Countable Systems of Differential Equations
COUNTABLE SYSTEMS OF DIFFERENTIAL EQUATIONS
A.M. Samoilenko and Yu.V. Teplinskii
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Utrecht · Boston, 2003
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Contents
PREFACE
vii
1. GENERAL CONCEPTS OF THE THEORY OF INFINITE SYSTEMS OF DIFFERENTIAL EQUATIONS 1.1. Theorems on Existence and Uniqueness of Solutions 1.2. Truncation Method 1.3. Solutions of the Linear System 1.4. Matrizant of a Linear System 1.5. Normal Autonomous Systems 1.6. Periodic Solutions
1 1 11 15 20 27 32
2. INVARIANT TORI 2.7. Green Function 2.8. Existence of a Smooth Invariant Torus 2.9. C^-Differentiability of the Invariant Torus 2.10. The Case of Infinitely Many Angular Variables 2.11. Theorem on Convergence of the Sequence of Invariant Tori . . . . 2.12. Invariant Tori of Nonlinear Systems 2.13. Exponential Attraction of Motions in a Neighborhood of the Invariant Torus of a System of Equations to Its Motions on the Torus
49 49 68 75 99 115 119
3. REDUCIBILITY OF LINEAR SYSTEMS 3.14. Erugin and Floquet-Lyapunov Theorems 3.15. Periodic Systems 3.16. Systems with Almost Periodic Coefficients 3.17. Quasiperiodic Systems with Unbounded Right-Hand Side 3.18. Decomposition of Countable Systems
151 151 155 163 175 189
ν
133
vi
4. IMPULSIVE SYSTEMS 4.19. Some Results of the Theory of Linear Systems 4.20. Integral Sets and Invariant Tori 4.21. Periodic Solutions for Impulsive Systems with Small Parameter 4.22. Approximate Solution of the Periodic Problem of Control
REFERENCES
197 197 207 231 252
271
PREFACE
The present book is devoted to the solution of various problems in the theory of differential equations in a space T t of bounded numerical sequences (called countable systems). In particular, we deal with the general theory of countable systems, the theory of oscillating solutions, and the theory of countable systems with pulse action. Our main attention is given to the generalization of the results of numerous authors obtained in recent years for finite-dimensional systems of differential equations to the case of systems from the analyzed class. It is clear that countable systems form a class of systems of differential equations in Banach spaces and, thus, many issues of the theory of equations of this sort are well known. At the same time, differential equations in the space 9JI have many specific features, which caused the necessity of development of the theory of these equations originated by A. N. Tikhonov and K. P. Persidskii [Tik, Perl, Per2]. As a result of the systematic investigation of countable systems, K. P. Persidskii [Per4] proved theorems on the existence and uniqueness of a solution of the Cauchy problem, developed methods for their approximate solution, and created the theory of stability of these systems. Extensive subsequent investigations of these systems were carried out by O. A. Zhautykov and K. G. Valeev [Zhal-Zha3, VaZ]. A great number of problems connected with the development of asymptotic methods and the investigation of oscillatory processes was solved by the followers of the scientific school of Ν. N. Bogolyubov and Yu. A. Mitropolsky. The proposed monograph is based on the original results obtained by the authors in [Satl-Sat8, STA, STL, STT1, STT2, Tepl-TeplO, TeAl-TeA3, TeT, TeL] and consists of four chapters. The first chapter is devoted to the presentation of the general concepts of the theory of infinite systems of differential equations and is partially based on the results obtained by K. P. Persidskii. vii
viii
Preface
The second chapter deals with the theory of covariant toroidal manifolds of linear and nonlinear countable systems similar to the theory of extensions of dynamical systems on a torus developed recently for finite-dimensional systems by Yu. A. Mitropolsky, A. M. Samoilenko, and V. L. Kulik [MSK2, SamlO, Sam8, SaK2, SaK3]. The third chapter contains results on the reducibility of linear systems with periodic and quasiperiodic coefficients. In the fourth chapter, the theory of pulse systems [SaPl-SaP3, Per] is generalized to the case of countable systems of differential equations. Throughout the book, we use double enumeration of formulas and statements (definitions, lemmas, theorems, etc.). The first number is the number of a given section and the second number is the ordinal number of a formula or statement (definition, lemma, theorem, etc.) in this section.
The Authors
1. GENERAL CONCEPTS OF THE THEORY OF INFINITE SYSTEMS OF DIFFERENTIAL EQUATIONS
In Chapter 1, we present various facts from the theory of countable systems of differential equations used in the subsequent chapters. The major part of these facts can be found in the works [Perl-Per4, Tik], which are well known to the experts in various fields of mathematics, physics, and engineering dealing with systems with infinitely many degrees of freedom. At present, there are several directions of investigation of these systems, including their general theory, systems of partial differential equations, characteristic numbers and the stability of solutions, averaging, differential equations in normed spaces, "multiperiodic" solutions, etc. [Bro, VaZ, Gor, DaK, DaKr, Erm, Zhal-Zha3, Res, Khal, Kha2, Kha]. In our book, we present only some results from the theory of systems of ordinary differential equations: theorems on the existence and uniqueness of solutions, properties of solutions of linear and nonlinear systems (including conditions for the continuous dependence of solutions on the initial data), and matrizants of linear systems and their properties. Moreover, we consider analogs of some properties of normal autonomous systems [Pon] and present the fundamentals of the numerical-analytic method aimed at finding periodic solutions of nonlinear countable systems [SaR].
1.1. Theorems on Existence and Uniqueness of Solutions Consider a system of differential equations of the form = fi(x,yi,y2,---,yn,···), ι
»= 1,2,....
(l.i)
2
General Concepts of the Theory of Infinite Systems
Chapter 1
Definition 1.1. A function / ( y i , y2, · • ·) is called strongly continuous i f , for any ε > 0, there exist No and δ > 0 such that the inequality \y[ — y"\ < δ, i — 1 , 2 , . . . , No, implies the estimate \f(y'i,y'2,···)-fivlvl
•••)]< e.
Theorem 1.1. Assume that the right-hand sides of the system of equations (1.1)
(a) are defined for any Vi e R\ i — 1 , 2 , . . . , and all χ G Tq = [xq, xq + a] C Ä1; (b) are strongly continuous in yi, y2, • • • for fixed χ and measurable in χ for fixed yi, i = 1 , 2 , . . . ; (c) satisfy the inequalities \fi(x,yuV2,
···)!
oo, is also uniformly bounded and equicontinuous and, hence, it also contains a convergent subsequence This process can be continued infinitely. We compose the table y[ai\x)
y^\x)
···
y{ e ->(s)
.··
y^\x)
y^\x)
.·.
y^\x)
···
y^\x)
y^\x)
·..
y^\x)
···
and rewrite the set of sequences row by row: y[ai\x)
y) ^ H· It is easy to see that the series ys(x) =y°s + (x + x0)y°s+l
+ ^
y°s+2 + ... ,
s = l,2,... ,
define the required solution. Thus, in particular, the solution of this sort that passes through the point (0; 0 , 0 , 0 , . . . ) e Η has the form ys{x) = 0, s = 1 , 2 , . . . . Consider a function
(θ,
χ = 0.
/ χ dsf{x) The vector yis(z) = — s , s = 1 , 2 , . . . , is also a solution of the system of dx equations (1.5) that passes through the point (0; 0 , 0 , 0 , . . . ) € H. The presented solutions have the same initial conditions but do not coincide, i.e., the theorem does not guarantee the uniqueness of a solution for certain initial conditions. The following theorem is presented here without proof: Theorem 1.2. Let the system of equations (1.1) be such that oo 1- | / η Ο , 2 / ι , · · · , 2 / ή > · · · ) - / η ( ζ , ! / ΐ , · · · , ί / η , · · · ) Ι < Σ Kni\y'i ~ y"\, Tl = 1,2,... ;
i=l
Section 1
Theorems on Existence and Uniqueness of Solutions
1
00
2. Σ Κ™ — Ai < A < oo, Kni = const > 0, A — const > 0 , n=1 const > 0 , ζ = 1,2,
Ai
=
77ien exists at most one solution (y\(x), 1/2(2), • • •) of the system of equations (1.1) satisfying given initial conditions and such that 00
< Β — const < 00 .
y^ \yn(x)\ n=l
We now consider a domain h defined by the inequalities 0 < χ < r, \ys\ < R°, s = 1 , 2 , . . . , and the domain Η defined in the example considered above. Assume that the right-hand sides of the system of equations (1.1) in the domain Η satisfy the following conditions: 1*. The functions fo, i — 1 , 2 , . . . , are continuous in χ at any point, i.e., fi(x + &x,yi,y2,...)
- fi(x,yi,y2,...)
-»· 0
as
Ax - » 0.
2*. The functions fi, i = 1 , 2 , . . . , satisfy the Cauchy-Lipschitz condition in variables y \ , y 2 , . . . , i . e . ,
\fi(x,y[,y'2,···)
- fi(x,y",V2,···)]
< 0 there exists a number g' — g(u) such that es(g') < v. We set g — g'. One can also find a number N(s, v) such that J'sn < α sup{|ui(x) - vltn(x)\,...,
\ug(x) - fg,n(z)|}
< ai/
for η > N(s, u). Then, for η > N(s, u), we have Jsn < (2αη + a)ν for any χ e σ. This means that, for any χ e σ, X
X
lim η—»oo Consequently, χ Us(x)=y°s+
J /e(r,ui(r),u2(r),...)dr,
s = 1,2,..
This yields us(x) = ys(x), s = 1 , 2 , . . . , i.e., for any fixed natural g, max(sup{|yi(:r) - f i , n ( x ) | , · · . , | y g { x ) IfciT
0
as η —» oo, which contradicts inequality (2.9). Theorem 2.1 is proved. Corollary 2.1. Assume that σ C [0, r] and the quantities XQ and σ are chosen so that the quantity 7 satisfies the condition 7 < R. Then Theorem 2.1 is true in the domain h.
Section 3
Solutions of the Linear
System
15
1.3. Solutions of the Linear System Consider a linear system of equations of the form dxs
(3.1)
dt
i=1
satisfying the following conditions [called conditions ( P ) ] : In the domain H, the functions pSi(t), p8{t) = \Psj(t)\ are continuous for all s, i = 1 , 2 , . . . , and satisfy the inequalities ps(t) < a(t), where a(t) is a continuous function of t. Assume that a sequence of vector functions (xn
(t)\
X21 ((r)||dr,
m
to
where we have introduced the following compatible norms of vectors and matrices: ||χ(-)(ί)||=8υρ{|^(ί)|,...}
and
||P(i)|| = sup £ i
\Pij(t)\.
3=1
By using the Gronwall-Bellman inequality, we arrive at the formula t
||x (m) (i)ll < c ^ e x p j J a ( r ) d r | } ,
tea,
m = 1,2,....
to
Consequently, (m)
X
(t)| < cß exp|| J o ( r ) dr |}
(3.4)
ίο for all t e σ. In expression (3.4), we fix the values of s and t. Without changing the absolute values of c\,..., Cm, we choose them to guarantee the validity of the inequalities cgxsg(t) > 0, g = 1 , 2 , . . . , m. Then ^m
Σ
Ν
t.
\ ^ c/3exp| / α(τ to
i = 1
whence
™
ι
t
< 0exp{ / a(r)dr|}. i = 1
(3.5)
to
Since the quantities s and t were chosen arbitrarily, inequality (3.5) is true for all s = 1 , 2 , . . . and t 6 σ.
Section 3
17
Solutions of the Linear System
By virtue of (3.5), we obtain
} M t ) | < Σ
Ν
x
\ si(t)\
i=1
< c/?exp{ / a ( r ) d r | } , xo
5 = 1,2,...,
and, consequently, t ||x(i)|| < cßexp|
Jα(τ)άτ
|},
t e σ.
(3.6)
This means that series (3.3) are absolutely convergent and, in view of the well-known Lebesgue theorem, can be integrated term by term. This yields m oo Y,cg(j2psk(t)xkg{t)) 9=1 fc=l
=
J2cgx'sg(t)
1= 1,2,...,«,....
(6.4)
Assume that the right-hand side of system (6.4) is defined for any real t and all χ = (χι, X2, • • ·) from a domain D. Moreover, for any fixed t and χ from their domains of variation, the function f(t,x) = (fi(t, χ), /2(i, x), •..) is a point of the space Wl. We represent the system of equations (6.4) in the form of a differential equation in the space SOt:
dx
— = /(*,*).
(6.5)
The solution of (6.5) is understood as a function x(t) = (xi(t),x2(t),...) defined for t from an interval (a, 6), continuously differentiable with respect to t in this interval, satisfying equation (6.5), and belonging to the space DJl for all
t
g ( a , 6).
Assume that the function f(t,x) is a continuous periodic function of t with period Τ that satisfies the inequalities
\f(t,x)\
< M, \f(t,x')-f(t,x")\ · · • is defined by the recurrence relations t
awt-i(f)
=
(l
-
~ ψ ~ )
τ+Τ
J
J
τ
t
(6.9)
Oim(s)ds,
m = 0,1,2,... . Then there exist positive
where
πι
2.
and ( ^
constants
) are, respectively, the integral and fractional parts of the
number —, such that
am+i(t)
/. By induction, we conclude that, for all m > 0, ( t , r ) G (—00,00) x (—00,00), and xq G D / , the functions xm(t,T,x0) belong to the set D. We now estimate the expression r m + 1 = |Zm+l(i,T,Zo) - Zmit.T.zo)!·
(6.24)
In view of inequalities (6.6) and relations (6.21), we obtain t
J
rm+1 (t) < ( l -
T+T
J
Krm dt +
Τ
Κrm
dt.
t
We set rm(t) Then, for am(t),
= K^MaUt).
(6.25)
we can write t
<Xm+i(t)
1 and ί G [τ, r + Τ]. Since, by virtue of (6.21), ai(i) 0 and t G [τ, τ + Τ}. Taking into account relations (6.25) and (6.27), we finally get r™+i (t) < K m M T m g r o 2 ( i - r ) ( l for m > 0 and τ < t < τ + T.
(6.28)
Section 6
Periodic
Solutions
39
By using the estimate for the quantity qm deduced in Lemma 6.1 and inequality (6.28), we obtain
Μ
rji \xm+k(t,T,x0)
- Xm(t,T,X0)\\
K—l (6 29)
2. Inequality (6.29) implies that the sequence xm(t, r , xo) converges to a function Xoo(t, τ, xo) uniformly for all t,r,xο from domain (6.22). The periodicity of the function χ^ί,τ,χο) in t follows from the periodicity of the functions Xm(t,T, Xo). Passing to the limit in equality (6.21) as m —> oo, we see that x<x>{t, τ, xo) satisfies equation (6.23). Lemma 6.2 is proved. We now estimate the deviation of the function Xoo{t, τ, xo) from xn(t, r , xo)· It follows from equation (6.23) that ι / χ , \Xoo{t,T,Xo) - Xo|
2, inequalities (6.28) and (6.29) imply that
whence it follows that (6.30)
40
Chapter 1
General Concepts of the Theory of Infinite Systems
for all η > 0, t, τ e (—oo, oo), and XQ G D — symbol:
0
for for
MT
Here, δ\η is the Kronecker
η φ 1, n = 1.
(
Theorem 6.1. Any point (τ, :ro) £ (oo, oo) χ ( D
MT\ — 1 has α Δ -constant
relative to a Τ-system specified in the domain D of the space VJl. Proof. Since the right-hand side of a T-system, i.e., the function f(t,x), satisfies the conditions of Lemma 6.2, the sequence of periodic functions (6.21) converges uniformly to the function Xoo(t, r, xo)· We set Λ(τ,χο) = f(t,Xoo(t,T}X0)). (6.31) Let us show that this function specifies the Δ-constant at a point r, XQ relative to a T-system defined in the domain D. To prove the last statement, it suffices to show that the equation dx
= f(t,x)
~dt
- u,
(6.32)
ueVJl,
has a T-periodic solution χ = x(t,r,x o) such that χ (τ, τ, χ ο) = XQ for a single value of the parameter u. Assume the contrary. Then there exist constants u\ and u2,u\ φ U2, such that the solution x(t, r, XQ, U) is periodic for U = u\ and u = 1*2For the difference x(t, r, £0,^2) — x{t, τ, XQ,UI), we obtain the identity t x(t,u2)
- x(t,ui)
= J{f(t,x{t,u2))
-
f{t,x(t,Ui))
+ f(t, x(t, Ui)) - f(t, x(t, u2))) dt.
(6.33)
This yields t |a;(i,U2) - x(t,ui)\
dt
< Κ τ
+
t - T
r+T J
\x(t,u2) -
x(t,Ui] \dt
(6.34)
Section 6
Periodic Solutions
41
and, finally, Ix(t,u2)
- x(t,ui)\
< Κnan(t)
Ix{t,u2) -
®(ί,«ι)|ο,
where η = 0 , 1 , . . . , \x(t, u2) — x(t, ui)|o = sup \x(t, u2) — x(t, wi)|, and an(t) t is a positive function defined by relations (6.26). By using the estimate for an(t), we obtain j^nrpn 'j* |a?(i,«2)-ar(i,«i)| < j \x{t, u2) - x(t, u i ) | 0 . At the same time, in view of the fact that the operator Q is completely regular, we can write \x(t, u2) - x(t, «ι)|ο < qn which is possible only for x(t,u2) rem 6.1.
Τ
"2) - x(t, ui)|0 ,
= x(t,ui).
0 < q < 1,
This contradiction proves Theo-
By using Lemma 6.2 and estimate (6.30), one can easily prove the following corollary of Theorem 6.1: Corollary 6.1. The solution χ — x(t,r,x 0) of a Τ-system passing through a MT point XQ Ε D — = Df att — τ is T-periodic if and only if the A-constant L· at the point r, XQ is equal to zero. This solution x(t, r, xo) satisfies the identity
x(t,r,x
0) =
Χοο^,Τ,Χο),
where Xoo{t, r, xq) is the limit of the sequence of periodic functions (6.21), and the following estimate is true: II x(t, τ, x0) - χn(t, τ, x0) II < qn
+
where η > 0 and t G (—00,00). This corollary relates the problem of the existence of periodic solutions of a T-system to the problem of zeros of the function Δ = Δ (τ, xo) and the problem of finding these solutions to the evaluation of the functions xn(t, r, XQ). The periodicity of the indicated functions reduces their evaluation to the determination of definite integrals and, hence, allows one to use computers.
42
General Concepts of the Theory of Infinite Systems
Chapter 1
The functions x n (£, τ, χo) can be especially easily found in the case where f(t,x) is a polynomial function with respect to χ whose coefficients are trigonometric polynomials. In this case, the expressions χ η ( ί , τ , χo) are also trigonometric polynomials and their evaluation reduces to elementary operations, such as the operation of multiplication of polynomials. If the functions xn(t, τ, XQ) are already determined, then we can find the Δ constant at the point R, XQ. Indeed, it is clear from the proof of Theorem 6.1 that the Δ-constant at a point (τ, xo) £ (—1oo, oo) χ Df is determined by using the formula Δ ( τ , ζ ο ) = f{t,Xoo(t,T,xo)). (6.35) If we substitute the function xn(t, r, xo) for XOO(T, τ, we get Δ η ( τ , χ ο) = f(t,xn(t,r,:c0)).
XQ)
in relation (6.35), then (6.36)
This function is regarded as the nth approximation to the Δ-constant. It is easy to see that Δ η —• Δ as η —» oo and, moreover, ||Δ(τ,χ0) - Δη(τ,*0)|| < ^ q
n + 1
+
M.
(6.37)
By using the computed values of the Δ-constant, one can solve, in many cases, the problem of the existence and location of points at which the Δ-constant is equal to zero and, hence, to solve the problem of the existence and determination of periodic solutions of the T-system. Thus, in finding the location of points at which the Δ-constant is equal to zero, one can use the following statement: Lemma 6.3. Let Δ'η(τ,χο) the Δ-constant
be the already calculated nth approximation to ( MT λ Δ (τ, xo) at the point (τ, xo) £ ( - o o , oo) χ ID —J — Df
and let | | Δ η ( τ , ζ 0 ) - Α'η(τ,χο) I f , in addition, a closed domain D\ C D
|| 0, 3.1 |A'„(r,*o)|| < , ( l +
q
, 3.1 Λ +
d +
3.1 „ + 1 ( I
d = sup ||x — xo||. xeDx
(„ + ^ )Μ +
Section 6
Periodic
43
Solutions
By using this lemma and the values of Δ-constants computed at finitely many points, one can select a subset of D\ containing all points at which the Δ-constant can be equal to zero and, hence, specify the set containing all possible points through which periodic solutions of the T-system can pass at t = τ. We now proceed to the proof of Lemma 6.1. Assume that the Δ-constant is equal to zero at a point r, x', x1 £ D\. Then ΙΚ(τ,χο)||
H i -
holds on the boundary Γ/^ of the domain D\.
+
l i ) Μ
(6.48)
46
Chapter 1
General Concepts of the Theory of Infinite Systems
Then Τ-system (6.47) possesses a Τ-periodic solution χ = x(t) such that χ (τ) e Di. For η = 0, one has Δο(χ) = f(t, x), and Theorem 6.2 establishes the existence of periodic solutions of system (6.47) relative to the averaged system
Proof. By virtue of estimate (6.37) and Lemma 6.4, the set
is contained in AD\. If this set contains the origin of the coordinate system Δ = ( Δ 1 , . . . , Δ η , . . . ) , then the set AD\ also contains the origin of coordinates. This property is sufficient for the T-system to have T-periodic solutions. Therefore, to prove the theorem, it suffices to show that 0
e
-
(_1_
+
M
.
Since the mapping An(x) is topological, the set AnD\ is a domain and, moreover, the boundary ΓauDi of the domain Δ η £>ι is the image of the boundary of the domain D\\ r AnD, = Δ
η
ΐν
(6.49)
By virtue of condition (i) in Theorem 6.2, the set Δ η £>ι contains the singular point of the system Δ = ( Δ 1 , . . . , Δ η , . . . ) . This point belongs to the set AnD\ — r if the distance between it and all points 2 of the boundary Γδ„0ι ° f the set AnD is not smaller than r. This means that the origin belongs to the set
whenever
( ϊ ^ + w) Μ·
IWI s
By virtue of relation (6.49), we get the identity An(x)
ζ
Section 6
Periodic Solutions
47
Therefore, inequality (6.50) takes the form j
n
i
^
W
I
I
S
^
^
+ ^ M .
(6.51)
This enables us to conclude that the T-system possesses T-periodic solutions, provided that inequality (6.51) is satisfied. Theorem 6.2 is proved.
2. INVARIANT TORI In the present chapter, our main attention is focused on the investigation of the invariant toroidal manifold of a countable system of differential equations either in the space DJl of bounded number sequences or in the products VJl χ Tm or ÜJI χ Too, where Tm and Too are ra-dimensional and countably-dimensional tori, respectively. In our presentation, we mainly deal with conditions for the existence of invariant toroidal manifolds for linear systems in SDT χ Tm and ÜJl χ Too, the perturbation theory of these manifolds for nonlinear systems, the properties of smoothness and stability of these manifolds, and the behavior of trajectories in small vicinities of invariant toroidal manifolds. Whenever possible, we try to reduce the analyzed problems to the case of finite-dimensional systems of differential equations with increasing dimensionality. The problems discussed in this chapter were studied in numerous works, among which we especially mention [SamlO, MSK2, BMS, Kull, Kul2, MiL, MiK2, Saml, Sam2, Sam5-Sam9, Saml3, S a K l - S a K 3 , SaPl].
2.7.
Green Function
Let ÜJI be the space of bounded number sequences χ = (x\, £2, · · ·) equipped with the norm ||x|| = s u p n { | x n | }. The metric induced by this norm turns Wl into a Banach space. Consider an infinite matrix A = 00
sup Σ
i
1
[aij]fj=i
such that
I CLij I = a < 00. It is easy to see that the operation of multiplication of
the matrix A by χ e 9JI specifies a linear bounded operator in the set DPI whose norm is equal to a . Indeed, ||Λχ|| = sup
whence
Σ
k=1
a
ikXk
< sup Σ
k=1
I|j4x|| . .. < a for ||a;|| ψ 0,
49
I aik I I xk I < a s u p | xk \ ,
50
Invariant Tori
Let us show that a =
sup IWI/o
\\Ax\
ber ε > 0 and find an element
ζ
Chapter 2
. We choose an arbitrarily small real num-
e 9Jt such that
\\Ax°\ χυ
> a - ε.
Obviously, there exists a number ρ such that oo
Σ IaPkI
>α~ε·
fc=l
We choose x° = (x®, x®, • • •) such that | | = | x\ \ — moreover, | apk x^ \ = apk x\ for k = 1 , 2 , 3 , . . . . Then sup i UCL
oo Σ aikx k=1
>
Χ Λ
Σ I apk k=1 U? I
=
= I x° I = - . . . and,
oo Σ a pkx k=1 x\
Σ \aPk k=l
>
a —ε.
This operator norm is, at the same time, the matrix norm associated with the vector norm in the space VJt. Consider a system of differential equations of the form (7.1) where ψ — (φι,...,
