On approximation of the solutions of linear differential equations in Banach spaces by a difference process

R O C Z N I K I P O L S E I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a : I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

Kr z y s z t o f No w a k

(Warszawa)

On approximation of the solutions of linear differential equations in Banach spaces by a difference process

In this paper we shall consider the difference equation

(*) x ( h + i ) - x ( h )

tjc

■A {t]c+l)®(tk+l) ?

where, for any t, A(t) is a linear, not necessarily bounded operator from a fixed Banach space X into itself.

Krein in [

] (chapter Y, §4) demonstrated, that under assumptions of Kato’s theorem (see p. 188 of this paper) the values of the solution of this difference equation approximate the solution of the differential equation

(Î) dx(t)

dt A(t)x(t).

Here we give a new demonstration of this theorem and, moreover, we prove that difference quotients on the left-hand side of (*) approxi­

mate the derivative of the solution of (*). The mean idea of reasonings was suggested to the author by J. Kisynski.

1. Fundamental solution of equation of the type dx(t)jdt = A(t)x(t).

Let X be a Banach space. For every te <

, T> let A(t) be linear operator, Rot necessarily bounded, with domain B(A{t)) and range R(t) contained in X and B[A(t)) dense in X.

We shall assume that B(A (<)) = Y is independent of t.

Consider the Cauchy’s problem

(1.1)

dx{t) dt a?(

)

= A(t)x(t)

= a>0,

for te (0, T } ,

where x0e X is given and x( ), the solution, is an X-valued function.

We call a fundamental solution of equation (*) two-parameter family

188 К. N o w a k

U{t, s), О < s < t < T, of linear bounded operators of the space X into itself, which satisfies the following three conditions:

(a) U(s, s) = 1 for 0 < $ < T,

(b) for every x e X the function (t, s) -> U(t, s)x is continuons in the sense of the norm in X on the triangle

AT = {(t , s ):

(c) U(t, s) Y a Y for any (£, s)e AT and if xe У, then the function {t, s) -> U{t, s)x is strongly continuously differentiable on AT and the following formulas hold:

— U(t, s)x = A{t) U(t, s)x, d

—— U(t, s)x = —U(t, s)J.(s)tf.

OS

The famous theorem of T. Kato concerning Cauchy’s problem of type (

.

) is as follows.

Th e o r e m

1 (T. Kato [1]). Let X be a Banach space, let t ->A(t) be a function defined on <

, T } which values are the infinitesimal generators of one-parameter strongly continuous semigroups of operators belonging to J£(X, X) (see for instance [1], chapter 1) such that

(

.

) ||exp(£A(i)')|| < eK* for Çe(

, +oo), where К is a constant independent of t.

Suppose that D{A(t)) = Y is independent of t, and that for every x e Y the function t - > A ( t ) x is strongly continuously differentiable on <

, T').

Under these assumptions there exists a unique fundamental solution of equation (*). I f x0e Y , then x(t) — U(t, 0 )x 0 is the unique strongly continu­

ously differentiable solution of (

.

).

2. Approximation of the fundamental solution. Under assumptions of Theorem 1 let tn(t), n =

,2 , .. . , be a sequence of functions piecewisely continuous on

, T ) , such that

< tn(t) T and tn( t ) - >t uniformly on <0,T>, as n -> oo. Let, for every n = 1 , 2 , ..., dn(tl s) and en(t, s) be functions defined on the triangle AT, with values in / о , — where К is the constant occurring in (1,2), such that ' '

(

.

) for every n =

,

, . . . and for every fixed tfe<

, T} the functions-

s - >dn(t, s) and s -> en(t, s) are piecewisely continuous on <

, t )T

(2.2) for every n = 1 , 2 , . . . and for every fixed se <0, T} the functions t -> en(t, s) and t -> dn(t,s) are piecewisely continuous on <e, T>, (2.3) lim sup en(t,s) = limsup dn(t,s) = 0

oo /dT n-yco AT

Assume that for every n = 1 , 2 , . . . there exists defined on AT strongly continuous function (t , s) Un(t, s) with values in L£ {X, X) such that

(2.4) Un(s, s) = 1 for every se <0, T>, (2.5) Un( t , s ) Y c Y for every (t , s ) e A T,

(

.

) for every ха X and for every fixed se ( 0 , T) the function r Un{t, s)x has the strong derivative piecewisely strongly continuous on <s , T'y and such that

— Un( t, s )x = A(tn {t))\L-en{ t,s)A(tn{t))]~ 1 TJn{t,s)x at all the points of its continuity,

(2.7) for every x e Y and for every fixed te <0, T} the function s -> Un(t, s)x has the strong derivative piecewisely strongly continuous on

, ty and such that

— Un{ t ,s ) x = - U n{t,s)A(tn{ s ) ) [ l - d n(t, s)A(tn{s))]~1x at all the points of its continuity.

Under the above assumptions the sequence of operator-valued func­

tions {t, s) -> TJn{t, s), n =

,

, . . . , is strongly and uniformly on AT convergent to the function {t, s) -> U(t, s).

Remark. Under assumptions of Theorem 1, by Hille-Yosida theorem (see [3], p. 249), for any A > К and te <0, T> the operator (Al —A

exists and belongs to ü?(X, X). Moreover, we have (2.8) ||(A 1-A (i))-

| | < ( A - A

for every A> K andie<0, T>. It follows, that in conditions (2.6) and (2.7) the operators (l — ek{t, s)A(tn(t ))])~1 and (l — dk{t,s)A(tn(s )))~1 have sense and belong to <£(X,X).

Pr o of of Theorem 2. We have

Un{ t , s ) x - U { t , s ) x = - U n(*, r)t7(r,s)a;|^

because TJn{t,t) = TJ{s, s) = 1. Since the functions (t, r) -> TJn{t, r)

and (r,

-> TJ{r,s) are strongly continuous and for fixed t, s and x0€ Y

190 К . N o w a k

the function r -> Un(t, r) U(r, s)x is piecewisely smooth it follows that if X€ Y, then

Г г=(

(2.9) Un{t, s)cc— U(t, s)x = —Un{ t , r ) ü { r , s ) æ

= — J * Э r) U(r, s) x]dT

s

u n(t, r) — Щr, S)æ + - — UJ t , r)U(r, s)x\dx dt

= ~ j [Un{t, r ) A{ r ) U{ r, s ) x - U n{t, r)A(tn{x)) x

a

x ( l — dn{t, r)A(tn(T))}-l U{T, s)x]dr t

= J Un(t, r)[A(tn{ r ) ) [ l - d n(t, т)А(гп(т)))-1 - Ai r] \ U{T, s) xdr

S t

=

J

S

*

+ J ^»(<, r)\A(tn{T) ) - A( r) ] U( T, s)xdr.

In order to estimate the last two integrals we shall need the following

Le m m a 1. ||

Un{t,

C for

t , s ) e A T and n =

where G is a constant independent of t, s and n.

Pr o o f Let s e<0, T> and x e Y be fixed; for t e ( s , T } put y{t) = e- 2E(i- s) Un(t, s)x, (p{t) = ||y(«)ll . .

- A n(t) = A(tni t ) ) ( l - e nit, s)A(tnit)))~1.

The function y it) as piecewisely smooth, satisfies the Lipschitz condition. Therefore <рЦ) satisfies the Lipschitz condition too. At any point te <s, T), at which the derivative y'it) exists, i.e. for every te <s, T>

except of a finite number of points, we have y i t - h ) = —hy'{t) + y{t) + o{h)

= —h ( —2Ky{t) + A nit)yit)) + у it) 4- o(h),

so that

We have

\\y(t-h)\\ > h 4- — I y ( t ) - A n(t)y(t)

M M ) M M )

+ o(h).

( l ~ e n(t, s)A(tn{t )))~1 and hence, by (

.

),

\ \Xx-An(t)x 11 > Ш --- -i-

II ^\ en{t, s) en{t, s) 7

—

l x\

[ A + M M ) en( t , s ) - K e 2 n (t,s)\

h i - ^ ( M ) ] M > ( A ~ 2 g ) M -

Therefore, for every t e ( s , T y except at most of a finite number of points,

<p(t-h) = ||y(J —й)|| > й

(2£ + т )

/ ( M A n(t)y(t) +

so that

(

2

10

0 (h) ^ §у ( Щ 0 {h) — <p(t)-\-o(h)

limsup A

<p(t)-<p(t-Jl) h

Because <p (t) satisfies the Lipschitz condition, its derivative exists almost everywhere and, by (

.

), <p'(t) <

almost everywhere. There­

fore (p{t) is non-increasing, and hence

e- 2 K(t-S)^ = ^ <p(s) = M i.e. |( Un(t, Mil < e2K(t~s) \\x[{

for every x e Y and (t, s) e AT. SinceX is dense in Y, it follows that Lemma

holds with C — e2KT.

We return to the proof of Theorem 2. From (2.9), by Lemma

, we obtain that for every xe Y and (t, s)e AT

(

2

11

\\Un{t, s ) x - U(t, Mil < CT sup ||[(1-<М т, «)-А(«эт,('г)))~

—l] x

0

<s<r<

' ' ’ J xA(tn{r))U(r, s ) x \ \ + C T sup \\[A(tn{ r) ) -A(r )]U(T, s) x\ \ . In order to prove that right-hand side of (

.

) convergas to zero as n -»

, we shall make use of the following criterion of uniform conver­

gence to a continuous limit:

192 К. N o w a k

Let X x and X 2 be Banach spaces and let t- ^ A ( t) and t A n(t), n = 1 , 2 , . . . , be functions defined on a compact set Z <=. Rm with values in Se{X1\ X 2).

Then for every x t X x the X 2-valued function t A {t)x is strongly continuous on Z and the sequence of X 2-valued functions t - ^ A n(t)x, n = 1 , 2 , . . . , is strongly convergent uniformly on Z, to the function t -> A {t)x if and only if for every sequence tn, n = 1 , 2 , . . . , of elements of Z , such that lim tn = t and for every sequence xn, n = 1 , 2 , . . . , of

П —> o o

elements of X x strongly convergent to x the sequence A (tn)xn converges strongly in X 2 to A(t)x.

This criterion implies immediately the following

Le m m a 2.

Let X x, X 2 and X 3 be Banach spaces and let Z a Rn be a compact set. Let t A{t) and t A n(t), n = 1 , 2 , . . . , be Y? (Xx; X 2)- vdlued functions defined on Z and l e t t - + B (t) and t -> Bn(t), n — 1 , 2 , . . . , be ££{X2, X z)-valued functions defined on Z. Suppose that for every xe X x the X 2-valued function t - >A( t ) x is strongly continuous on Z and the se­

quence of X 2-valued functions t ->■ A n(t)x, n = 1 , 2 , . . . , converges strongly, uniformly on Z, to t - > A( t ) x . Suppose furthermore that for every y e X 2 the X z-valued function t -> B (t) у is strongly continuous on Z and the se­

quence of X z-valued functions t - » Bn(t)y , n = 1 , 2 , . . . , converges strongly, uniformly on Z , to t - > B( t ) y .

Then, for every xe X x, the X z-valued function t -> B(t) A(t)x is strongly continuous on Z and the sequence of X z-valued functions t Bn(t)An(t)x,

•n = 1 , 2 , . . . , converges strongly, uniformly on Z , to t B(t)A(t)x.

Now let us treat Y as a normed linear space with the norm (2.12) INI

— IN Ix ~ H I^ (O)llx* °°€ ^ .

Because all the operators A(t), t*. <0, T), as infinitesimal generators of strongly continuous semigroups, are closed, and since they have common domain Y, it follows that Y under norm (2.12) becomes a Banach space.

The operator valued function t -> A{t), by graph theorem, is now an J?(Y; X )-valued function strongly continuously differentiable on <0, T}

and therefore, by Banach-Steinhaus theorem, continuous on <0, T) in the sense of the norm in

S?( Y; X). It follows that for any fixed л > К the J?( Y, X )-valued function t (Al — A(t)) is also continuous on <0, T) in the sense of the norm in j5f(Y; X) and, because then (Al — X^))-1 exists and is defined on the whole X, by closed graph theorem and by an application of the Neumann series, we see that t -> (Al — X(£))-1 is an JT{X; Y)-valued function continuous on <0, T > in the sense of the norm

in Y).

Because U ( t , s ) Y c z Y and U(t, s)e JT(X, X), by closed graph

theorem, ü{t , s ) \Y e Y, Y) for every { t , s ) t A T.

From (b) and (c), it follows that for any x e Y and X > K the X-valued function t, s -> (Al — A(t)) U(t, s)x is strongly continuous on AT.

Because U(t, s)

=

—- A(t)) U(t, s)|F it follows that t, s -> U(t, s)|F is an <S?(Y, Y )-valued function strongly continuous on AT.

By Lemma 2, this implies immediately that

(2.13) for every xe Y the sequence of X-valued functions t, s -> A(tn(t)) U { t , s ) x , n = 1 , 2 , . . . , converges strongly, uniformly on AT, to the Х -valued strongly continuous function t, s -> A(t)U(t, s)x, and consequently the second term on the right-hand side of (

.

) converges to zero as n

Moreover, from (2.13) and from Lemma

we see that, in order to show that first term on the right-hand side of (

.

) converges to zero as n

it suffices to prove that

(2.14) lim sup jj (1 — e A^ f j^ x — a?|| x = 0 for every xe X . ejO

Because, by (

.

)

for «do, L )

in f£{X, X) and because Y is dense in X, we see that it is sufficient to prove (2.14) for every xe Y.

But if xe Y, then

(1

— eA{t))~lx — x =

— eA(t))~l A(t)x so that

|| ( l - e A i t ^ x - x W < ^ - ^ -•|lX(t)a?ll

and, since \\A{t)x\\ is bounded for t e ( 0 , T ) by continuity of t - ^ A ( t ) x , (2.14) follows. This completes the proof of Theorem 2.

3. Convergence of the difference process and some other corollaries to Theorem 2.

Theorem 3.

Under the assumptions of Theorem

let U(t,s) be the fundamental solution of the equation

dx(t)

dt A(t)x(t)

and let xe X. Let 7 tn:

= t(0n) < 4n) < ... < = T, n =

,

, .. ., be a normal sequence of partitions of the interval <

, T'y, such that max (t ^ —

-i) <

IK, where К is the constant occurring in (1.2).

13— R o e z n i k i P T M — P r a c e M a t e m a t y c z n e X V I I

194 К . N о w a k

For every n = 1 , 2 , . . . , let {xn(tl^)? xn(t{n)), . xn(t^)} be solution of the difference equation

(3.1)

Ж 4 ”’ -- (4+l) (4+l) Î ^ -- 0, 1, ..., hn 1 such that

Then

xn( 0) = x .

lim max ||#n(4n)) — Z7(4n)? 0)a?|| = 0.

R em ark. The solution of the difference equation (3.1) exists if max (4n) — t*_i) < 1/-ЙГ, where К is. the constant occurring in (1.2),

k —\,. .. ,k n

and has the form

®»(4+i) = P - ^ i + i - ^ ^ - ^ f c + i ) ) X ( 4 ) for fc = 0 , 1 , . . . , hn — 1.

P ro o f of T heorem 3. From the above remark we see that

®»(4n)) = Cr»(4n)> 0)a?, h = 0, ..., 1 cn, n = 1, 2, ..., where

Un(t, 8 ) = ( 1 - ( « - * ) А ( ^ О ) - 1 if 4n) ^ $ < < < 4+i and

Unit, 8)

(1

(^ir}

^irii)-^

.. .(1 - ( 4 % - 4 ï i M (4?2))_I(1 - (4+! - *) -4. ( 4 1 ) ) '1 if 4n) < s < 4+i and 4n> < t < 4+ij :P > If easy to verify, that for such sequence t, s Un(t, s), n = 1 , 2 , . . . , of*

9?(X; X)-valued functions defined on zJr he assumptions of Theorem 2 are satisfied with

tn(T) = T, tu(t) = 4”>, for и <ф>4"Л).

f < T 4 . \ T ~ s a * * < e . , D ,

“ ’ U - 4 Ï - . и » < < £ - „ I * - * if f>*«<4n)>4+i)>

*" I « - i p ii ^<4"l>4+i)>s < 4 n), d„(T,>) = T - * if s . <4;»-!, T ),

<4(M ) = t — s

f(n) _ о

г&+1 *

if <, S£ <4”, ,4+i)>

if se<4"l, 4 ï . ) , o 4 ? 1

Therefore Theorem 3 follows from Theorem 2.

There are some other sim ilar examples to Theorem 2.

Ex a m p l e

1. Under the assumption of Theorem 1, let лп: 0 =

< . . . < 4 " > — T, n = 1 , 2 , . . . , be a norm al sequence of partitions of the interval <0, T). F o r any n = 1 ,2 , . . . and { t , s ) e A T put

Un{t, s) = exf >( ( t - s) A{ t P) ) if 4T\ < « < t < 4+i»

Un{t, s) = e x p ( ( « -

n)) J .(

n)))e xp ((

№ > -

w>

) J .(

n

1)) ...

... exp ((4+2— 4+i) A (4+i)) exp ((4+i — s) A (tp)) if 4n) < s < 4+i» ^ < 4 + i’ P > k. Then the assumption of Theorem 2 are satisfied w ith tn(t) = 4W) for U (4 ^4 + i) and dn(t, s) = en(t, s) == 0.

Relation (2.8) U { t , s ) = lim Un(t, s) strongly, uniform ly on AT gives,

n — HX>

in this case the famous K a to ’s representation of the fundam ental so­

lution in the form of so-called “m ultiplicative in tegral” .

Ex a m p l e

2. Under assumptions of Theorem 1 let tn(t) = t and dn(t, s)

— en(t, s) = 1 /n. Then, fo r every n > K, where К is the constant from assumptions of Theorem 1, the unique

t, s Un(t, s) satisfying (2.4)-(2.7) m ay be constructed b y solving the integral equation

Un(t,s) = 1 + j A( r) ( l — ^ A (

)\ Un(t, s)dx

S '

by common method of successive iterations. R elation (2.8) in this, case gives the w ell-know n approxim ation form ula of Yosida.

Ex a m p l e

3. Le t A(t) = const = A be an infinitesim al generator of a strongly continuous semigroup of operators belonging to

such that ||exp(L4)|| < MeKt. Let

/ t — s \~n

Un{t, s ) = | l---—— AJ , n = [ Y ] - f 1, [ H ] + 2, ...

Then the assumptions of Theorem 2 are satisfied w ith tn{t) == t and dn(t, s) ÿ_§

==en(t,8) = - --- . From relation (2.8) we obtain in this case the form ula

of H ille

exp(L4) = lim f l---A

n—>oo \ Vt

strongly, almost u n iform ly on <0, oo).

I t must be pointed out, that in above three examples we have proved relation (2.8) b y means of Theorem 2 and therefore assuming the existence

°f the corresponding fundam ental solution U(t, s), because our proof of

196 К . N o w a k

Theorem 2 was based on the theorem of Kato (Theorem 1 of this paper).

On the other hand Hille, Yosida and Kato in these situations have proved directly the convergence of corresponding sequences Un{t 1 s), obtaining in this manner the proofs of existence of the fundamental solution U(t, s).

4. Approximation of derivative of the solution of equation (*).

Th e o r em 4.

Under the assumptions of Theorem

let x e Y . Then lim max

n— x

к= 0 , 1 ,...,кп— X dt U(t, 0)x t=An) 1 lk+l = 0.

Proof . Let us treat Y as a Banach space, in the same manner as in the proof of Theorem 2, and let Un{t,s), n — 1, 2, (t, s)e AT, be defined as in the proof of Theorem 3. Then

and

so that

^(4+1) f(n) __/(«)

lk+l lk A (i% ) U n( t l , 0 )x

dt U(t, 0)x 4 (iff,) ^(4+1. 0)*

41 - 4 " ' where

< M II Un (41 , 0) SO - , o ).*|| r ,

M = sup WAitjW&ir'X)- 0

dt U(t, 0 )x\ (n)

*+i

Therefore we need only to prove that

(4.1) lim max \\Un( t \ 0 ) x - U ( t P , 0)x\\r = 0.

h—^{x »} k~l ,...,kn

We shall prove (4.1) considering the operators

F #(l, 8) = ( i l - l ( l n(l)| Un(l, e ) ( l l - 4 ( l ft(e)j) ( t , S ) e A T ,

n = 1 , 2 , . . . , where A > К is fixed. These operators well defined, because the range of Un{t, s) is always contained in Y. For every n = 1 , ... put

<Pn(T) = T, 9n(t) = if U <41 41 ), к = o , f c „ - l . We say that (4.1) follows from the fact that

(4.2) for every fixed x e X the sequence t -> Vn{<pn{t), 0) of X-valued functions converges strongly uniformly on <0, T} to a strongly continuous limit.

Indeed suppose for a moment that (4.2) is true. Then, because the

J2?(X, Y )-valued function t -> (Al — A(i))-1 is continuous on <0, T)

in the sense of the norm in J£{X, Y), it follows that, as n -> oo, then for every xe Y the sequence of Y -valued functions

t -> Un(<pn(t), 0)a? = [Я1 - A [ t n(cpn{t)))-l Vn{cpn{t), 0)-(Я1-А(^п)))]а?

converges strongly, uniformly on <0, T}, to a strongly continuous limit.

On the other hand it follows from the proof of Theorem 3 that Un(ç>n(t), 0) x converges in the sense of the norm in X to JJ(t, 0 )x. It follows that the sequence t -> Un(<pn{t), 0 )x, n =

,

, . . . , converges also in the sense of the norm in Y to t U(t, 0)x, which is an Y -valued function strongly continuous on <

, Tf. But this immediately implies (4.1). Therefore, in order to complete the proof, we need only to prove (4.2). We have

r„( f i , o ) - p „ ( f i , o )

= Y I V J f i , f i ’) r n( t ' , о ) - и М f i \ ) о)]

г=1

= Y U« № ' *f-i) — я» ( f i . f i ) ] TM!-’.,

).

i=l

Because

у « а л), f i ) = ( i - ^ ( f i ) H i - ( < s " ) - f i ) ^ ( f i ) ) - i - ( i - ^ ( f i

= f i X i - ^ t f i ^ i - ^ Æ 1,)) -1, we obtain

^ ( f i , o ) - p , ( f i , o )

= f i ) tr„ (fi, f i ) [ ( i - ^ ( f i ) ) ( i - ^ ( f i i ) ) '

- i ]

)

<=l

к

г=1

к ^Лп)

= - tf.Cfi, f i j ) / -^

^ - <г^(

—- A ( f i

T^„(fi, o)

i = l ,(n)

f i i Therefore

(4.3) Tn( f i - I 7 . ( f i , 0)) f i

- f U n ( t £ \ < p n ( r ) ) ^ ^ - ( l - A { < p n ( T ) ) )

1 0)dr.

198 К . N o w a k

Now for any n = 1 , 2 , . . . and ( t , s ) e A T put

and, for a fixed n, consider the integral equation t

(4.4) X n(t) — + j K n(t, z ) X n{cpn(z))dz, { t , s ) e A T.

о

This equation has unique ^ ( X ; X)-valued solution t -> X n(t) defined on <0, T>, and this solution may be obtained by the common method of successive interations. From (4.3) we see that

for every n = 1 , 2 , ... and (t,s)e AT. Therefore, by the criterion, for­

mulated on p. 192, (4.2) will be proved if we shall show that

(4.5) for any fixed oceX the sequence t -> X n(t)æ, n ' = 1 , 2 , . . . , of X-valued functions converges strongly, uniformly on <0, T>, to a strongly continuous limit.

dA(s)

We are going to prove (4.5). Because s - > —- — is an J ? ( ¥ , X) -

(л/Ь

valued function strongly continuous on <0, T> and s -* (A l-J .(s))-1 is an (X, Y)-valued function strongly continuous on <0, T>, it follows from Theorem 3 and from Lemma 2, that

(4.6) the sequence t, s -» K n(t, s), n — 1 , 2 , . . . , of

5?(X, X)-valued functions converges strongly, uniformly on An, to the strongly continuous limit

It follows by Banach-Steinhaus theorem, that there is a constant N > 0 such that

for every ( t , s ) e A T and n = 1 , 2 , . . .

Solving the integral equation (4.4) by successive iterations we obtain that, for any n = 1 , 2 , . . . ,

°) =

(4.7) \\Kn( t ,s ) \ \ ^ N and \\E (t,s )\\^ N

oo

(4.8)

m = 0

where

о

Similary, the f£{X, X)-valued solution X(t) of the integral equation t

X(t) = U(t, 0) + / K( t, r ) X( r) d t

,

has the form

OO

X{t) = V Xoo mit),

*i. / ’

m=о where

t

(4.10) X ^ { t ) = 17(*,0), -“^co.m+l (t) = J K ( t , r ) X 00 tin{x)dx.

0

From Theorem 3 and from (4.6), (4.9) and (4.10), by an induction in m, we infer that

(4.11) for every fixed m = 0 , 1 , . . . and xe X the sequence of X-valued functions t ^ X nm(t)x, converges strongly, uni­

formly on <0, T}, to the strongly continuous limit t -> X^^ft).

On the other hand from Lemma 1 and from (4.7), (4.9) and (4.11) we obtain that

(Nt)m

(4.12) \\X f -

ml

Now, (4.5) follows immediately from (4.8), (4.11) and (4.12) and therefore Theorem 4 is proved.

References

[1] T. K a t o , Integration o f the equation o f evolution in a Banach space, J. Math. Soc.

Japan. 5 (1953), p. 208-234.

[2] S. G. K r e in , L in ea r d ifferential equations in B a n a c h space (inrussian), Moscow 1967.

[3] K. Y o s id a , F unctional analysis, Springer, 1965.