.V. Polynomial tempered distributions . . . . . . . . . . . . . . . . . . . . . . 123

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.2, 2

2010

..

L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

.., .., ..

. . . . . . . . . . . . . . . . . . . . . . . . . . 10

..

. . . . . . . . . . . . . . . . . . . . . . . . . 21

.., .. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

.. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

.., .., .. -

. . . . . . . . 48

.., .., ..

. . . . . 55

.., ..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

.. . . . . . . . . . 83

..

. . . . . . . . . . . . . . 101

. . . . . . . . . . . . . . . 111

. ., ..

. . . . . . . . . . . . 116

.. . . . . . . . . . 123

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i i Carpathian Mathematical

ii. .2, 2 Publications. V.2, 2

517.52

..

I I

I L1

.. i i iL1// i i ii. 2010. .2, 2. C. 49.

, iL1

i ii ii i .

i i x i X

M X d(x, M) = x M = inf{x y : y M}.

y M x, x y = d(x, M), i

i x i M PM(x). M i (AP-), PM(x) =

i x X\ M.

X i X. , i f X

i M X, i x M , f(x) = sup f(M).

(M) i ii, i i M,

(M) = {f X : x M | f(x) = sup f(M)}.

1972 i . i A. [5] , ii

i M X i , i M i

i i B,

(A) (B) = {0}.

[1-6] , i i -

i . , , :

c0, c, c(X) ( i X), L. i

2000 Mathematics Subject Classification: 46E30, 46B28.i i : i L1, i , i i.

c .., 2010

8/6/2019 T2 N2

5/132

i i i L1 5

. : i i i L1

i ?

M i X K, 0 M, i

, x X { K : x M}

K.

, [2] i i i L1[1, 1].

i i ii [2] i , i w-

i ii ii L1 i i -

i i L1.

1 ii

ii,

(BL1) = {f L : ({t [0, 1] : |f(t)| = f}) > 0},

BL1 L1.

i i, L1, i i i

L1 L. , y L,

y(x) =

[0,1]

y(t)x(t)d i x L1.

i A ii i X sp(A)

ii A.

ii i .

1.1. (yn)n=1 ii i yn L, (an)n=1

i ii an R, an = 0, (zn)n=1 ii i zn L,

yn = yn + anzn n N, ii (zn)n=1

:

(i) i A [0, 1] (A) > 0, in

k=1

bkzk

A, bk = 0 i 1 k n.

i sp{yn : n N (BL1)} = {0}.

. i i h : [0, 1] R, h sp{yn : n N} (BL1). i

h (BL1), A = {t [0, 1] : |h(t)| = h} , (A) > 0. , (A+) > 0, A+ = {t [0, 1] : h(t) = h}.

i , i h sp{yn : n N}, h =n

k=1

(kyk + kakzk).

n

k=1

kyk(t) +n

k=1

kakzk(t) = h

i t A+. n

k=1

kakzk(t) = h n

k=1

kyk(t) = g(t). i i g(t)

, i c R i B A+ (B) > 0, g(t) = c

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8/6/2019 T2 N2

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i i i L1 7

i xnk+1 L, xnk+1 2k i ynk xnk+1 k+1.

i

yn k+1

i=n

xni ynk xnk+1 + k+1nk+1 2k + k+1,

(ii). unn = 2xnn i unk = 2

kxnk k > n. i ii : N

{(n, k) : n N, k n} i zm = u(m).

N i i i :

M1 = {1(n, n) : n N}, M2 = N \ M1.

i ynw

0 i yn xnn 2n, xnnw

0, i unnw

0, , , i-

i (zm)mM1 w-i . , i xnk

14k

= k, unk 12k

i n N i k > n. , i 2.2, ii (zm)mM2

i . zm w

0 m .

, i yn k

i=n

xni 2k, yn =kn

xnk. , k

i=n

xni =

12

unn +k

i=n+1

12i

uni cc{uni : i n} = cc{zn : n N} i n N i k > n, ,

yn cc{zn : n N}w

n N.

, i i xnk i i i mnk ii, -

ii i xnk (i) 1.1. -

ii (zn)

n=1

. , sp{zn : n N} (BL1) = {0}.

1.4. X i, (yn)n=1, (zn)

n=1 ii

ii ii yn, zn X, i {yn : n N} sp{zn : n N}

w

i supnN

|yn(x)| > 0 i x X. i supnN

|zn(x)| > 0 i x X.

. supnN

|zn(x)| = 0 x X, zn(x) = 0 i n N.

, L = {y X : y(x) = 0} w- X. {yn : n

N} sp{zn : n N}w

L. , yn(x) = 0 i n N, i.

2

1. X i, B X, (yn)n=1 w

-

i ii ii yn X, sp{yn : n N} (B) =

{0}. i M = {yn : n N}o -.

. , (M) sp{yn : n N}.

y0 = 0, y0 (M). i ,

supxM

y0(x) = | supxM

y0(x)| = 1. i y0 (M), i x0 M,

y0(x0) = 1.

8/6/2019 T2 N2

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8 ..

i |y0(x)| 1 i x M, y0 Mo = {yn : n N}

oo. i

i, {yn : n N}oo w-

{yn : n N}, y0 cc{yn : n N}w

= B.

N1 = {n N : |yn(x0)| 0 ix X,

M i .

. i 1.3 i ii (zn)n=1 zn L,

znw

0, sp{zn : n N} (BL1) = {0} i {yn : n N} cc{zn : n N}w

.

B = cc{zn : n N}w

. , M = Bo = {zn : n N}ooo = {zn : n N}

o. i,

i 1 M - L1.

, M i . , i -

i cc{zn : n N}o i , sup

nN|zn(x)| > 0 i

x X. , i supnN

|yn(x)| > 0 i x X, i 1.4

supnN

|zn(x)| > 0 i x X.

, , i,

AP- M i L1 , i L1

i i c0.

2.1. X i i, (yn)n=1 w

-i i-

i i yn X i M = {yn : n N}

o. i, M , X i

c0.

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i i i L1 9

. T : X c0

X xT

(y1(x), y2(x),...) = T x.

i ynw

0, yn(x) n

0 i x X, , , T x c0 i x X.

i, T ii. , T .

, i ii i ii (yn)n=1

. i , yn 1 -

n N. i x X. i T xc0 = (y1(x), y2(x),...)c0 = supn

|yn(x)|

supn

yn x x. , T 1 i T .

C > 0 , x C x M = {z X : supn

|yn(z)| 1} =

{z X : T zc0 1}. T zc0 1C

z z X, T

. , T i .

i

1. ..

// . . 1983. T.33, 4. . 549558.

2. Balaganskii V.S. Antiproximinal sets in the space of continuous functions, Math. Notes, 60, 5 (1996),

485494.

3. Cobzas S. Antiproximinal sets in the Banach space c(X), Comment. Math. Univ. Carolinae, 38, 2 (1997),

247253.

4. Cobzas S. Antiproximinal sets in the spaces c0 and c, Math. Notes, 17 (1975), 449457.

5. Edelstein M., Thompson A.C. Some results on nearest points and support properties of convex sets in c0,

Pacific J. Math., 40 (1972), 553560.

6. Klee V. Remarks on nearest points in normed linear spaces, Proc. Colloquium on Convexity, Copenhagen,(1965), 168176.

i i i i. . ,

ii,

i 20.09.2010

Balan V.A. Construction of radially bounded antiproximinal sets inL1, Carpathian Mathema-tical Publications, 2, 2 (2010), 49.

It is shown that in L1 space the polar of the weakly convergent sequence contains radially

bounded completely convex set.

..

L1 // . 2010. .2, 2. C.

49.

, L1

.

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517.51

.., .., ..

I I

I

.., .., .. i i

i// i i ii. 2010. .2, 2. C. 1020.

i : i i ii L

C(Y) i Y i i i i X -

i, i i f : X Y R i -

ii i i fn : XY R, fxn = fn(x, ) L

i n N, x X, i fxn fx Y x X? :

i C(Y) , i f : XY R

i i fn .

1. i if : R2 R

i fn : R2 R, i o ii i i, . i i i [10], i , i i f : R2 R . . i[15] , i i i i. i [1] i , i i f : [0, 1]2 R i ii i iii i i fn : [0, 1]2 R, fxn (y) = fn(x, y) f(x, y) = f

x(y)

[0,1] n x [0, 1].

i i, i i ii i i, i, , , i i i i - i. i i i [12, 3, 2], i [5], -i, i . , i ii (25 2009), ii i i i i i i ii (15

2000 Mathematics Subject Classification: 54C30, 65D15.i i : i i i i, i i -

i.

c .., .., .., 2010

8/6/2019 T2 N2

11/132

i i i 11

2009) i ii i i ii i i i [7, 4], i i.

i i , i [7], i , [7, 2].

2. i Y C(Y) i i - i i g : Y R, Cp(Y) i,i i ii, i qy(g) = |g(y)|, y i Y [8, c.30].

X i i. i f : X Y R i (x, y) X Y fx(y) = fy(x) = f(x, y). i f i, ifx : Y R i fy : X R i i x Xi y Y. i i i i f : X Y R CC(X Y).

i if : XY R iii i : X RY, (x) = fx x X. , i fi i f.

1. X i Y ii i i f : X Y R. i i ii:

(i) f CC(X Y);

(i) (X) C(Y) i i : X Cp(Y) .

. (i)(ii). f CC(X Y). i x X (x) = fx C(Y), , (X) C(Y). , i : X Cp(Y) i i x X. i x0 X i i V g0 = (x0) i Cp(Y), i g Cp(Y),

maxk=1,...,n

|g(yk) g0(yk)| < ,

y1, . . . , yn Y i > 0. i i i fyk k = 1, . . . , n i ix0, i i U x0 X, i x U i k = 1, . . . , n ii |fyk(x) fyk(x0)| < . x U i g = (x). i

k = 1, . . . , n

|g(yk) g0(yk)| = |fx(yk) f

x0(yk)| = |fyk(x) fyk(x0)| < ,

, g V. , (U) V, i i i i x0.

(ii)(i). (X) C(Y) , fx C(Y) x X. -, y Y i y : Cp(Y) R, y(g) = g(y) i. i i : X Cp(Y) i ii fy : X R, fy = y , (y )(x) = y((x)) = fx(y) = fy(x).

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12 .., .., ..

3. Y Cu(Y) i i(C(Y), ), ii C(Y), ii

g = maxyY

|g(y)|.

i i X i Y C(X Y) i i

i f : X Y R, i, i i X Y. , C(X Y) CC(X Y).

2. X i i, Y i i - if : X Y R. i i ii:

(i) f C(X Y);

(ii) (X) C(Y) i i : X Cu(Y) .

. (i)(ii). f C(X Y). i f CC(X Y), , (X) C(Y) 1.

, i : X Cu(Y) . x0 X i > 0.i i f , y Y iii Uy x0 X i Vy y Y,

|f(u, v) f(x0, y)| 0

xq(l0, , l1, )

,q

jZ

2qj(01) supk

min(2k0 , 2j(01)+k1)xkC

q

jZ

2qj[0(1)+1]xjqC = cx

ql,q

.

, (l0, , l1, ),q l

,q . x l

,q ,

ii

xq(l0,1 , l

1,1 ),q

jZ

2qj(01)

kZ+

min(2k0, 2j(01)+k1)xkCq

ckZ

2kq xkqC = cx

ql,q

c > 0. l,q (l0,1 , l

1,1 ),q.

ii i (l0,1 , l1,1 ),q

l0,q0 , l

1,q1

,q

(l0, , l1, ),q 1 q, q0, q1 . i l

,q

l0,q0 , l

1,q1

,q l,q , i i-

i (1) .

i Eq (C) i i i lq =

(k) : k

C, (k)lq =

kZ+k

qC

1/q<

. ii ii I: Eq (C

)

x (k := kxk) l

q . K-i ii

lq0, l

q1

,q

i

K(t, I(x), lq0 , lq1

) infx=x0+x1

I(x0)lq0 + tI(x1)lq1

IEq0(C)lq0K

tIEq1(C

)lq1

IEq0(C)lq0

, x, Eq0(C), Eq1(C

)

.

i = tIEq1(C)lq1

I1Eq1(C)lq0

,

I(x)(lq0 , lq1),q

IEq0 (C)lq0

IEq1(C

)lq1

IEq0(C)lq0

x(Eq0(C), Eq1 (C

)),q

.

ii i i I1, i

I,

I(x)(lq0

, lq1),q

= x(Eq0

(C),Eq1

(C)),q

i x Eq0(C), Eq1(C),q .

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i i 25

i i Eq (C, C),q i i

l,(,)q, =

x := (xk)kZ+ : x Eq (C

, C),q

xl,(,)q,

= xEq (C,C),q . -

i i ii i

lq0, lq1

,q

= l(,)q, ,

l(,)q, =

(k) : k (C, C),q , (k)l(,)

q,

=

kZ+k

q(C,C),q

1/q<

, 1/q =

(1 )/q0 + /q1 [10, 1.18.1], ii (2).ii (3) (1) i 4.7.2 [7].

x [Eq0(C), Eq1(C

)], f(z) F(Eq0

(C), Eq1(C)) i f() = x. i

g(z) =

IEq0(C

)lq0

IEq1(C)lq1

zIf(z) F(lq0, l

q1

), g() = Ix.

i

g(z)F(lq0 ,lq1 )

I(1)Eq0(C

)lq0I

Eq1(C)l

q1

f(z)F(Eq0(C),Eq1(C

))

Ix[lq0 ,lq1 ]

I(1)Eq0(C)lq0

IEq1 (C)lq1

x[Eq0(C),Eq1(C

)] .

ii i i I1,

I(x)[lq0 ,lq1 ]

= x[Eq0 (C),Eq1 (C

)] i x [Eq0

(C), Eq1(C)].

i Eq [C, C] i i l

,[,]q, =

x := (xk)kZ+ : x Eq [C

, C]

xl,[,]q,

= xEq [C,C ] . ii (4)

i i ii

lq0, l

q1

= l[,]q, , l

[,]q, =

(k) : k

[C

, C

], (k)l[,]q, =

kZ+ kq

[C,C ]1/q

<

, 1/q = (1 )/q0 + /q1 [10, 1.18.1].

Eq (C, C),q, E

q [C

, C] i, i, i (1) i (2), i-

i i Eq0(C) i Eq1(C

).

2 ii

0 < , < , 1 q , 0 < . ii

E,q, (C) =

x C : xE,q, (C) =

0

tE(t, x)dtt

1/

<

,

E,q,(C) =

x C : xE,q,(C) = sup

t>0tE(t, x) <

,

E(t, x) = inf yEq (C)t

x yC, y Eq (C

), 0 < t < .

E,q, (C)

, 0 < < 1 i E,q, (C) i x

E,q, (C). i

7.1.7 [7], = 1/( + 1) i = r (0 < r ) ii

E,q, (C) = Eq (C), C,r

. (5)

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i i 27

. i ii [7, 3.11.5], = (1 )0 +1,

0 = 1/(0 + 1), 1 = 1/(1 + 1) i = r E,0q,0 (C

)0, E,1q,1 (C)1

,=

E,q,r (C)

. (12)

i [7, 3.11.6], E,0q,0 (C

)0, E,1q,1 (C)1

,=

E,0q,0 (C), E,1q,1 (C

)

,r, (13)

= 1/. (12) i (13) = (1 )0 + 1

E,0q,0 (C), E,1q,1 (C

)

,r=

E,q,r (C), i (9).

0 < <

xE,0q,0 (C

),E,1q,1 (C

),

0tK(t, x; E,0q,

0

(C), E,1q,1

(C))dt

t

1/

supt>0

tK(t, x; E,0q,0 (C), E,1q,1 (C

))

(1 /) c x

E,0q,0 (C

),E,1q,1 (C

),

,

i (10). E,q, (A) E,q,(A) i i-

i K(t, x; E,0q,0 (C), E,1q,1 (C

)) c1txE,q, (C), x E

,q, (C

).

I ii K(t, x; E,0q,0 (C), E,1q,1 (C

)) t xE,1q,1 (C), x E,1q,1 (C

),

xE,0q,0

(C),E,1

q,1(C)

1,1

= 1

0

t1K(t, x; E,0q,0 (C), E,1q,1 (C

))dt

t

+1

t1K(t, x; E,0q,0 (C), E,1q,1 (C

))dt

t c xE,1q,1 (C)

+

supt>0

t0K(t, x; E,0q,0 (C), E,1q,1 (C

))

1

z(10)dz

z

c1 xE,0q,0 (C

),E,1q,1 (C

)0,

,

i

E,0q,0 (C), E,1q,1 (C

)

0,

E,0q,0 (C), E,1q,1 (C

)

1,1. i i

(10) (11).

0 < < 1, 0 Re < Re < , 1 q i i Y i

i C, (C, C),q [C, C]. ii i (E

(n)q (Y)),

ii i ii ((n))nZ+ i, limn

(n) =

i i i i Y

l[E(n)q (Y), Y] =

nZ+yn = y Y : yn E

(n)q (Y)

yl[E

(n)q (Y),Y]

= inf y=

ynnZ+

ynY, ii i .

i Eq(Y) := >0

Eq (Y).

8/6/2019 T2 N2

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28 .I.

3. i ii

l

E(n)q (Y), Y

= Eq(Y), (14)

Y.

. i i i i l1 =

nZ+yn =y : yn E

(n)q (Y)

yl1 = inf

y=

yn

nZ+

ynE(n)q (Y), ii i

. i i l1[E(n)q (Y)] =

nZ+

yn = y :

yn E(n)q (Y)

y

l1[E(n)q (Y)]

=

nZ+ynE(n)q (Y). i i, i

E(n)q (Y) i. ii yl1 yl1[E(n)q (Y)] , i

l1[E(n)q (Y)] y y l1

i l1 - l1[E(n)q (Y)] , i .

- x E(n)q (Y) xl1 xE(n)q (Y).

E(n)q (Y) E(n+1)q (Y),

i ii i xl1 limn

xE(n)q (Y)

= xY.

> 0 i i Y

yn ,

yn = y yl1

nZ+ynE(n)q (Y)

nZ+ynY. i , - i Y

yn = y

nZ+ynY

nZ+ynE(n)q (Y).

i yl[E

(n)q (Y),Y]

yl1

nZ+ynY + , yl1 = yl[E(n)q (Y),Y]. ,

i l

E(n)q (Y), Y

i.

- y Eq(Y) i -

Y y =

yn , yn E(n)q (Y), ii

l

E(n)q (Y), Y

= Eq(Y) . , yY

nZ+

ynY - -

, yY yl[E(n)q (Y),Y] i y Eq(Y) i ii i.

3

i L() (1 < < ) i 2l

A : C1 u ||2l

a(t)Du(t) L(), a C

() (15)

C1 :=

u W2l () : Bju(t)| = 0; j = 1, . . . , l

, W2l ()

i i Bj =

||kjbj,(t)D

, bj,(t) C(), 0 k1 < k2 < . . . < kl,

i i. i ,

A . i [10, 4.9.1],

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i i 29

0 A + I . i i i

i A i A + I i [9], i ,

A . i [10, 5.4.3], A

(A) = {n}nN, limn

|n| = , i i i Rn

n C ii. i Lin ii i,

i i Y i i i (C, C),q [C, C] (0 Re < Re < ,1 q < , 0 < < 1).

4. 1 q0, q1 < , 1/q = (1 )/q0 + /q1, m, n N, = 0, 1, i

ii Eq(Y) = Y. i

Y =

nZ+

un = u : un Lin

Rk : |k| < min

(n)1

m+1 , (n)1

n+1

, (16)

= m0(1 ) + n0, = m1(1 ) + n1.

. i 1 [9], 1 q < i m N, Eq (C

m

) = Lin

Rn :|n|

m+1 <

. 1.2, 1.15.3 [10] i-

ii i Eq (Cm),

Eq (C) =

Eq0(C

m0), Eq1(Cn0)

= Lin

Rn : |n| < min

1m0+1 ,

1n0+1

, (17)

Eq (C) =

Eq0(C

m1), Eq1(Cn1)

= Lin

Rn : |n| < min

1m1+1 ,

1n1+1

, (18)

= m0(1 ) + n0, = m1(1 ) + n1. 1.2, i

i (17) (18), Eq (Y) = Lin

Rn : |n| < min

1m+1 ,

1n+1

.

ii (16) (14).

ii a A , i i [9] 1 q < i

i m N

Eq(Cm) :=

>0

Eq (Cm) =

u Exp (Cn)| : BjA

ku| = 0;j = 1, . . . , l; k Z+

,

Exp (Cn) i i i i i Cn. Y =

nZ+

un = u : un Exp (Cn)| ; BjA

kun| =

0;j = 1, . . . , l; k Z+

.

i

1. .., .I. i i

i // . . . 1995. . 47, 5. . 616628.

2. .., .. // .

1997. .9, 6. . 90108.

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30 .I.

3. .., .. - -

// . . . 1989. . 44, 3. . 5591.

4. .I., .. i , ii i

// . . 2007. 12. . 1622.

5. ., . Ii i i -

i i // i i. -. i .-. 2005. . 64. C. 99106.6. .. // . . 1983. . 27,

9. . 791793.

7. Bergh J., Lofstrom J. Interpolation spaces. An introduction, Springer, Berlin, 1976. 247 p.

8. Dunford N., Schwartz J.T. Linear operator. Part I: General theory. Intersci. Publishers, New York,

London, 1958.

9. Lopushansky O., Dmytryshyn M. Operator calculus on the exponential type vectors of the operator with

point spectrum // Chapter 12 in book General Topology in Banach Spaces. Nova Sci. Publ., Huntington,

New York. 2001. P. 137145.

10. Triebel H. Interpolation theory. Function spaces. Differential operators, Springer, Berlin, 1995. 664 p.

i i i. . ,

I-i,

i 15.10.2010

Dmytryshyn M.I. The spaces of exponential type vectors of complex degrees of positive operators,

Carpathian Mathematical Publications,2

, 2 (2010), 2130.New classes of interpolation spaces of exponential type vectors of complex degrees of posi-

tive operators are defined. Properties of the approximation spaces generated by the considered

interpolation spaces are investigated. An example of application of constructed theory to the

regular elliptic boundary problems is considered. In the example exponential type vectors co-

incide with root vectors. On the other hand, for operators with constant coefficients the set of

exponential type vectors is subclass of whole functions of exponential type.

.. -

// . 2010. .2,

2. C. 2130.

- . -

, -

. ,

,

-

.

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517.98

..1, I..2

I I

.., I.. i-

i // i i ii. 2010. .2, 2. C. 3138.

i i i i

L1[0,) L[0, ) L[0, 1] i .

X, Y i K, K = R K = C. ,

i P : X Y n-i i, i

n-ii i A : Xn Y , P(x) = A(x , . . . , x). i m

i n-i ii, n = 1, . . . , m . G i i i i X. i f

X i G ( G-,) f((x)) = f(x)

G. , X = p (1 p < ) i G = G

i . i G i i p :

i=1

xiei

=

i=1

xie(i),

{e1, e2 . . . , } i p. ii G-i i

p .i f i Lp[0, 1] , f(x) = f(x)

i , i ii ii [0, 1], i i

i.

I , X = p i G = G i, -

i i,

i : (x1, x2, . . .) (x1, . . . , xi1, 0, xi, . . .). (1)

2000 Mathematics Subject Classification: 46-02, 46E30, 46J20.i i : , i i ii i.

c .., I.., 2010

8/6/2019 T2 N2

32/132

32 .., I..

G-i i .

i i i p i Lp[0, 1], 1 < p < ,

i i [12]. i

ii i i [7, 1]. , i [7]

ii

-ii i - ii. [1] i -

i i p. i i i

i [2, 3, 4, 14, 15].

i [7] , , i ii , i

Fk(x) =i=1

xki , k = p, p + 1 . . . , (2)

i i ii i p,

p i , i i p. , - i P p + n 1, n 1 i i q Cn,

P(x) = q(Fp(x), . . . , F p+n1(x)).

Gk(x) =

10

xk(t)dt i i i i

Lp[0, 1], k = 1, . . . , p . i [7], i i Lp

ii Gk, k p.

i i i [8, 9, 12]. , i [8] (-

2.1) . , i i i

Fk1,...,kn(x) =

i1

8/6/2019 T2 N2

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i i 33

xE = max{xL1[0,), xL[0,)}.

Ps(E) i ii i E,

i i ii i [0, ).

, , Rk(x) =

0

xk(t)dt.

1.1. i Rk i E k = 1, 2, . . . , i ii {Rk} i Ps(E).

. i i E -ii i i-

i [0, ) (. [10, .118] ), ii ii {Rk}

Ps(E) i [7], 2.12.

i, P Ps(1), P = q(F1, . . . , F m), - iii Q Ps(E), Q = q(R1, . . . , Rm). , ii i

i ii.

(a1, . . . , an, . . .) 1

fa(t) =k=1

akk(t), (4)

k = [k 1, k), k = 1, 2, . . . , i k(t) =

1, t k;

0, t / k.

ifa = max

k

|ak|, supk

|ak|

=k

|ak| = a.

(Q) Q Ps(E) ii i i i

{fa, a 1}.

,

Rn

k=1

akk(t)

= Fn

k=1

akek

, (5)

Q(fa) = q(R1(fa), . . . , Rm(fa)) = q(F1(a), . . . , F m(a)) = 1(Q)(a).

, = 1.

1.2. i ii P Ps(1), Q = (P) Ps(E) iii:

P Q.

. P i 1, P = q(F1, . . . , F m) i Q =

(P) = q(R1, . . . , Rm).

, , P = c. i > 0 i x 1, x = 1,

c |P(x)| c + .

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34 .., I..

F1(x) = c1, . . . , F m(x) = cm. i c |q(c1, . . . , cm)| c + .

i fx E (4), fx = x = 1 i x i, i

R1(fx) = c1, . . . , Rm(fx) = cm.

i ,

|Q(fx) c| = |q(c1, . . . , cm) c| = |P(x) c|

i ,

P Q.

1.2 , .

1.1. H1(X), H2(Y) i i i -

X, Y, i T : H1(X) H2(Y) i, T i P1(X) P2(Y), P1(X), P2(Y)

i i ii H1(X) i H2(Y) ii. i T ii

H1(X) i H2(Y).

. , T : H1(X) H2(Y) i. i i i

i f, T(f) = 0. f =k=0

Pk = 0, Pk k-i i,

k = 0, 1, . . . . i, i T, T(f) =

k=0T(Pk) = 0. , i

, T i H1(X) H2(Y). , T -

i i. i i

, , T1 .

1. Hbs(E) Hbs(1), i i.

. i =

m=0

Qm, Hbs(E) i =

m=0

Pm, Hbs(1), Pm = (Qm).

i

m=0 Qm i, lim supm

Qm1/m =

1

RE()= 0,

RE() i ii E.

1.2 , Pm Qm m. i

1

R1():= lim sup

mPm

1/m lim supm

Qm1/m =

1

RE()= 0,

i, , R1() = .

8/6/2019 T2 N2

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i i 35

, ,

m=0

Qm Hbs(E), ,

m=0

Pm Hbs(1), i

Hbs(E) Hbs(1), i i i

ii i . , -

i i i .

, i, ,

1.1, , Hbs(E) i Hbs(1).

i 1.1. Mbs(E) Mbs(1).

2 L[0, 1]

2. Hbs(E) Hbs(L[0, 1]), i i.

. i L[0, 1] i E : = , t [0, 1];

0, t [0, ) \ [0, 1].

T : Hbs(E) Hbs(L[0, 1]), T(f)() = f(). , T i, |T(Qn)| Qn

n-i i Qn i, i , T(Rn) = Gn. , T

ii 1.1.

i Hbs(E), i Hbs(L[0, 1]) ii i. i n N, i i (a1, . . . ,

an) Cn i k = [

k1n

, kn

), k = 1, . . . , n .

i i, i fa(t) :

fa(t) =n

k=1

akk(t),

k(t) = 1, t k,

0, t / k

.i

Gm(fa) =

10

n

k=1

akk(t)

mdt =

10

n

k=1

amk k(t)

dt =

nk=1 a

mk

n=

Fm(a)

n.

f L[0, 1] fn(t) =n

k=1 aknn(t) ii i-

i, fn f n . i

Gm(f) = limn

nk=1 a

mkn

n.

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36 .., I..

2.1. i (1, . . . , n) Cn i i if(t) L[0, 1],

Gk(f) = k.

. i (. [1]), i x = (x1, . . . , xn, . . .) p , Fk(x) = nk.

i Gk(f) =Fk

n

.

3 i ii i E

E i 0 s1 < s2 ii [s1,s2] i Hbsbs (E)

:

[s1,s2]((t)) = ([s1,s2](t)) :=

(t), t [0, s1),

0, t [s1, s2],

(t s2 + s1), t (s2, ).

3.1. i f E , f(t) = f([s1,s2](t)) -i [s1,s2], 0 s1 < s2.

, [s1,s2] i, (1).

i (3), i Fk1,k2 i

Fk1,k2(x) =

i,j=1

xk1i xk2j+i,

i k1 = k2, i . , -

ii E i i.

3.1. i

R1,1() =

0

0

(t)(t + s)dsdt

, .

i i .

3.1. i (t) = exp(t) i i i1,3 : 1 3, k = [k 1, k), k = 1, 2, 3. ,

R1() =

0

exp(t)dt =1

exp(t 2)dt +

2

exp(t)dt +

3

exp(t + 2)dt +

3

exp(t)dt,

R1() = 1,3(R1()). ,

R1,1() =

0

0

exp(t)exp((t + s))dsdt =

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i i 371

1

exp(t 2)exp(t s 2)dsdt +

2

2

exp(t)exp(t s)dsdt+3

3

exp(t +2)exp(t s + 2)dsdt +

3

3

exp(t) exp(t s)dsdt = 1,3(R1,1()).

, i R1,1() .

(a1, . . . , an, . . .) 1 i fa(t) (4).

i R1,1 Pbsbs (E) ii i i i

{fa, a 1}, , i ii, .

3.2. i i .

. ,

R1,1(fa) =

0

0

k=1akk(t)

k=1akk(t + s)dsdt =

1

2

i=1

a2i +i

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38 .., I..

5. Dineen S., Complex Analysis in Locally Convex Spaces, North-Holland, Mathematics Studies,

Amsterdam, New York, Oxford, 57(1981).

6. Dineen S., Complex Analysis on Infinite Dimensional Spaces, Monographs in Mathematics, Springer,

New York, 1999.

7. Gonzalez M., Gonzalo R., Jaramillo J., Symmetric polynomials on rearrangement invariant function

space, J. London Math. Soc. (2) 59, (1999) 681697.8. Gonzalo R., Multilinear forms, subsymmetric polynomials and spreading models on Banach spaces, J.

Math. Anal. Appl. 202 (1996), 379397.

9. Hajek P., Polynomial algebras on classical Banach spaces, Israel J. Math. 106 (1998), 209220.

10. Lindestrauss J., Tzafriri L., Classical Banach spaces II, Springer-Verlag, New York, 1977.

11. Mujica J., Complex Analysis in Banach Spaces, North-Holland, Amsterdam, New York, Oxford, 1986.

12. Nemirovskii A.S., Semenov S.M., On polynomial approximation of functions on Hilbert space, Mat. USSR

Sbornik 21 (1973), 255277.

13. Novosad Z., Zagorodnyuk A., Polynomial automorphisms and hypercyclic operators on spaces of analytic

functions, Arch. Math. 89 (2007), 157166.

14. Zagorodnyuk A., Spectra of algebras of entire functions on Banach spaces, Proc. Amer. Math. Soc. 134

(2006), 25592569.

15. Zagorodnyuk A., Spectra of algebras of analytic functions and polynomials on Banach spaces, Contemp.

Math. 435 (2007), 381394.

1 i i i. . ,

I-i, 2 I i ,

i,

i 17.09.2010

Zagorodnyuk A.V., Chernega I.V. Spectra of algebras of symmetric and subsymmetric analytic

functions, Carpathian Mathematical Publications, 2, 2 (2010), 3138.

Algebras of symmetric and subsymmetric analytic functions of bounded type on spaces

L1[0,) L[0,) and L[0, 1] and their spectra are investigated.

.., .. // . 2010. .2, 2.

C. 3138.

L1[0,) L[0,) L[0, 1] -

.

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515.12+512.58

Zarichnyi I.M.

CHARACTERIZATION OF THE MACRO-CANTOR SET UP TO COARSE

EQUIVALENCE

Zarichnyi I.M. Characterization of the macro-Cantor set up to coarse equivalence, Carpathian

Mathematical Publications, 2, 2 (2010), 3947.

We characterize metric spaces that are coarsely equivalent to the macro-Cantor set 2

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40 Zarichnyi I.M.

Definition 1. A multi-valued map : X Y between metric spaces X, Y is called

macro-uniform, if () is finite for each < ;

a coarse equivalence, if (X) = Y, 1(Y) = X and both multi-valued maps and

1 are macro-uniform.

Two metric spaces X, Y are called coarsely equivalent if there is a coarse equivalence

f : X Y. In particular, the macro-Cantor set is coarsely equivalent to the macro-Cantor

cube

2 i}.

In the sequel, for a metric space (Y, ) and a subset C Y by U(C) we denote the

-neighborhood of C in Y. For any nonempty sets A, B Y we put

dist(A, B) = inf{(a, b) | a A, b B}.

The following is a characterization theorem for the macro-Cantor set.

Theorem 1. A metric space (Y, ) is coarsely equivalent to the macro-Cantor set if and

only if there exist numbers a > 0, n N and monotonically increasing divergent sequences

(ai)iN, (ni)iN of real and natural numbers respectively, such that the following holds: for

every i the set Y can be written as the disjoint union of a countable family of sets {Yj}jN,

such that for every j, k N

diam(Yj) ai, dist(Yj , Yk) > ai1 and the set Yj can be coveredby2ni+n sets and cannot be covered by less than 2ni sets of diameter not exceeding a.

Proof. Without loss of generality we can assume that ni1 ni2 n > 2 for every i.

Necessity. Let a metric space (Y, ) be coarsely equivalent to the macro-Cantor set X.

Then consider a multi-valued map f: X Y from the definition of coarse equivalence and

define sequences {ai}iN, {bi}iN in the following way.

Put b1 = 1. Suppose that we have defined b1, . . . bi and a1, . . . , ai1. From the definition

of coarse equivalence for f there exist natural numbers ai > f(bi) and bi+1 > f1(ai).

Let i N. We can represent X as the union X =jN

Xij, where diam(Xij) = bi,

Ubi(Xij) = X

ij for all j N. Then for all i, j N define Y

ij = f(X

ij). It is easy to see that

Y =jN

Yij , diam(Yij ) = ai for all j N, i N.

Since for all i > 1 dist(Xij, Xik) > bi, i, j N, we easily obtain that dist(Y

ij , Y

ik ) > ai1,

j,k N.

Then we can see that for any i,j,k N, i < j, there exists a unique l N such that

Yik Yjl .

Note, that for all i > 1, j N, the set Yij can be written as the union Yij =

2bib1k=1

Y1lk . The

set Yij can be covered by at most 2bib1 sets of diameter a1.

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Characterization of the macro-Cantor set up to coarse equivalence 41

Similarly, Yij =2bib2k=1

Y2lk . Since the distance between the sets Y2p and Y

2q is greater then

a1, the set Yij cannot be covered by less than 2

bib2 sets of diameter not exceeding a1.

The necessity is proved.

Sufficiency. Let Y be a metric space, numbers a > 0, n N, and monotonically

increasing sequences (ai)iN, (ni)iN, of real and natural numbers respectively are from theconditions of the theorem.

Let X denotes the macro-Cantor set and

Xij = {x = (x1, x2, x3, . . . ) | xi = 1, xi+1 = 2, . . . , } ,

where

j = 1 + 1 + 2 2 + 22 3 + + ap 2

p1 + . . . ,

k {0, 1}. It is easy to see that diam(Xij) < 2

i, and for every a < b, c, d, either Xac Xbd or

Xac Xbd = .

We will use that (A) is the minimal natural number k such that A can be written as

the disjoint union of k balls of diameter not exceeding a.

For every natural i let Y = Yi1 Yi2 . . . be a decomposition such that for every natural

j, k, diam(Yij ) ai, dist(Yij , Y

ik ) > ai1 and 2

nj (Yjk ) 2nj+n for every natural j, k. Let

jmax = maxk

(Yjk ).

We assume that

Yk1 = Yt1 Y

t2 Y

tr1

,

Yk2 = Ytr1+1 Y

tr1+2 Y

tr2

,

. . .

for every natural k, t, k > t.

Step a). Consider a sequence of real numbers (k) such that 1 < k < 2,kN

k = 2,

0 = 1.

Let us construct sequences of natural numbers (ci)iN, (di)iN by induction. Let d1 = 1,

and let ci > di be such that

1 +2ndi+n 2nd1+3n+2 8

2nci 2i+1,

1 2ndi

2nd1+3n+2

2nci+n 8 1

2i+1,

(a(2i + 1))

di > ci1 and for any t > 2ndi ,

t + ci1max t 2i,

t ci1max t 1

2i.

(a(2i))

Let us consider the following conditions for a multi-valued function f : A B:

f(aci1) ndi , f1(ndi) aci . (f(i))

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42 Zarichnyi I.M.

Let p = 2nd1+3n+2.

Step b). During this step for every natural i we have to construct a multi-valued surjective

function fi(A, B)(x) : A B, which maps the set A Y into B X. Here A =

Ydil1 Ydilp

, B = Xndik , 1 p p

. Also the function fi must satisfy conditions (f(1)),

(f(2)), . . . , (f(i)).

Fix i N, and let A, B be sets. Let us construct fi(A, B) by induction. Without loss ofgenerality we can assume that

A = Ydi1 Ydip , B = X

ndi1 .

It is easy to see that

p 2ndi (A) p 2ndi+n.

Base of induction. Let A01 = A.

Step j, j {1, . . . , i 1}. We see that the set Y is written as a disjoint union of sets

Y = Aj11 A

j1

2ndi

ndij+1

such that for any k {1, . . . , 2ndindij+1}

(A)

2ndindij+1

2(j1)t=0

1

t (Aj1k )

(A)

2ndindij+1

2(j1)t=0

t,

the set Aj1k is the union of sets from the family {Ycijq }, and it is assumed that

fi(A, B)(Aj1k ) = X

ndij+1k , f

1i (A, B)(X

ndij+1k ) = A

j1k .

Consider the set Aj1k . We can represent it as the union

Aj11 = Ycijk1

Ycijks

.

Now write the set Xndij+1k as the disjoint union of 2

ndij+1ndij = sets, Xndij+1k =

Xndije1 X

ndijef . We have to divide these sets between the sets Y

cijkr

.

Let : {e1, . . . , ef} {k1, . . . , ks} be a surjective function. Note that > s. Let

(kr) = |{et|(et) = kr}|. We have to find

that minimizes the difference of(Y

cijkr

)

(kr).

We see thats

l=1

(Ycij

kl) = (Aj1

k

). Thus there is such that, for every l {1, . . . , s},

(Aj1k )

((kr) 2) (Y

cijkr

) (Aj1k )

((kr) + 2),

(Aj1k )

(1

2

(kr))

(Ycijkr

)

(kr)

(Aj1k )

(1 +

2

(kr)).

Now let us look at (kr):

(Ycijkr

) f

(Aj

1

k )

1

4 (kr),

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(kr) (Y

cijkr

)

(Aj1k ) 2

2ncij 2ndij+1 2ndi

2ndij p 2ndi+n 2ndij+1 4

2ncij

2ndij+n 4p.

Then by condition (a(2(i j) + 1))

(1 + 2(kr)

) 1 + 2ndij+n

4p2ncij

2(ij)+1,

(1 2

(kr)) 1

2ndij p

2ncij+n 4

1

2(ij)+1.

We see that

(A)

2ndindij

2i1t=2(ij)+1

1

t

(Ycijkr

)

(kr)

(A)

2ndindij

2i1t=2(ij)+1

t.

Now consider the set Ycijkr and (kr). The sets X

ndijo1 , . . . , X

ndijo(kr) are mapped to the set

Ycijkr

. Represent the set Ycijkr

as the disjoint union of sets

Ycij1q1 Ycij1ql

.

Now map them into the sets Xndijo1 , . . . , X

ndijo(kr)

to minimize the difference of (Ajk), where

Ajk = f1(X

ndijk ).

This can be done so that

(Ycijkr

)

(kr)

cij1

max (A

j

k)

(Ycijkr

)

(kr) +

cij1

max .

We see that(Y

cijkr

)

(kr) p 2ndij

1

2 2ndij1,

then by condition (a(2(i j)))

(Ycijkr

)

(kr)+ cij1max

(Ycijkr

)

(kr) 2(ij),

(Ycij

kr)

(kr) 12(ij)

(Y

cij

kr)

(kr) cij1max .

Thus,

(A)

2ndindij

2i1t=2(ij)

1

t (Ajk)

(A)

2ndindij

2i1t=2(ij)

t.

Assume that f(Ajk) = Xdijk , f

1(Xdijk ) = A

jk. As a result the following condition

(f(i j)) holds:

f(acij1) ndij f1(ndij) acij .

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44 Zarichnyi I.M.

After step (i 1) we obtain our function. The set A is written as the union of the family

of sets {Ai1k }. For every k

0 < p 2nd1 (A)

2ndind12i1t=2

1

t (Ai1k ),

therefore A(i1)

k is nonempty. For every k and for every x A(i1)k let f(A, B)(x) = X

nd1k .

Note, that the constructed function satisfies conditions (f(1))(f(i)).

Step c). Now we have a sequence (fi) of functions. We have to construct function from

Y to X. We can write Y as the union of the sets

Y1 = Yd11 , Yi = (Y

di1 Y

di2n ) \ Yi1.

Let X = X1 X2 . . . , where X1 = Xnd11 , and Xi = X

ndi1 \ X

ndi11 for i N \ {1}. For

every i we have to map (Ydi1 \ Ydi11 ) into Xi.

Consider step i. We see that Yi = Ydi1l1 Y

di1lt . Also 2ndi+n (Yi) 2ndi+2n and

2ndi1 (Ydi1j ) 2

ndi1+n. Then 2ndindi1 t 2ndindi1+2n.

We have Xi = Xndi1 \X

ndi11 = X

ndi1m1 X

ndi1mu . It is easy to see that u = 2

ndindi1 1.

Also u < t < up.

Now we can write Yi as the disjoint union of sets Yi = Y(i,1) Y(i,u), where every

Y(i,k) is the union of w sets of the family (Ydi1lr

), 1 w p.

Now for every k {1, . . . , u} using the function fi(Y(i,k), Xndi1mk ) we shall map the set

Y(i,k) into the set Xndi1mk .

The last theorem can be reformulated.

Theorem 2. A metric space (Y, ) is coarsely equivalent to the macro-Cantor set if and

only if there exist numbers a > 0, n N and monotonically increasing divergent sequences

(ai)iN, (ni)iN of real and natural numbers respectively, such that the following holds: for

every i the set Y can be written as the disjoint union of a countable family of sets {Yj}jN,

such that for every j, k N, diam(Yj) ai, dist(Yj , Yk) > ai1 and the set Yj can be covered

by n ni sets and cannot be covered by less than ni sets of diameter not exceeding a.

Now using Theorem 2 we can prove its more general version.

Theorem 3. A metric space (Y, ) is coarsely equivalent to the macro-Cantor set if andonly if there exist monotonically increasing divergent sequences (ai)iN{0} of reals, (ni)iNand (mi)iN of naturals, such that the following holds: for every i the set Y can be written

as the disjoint union of a countable family of sets {Yj}jN, such that for every j, k N

diam(Yj) ai, dist(Yj, Yk) > ai1 and the set Yj can be covered by mi sets and cannot be

covered by less than ni sets of diameter not exceeding a0.

Proof. To prove this theorem we will show that for the space Y the conditions from Theorem

2 hold true. We will construct monotonically increasing sequences (bi)iN and (ki) of real and

natural numbers respectively, such that for all i N the set Y can be written as disjoint union

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of a countable family of sets {Zij}jN, such that for all j, l N diam(Zij) bi, dist(Z

ij, Z

il ) >

bi1 and the set Zij can be covered by ki sets and cannot be covered by less than k ki sets

of diameter not exceeding b0 = a0.

By the formulation of the theorem, for all i N set Y can be written as disjoint union

of a countable family of sets which we will denote by {Yij }jN.

Define k = max{3, [m1n1 ] + 1}.

Base of induction. Put b1 = a1, k1 = n1, t1 = 1. For all j N, let Z1j = Y

1j . It is easy

to see that, for the family {Z1j }jN, all conditions hold.

i-th step of induction, i > 1. We have a natural number ti1 and a real number bi1 such

that bi1 = ati1. We have to find numbers bi > bi1 and ki, and write Y as disjoint union of a

countable family of sets {Zij}jN, such that for all j, l N diam(Zij) bi, dist(Z

ij, Z

il ) > bi1

and the set Zij can be covered by k ki sets and cannot be covered by less than ki sets of

diameter b0.

Consider the family of sets {Yti1+1j }jN. The mutual distances between the distinct

elements of this family are at least ati1

. Every of these sets can be covered by nti1+1

sets and cannot be covered by less than mti1+1 sets of diameter not exceeding b0. Put

ki = mti1+1.

Take a number ti, such that nti > mti1+1. Consider the family of sets {Ytij }jN. The

diameter of each of them is less than ati . Put bi = ati. Each of them can be covered by mtiand cannot be covered by less than nti of sets of diameter b0.

For all u N consider the set Ytiu . This set can be represented as disjoint union of a

finite number of sets from the family {Yti1+1j }jN. Without loss of generality we can write

Ytiu = Yti1+11 Y

ti1+12 Y

ti1+1v . Each of the sets Y

ti1+1p , p {1, . . . , v}, can be covered

by mti1+1 sets of diameter b0. The set Ytiu cannot be covered by less than nti > mti1+1 sets

of diameter b0.Put p0 = 0. There exist numbers p1, p2, . . . , pq, 0 = p0 < p1 < p2 < < pq < v, such

that the sets Yti1+1pl1+1

Yti1+1pl1+2

Yti1+1pl can be covered by 2 mti1+1 and cannot be

covered by less than mti1+1 sets, and the set Yti1+1pq+1 Y

ti1+1pq+2 Y

ti1+1v can be covered

by mti1+1 sets of diameter b0. Then define

Ziu1 = Yti1+11 Y

ti1+12 Y

ti1+1p1

,

Ziu2 = Yti1+1p1+1 Y

ti1+1p1+2 Y

ti1+1p2

,

. . .

Ziu,(q1) = Yti1+1pq2+1

Yti1+1pq2+2

Yti1+1pq1 ,

Ziu,(q) = Yti1+1pq1+1

Yti1+1pq1+2

Yti1+1pq Yti1+1pq+1 Y

ti1+1u .

Put u = q.

It is easy to see that the sets Ziur, r {1, . . . , q 1}, can be covered by 2 mti1+1 k kisets and cannot be covered by less than mti1+1 = ki sets of diameter b0. The set Z

iuq cannot

be covered by less than mti1+1 = ki, can be covered by 3 mti1+1 k ki sets of diameter

b0. Also the diameters of these sets are less than ati = bi, and their pairwise distances are

less than ati1 = bi1.

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46 Zarichnyi I.M.

So we represent the set Y as disjoint union of a countable family of sets, Y =

{Zip,q|p

N, q = 1, . . . , p}, for which all conditions are true. Now we can enumerate these sets by

naturals and we shall represent Y as the disjoint union of the family {Zij}jN.

Definition 2. A metric space(X, d) is called asymptotically zero-dimensional if for alla > 0

there exists a uniformly bounded a-disjoint cover of X.

A cover U of metric space X is called

uniformly bounded if its mesh sup{diam U : U U} is finite.

a-disjoint if dist(A, B) > a for every A, B U.

Theorem 4. A metric asymptotically zero-dimensional space (X, ) is coarsely equivalent

to macro-Cantor set if and only if there exists number a > 0, and the following conditions

are true:

1) for every n N there exists r N, such that for any x X the r-ball Ur(x) cannot

be covered by less than n balls of radius a,2) for every r N there exists m N, such that each r-ball Ur(x) can be covered by m

balls of radius a.

Proof. Necessity. By the Theorem 3 there exist monotonically increasing sequences

(ai)iN{0} of reals, (ni)iN and (mi)iN of natural numbers. Put a = a0. We will show

that conditions 1) and 2) are true.

a) Consider an arbitrary natural n. Then there exists j, such that nj > n. Put d = aj+1.

It is easy to see that condition 1) is true.

b) Consider an arbitrary number d. Then there exists such j, that aj > d. Put m = mj+1.

Easy to see that condition 2) is true.Sufficiency. Suppose that (X, ) is a space and conditions 1) and 2) are true. We shall

construct by induction monotonically increasing sequences (ai)iN{0} of real, (ni)iN and

(mi)iN of natural numbers to satisfy conditions of the Theorem 3.

Base of induction. Put a0 = a, m0 = 1.

i-th step of induction, i N. Put ni = mi1 + 1. By condition 1) for number ni there

exists d. By definition of asymptotic dimension zero, for the space X there exists a totally

bounded ai1-disjoint cover. Let b be the mesh of this cover. Put ai = max{b,d,ai1 + 1}.

It is easy to see that this sequence satisfies the conditions of the Theorem 3.

It is well known that every zero-dimensional compact metric space without isolated pointsis homeomorphic to the Cantor set. In our characterization the first condition is an analogue

of space without isolated points in metric geometry.

Applying Characterization Theorem 4 one can easily prove the next corollary.

Corollary 1. For every n N the hyperspace expn(2

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References

1. Dranishnikov A., Zarichnyi M. Universal spaces for asymptotic dimension, Topol. Appl., 140, no.2-3

(2004), 203225.

2. Roe J. Lectures in Coarse Geometry, University Lecture Series Vol. 31, American Mathematical Society:

Providence, Rhode Island, 2003.

3. Engelking R. General Topology, PWN, Warsaw, 1977.

Institute for Applied Problems of Mechanics and Mathematics,

Lviv, Ukraine

Received 22.10.2010

i I.. i - i i-

i // i i ii. 2010. .2, 2. C. 3947.

i i, i -

2

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517.948:517.946

.I., .., ..

I I I

I I

.I., .., .. i i i -

i i // i i ii. 2010.

.2, 2. C. 4854.

i i i i -

i i, i ii i-

i i .

i i i

, i i i i i

i i ii i. I .. i i i i i,

i i i i i-

. i i ii i -

i (. [2, 3]). [7] i

i i i i i , i

i i i -

i i i i

i ii ii . ii ii

i i i i i

x(t) = g(t, x(t)) h(t, x(t)), (1)

x(t0) = x0. (2)

i i i iii ..

[3] i i (1) ,

i (1) , -

i.

2000 Mathematics Subject Classification: 34A45.i i : i , iii, ii.

c .I., .., .., 2010

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i i i ... 49

1 i

i (1), (2) g(t, x(t)), h(t, x(t)) i -

i t [t0, T], x S(x0, M) = {x||x x0| M, x R1} (R1 i

). i (1), (2) i i

[t0, T] i. B1 i t [t0, T],

y, z S(x0, M) i a1(t,y,z), 1(t,y,z), b1(t,y,z), 1(t,y,z), i ii-

t [t0, T], y, z S(x0, M) :

(a1(t,y,z) + 1(t,y,z))(z y) g(t, z) g(t, y), (3)

(b1(t,y,z) + 1(t,y,z))(z y) h(t, z) h(t, y).

i a1(t,y,z), 1(t,y,z), b1(t,y,z), 1(t,y,z) y, -

z i a1(t,y,z) 0, 1(t,y,z) 0, b1(t,y,z) 0, 1(t,y,z) 0. (3) iii. -

ii i u(t), v(t),

u(t0) x0 v(t0) (t [t0, T]), (4)

u(t) v(t), (5)

u(t) g(t, u(t)) h(t, v(t)), (6)

v(t) g(t, v(t)) h(t, u(t)) (t [t0, T]),

u(t), v(t) S(x0, M) t [t0, T]. [u(t), v(t)] = {x(t)|u(t)

x(t) v(t)}. ii

y0(t) = u(t), z0(t) = v(t) (7)

yn+1(t) = a1(t, yn(t), zn(t))(yn+1(t) yn(t))

b1(t, yn(t), zn(t))(zn+1(t) zn(t)) + g(t, yn(t)) h(t, zn(t)), (8)

zn+1(t) = (a1(t, yn(t), zn(t)) + 1(t, yn(t), zn(t)))(zn+1(t) zn(t))

(b1(t, yn(t), zn(t)) + 1(t, yn(t), zn(t)))(yn+1(t) yn(t)) + g(t, zn(t)) h(t, yn(t)),

yn+1(t0) = zn+1(t0) = x0. (9)

1. B1 i i i u(t), v(t) ii (4)(6). i ii (7)(9) i ii-

yn(t) yn+1(t) zn+1(t) zn(t), (10)

n = 0, 1,...,t [t0, T]

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50 .I., .., ..

. (7) i ii (5) , y0(t) z0(t). I (6) (8)

n = 0 ,

y1(t) y

0(t) a1(t, y0(t), z0(t))(y1(t) y0(t)) + b1(t, y0(t), z0(t))(z0(t) z1(t)),

z0(t) z

1(t) (a1(t, y0(t), z0(t)) + 1(t, y0(t), z0(t)))(z0(t) z1(t)) + (b1(t, y0(t), z0(t))+

1(t, y0(t), z0(t)))(y1(t) y0(t)).

i ii ii [2] , t [t0, T]

y1(t) y0(t), z1(t) z0(t).

i , (8) (3)

z1(t) y

1(t) = (a1(t, y0(t), z0(t)) + b1(t, y0(t), z0(t)))(z1(t) y1(t))

(a1(t, y0(t), z0(t)) + b1(t, y0(t), z0(t)))(z0(t) y0(t))

1(t, y0(t), z0(t))(z0(t) z1(t)) 1(t, y0(t), z0(t))(y1(t) y0(t)) + g(t, z0(t))

g(t, y0(t)) + h(t, z0(t)) h(t, y0(t))

(a1(t, y0(t), z0(t)) + b1(t, y0(t), z0(t)) + 1(t, y0(t), z0(t)) + 1(t, y0(t), z0(t)))(z1(t) y1(t))+

1(t, y0(t), z0(t))(y1 y0) + 1(t, y0(t), z0(t))(z0(t) z1(t))

(a1(t, y0(t), z0(t)) + b1(t, y0(t), z0(t)) + 1(t, y0(t), z0(t))+

1(t, y0(t), z0(t)))(z1(t) y1(t)).

ii ii, , t [t0, T]

y1(t) z1(t).

, ii (10) i n = 0. i

n = k 1, n = k i (8), (9) B1 :

yk+1(t) y

k(t) = g(t, yk(t)) g(t, yk1(t)) + h(t, zk1(t)) h(t, zk(t))+

a1(t, yk(t), zk(t))(yk+1(t) yk(t)) + b1(t, yk(t), zk(t))(zk(t) zk+1(t))

a1(t, yk1(t), zk1(t))(yk(t) yk1(t)) b1(t, yk1(t), zk1(t))(zk1(t) zk(t))

a1(t, yk1(t), yk(t)) + 1(t, yk1(t), yk(t))(yk(t) yk1(t))+

b1(t, zk(t), zk1(t)) + 1(t, zk(t), zk1(t))(zk1(t) zk(t))+

a1(t, yk(t), zk(t))(yk+1(t) yk(t)) + b1(t, yk(t), zk(t))(zk(t) zk+1(t))

a1(t, yk1(t), zk1(t))(yk(t) yk1(t)) b1(t, yk1(t), zk1(t))(zk1(t) zk(t))

a1(t, yk(t), zk(t))(yk+1(t) yk(t)) + b1(t, yk(t), zk(t))(zk(t) zk+1(t)),

zk(t) z

k+1(t) = g(t, zn1(t)) g(t, zn(t)) + h(t, yn(t)) h(t, yn1(t))+

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i i i ... 51

(a1(t, yk(t), zk(t)) + 1(t, yk(t), zk(t)))(zk(t) zk+1(t))+

(b1(t, yk(t), zk(t)) + 1(t, yk(t), zk(t)))(yk+1(t) yk(t))

(a1(t, yk1(t), zk1(t)) + 1(t, yk1(t), zk1(t)))(zk1(t) zk(t))

(b1(t, y

k

1(t), z

k

1(t)) +

1(t, y

k

1(t), z

k

1(t)))(y

k(t) y

k

1(t))

(a1(t, yk(t), zk(t)) + 1(t, yk(t), zk(t)))(zk(t) zk+1(t))+

(b1(t, yk(t), zk(t)) + 1(t, yk(t), zk(t)))(yk+1(t) yk(t)),

zk+1(t) y

k+1(t) = g(t, zn(t)) g(t, yn(t)) + h(t, zn(t)) h(t, yn(t))

1(t, yk(t), zk(t))(zk(t) zk+1(t)) 1(t, yk(t), zk(t))(yk+1(t) yk(t))+

(a1(t, yn(t), zn(t)) + b1(t, yn(t), zn(t)))(zn+1(t) yn+1(t))

(a1(t, yn(t), zn(t)) + b1(t, yn(t), zn(t)))(zn(t) yn(t))

(a1(t, yk(t), zk(t)) + b1(t, yk(t), zk(t)) + 1(t, yk(t), zk(t))+

1(t, yk(t), zk(t)))(zk+1(t) yk+1(t)).

ii, ii ii, -

,

yk+1(t) yk(t) 0, zk(t) zk+1(t) 0, zk+1(t) yk+1(t) 0.

, ii (10) i n = k. i -

ii, .

i ii ii (7)

(9) i y(t), z(t) (y(t), z(t))

i

y(t) = g(t, y(t)) h(t, z(t)), (11)

z(t) = g(t, z(t)) h(t, y(t))

y(t0) = z(t0) = x0. (12)

i, ii {yn(t)} {zn(t)}, i ii

(7)(9), i ii, y(t) -

z(t) i [t0, T] (y(t), z(t)) i

(11), (12). ii i i i i i-

{yn(t)}, {zn(t)} i. ,

i , ii i i

i (1), (2).

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i i i ... 53

(a1(t, yn(t), zn(t)) + 2(t, yn(t), zn(t))(zn(t) yn(t))+

(b1(t, yn(t), zn(t)) + 2(t, yn(t), zn(t)))(zn(t) yn(t))+

(a1(t, yn(t), zn(t)) + b1(t, yn(t), zn(t)))(zn+1(t) yn+1(t))

(a1(t, yn(t), zn(t)) + b1(t, yn(t), zn(t)))(zn(t) yn(t)) =

f(2)n (t)(zn+1(t) yn+1(t)) + f(1)n (t)(zn(t) yn(t)),

i (13). i (14) (13),

ii ii [6].

1.1. ii (7)(9) -i a1(t,y,z), b1(t,y,z), 2(t,y,z), 2(t,y,z). , ,

2(t,y,z) = 3(t,y,z)(z y), 2(t,y,z) = 3(t,y,z)(z y)

( 0),

3(t,y,z), 3(t,y,z) i ii t [t0, T], y z , y , z S(x0, M), i (14),

f(1)n (t) = (3(t, y(t), z(t)) + 3(t, y(t), z(t)))(zn(t) yn(t)). (16)

i (15) (14), > 0 ii (7)(9)

ii ii, = 1 > 1 ii (7)(9)

i, ii, ii. ,

a1(t,y,z) =g(t, y)

y, b1(t,y,z) =

h(t, y)

y,

2(t,y,z) =1

2

2g(t, y)

y2(z y), 2(t,y,z) =

1

2

2h(t, y)

y2(z y),

ii = 1, i i i-

ii .. i [2, 3]. , i , i i g(t, x)

h(t, x) i , , i i ,

i i i i ii.

1.2. iii (7)(9) i i .. [2, 3] i ii ii i-

i g(t, x) h(t, x).

i

1. ii .. i . .: , 1981. 224.

2. .., .. . : , 1977. 741 .

3. .. i ii .. i i -

i i // .. .. 1996, 4. . 303-306.

4. .. . :

, 1986. 260 .

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54 .I., .., ..

5. ., . ii iii i i ii //

i i i 2009. .294295.

6. . - . . : , 1954. 589 .

7. . . : , 1975. 443 .

8. .., .. i i i ii i // i i-

ii i. . i. .119 1977.

9. Kaczorek T. Polinomial and rational matrices. Applications in dynamical systems theory, Springer:

Communications and Control Engineering.Dordrecht, 2007 503 p.

i i i. . ,

I-i,

i 06.10.2010

Kopach M.I., Obshta A.F., Shuvar B.A. An application of analogues of two-sided Kurpels

methods to ordinary differential equation, Carpathian Mathematical Publications, 2, 2 (2010),

4854.

An analogues of two-sided Kurpels methods of approximate solution of ordinary differential

equation that give possibility to get above-linear convergence in the case of nondifferential right

part are constructed and investigated.

.., .., .. -

// -

. 2010. .2, 2. C. 4854.

-

, -

.

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56 .I., .., ..

x = (x1, . . . , xn1) Rn1; (x, t) = (x, xn, t) Rn+1T ; (x

, t)

RnT; (x, y) =n

i=1

xiyi; |x|2 = (x, x) =n

i=1

x2i ; |x|2 = (x, x) =n1i=1

x2i .

Rn D S. , D -i i i D1 i D2 S1, S S1 = . -

D1 i S

1,

D2 i S

2= S

S1.

(x) = (1(x), . . . , n(x)) (1)(x) = ((1)1 (x), . . . ,

(1)n (x)) i -

ii i i D2 x S x S1ii. Dm = Dm Sm, m = Dm (0, T), m = Sm [0, T], m = 1, 2,D = D S, = S [0, T].

i i: Drt i Dpx

i t r i - i x p ii, r, p ii ii ; Dt t ; Di xi ; Dij

2

xixj, i, j = 1, . . . , n; =

(D1, . . . , Dn), i (1)i (i = 1, . . . , n) i i

S S1 ii, i =

nk=1 ikDk, ik =

k

i ik, (1)

i =

nk=1

(1)

ik Dk, (1)

ik =

ki (1)i (1)k , ki . i [6] Hl+,(l+)/2(R n+1T ), H

l+,(l+)/2(m), Hl+(Rn) (l = 0, 1, 2; m = 1, 2; (0, 1) i) H2+. i i Hl+,(l+)/2(B), i ( l = 2 i t) t = 0 H

l+,(l+)/2(B), ||w||Hl+(B) ||w||Hl+,(l+)/2(B) -iw Hl+(B) Hl+,(l+)/2(B), B Rn R n+1T m, m = 1, 2,ii. C, c i i, i i (x, t), i i .

2 ii i

i R n+1T ii ii ii

Lsu n

i,j=1a(s)ij (x, t)Diju +

n

i=1a(s)i (x, t)Diu + a

(s)0 (x, t)u Dtu, s = 1, 2. (1)

, ii i L1 i L2 i Rn+1T i i

:

(A1)n

i,j=1

a(s)ij (x, t)ij 0s||2, a(s)ij = a(s)ji , 0s > 0, s = 1, 2, (x, t) R n+1T , Rn;

(A2) a(s)ij , a(s)i , a

(s)0 H,/2(R n+1T ) , s = 1, 2, i, j = 1, . . . , n.

(A1), (A2) i (..) Gs(x, t; , ) (0 < t T, x, Rn) Ls, s = 1, 2 [6]:

Gs(x, t; , ) = G(,)0s (x, t; , ) + G1s(x, t; , ), s = 1, 2, (2)

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G(,)0s (x, t; , ) = G

(,)0s (x

, xn n, t ) = (2

)n(det As(, ))1/2(t )n/2

exp

(A

1s (, )(x ), x )

4(t

)

, s = 1, 2, t > , (3)

As(, ) =

a(s)ij (, )

n

i,j=1, A1s (, ) =

aij(s)(, )

n

i,j=1 , i

As(, ), G1s i , i "" i, i G(,)0s (3)

t + 0 i Gs 0 , t . i i i i C i c, i Gs i G1s 0 < t T, x, Rn, 2r + p 2, i i

|Drt DpxGs(x, t; , )| C(t )(n+2r+p)/2 expc |x |

2

t

, (4)

|Drt D

pxG1s(x, t; , )

| C(t

)(n+2r+p)/2 expc

|x |2

t . (5) ii i :

u(1)s (x, t) =

t0

d

S1

Gs(x, t; , )Vs(, )d, (x, t) R n+1T , s = 1, 2, (6)

u(0)2 (x, t) =

t0

d

S

G2(x, t; , )V0(, )d, (x, t) R n+1T , (7)

Vs, s = 1, 2, V0 i ii 1 i ii i. i i (4), (5), iu(1)s , u

(0)2 i R

n+1T , i

Lsu(1)s = 0, s = 1, 2, R n+1T \1, Lsu(0)2 = 0, R n+1T \ i u(1)s (x, 0) = 0,

u(0)2 (x, 0) = 0.

(x, t) 1 (x, t) i N(s)(x, t) =(N(s)1 (x, t), . . . , N

(s)n (x, t)), N

(s)i (x, t) =

nj=1

a(s)ij (x, t)(1)

j (x), i = 1, . . . , n, s = 1, 2, N(x, t) =

(N1(x, t), . . . , N n(x, t)), Ni(x, t) =n

j=1

a(2)ij (x, t)j(x), i = 1, . . . , n. Vs H

,/2(1),

s = 1, 2, V0

H

,/2(), u(1)s

H

1+,(1+)/2(s), s = 1, 2, u(0)2

H

1+,(1+)/2(2) (.

[2], [6], [14]) i i i u(1)s , s = 1, 2, u(0)2

([6, . 459])

u(1)s (x, t)

N(s)(x, t)=

t0

d

S1

Gs(x, t; , )

N(s)(x, t)Vs(, )d + (1)s11

2Vs(x, t), (x, t) 1, (8)

u(0)2 (x, t)

N(x, t)=

t

0

d

S

G2(x, t; , )

N(x, t)V0(, )d 1

2V0(x, t), (x, t) . (9)

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58 .I., .., ..

I i i i (8) ii (0 < t T,x, S1) Gs(x, t; , )N(s)(x, t)

C(t )(n+1)/2 expc |x |

2

t

, (10)

i i G2(x,t;,)N(x,t)

i (9) 0

< t

T, x,

S.

.. Gs, s = 1, 2, (2) ii i,i i i i i i . i

u(2)s (x, t) =

Rn

Gs(x, t; , 0)s()d, (x, t) R n+1T , s = 1, 2, (11)

i i

u(3)

s (x, t) =

t

0

dRn

Gs(x, t; , )fs(, )d, (x, t) Rn+1

T , s = 1, 2, (12)

s() i fs(, ), s = 1, 2 i i. , s i Rn, fs H,/2(R n+1T ), i (. [6, . IV, 14]), i u(2)s ,u(3)s , s = 1, 2, i R n+1T , i Lsu

(2)s = 0, Lsu

(3)s = fs, s =

1, 2, R n+1T i i u(2)s (x, 0) = s(x), u

(3)s (x, 0) = 0, x Rn, s = 1, 2.

u(2)s H2+,(2+)/2(R n+1T ), , s H2+(Rn), i u(3)s H2+,(2+)/2(R n+1T ).

3 i

i Es, s = 0, 1, 2, i i i -i i i , i . 4 - i. , i i i- i i i-i E, [2, 14]. E0, - L2 i u

(0)2 . , S = R

n1, , = RnT. RnT i :

L

2 =

n1i,j=1

h(2)ij (x

, t)Dij Dt, (13)

h(2)ij = a(2)ij a(2)in a(2)nj

a(2)nn

1

, i, j = 1, . . . , n 1. H2(x, t; , ) (0 < t T, x, Rn1) .. L2, i

i RnT. ((x, t) RnT)

E0(x, t) = 2

t

t0

(t )1/2d

Rn1

H2(x, t; , )(, )d

t=t

. (14)

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3.1. ([2], [14]). E0 ii , ii H

1+,(1+)/2(R nT) i H

,/2(R nT). i

E10 : H

,/2(R nT) H

1+,(1+)/2(R nT).

3.1. 3.1, ,

E0 i i H 2+,(2+)/2(R nT) i H 1+,(1+)/2(R nT) i i E10 : H

1+,(1+)/2(R nT) H

2+,(2+)/2(R nT).

, E0 i ()(. [2, 10, 14]), :

E0(x, t)u(0)2 = (A2(x, t)(x), (x))1/2 V0(x, t)+t

0

d Rn1

K20(x, t; , )V0(

, )d, V0 H

,/2(R nT), (15)

K20 i (10), i i |x |2 i |x |2.

, S i D , S = {x Rn|xn = F(x)}, i F

F H2+(Rn1). (16)

i - i v(x, t) = S [0, T] v(x, t). i

E0 ii (14), i

H2(x, t; , ) = H2(x, t; , )|xn=F(x),n=F() .. (13) ii

h(2)ij = a(2)ij a(2)in a(2)nj

a(2)nn

1, (17)

a(2)ij = a(2)ji = a

(2)ij , i, j = 1, . . . , n 1, a(2)in = a(2)ni = a(2)in

n1k=1

a(2)ik Fk, i = 1, . . . , n 1,

a(2)nn = a(2)nn

2

n1k=1

a(2)kn Fk +n1

k,l=1

a(2)kl FkFl, Fk =F

xk

, k = 1, . . . , n

1.

i E0 i u(0)2 , ii- (15).

. , S i D - i H2+. , (. [2, 6]), S H2+, x0 i Ox0 , SOx0 i (ii) i {y} = {y1, . . . , yn},i x0, i

yn = Fx0(y), y

Bx0, Fx0

H2+(Bx0),

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60 .I., .., ..

Bx0 = {y Rn1 | |y| < d}, d > 0 i x0, ii sup

xS||Fx(y)||H2+(Bx).

D =

x D | infx0S

|x x0| <

, ( > 0), i -

(0 < < d/2). i (. [9]) i i i

x(m) S, m = 1, . . . , N , i D(m)j , j = 1, 2, i i {y}, i x(m), i

D(m)j = {y Rn | |y| < j, |yn Fx(m)(y)| < j}, j = 1, 2,

i i :

1) i = (), D mD(m)1 ;

2) i i N0 ( i ), - N0 + 1

D(m)

2 i. i, i

mD(m)2

D, i

(m) C(Rn) :0 (m) 1, (m) = 1 x D(m)1 , 0 x Rn\D(m)2 ,

supp((m)) D(m)2 ,N

m=1

(m)(x) = 1, x D, (18)

i

(m) = (m)

Nm=1

(m)

21, S

(m)j = D(m)j S, j = 1, 2. (19)

{y, t} i i x(m). {x} i {y} iii

Y = C(m)(X X(m)), C(m) , X i Y i x1, . . . , xn i y1, . . . , ynii. S

(m)

2,0 i S(m)

2 yn = 0. i

y

S(m)

2 i y

S(m)

2,0 i iii. , S(m)

2 i i {y} i x(m) i

yn = Fx(m)(y) = F(m)(y), y S(m)2,0 , (20)

F(m) i H2+(S(m)

2,0 ).

(17) ih(2,m)ij (y, t), (y, t) S(m)2,0 [0, T],

i, j = 1, . . . , n 1, m = 1, . . . , N , i i i a(2)ij , i, j = 1, . . . , n, a(2,m)ij , a

(2,m)ij i A

(m)2 (y

, t) = C(m)A(m)2 (y

, t)(C(m))T,

A(m)2 (y

, t) = A2(x(y), t). , i h(2,m)ij (y

, t)

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(13), i S(m)2,0 (0, T] ii i L(m)2 . -

ii Rn,(m)

T = Rn1,(m)[0, T] i -

(1), (2) i H(m)2 (y, t; , ) (0 < t T, y, Rn1,(m)) .. L

(m)2 .

i i

H(m)2 (x, t; , ) H(m)2 (y(x), t; (), ), 0 < t T, x, S(m)2 ,

H2(x, t; , ) =N

m=1

(m)(x)(m)()H(m)2 (x, t; , )(m)n (()), 0 < t T, x, S,

(m)(y) = C(m)(m)(y), (m)(y) = (x(y)) i, , H

l+,(l+)/2(),

l = 1, 2, (x, t)

E(m)0 =2

t

t0

(t ) 12 dS

(m)(x)(m)()H(m)2 (x, t; , )(m)n (())(, )d

t=t

,

E0(x, t) = 2

t

t0

(t ) 12 dS

H2(x, t; , )(, )d

t=t

=

Nm=1

E(m)0 (x, t). (21)

E0 . i (21) i-ii

E(m)0 , m = 1, . . . , N , . i , 3.1 3.1. .

3.2. E0, (21), ii -, i i H

l+,(l+)/2() i H

l1+,(l1+)/2() , l = 1, 2.

iE0 = 0 i H

l+,(l+)/2() i

= 0.

i E0 i u(0)2 (7). ii- (15), V0 H ,/2()

E0(x, t)u(0)2 = (A2(x, t)(x), (x))1/2 V0(x, t) +t

0

d

S

K20(x, t; , )V0(, )d, (22)

, K20(x, t; , ) i i 0 < t T, x, S, i (10).

i E0, i i- i Es, s = 1, 2, i i S1, Ls, s = 1, 2, i u

(1)s , s = 1, 2, (6).

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ni,j=1

(1)ij (x, 0)

(1)i

(1)j 2 + (

(1)(x, 0),2) ((x, 0),1) + (1)0 (x, 0)2+

0(x, 0)1 n

i,j=1

a(s)ij (x, 0)Dijs

ni=1

a(s)i (x, 0)Dis a(s)0 (x, 0)s fs =

(x, 0) + (s 1)Dtz(x, t)|t=0, x S1, s = 1, 2,n

i,j=1

ij(x, 0)ij2 + ((x, 0),2) + 0(x, 0)2 n

i,j=1

a(2)ij (x, 0)Dij2

ni=1

a(2)i (x, 0)Di2 a(2)0 (x, 0)2 f2 = (x, 0), x S. (30)

i .

. ii i Ls, s = 1, 2, i L4, L5 (1), (2) i (1), (2) ii, S i S1 i fs, s, s = 1, 2,

z, , (23)(27) i (28), (29). i i (30)

(23)(27)

us H2+,(2+)/2(s), s = 1, 2, (31)

i

2

s=1

usH

2+,(2+)/2

(s) C

2

s=1

fsH

,/2

(Rn+1

T )+

2s=1

sH2+(Rn) + zH2+,(2+)/2(1) + H,/2(1) + H2+,(2+)/2()

. (32)

. i (23)(27) i

us(x, t) =3

m=0

u(m)s (x, t), (x, t) s, s = 1, 2, (33)

u(0)1 0, iu(0)2 , u(1)s , u(2)s , u(3)s , s = 1, 2, i (6), (7), (11),(12). i (33) i iVm, m = 0, 1, 2, ii u(0)2 , u

(1)s , s = 1, 2. ii, . 2, ,

i ii Vm, m = 0, 1, 2, i, u(x, t) (25), (26), (27), i (30), (31) ii (32).

a priori, Vm, m = 0, 1, 2,

V0

H

,/2(), Vs

H

,/2(1), s = 1, 2, (34)

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64 .I., .., ..

i (27). i i-i (27), i i , i ii - i, i i i. i, ii

((x, t),u2) =n

i=1i(x, t)iu1 + (x, t)

u2(x, t)

N(x, t) ,

i = Di i(N,)n

k=1

NkDk, i = 1, . . . , n, i S,

(x, t) =((x, t), (x))

(N(x, t), (x)).

i (27) i ((x, t) \S):

L5u(x, t) n

i,j=1

ij(x, t)iju2 +

ni=1

i(x, t)iu2 + 0(x, t)u2 Dtu2 = 0(x, t) (35)

L5u(x, t) n

k,l=1

kl(x, t)Dklu2 +n

k=1

k(x, t)Dku2 + 0(x, t)u2 Dtu2 = 0(x, t), (36)

kl(x, t) =n

i,j=1 ij (x, t)ik(x)jl (x), k, l = 1, . . . , n ,k(x, t) = k(x, t) (x, t)Nk(x, t)

ni,j=1

ij(x, t)i(j(x)k(x)),

0(x, t) = (x, t) (x, t)u2(x, t)N(x, t)

. (37)

, i (35), (36), i 0, i- i i i u2, ,

(33). i u(0)2 (x,t)

N(x,t) (9),

u

(1)

2

(x,t)

N(x,t) i ii

u(1)2 (x, t)

N(x, t)=

t0

d

S1

G2(x, t; , )

N(x, t)V2(, )d, (x, t) , (38)

G2(x,t;,)N(x,t)

, (28), i (0 < t T,(x, t) , (, ) 1)

G2(x, t; , )

N(x, t)

C exp|x |2

t (39)

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C, i d0. i (39) i i i i (38).

(36) i i \S. ii, , (34), (37) ii (.. 2), ii kl, k, k, l = 1, 2, . . . , n, 0 0

H,/2

(). i (. [1, 2, 9, 14]), u2 i,

u2(x, 0) = 2(x), x S, (40)

ii

u2H2+,(2+)/2() C 0H,/2() + 2H2+(S) . (41)

i i i i (36). i L5 .., i (x, t; , ) (0 < t T, x, S). i, i, i .. i i, i i - E0 (. . 2). i , i ii ix i t i i (4) (5), i ii n n 1.

, , S = Rn1. i = Di,i = Di, i = 1, . . . , n 1, n = 0, i (36) ii -i i ii, = RnT, :

L5u n1

k,l=1

kl(x, t)Dklu2 +

n1k=1

k(x, t)Dku2 + 0(x

, t)u2 Dtu2 = 0(x, t), (42)

k(x

, t) = k(x, t) n(x, t) a

(2)in (x

, t)

a(2)nn (x, t)

, 0(x, t) = (x, t) n(x

, t)

a(2)nn (x, t)

u2(x, t)

N(x, t).

I .. L5 (1), (2).

, S H2+, S = {x Rn|xn = F(x)}, i F(x) (16). , i [10], , [9]:

(x, t) (z, t), zi = xi, i = 1, . . . , n 1, zn = xn F(x),

S i S = {z Rn|zn = 0}. , i i i i

u2(x, t) = u2(x

, F(x), t),

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66 .I., .., ..

ii .. (x, t; , ) (0 < t T, x, Rn1), i (36), i Dnu2 = 0, Dknu2 = 0, k = 1, . . . , n,kl(x

, t) = kl(x, F(x), t), k = 1, . . . , n 1, k(x, t) = k(x, F(x), t), k = 1, . . . , n,

0(x, t) = 0(x

, F(x), t), 0(x, t) = 0(x, F(x), t):

L5u n1

k,l=1

kl(x, t)Dklu2 +

n1k=1

k(x, t)Dku2 + 0(x, t)u2 Dtu2 = 0(x, t), (43)

ii kl(x, t),k(x

, t) 0(x, t) i (37), i

Bn1(x, t)) =

kl(x

, t)n1

k,l=1

i , ii , i-ii (1).

i (43) .. i

L5u = 0, (x, t) S (0, T), (44)

L5u i i (35) (36). i i

P0(x, t; , ) = P

(,)0 (x

, t ) = (2)(n1)(det Bn1(, ))1/2(t )(n1)/2

exp

Bn1(

, )1

(x ), x

4(t )

, 0 < t T, x

,

R

n1,

0(x, t; , ) =

(,)0 (x

, t) = P0(x, t; , )n(), 0 < t T, x, Rn1.

i

P0(x, t; , ) = P(,)0 (x

, t ) = (2)(n1)(det Bn1(, ))1/2(t )(n1)/2

exp

Bn1(, )1

(x

), x

4(t ) , 0 < t T, x, S,

0(x, t; , ) = (,)0 (x

, t ) = P0(x, t; , )n(), 0 < t T, x, S.

i, i P0(x, t; , ) .. i

n1

k,l=1kl(

, )Dklu2(x, t) Dtu2(x, t) = 0, (x, t) R nT,

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i i (, ) R nT ii. i 0 .. i (44), .. i i [6, . IV, 11].

i, ii S H2+, , i .. S, -

i (18). i (36) ii i {y, t} i x(m) i:

L5,(m)u(m)

nk,l=1

(m)kl (y, t)Dklu

(m)2 +

nk=1

(m)k (y, t)Dku

(m)2 +

(m)0 (y, t)u

(m)2

Dtu(m)2 =

(m)0 (y, t), (y, t) S (0, T], (45)

u(m)2 (y, t) = u2(x(y), t), (m)kl (y, t), k, l = 1, . . . , n, i

B(m)(y, t) = C(m)B(m)(y, t)(C(m))T, B(m)(y, t) = B(x(y), t) = (ij(x(y), t))ni,j=1 ,

(m)k (y, t), k = 1, . . . , n,

(m)(y, t) = C(m)(m)(y, t), (m)(y, t) = (x(y), t),

(m)0 (y, t) = 0(x(y), t), (m)0 (y, t) = 0(x(y), t) =

(m)(y, t) (m)(y, t) u(m)2 (y,t)

N(m)(y,t),

(m)(y, t) = (x(y), t), (m)(y, t) = (x(y), t) = ((m)(y,t),(m)(y))

(N(m)(y,t),(m)(y)), N(m)(y, t) = C(m)N(m)(y, t),

N(m)(y, t) = N(x(y), t) i, i i A(m)2 (y, t).i , , (y, t) S(m)2 [0, T], i (45)

i i

u(m)2 (y, t)

yS(m)2

= u(m)2 (y, F(m)(y), t) = u(m)2 (y

, t),

F(m)(y) (20),

L5,(m)u(m)

n1k,l=1

(m)kl (y

, t)Dklu(m)2 +

n1k=1

(m)k (y

, t)Dku(m)2 +

(m)0 (y

, t)u(m)2

Dtu(m)2 =

(m)0 (y

, t), (y, t) S(m)2,0 (0, T]. (46)

ii (46) (m)0 (y, t) =

(m)0 (y

, F(m)(y), t), (m)kl (y

, t) =(m)kl (y

, F(m)(y), t),

(m)k (y, t) = (m)k (y

, F(m)(y), t), k, l = 1, . . . , n 1, i (37), iii i {y, t}.

i i i

m(y, t; , ) =

Pm(y, t; , )(m)n (

), 0 < t T, y, S(m)2,0 ,

i Pm(y, t; , ) .. ii i i i i-i

n1

k,l=1(m)kl (

, )Dklu(m)2 (y

, t) Dtu(m)2 (y, t) = 0.

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i - 69

t0

d

S

(x, t; , )

(, ) (, )

3j=2

u(j)2 (, )

N(, )

d. (50)

i i Vm, m = 0, 1, 2, (26). i , ii (26), i i ,

i ii i, i i i ii:

((1),u2) =n

i=1

(1)i (x, t)

(2)i u2 + 2(x, t)

u2(x, t)

N(2)(x, t),

(,u1) =n

i=1

i(x, t)(1)i u1 + 1(x, t)

u1(x, t)

N(1)(x, t), (51)

(s)i = Di

(1)i

(N(s),(1))

n

k=1

N(s)k Dk, i = 1, . . . , n, s = 1, 2, i ii

S1,

1(x, t) =((x, t), (1)(x))

(N(1)(x, t), (1)(x)), 2(x, t) =

((1)(x, t), (1)(x))

(N(2)(x, t), (1)(x)).

(51) (25), (26) i:

L4u(x, t) n

i,j=1

(1)ij (x, t)

(1)i

(1)j u2 +

ni=1

(1)i (x, t)

(2)i u2

ni=1

i(x, t)(1)i u2+

(1)0 (x, t)u2 Dtu2 = (x, t), (x, t) 1\S1, (52)

L4u(x, t) n

k,l=1

(1)kl (x, t)Dklu2 +n

k=1

(1)k (x, t)Dku2 + (1)0 (x, t)u2 Dtu2 = (x, t), (53)

(1)kl (x, t) =

ni,j=1

(1)ij (x, t)

(1)ik (x)

(1)jl (x), k, l = 1, . . . , n ,

(1)0 (x, t) = (1)0 (x, t) + 0(x, t),

(1)k (x, t) =

(1)k (x, t)k(x, t)+

2s=1

(1)s1s(x, t)N(s)k (x, t)n

i,j=1

(1)ij (x, t)i(

(1)j (x)

(1)k (x)),

(x, t) = 0(x, t) +2

s=1

(1)s1s(x, t) us(x, t)N(s)(x, t)

,

0(x, t) = (x, t) 0(x, t)z(x, t) +n

i=1

i(x, t)(1)i (x, t)z(x, t). (54)

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70 .I., .., ..

i i , (52) (53) i i 1\S1. ii, , (34), (54) ii (. . 2), ii H,/2. i u2 i, -

u2(x, 0) = 2(x), x S1, (55) ii

u2H2+,(2+)/2(1) CH,/2(1) + 2H2+(S1)

. (56)

i (53), (55) i

u2(x, t) =

S1

1(x, t; , 0)2()d t

0

d

S1

1(x, t; , )(, )d, (x, t) 1, (57)

1(x, t; , ) .. L4, i . i , i u2 1:

ii (33), s = 2, (x, t) 1, ii (57). -i i i , (6)(8), i, ii i Vm, m = 0, 1, 2. i Vm -, (57) (25). i i ii i Vm, m = 0, 1, 2, :

t0

d

S(m)

Gm(x, t; , )Vm(, )d +

2l=0

t0

d

S(l)

Kml(x, t; , )Vl(, )d = m(x, t),

(x, t) (m), m = 0, 1, 2, (58) (m) = S(m) [0, T], m = 0, 1, 2.

K10(x, t; , ) =

t

ds

S1

1(x, t; , s)2(, s)G2(, s; , )

N(2)(, s)d,

K20(x, t; , ) = K10(x, t; , ) + G2(x, t; , ),

Kml(x, t; , ) = (1)l112

l(, )1(x, t; , )+

t

ds

S1

1(x, t; , s)l(, s)Gl(, s; , )

N(l)(, s)d, m, l = 1, 2,

m(x, t) =

S1

1(x, t; , 0)2()d 3

l=2u(l)m (x, t) + zm(x, t)

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i - 71

t0

d

S1

1(x, t; , )

0(, ) +

2i=1

3j=2

(1)i1i(, ) u(j)i (, )

N(i)(, )

d, m = 1, 2,

z1(x, t) z(x, t), z2(x, t) 0, (59) K0m, S(m), m = 0, 1, 2, 0 (50).

Kml, m, l = 0, 1, 2, (50) (59) ii (4), i i n, r i p ii n1, 0 i 0, i m, m = 0, 1, 2, , (30) ii,

m H

2+,(2+)/2((m)), m = 0, 1, 2.

i (58) i i I . ii(22), 3.2 , i Em, m = 0, 1, 2, ii i (58) i i-

i i II ((x, t) (m)

, m =0, 1, 2):

Vm(x, t) +2

l=0

t0

d

S(l)

Rml(x, t; , )Vl(, )d = m(x, t), m = 0, 1, 2, (60)

m(x, t) =

Am(x, t)

(m)(x), (m)(x)1/2 Em(x, t)m,

A0(x, t) A2(x, t), (0)

(x) = (x), (1)

(x) = (2)

(x),

Emm H

1+,(1+)/2((m)), m H

,/2((m)), Rml, m, l = 0, 1, 2,

ii (10). i (60) i Vm,

m = 0, 1, 2. i , , Vm, m = 0, 1, 2, (34). , i (31),

i (32) i i (23)(27). , (33), (60) i us,s = 1, 2, i i:

Lsus(x, t) = fs(x, t), (x, t) s, s = 1, 2,

us(x, 0) = s(x), x Ds, s = 1, 2,us(x, t) = v1(x, t) + (2 s)z(x, t), (x, t) 1, s = 1, 2,

u2(x, t) = v(x, t), (x, t) , (61) i

s(x) = v1(x, 0) + (2

s)z(x, 0), x

S1, s = 1, 2,

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517.956.4

.., ..

.., ..

// . 2010. .2, 2. C. 7482.

,

. , -

,

, .

, -

. ()

, t, [1, 2].

(...), -

.

.

1 . .

x = (x1, x2) R2. (0,T] = {(t, x), x R

2, 0 t T}, 0 < t, (t, x; , ) =

(x1 1)2(4(t ))1 + 3(x2 2 + (x1 + 1)(2(t ))

1)2(t )3, n N.

[1], [2]:

tu(t, x) x1x2u(t, x) =n

r=1

[ar2 (t, x)2x1

+ ar1 (t, x)x1 + ar0 (t, x)]ur(t, x),

= {1,...,n}, (t, x) (0,T], (1)

u(t, x)|t= = u0(x), 0 < t T, (2)

2000 Mathematics Subject Classification: 42A38, 46H30. : , , -

.

c .., .., 2010

8/6/2019 T2 N2

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75

tw(t, x) =n

r=1

[ar2 (t, x)2x1

+ar2 (t, x)x1 +ar0 (t, x)]wr(t, x) -

[0,T], (x2 ); .., 1 R1, det{k=2

ak(t, x)(i1)2

I} = 0 j(t, x), j = {1,...,n},

Rej(t, x) 0, (t, x), j = {1,...,n}, I ,1 = i, ak = (ark )nr,=1. (1) :A) arj (t, x), j = {0, 1, 2} ,

, (t, x) [0,T];) ar,j = a

,rj , j = {0, 1, 2}, {r} {1,...,n};

B) |arj (t, x)arj (t, x)| K(|x1x1|1 +|x2x2|2), {(t, x), (t, x)} [0,T], 1 (0, 1],2 (13 , 1], K > 0 , t,x,x;

) , jx1arj (t, x), j = {0, 1, 2}, -

). ... (1). ... (1)

E(t, x; , ), {(t, x), (, )} [0,T], t > ,

-

u(t, x) =

t

d

R2

E(t, x; , )f(, )d

f(t, x) -

tu(t, x) x1x2u(t, x) =2

k=0

ak(t, x)kx1

u(t, x) + f(t, x), (t, x) (,T]. (4)

1. (1) ), ), ), ...

E(t, x; , ), t > . ), ... (1) E(t, x; , ), t > , E(t, x; , ) = E(t, x; , ),

|kx1E(t, x; , )| Ck(t )4+k2 exp{c(t, x; , )}, 0 k 2, (5)

|x2E(t, x; , )| C1(t ) 72 exp{c(t, x; , )}, (6)|h1kx1E(t, x; , )| Ck(t )

4+k+12 |h1|1 max[exp{c(t, x; , )},

exp{c(t, x; , )}], x = (x1 + h1, x2), (7)|h2x2E(t, x; , )| C1(t )

7+322 |h2|2 max[exp{c(t, x; , )},

exp{c(t, x; , )}], x = (x1, x2 + h2), (8) c, Ck, C1 , 1, 2, sup

[0,T]

|ak(t, x)|, T, - a2(t, x).

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79

(, ; , )d = Z0(t, x; , , (t, )) + V. (24)

(t, x; , ) , E(t, x; , ), t x, t > (1). , (t, x; , ) t >

,

|(t, x; , )| C(t )632

2 exp{c(t, x; , )}, (25)

|h2(t, x; , )| C|h2|

2(t ) 63

22 max[exp{c(t, x; , )},

exp{c(t, x; , )}], 2 < 2, 2 = 2 2. (26) . t x1x2

2k=0

ak(t, x)kx1

E(t, x; , ), (24), - (23), (t, x; , ) e

(t, x; , ) = K(t, x; , ) +

t

d

R2

K(t, x; ,, )(, ; , )d,

K(t, x; , ) =2

k=1

[ak(t, x) ak(t, x1; (t, ))]kx1Z0(t, x; , , (t, ))+

a0(t, x)Z0(t, x; , , (t, )).

(23)

|K(t, x; , )| C1(t )632

2 exp{c(t, x; , )}, (27)

(t, ; , ) =

m=1

Km(t, ; , ), (28)

K1(t, x; , ) = K(t, x; , ),

Km(t, ; , ) =

t

d

R2

K1(t, x; )Km1(, ; , )d.

(28).

|K2(t, x; , )| A21t

d

R2

[(t )( )] 2322 exp{c(t, x; , )}

(t )2 exp{c(, ; , )}( )2d A21B(32

2,

322

)

(

C

)(t

)

6622 exp

{c(t, x; , )

}, t > , (29)

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80 .., ..

, ,

|Km(t, x; , )| Am1

m(322

)

(3m22

)(t ) 3m262 exp{c(t, x; , )}. (30)

(30) (28) t

> 0 (t, x; , ) (25). (26). (t )3/2 < |x2 x2 | = |h2|

(25). |x2 x2 | (t )3/2.

|h2K(t, x; , )| 2

k=1

|h2ak(t, x)||kx1Z0(t, x; , , y(t, ))| +2

k=1

|ak(t, x)

ak(t, x1, (t, ))||h2kx1Z0(t, x; , , (t, ))| + |a0(t, x)||h2Z0(t, x; , , y(t, )|+

|h2a0(

t, x)||

Z0(

t, x;

, , (

t, )|

M[|

h2|2

(t

)

3

+ |x2

2+

1(t )|2|h2|(t )9/2 + |h2|(t )7/2 + |h2|2(t )3]exp{c(t, x; , )} |h2|2 (t )(632 )/2 exp{c(t, x; , )}. (31)

h22x1

Z0(t, x; , , (t, )) -

, . -

; |x2 x2| (t )3/2 1 1, 1 + |x22+(x1+1)(2(t))1|

(t)3/2

|x22+(x1+1)(2(t))1(t)3/2

+ x2x2(t)3/2

| |x22+(x1+1)(2(t))1|(t)3/2

+ 1.

(25), (31) h2(t, x; , ):

h2(t, x; , ) = h2K(t, x; , ) +

t

d

R2

h2K(t, x; , )(t, x; , )d.

:

t

d

R2

|h2K(t, x; , )||(t, x; , )|d =t|h2|2/3

+

t

t|h2|2/3

= I1 + I2,

I1 t |h2|2/3, .. t |h2|2/3, , (31),

I1 C(t )3(2+

2)

2 |h2|2 exp{c(t, x; , )}. (32) I2:

t

|h2|2/3

d

R2

|K(t, x; , )||(, ; , )|d+t

|h2|2/3

d

R2

|K(t, x; , )| |(, ; , )|d,

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82 .., ..

R1

x2Z0(t, x; , ; y(t, ))|y2=x2x1(t)d2 = 0,

x2

R1

Z0(t, x; , ; y(t, ))d2 =

R1

[x2Z0(t, x; , ; y(t, ))

x2Z0(t, x; , ; y(t, ))|y2=x2x1(t)]d2. Z0(t, x; , ; (t, )) (35), (36)

x2V, kx1

V, 0 k 2, , E(t, x; , ) Z0(t, x; , ; (t, )).

, ... (1) [3, c.

91]. -, ...

(1), , .

1. .. i // . . .2009. .12, 3..1650

1563.

2. .. i ii, i

i // . . 2007. 102, 0.9 . . 2007, 12.

3. .. . .: , 1964. 443.

. . ,

-,

26.10.2010

Malytska H.P., Burtnyak I.V. Method of parametris for the ultraparabolic systems, Carpathian

Mathematical Publications, 2, 2 (2010), 7482.

We consider systems of ultraparabolic equations which generated equation of diffusion with

inertia. Using the modified method Levy, we constructed the fundamental matrix of solutions

of this system of Kolmogorov equations of second order, and got estimations of derivatives,

included in the system.

.., I.. i //

i i ii. 2010. .2, 2. C. 7482.

i i, i i -

i ii. i i, -

i i , i-

i, .

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515.12+512.58

Nykyforchyn O.R.

ATOMIZED MEASURES AND SEMICONVEX COMPACTA

Nykyforchyn O.R. Atomized measures and semiconvex compacta, Carpathian Mathematical

Publications, 2, 2 (2010), 83100.

A class of atomized measures on compacta, which are generalizations of regular real-valued

measures, is introduced. It has also been shown that the space of normalized (weakly) atomizedmeasures on a compactum is a free object over this compactum in the category of (strongly)

semiconvex compacta.

Introduction

It has been known for a long that the space P X of probability measures on a compactHausdorff space X with the weak topology is a convex compactum, i.e. it can be embedded

into a locally convex topological vector space as a compact convex set. Moreover, it is a free

convex compactum [3] over X, i.e. it contains X as a closed subspace so that each continuous

mapping from X to a convex compactum Kcan be uniquely extended to an affine continuous

mapping from P X to K.

Some applications require the class of convex compacta to be extended to the class of

so-called semiconvex compacta [8]. The goal of this work is to show that free semiconvex

compacta can also be obtained as spaces of special measures, which we call atomized.

We use the following terminology and denotations : I = [0, 1] is a unit segment, R+ =[0;+), R+ = (0; +), Q+ = Q (0;+). A compactum is a (not necessarily metrizable)

(bi)compact Hausdorff topological space.

For basic definitions and facts of the category theory cf. [7]. The category of Tychonoff

spaces Tych and the category of compacta Comp consist of all Tychonoff spaces and all

compact Hausdorff spaces, respectively, and their continuous maps.

2000 Mathematics Subject Classification: 18B30, 54B30.Key words and phrases: Semiconvex compactum, measure, atom, monad.Partially suported by the grant 25.1/099 of State Fund of Fundamental Research of Ukraine, and by Inter-

governmental Programme of Scientific and Technological Cooperation, project M/95-2009.

c Nykyforchyn O.R., 2010

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84 Nykyforchyn O.R.

1 Atomized measures

In the sequel X is a compactum, Exp X is the collection of all closed subsets of X,

and exp X = Exp X \ {}. Each regular real-valued additive measure on X is uniquely

determined by its values on closed subsets of X, hence the following is equivalent to the usual

definition:

Definition. A function m : Exp X R is a regular additive measure on X if, for all

A, B Exp X:

(1) m() = 0;

(2) A B implies m(A) m(B) (mo

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