A general model of evolutionary processes. Exponential dichotomy—I

Nonlinear Analysis, Theory, Printed in Great Britain.

Methods

& Applications,

Vol. 21, No. 3, pp. 207-218,

1993. 0

0362-546X/93 $6X9+ .oO 1993 Pergamon Press Ltd

A GENERAL MODEL OF EVOLUTIONARY PROCESSES. EXPONENTIAL DICHOTOMY-I J. APPELL,? V. LAKSHMIKANTHAM,~ NGUYEN VAN MINH$ and P. P. ZABREIKO§ TMathematisches Institut, Universitat Wurzburg, Am Hubland, D-97074 Wurzburg, Germany; $ Department of Applied Mathematics, Florida Institute of Technology, Melbourne, FL 32901-6988, U.S.A.; and 0 BGIJ Minsk, Matematicheskij Fakultet, Pl. Nezavisimosti, 220050 Minsk, Belarus (Received

1 February

Key words and phrases:

1992; received for publication

26 January

1993)

Flow, exponential dichotomy, dynamical system, spectrum.

1. INTRODUCTION IN MANY problems of physics and mechanics determined from the following model

the evolution

t x(t) - x(s) =

of evolutionary

processes is

f

du +

A(u)x(u)

Is

rs

B(u)x(u)Hdu)

(1)

where x(t) is the state of the process at t E J, J is an interval in IR,A, B are matrix functions, and ,Uis a measure on R. Mathematically, the model (1) can be stated in a more general form as follows. Suppose that X is a given Banach space. We denote by B(J, X) the space of all bounded X-valued functions on J, where J is an interval in R. Suppose that D(J, X) C B(J, X) has the following property: if x E D(J, X) then xl1 E D(Z, X) for every interval Z c J. Consider a family of operators rI s, f : mb, t1, XI + x. (2) Assume that for every pair s, t (s < t) D([s, t], X) is a Banach space, and I-L,, E G(D([s, t1,n

m

(3)

Consider the equation x(t) - x(s) = K,,a,,,x

(4)

where x: J + X, and os, f denotes the restriction on the interval [s, t]. As another example of (4) we consider the following. Suppose that KI E $(B(J, X), X). Put D(Z, X) = B(Z, X) for every interval Z C J, and let K,,as,,x

= ~&,~lX

where xrs, t1 is the characteristic function of [s, t]. It is easily seen that the operators corresponding to (1) and in (5) satisfy the so-called consistence condition K,Ps,,x

= K,ps,,,x

+ *** + Kn.tq,,x

(5) II,, f

(6)

wheres c t, -c --. c t,, < t, and x E D(J, X). For arbitrary II,, I we shall deal with the existence of solutions of (4) on J in another paper. In the present paper we restrict ourselves to the case 207

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J. APPELLet al.

where II,,, satisfies (10) below. This guarantees the existence and uniqueness of continuous solutions to (4). We shall be concerned with the notion of exponential dichotomy and characterize it in terms of spectral properties of the generating operator. Note that our result is still true for a more general model which we call below “flow”. 2. STATEMENT

OF THE

PROBLEM

We denote by B(r, t) the set of bounded functions from [r, t] + X, where X is a real Banach space. Suppose that on every B(r, t) we are given a bounded linear operator II r, f : B(r, t> -+ x.

(7)

We shall always assume that the family of operators II,,, consistence condition, i.e. for every partition r < t, < t2 c -.K,,o,,x

= K,QJ,JIX

+ Q,,,*ot,,t,x

r, t E J (r < t) satisfies the have

< t, < t we

+ ***+ IL”,ior,,tx

(8)

where x is any bounded function defined on an interval J > [r, t]. Definition

1. We say that x E B(a, b) is a solution of the equation x(0 - x(r) = IL, to,, tx

(r < 0

(9)

if (9) holds for all r, t E [a, b], r < t. From now on, for convenience, we shall write II, tx instead of II,, ror, tx whenever this does not cause misunderstanding. Definition 2. A function x defined on (a, b), (a, b] or [a, b) C J is said to be a solution of (9) if for all T, t E (a, b), (a, b] or [a, 6), respectively, ~,,~x is a solution of (9). AN EXISTENCE AND UNIQUENESS THEOREM.

Assume on J

+rIlIYI,,/l<

C iR

the following inequality

K = const. < fw

holds for all t, r E J, t > r. Then for every to E J, x0 E X there exists a solution x of (9) on J satisfying x(&J = x0. In addition this solution is unique and continuous on J. Proof. Without loss of generality we suppose that J is a compact interval, say [to, T]. So we have x(0 = x0 + l-Jo,,&

t E [to, Tl.

(11)

Consider the operator S: B(i, , T) -+ B(to , T) defined by GM)

= x0 + &,x.

(12)

We have @x)(t) - (Sx)(s) =

_y; i

1,s

ifs < t, if t < s.

209

Exponential dichotomy-I

Thus, NW) where

IlxL = ;t~ Ilx~ll, hence

- (W(s)ll 5 It -

4a4J

(13)

S: B(t,, T) + B(t,, T). Furthermore,

WW,,

T)) C W,,

Tl, X) c WI,

n,

where C([t,, , T], X) denotes the space of continuous functions from [to, T] into X. For proving the theorem we need the following lemma. LEMMA1. Let all conditions llullJ = 1 one gets

of the theorem be satisfied. Then for every u E B(t,,, T) with

IlrI,,,(Kfu)ll wheref(t)

5

qfK2

(14)

= t - r.

Proof of lemma 1. From (8) we obtain

IlI-L,,Kf@ll5 Ill-b,,,Kf~)ll+ ... + Il~tn,tKf~)ll~ where r < t, < ... < t, < t. Consequently n+l

IIn,t(Kfu)II5 K2 C f(ti) Ati

(15)

i=l

where Ait, = t, - r, Ati = ti - ti_l, At,,,, Iln,,,(Kfu)ll

I K2

= t - t, . It is easily seen that from (15) we get ‘(r Ir

- r) dr = K2v.

(16)

This completes the proof of lemma 1. By induction one can show without difficulty that

Iln,,,(~T,,El(n,,,-,(Kfu)) . ..)I15

(t-nr;nKn.

Now we continue to prove the theorem. Taking into account (16) and the definition of S we have

(IS”p - S”lyIIJ I

““,I toY IIql-

JyllJ.

As K”(T - t,)“/n! + 0 as n -+ 03 there exists m such that S” is a contraction. This implies that there exists a unique fixed point Z E B(t, , T) of S”. It is clear that 3 is also a unique fixed point of S. In addition, Z is continuous on J. From (8) we can conclude easily that .? is a solution of (9) satisfying i?(to) = x0. This completes the proof of the theorem. LEMMA2. Under the assumptions of the above theorem the inequality

IIW, Al 5 llxllew(Klt- ~1) holds for all t, s E J, where X(t, s)x denotes the solution of (9) satisfying X(s, s)x = x.

(17)

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J. A~PELL

et al.

Proof. Suppose that t > s. From (8) we get Iln,,,(x(<,S)X)II

I

i K(ti+l -

ti)

i=l

Ilx(~9~)Xll~

suP

CE[fi~ti+ll

(18)

Since X(& s)x is continuous in r we get llI-L,t(X(L @x)115 K

’ IlX(L Gxll Gf. ss

(1%

IlW, Gil 5 lbll + K

’ IlX(tv WI1 dt-. ss

(20)

so

By applying Gronwall’s inequality we obtain

II-W Ml 5 llxllewW(t- s))

(t, s E J, t > s).

(21)

In the same way, we can prove (17) for s > t. From now on we assume that X is a real Hilbert space. Put X(t, s)x = x(t). We have
- = (x(0 - x(s),X(O) +
=

I -
tit1 + x(s)),

for s < t, for t -c s.

(22)

Thus, from the Cauchy inequality we get

I lltiol12- llx(~)l121 5 (Ilmll + llX(~mll~s,txll. Taking into account (19) we obtain, for t > s, I Ilx(t)II -

h(s)II 1 5 K

’ IlWll ss

dC.

(23)

So, for t 2 s, Ilx(t)Ii 2 Il.WII

- K

’ Il-MIt ss

d<.

(24)

By applying Gronwall’s inequality we get

II-W,Ml 2 llxllew(-W - ~1).

(25)

In the same way, for t < s we can prove that

Ilx(t,Ml 2 llxllev(--K(s - 0).

(26)

Combining (25) and (26) we have

IlWt,hd 2 llxllew(-Kit - 4).

(27)

Thus, under the assumptions of the above theorem we get a family of operators X(t, s), t, s E J, having the following properties: (i) X(t, s) E .C(X) (t, s E J), (ii) X(t, t) = Z (t E J), (iii) X(t, r)X(r, s) = X(t, s) (t, r, s E J), (iv) X(t, s) is continuous with respect to t, s.

Exponential

211

dichotomy-I

The properties (i)-(iii) are clear. We shall prove property (iv). In fact, we note first that it suffices to prove that X(t, s) is continuous with respect to t, because Jqt, s) = iqt,

0)x-‘(s,

0).

For every t’ E J we have IlX@,4 - XV, @II 5

IIU - W’, O)IIIlm dll*

Let t’ - t = s, where t’ > t without loss of generality. We have _qt + s, t)x - x = rI,,,+,x
+ s, l)x

(28)

for every x E X. Hence,

IlX(t+ s, t)x - XII5 K,zyp,,IIW + t, 0-4l5 KS expWs)llxll

(2%

for some D > 0. So we have for all t, t’, s E J Il_X(t’,t) - III 5 K/t - t’ID exp(KIt - t’l).

(30)

This proves (iv). Definition 3. We call every family of linear operators flow on J.

X(t, s), t, s E J, satisfying

(i)-(iv) a

Definition 4. A family of bounded linear operators T”, s E iR, is said to be a continuous linear dynamical system if it has the following properties: (i) To = Z, (ii) TS+’ = TST’ (s, t E R), (iii) s + TS is continuous. For example, let X(t, s) be a flow on R, and suppose that there exists an operator T such that X(t, s) = T’-‘. Then we say, as in the theory of differential equations, that the flow X(t, s) is autonomous. 3. HYPERBOLIC

CONTINUOUS

LINEAR

DYNAMICAL

SYSTEMS

LEMMA3. Suppose TS, s E iR, is a continuous linear dynamical system. Then we have fim ln! TS\I -=lnr S-+rn s

(31)

where r is the spectral radius of T = T’. This lemma is easily shown in the usual well-known way (see e.g. [l]). Definition 5. A continuous linear dynamical system TS, s E IR, is said to be hyperbolic if the spectrum a(T) of T does not intersect the unit circle. THEOREM1. Suppose that TS, s E IT?is a hyperbolic continuous linear dynamical system. Then X splits into a direct sum of two invariant closed subspaces X, , X2 such that there exist positive

212

J. APPELL et al.

constants D and Q!having the following properties

IIWI 25D ew(-~dlxll

(XEXl,S>O),

(32)

II~3ll 5 ~ew(-4ll_dl

(YE-G,S
(33)

Conversely, if X = X, @ X,, where X1 and X, are invariant closed subspaces of TS, s E IR, such that (32) and (33) hold, then TS, s E IRis hyperbolic. Proof.

Set p =

s

&. s’ R(I,

T) dl

where R(l, T) is the resolvent of T. Then P is a projection (see [2]). We define X, and X, to be Im P and Ker P, respectively. It may be noted that X, and X, are left invariant for TS, where s = p/q, p, q E 72. From the continuity of s + TS and the fact that

s

R(1, TS) dL

P=&

s’

depends continuously on s, s # 0, it follows that X, and X, are left invariant for Ts for every s E IR. In addition, a(TS I X,) and a(T-” IX,) are contained in the open unit disk for all s > 0. Now we shall show that there exists a norm in X, which is equivalent to the given norm such that ItT” 1XIII 5 4’

(35)

where q is any fixed positive constant in (rl, l), rl denoting the spectral radius of T IX, . In fact, set llxlll =

I= y 0

ds.

The integral on the right-hand side is convergent, as may be shown by using (31). We have

11 TSxlll=

mliT’:“xli 0

dt

=

qs

4

s

m11 Tysx”

0

dt

sco

11 T’xll

I

qs

4

0

dt

=

qsllxll

1,

d

This implies (35). Similarly, in X, there exists a norm II* II2 such that

11 T-’ 1&II,

5

q”

(37)

for all s > 0, where q’ is any fixed constant in (r2, l), r, being the spectral radius of T IX,. Finally we define II - 11’by the formula

llxll’= (Ilx# + Ilx&>“’ where x = x1 + x2, x1 E Xi, x2 E X, . Now we prove that II - II1 is equivalent to II* II in X1. In fact, we have

IIXIII5

mll“d;“ll drsllxll 0

(38)

213

Exponentialdichotomy--I

for all x E X1. On the other hand, since lim TS = I, for a fixed positive constant E there s+O

exists 6 > 0 such that ))T+ - III < E

(0 < s < 6).

Thus, 1 + E z 11TmSII 1 1 - E for 0 < s < 6. We have

MI12 1: y

ds2 I* ll’y

dsllxll2 &

1: f 11x11,

(39)

0

Putting ‘6

1 a=(1

!0

4-s b,

b=

m IIT’IX,II h i’0

qs

we conclude from (38) and (39) that

allWI 5 IIWI

5 dlxll 15 Wbll

(40)

for all s > 0, x E X1. Thus,

IIT”xll 5 @/aWlbll

(41)

((T-“x(1 5 (b’/a’)qrS/[xI(

(42)

for all x E X, , s > 0. Similarly,

for all x E X, , s > 0. This completes the proof of the first part of theorem 1. The second part is easily shown by using the spectral radius theorem. 4. EXPONENTIAL DICHOTOMY OF FLOWS Definition 6. A flow X(t, s) E C(X), t, s E IT?,is said to have an exponential dichotomy if there exist a projection P: X + X, and positive constants D and (Yhaving the following properties ((X(t)PX-l(s)!\

d D exp[-ol(t

- s)],

if t 1 s,

(43)

IlX(t)(Z - P)X-‘(s)jl

I D exp[ -a(s

- t)],

ifs 2 t;

(44)

here X(t) denotes X(t, 0). PROPOSITION1. A continuous linear dynamical system TS, s E Ii?,has an exponential dichotomy if and only if T is a hyperbolic linear operator. This proposition is a corollary of theorem 1. Now we are going to characterize the exponential dichotomy in terms of linear dynamical systems generated by the corresponding flows. LEMMA4. Let C,(R, X) denote the space of bounded uniformly continuous functions R + X with norm I/XII= sup Ilx(t))I. Then CJR, X) is a Banach space. Proof. It suffices to show that if (v,, n E IN) is a Cauchy sequence in C,(lR, X), then there exists u E C,(R, X) such that u, + u as n -+ co. In fact, since the space C&R, X), the space of bounded continuous functions R -+ X, is a Banach space, we see that u,, + u as n -+ 43

214

J. APPELL et al.

where v is some element of C&R, X). Now suppose that there exists a’ > 0 such that for any 6 > 0 one can find I,, ti E R with Its - ti 1 < 6 such that IIv(t,) - v(t;)II 2 E’. As v, + v, there exists N E iN such that sup/I UN(t) - v(t)11 < E ‘/4. So we have e’ 5

IIv(t*)- v(G)II5 lb&) - %&)ll + II+&) - h.&)Il + IIVdf;)- W,‘)ll.

Thus, E’ I &‘/2 + IIv,(t,)

- V&g/.

From the uniform continuity of v, it follows that there exists 6 > 0 such that for t, t’ E R with It - t’j < 6 one gets < &‘/2.

IIUN(f) - q&‘)ll

This contradicts our assumption, completing the proof of lemma 4. Let X(t, s), t, s E I?, be a flow; we shall define a linear dynamical system associated with it as follows. Let v E C,(lR, X) and put (T%)(t)

= X(t, t - s)v(t - s)

(t E IF?).

(45)

LEMMA5. Suppose that X(t, s) E S(X), t, s E R, is generated by (9) where (10) holds. Then T” E ~(C,(R, X)), s E R, is (Ylinear dynamical system. Proof.

First we shall show that T’v E C,(R, X). Indeed, we have (T%)(t)

- (T%)(t’)

= X(t, t - s)v(t - s) - X(t’, t’ - s)v(t’ - s).

Thus, II

- (T”v)(t’)ll

I

jlX(t, t - s)v(t - s) - X(t’,

t’ - s)v(t - s)ll

+ Ilx(t’, t’ - s)v(t - s) - x(t’, 5

Ilmt,t - 4 - xt - 411llvll + Ilx(t’, t’ - s)ll [Iv@ - s) - v(t’

t’ - s)v(t’

- s)ll

- s)ll.

(46)

From (17) it follows that IlX(t’, t’ - s)ll s exp[Klsl]. On the other hand, setting t’ - t = p and assuming without loss of generality that p L 0 we get

Ilm + P, t + P - 4 - mt, t - @II = Ilx(t + p, t)X(t, t + p - s) I

X(t, t + p - s)X(t + p - s, t - s)((

11X@+ p, t)X(t, t + p - s) - x(t,

t + p -

s)ll

+ Ilx(t,t+p--s)-X(t,t+p-s)X(t+p-s,t-s)ll

5 Ilmt+ P, 0 - III Ilm t + P - AI + 11X@,t + p - s)(l IIZ -X(t + p -

s, t - s)ll.

Now we estimate IIZ - X(t + p, t)ll. We have X(t + p, t)x - x = n,,,+,X(t

+ p, t)X.

(47)

215

Exponential dichotomy-l

So, according to (lo), IN

+ P, 0x - XII 5 KPO~pJX(t

+ r,

oxll*

(48)

In virtue of (17) we obtain

IlW + P, Ox- XII 5 0 wWp1 Ml.

(49)

[(~(t + p - s, t - s) - III I Kp exp[Kp].

(51)

This means that Similarly,

From (47), (50), (51) it follows that lim ((X(t + p, t + p - s) - X(t, t - s)(( = 0,

p-0

(52)

uniformly in t. Since u is uniformly continuous we see that from (46) TSu is also uniformly continuous. It is obvious that T”+’ = TST’ for all t, s E IR. This completes the proof of lemma 5. In general we do not know if the map s + TS is continuous.

LEMMA 6.Let X(t, s), t, s E R be the flow generated by (9) under the condition (lo), and let dim X < co. In addition, assume that T” is hyperbolic. Then the flow X(t, s) has an exponential dichotomy. Proof. We can choose in C,(R, A’) another norm II * II’ which is equivalent to the given norm and satisfies

IIT%lI’

< 4 < 1,

II%+

< 4,

where ES = Im P and E” = Ker P with

P=&

is’

R(A,T’) dil

(see [3, pp. 80-811). To continue the proof of lemma 6 we claim that ES and E” are C,(R, R)modules, i.e. if u E ES (or EU) and h E C,(lR, IR) then hu: t + h(t)u(t) belongs to E” (or E”, respectively). In fact, from the spectral radius theorem it follows that x E E” if and only if lim )ITnxlll’n < 1. n*+m So we have 1)T”huj( “,I 5 I[h[II” * 11 T”ull 1’n, hence, hu E ES. Similarly, we can prove that E” is a module over C,(R, IR). Set ES(t) = (x E XI x = u(t) for some u E E”J

(t E M,

E”(t)=(x~X~~=u(t)forsomeuczE~J

(t E IR).

216

J. APPELL et al.

We shall show that X = E”(O) 0 P(O). First, it is clear that ES(O) + E”(O) = X; thus, we have only to show that ES(O) II E”(0) = (0). In fact, suppose x E:ES(O) fl P(O), x # 0. Using the technique developed in [3, p. 216; 41 we can show the existence of a function liei = jJ pi fi

V= i i=l

i=l

where ei (i = 1, . . . . k) belongs to E” and fi (i = 1, . . . . k) belongs to E”, (e,(O), . . . . e,(O)) generates E”(O), (fi(0) ,...,fk(0)J generates EU(0), Ai, pi are equal to zero outside an e-neighbourhood of 0 and continuous on the whole line and j,

lAitt)l

>

Ov

j,

IPiCt)l

>

O

for t sufficiently small. This implies that v E E” fl E”, contradicting and EU. Now for every x E ES(O) we shall show that IlX(t, t’)xll I D exp[-a(t

- t’)]

the definition

(t L t’)

where D and CYare positive constants independent of t and t’. Suppose t&(t), sequence of continuous functions having the following properties: (i) lpk(t)I I 1 for all < E IR, pk(t) = 1, t fixed; (ii) Pk(O = 0 forIt - tl 1 l/k. Let x belong to ES(t - n). We have

IIwh4’ 5 dIlPk4I where v E ES, v(t - n)

= x. Suppose

of ES

k E tid) is a

(53)

that a, b are those constants which satisfy

alI4 5 llvll’5 bllvll

(vE G(R XI),

then (54) hence, II Tnpkvll

i

@/a)dl

Pk

(55)

d.

Note that a and b are independent of t and n. We have

sup Ilx(tyt -

IE-tb s l/k

n)pk(t

- n)v(< - dl

5

(b/a)q” ,,_;;“,p,,k

II Pk(t)v(t)\l.

By definition, we see that

IlW, t - Wll 5 (b/4q”llxll,

(56)

as k + co. Note that T’ is hyperbolic for every rational number t E iR and leaves ES and E” invariant. In addition a( T’ lES) and o(TWfjEu) are contained in the open unit disk for t > 0. Moreover, we have E”(t) = X(t)E”(O),

E’(t)

= X(t)E”(O)

(57)

217

Exponential dichotomy-I

for every rational number t E R. From the continuity of X(t, s) it follows that (56) is true for all t E IR and x E X(t - n)E”(O). So, the estimate IlX(t, t’)x[I = IlX(t, t - [t - t’])X(t - [t - t’],s)xll I (b/a)q[‘-f’lIIX(t holds for all t, t’ E R with t 2 t’ and all x E iY(t’ of t’ - t. Taking into account (17) we have

- [t - t’],s)xll

(58)

where [t - t’] denotes the integer part

IlX(t, t’)xJI 5 (b/a)q[‘-“I

exp[K] llxll.

(59)

Finally,

IW, Oxll 5 D ev[-4

- 01 llxll

(60)

where D = (b/a) exp[K], 01 = -1n q. Similarly, we can show that IlX(t, t’)xll 5 D’ exp[-a(t’

- t)]l/xll

(61)

for t < t’ and x E X(t’)ES(0). Combining (17), (60) and (61) we can show that X(t, s), t, s E IR has an exponential dichotomy (see [l, pp. 233-2371). This completes the proof of lemma 6. Combining the above lemmas we arrive at the following theorem. THEOREM 2.

Assume that dim X < 03. Then the flow X(t, s), s, t E R generated by (9) under the condition (IO) has an exponential dichotomy if and only if T’ is hyperbolic. Now we are in a position to prove the main result of the paper.

THEOREM 3.

Let dim X < cc). Suppose that the operator given in (9) satisfies the condition (IO), and that the flow X(t, s) generated by (9) has an exponential dichotomy. Then there exists a positive constant E such that every equation x(t) - X(T) =

I=iir,tX,

(62)

where

also has an exponential dichotomy. Proof. It may be noted that (62) satisfies all conditions of the existence and uniqueness theorem. Denoting by &t, s), t, s E R the corresponding flow, we have IlTqt + s, t)x - z(t

+ s, tjxll

= llrIt,,+,X(t

+ s, t1-Y-

fiit,t+s~ct +s, oxll

5 IInt,t+,m + s, ox - m,t+At + s9oxll + IInt,t+& + s, ox - fiit,t+sJw+ s, oxll 5 llnt,t+s (X(t

+ s, t)x - 2(t

+ &SK exp[(K + e)sl llxll.

+ s, t)x)ll

(64)

J. APPELLef al.

218

Taking into account (19) and applying Gronwall’s inequality we get IlX(t + S, t)x - 2(t + S, 0x11 4 &SK exp[(K + e)~]I/xll.

(65)

Consequently, Il~(t, t - 1) - &t,

t - l)ll 5 EK exp[2K + E].

(66)

In other words, IIT’ - F’II I EK exp[2K + E]. Finally, if E is sufficiently the assertion.

small then f* is hyperbolic.

(67)

Now applying theorem 2 we obtain

REFERENCES DALECKIIJu. L. & KREINM. G., Stability of Solutions of Differential Equations in Banach Space. Nauka, Moscow (1970) (in Russian). DUNFORDN. & SCHWARZJ. T., Linear Operators. General Theory. Interscience, Leyden (1963). NITECKIZ., Differentiable Dynamics-an Introduction to the Orbit Structure of Diffeomorphisms. The MIT Press, Cambridge, Massachusetts (1971). KATOKA. B., Dynamical systems with hyperbolic structure, 9th Summer Mathematical School, Kiev, pp. 125-211 (1972) (in Russian).