AD-Ai53 527 ABOUT PRODUCT FORMULAS WITH VARIABLE STEP-SIZE(U)

7j/i 7WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTERH PIERRE ET AL. JAN 85 IRC/TSR-2783 DAAO29-80-C-004.

UNCLASSIFIED F/G 2/1 NE-17hEi

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MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU Of STANDARDS 1963 A

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MnNRC Tt1clica~d buiruuiiry Report #2783

S ABOUT PRODIUCT F'ORMULAS W iTI

Uv) VAkiABIA. STIP-SIZE

Michel Pierre and Mounir Hihani

I

IA

* Mathematics Research CenterUniversity of Wisconsin-Madison

* 610 Walnut StreetMadison, Wisconsin 53705

January 1985

C) (Ikceivcti Auqjust 3, 1984)

LLJ

Cm, Approved for public release* ~Distribution unlimitedE T C

F_

Sponsored by MAY9 18

U. S. Army Research Office

P. u. box 12211DResearch TriangleNorth Carolina 27709

41

-. -. r 4 .;2on For

0; 7,&I -

UNIVERSITY OF WISCONSIN-MADISON .tlleti --MATHEMATICS RESEARCH CENTER 3thicatio

ABOUT PRODUCT FORMULAS WITH VARIABLE STEP-SIZE -tributlon/ -S--

Michel Pierre* and Mounir Rihani* .vAribu1tity Code

"' Avhiland/orTechnical Summary Report #2783?Dst special

January 1985

ABSTRACT 0 -A continuous semigroup of linear contractions (S(t)) t > 0 on a Blnach I

space X can be generated through the product formula

(1) S(t)x - lim U(!)nx Vx e X Vt > 0 Inn. ,

where (Ulhl)h>O is any family of contractions on X whose right-derivativeat h = 0 coincides with the infinitesimal generator A of S(t), that is

(2) Vx e D(A) i U(h)x = Ax (= limxS(h th+O t+t

Actually formula (1) extends to the case when S(t) is a semigroup ofnonlinear contractions "generated" by an m-accretive operator A andcondition (2) may also be weakened to

(3)I - U(h) h A in the sense of graphs.h

We look here at the same formula (1) when the regular step-size t/n isreplaced by a variable step-size, namely

(4) lim U(hnn) U(hnn ' ) "'" U~ h x = S(t)x Vx e D(A)

n +0 N N-i_

where . hn t and lim( max h )=O.l~n 1 n- ( n 1

14i<N in+co 14i N

Surprisingly, it turns out that (4) fails under assumption (3); weexhibit a counterexample. However, we prove that (4) holds true under thestonger assumption (2).

AMS (MOS) Subject Classification: 47H20, 65J15, 35R15.Key Words: Product formulas, nonlinear semigroups, accretive operators,

approximation scheme, stability and convergence.Work Unit Number 1 - Applied Analysis

* Department of Mathematics, University of Nancy I, B.P. 239, 54502-

Vandoeuvre, Les-Nancy, France.

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

0-'

SIGNIFICANCE AND EXPLANATION

Many numerical schemes are based on the following abstract productformula: given an evolution equation which can be written in the abstract

torm

du-(t) + Au(t) = 0 on (O,T)atu(O) u0 given

with A being a suitable operator on a functional space, then under adequateassumptions the solution u is given by the formula

u(t) lim U(1)nu

where (U(h)l h > 0 is a family of mappings whose right-derivative at h = 0

coincides with A, that is

x - U(h)xYx e D(A) lim = Ax.Im h

h+ h

This a natural extension of the exponential formula

U(t)= lr I A)-n un 0n+w

Above product formula provides an extensively used tool for the Lie-Trotterformulas insuring the convergence of various alternating direction methods forevolution equations.

In view of numerical applications, it is important to relax the restric-

tion that the time step-size be regular in the above schemes when, for instance,

coupled with path control technics or even for stability requirements.

The goal of this paper is to decide whether the Chernoff formula is validwhen the step-size is variable. It turns out that under the usual assumptions

instability and lack of convergence can appear as soon as the step size isslightly perturbed. We give here an example in that direction. However, weprove that slightly(stronger assumptions can prevent these pathological

situations. - .

*The responsibility for the wording and views expressed in this descriptivesummary lies with MRC, and not with the authors of this report.

. . .

.' -

ABOUT PRODUCT FORMULAS WITH VARIABLE STEP-SIZE

Michel Pierre* and Mounir Rihani*

1. Introduction.

Given (S(t))t.0 a continuous semi-group of linear contractions in a

9anach space X, it can be generated via the following exponential formula:

(1) Vt > 0; Vx e X, S(t)x = lim (I + 1 A)-nxw. n+00

where -A denotes its infinitesimal generator and I is the identity. More

generally, one can replace in (1) the resolvents (I + h A)- by a family of

contractions (U(h))h)0 from X into X satisfying

(2) vx e D(A) lim x - U(h)x = Axhh+O

Thus, it has been proved (see [11], [3)) that (2) implies

(3) Vt > 0, Vx e x, lim u(.)nx = S(t)xnn+00

Actually, as proved in [7], [4], the condition (2) can be weakened to assuming

that

I-U(h) -1(4) Vx e X, VX > 0, lim II + Xh )1 = (I + XA) - xh+0

which is equivalent to saying that the graph in X x X of the operator

h-'(I-U(h)) converges to the graph of A as h tend to 0.

The implication (4) => (3) turns out to have a nonlinear version as

proved by Br6zis - Pazy [2]. Namely, let A be an m-accretive operator in

X, that is A C X x X and

* Department of Mathematics, University of Nancy I, B.P. 239, 54502-

Vandoeuvre, Les-Nancy, France.

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

-i-"4€-:--'~~~~~~~~~~~~~~..... ..-.-.-...-........ .".... ." ....,. T 2- i"'.----, ,- ~ - - -.. .. ..*" - - .. .. . . ., . . . . . . . ' L 'I

(5) VX > 0, (I + AA) is a contraction everywhere defined on X.

Let (S(t)t)0 be the nonlinear semigroup generated by A in the sense of

Crandall - Liggett [51, that is

t -nl(6) Vt ) 0, Vx e D(A), S(t)x = i (m (I +t A)-x

n+00

Then, if (U(h) Jh>O is a family of contractions from X into X satisfying

(4), we have the product formula

(7) Vt 0, Yx e D(A), lim U( txn = S(t)xn

The purpose of this paper is to look at the formula (7) when the regular

tstep size - is replace by a variable step-size. In particular, under the

n

same assumptions as above does one have

(8) lim U(hN U(h n U(h )x = S(T)xn * n n

when

Nn

hn T, lim ( max h) =0?..'-- i=1 n+ 0 1 i0

n

We give here a negative and a positive answer to that question:

a) given A an m-accretive operator the condition (4) does not imply

(8) as proved by a counterexample described in paragraph 3.

*O - ) if A is a single-valued m-accretive operator and if U(h)

satisfies the stronger assumption (2), then (8) holds.

Acknowledgements. The authors thank Mike Crandall for several helpful

discussions on that subject.

-2-

*- ." .. . - . . . . . . , - . *. . . - . ' ' ° . ' . "

2. The positive result.

Let X be a Banach space with its norm denoted by "I and let A be a

single-valued, accretive operator on X, that is

(9) VX > 0, Vx, x e D(A), ix- X1 Ix- x + X(Ax A)1 •

For h e (0,d) let U(h): D(A) - D(A) satisfy

(10) Vh e (O,h0 ), Vx,x e D(A) iU(h)x - U(h)xI I ix - x

(11) Vx e D(A), lim x - U(h)x Axh 0 h

Given T > 0, 0 a subdivision of (0,T):

(12) 0 - t0 < t I < < N_ 1 < T ( tN, I1 max (ti - ti )

and z: [0,T[ X such that

(13) Vi = 1,...,N, Vt e ]ti1 l, ti], z(t) = zi e X,

let us assume there exist x0, X1l,...xN in D(A) such that

(14) Vi = 1,...,N , xi - U(hi)xi_1 i hizi

where hi ti - ti_ 1 . We then define Vzx by

(15) V1,z,x0 (0) = x0, Vi 1,...,N, Vt e ]ti-1 ti], Vo,z,x (t) = xi

Remark. If z 0, the relation (14) always defines a sequence (xi) such

that xi = U(hi)U(hi_1),...,U(hl)x0 which are the expressions we are mainly

interested in. However, many situations involve perturbed schemes like (14)

where z is not zero but of small norm in LI(0,T; X) and our method appliesI-

as well to this more general situation.

-3-

-6" - ::" ": : . : . " : .: .: : i .- : -: ~:: -:.i:i : :ii: -i~ :::j:i:: , .: .. : ::: .. - ": : ::::-* :"-'-'. -'- -'-" ". .

Our goal is to describe the behavior of v whenn n n

a ,z ,x

i n = 1r fT Izn(t)Idt = lr Ix n - x0 1 .lim I0n = lm f0 d= iIX 01 .

n+W n+Q n+0

We expect it to tend to the solution (if it exists) of

(16) du + Au = 0 on (0,T), u(O) = x 0 e D(A)dt"

By solution of (16), we mean a function u e C([0,T]; X) such that there

exists a sequence of step-functions up [0,T] + X satisfying

(17) Vj = 1,...,NP, Vs e ]sP sPI, uP(s) =u p

where

(18) 0 s p < sl < ... < s p < T Sp , P= sP-0 1 _ uI j -1

N3 3-1

(19) + Au p wp (u p uP(0))

N p ,.

(20) 0 = lim sup IuP(s)- u(s)l = lim I pLIwI - lim max Ipp-= <s4T p+C* j=l 3p+- I4jc.NP I

We will refer to such a solution as a mild solution of (16) (see [1]).

Theorem 1. Let u be a mild solution of (16). Let v = v be an n nYZ ,X

* sequence of functions defined by (12) - (15) where

rnj t Ini fT jzn(t)fdt = lim Ixn x0 j = 0lICY ,o n , = i lira 00o

n n+O n+ n+-

Then, if (9), (10), (11) hold

lim sup lu(t) - vn(t)l = 0

n- Ot<T

-4-

,,,,,,,,,,,,,,,,,,,,,,,,,,-'f-f. " -.ft.". ..- .... ..... ..- .- ". ."-t i

• - ' " - " - " ," "'-.. ft ±. . - ".'" " ," " - , ' . " - ' " f. "', . • -" t- " . - 't . ".- " - - ." • - - "'. " . " . " " " . " - - " . - "'t,

In the case when the range of I + XA contains D(A) for all X > 0,

thanks to Crandall - Liggett [5], one knows that -.

(21) Vx e D(A), Vt > 0, S(t)x = lim (I + -nx exists,n

and S(t) is a semigroup of nonlinear contractions from D(A) into itself.

Moreover, by definition, t + S(t)x 0 is a mild solution of (16) for all

T > 0.

In that situation, as an immediate corollary of Theorem 1, we have

Proposition 1. Assume A satisfies (9), (21) and that (10), (11) hold. Then

if (hn n is a sequence such that1ii<N n

Nn(22) lim (max h 0, hn = t

nn n-O in 1

we have for all x0 e D(A)

(23) lim U(h n ) U(hn 1 ) "'" U(hln)x 0 S(t)x0

n+* N N

and the convergence is uniform in t on bounded intervals.

The proof of Theorem 1 is based on the next lemma which is an adaptation

of technics used in [6] and [8).

Lemma 1. Let up: [0,T] + X satisfying (17) - (19). Then, for all £ > 0

there exists n = n (c,p) > 0 such that for all functions v .

satisfying (12) - (15) with 10l < n we have

Vy e D(A), Vi = 0,...,N, Vj = ,...,NP

- - - p)2 + j lt + 10p, sp 1/ 2,.1Ixi uPI ' Ix0 + lu0 - yl + {(t i i1 J 1,

(24) i

+) h( +pwIk IkI + E) + k k'

k=1 k=1

-5-

-- " -" -- . -"- ,

*~* V;:.,. ...- 4. ** *<-. . : .>- 1 25i

where a'P I = max ZP and

(25) M (y) = max {IAyl, max h 1 1y - U(hi)ylO1(iN

Proof. Lec us denote Ah = h-1 (I-U(h)). By (11), for £ > 0 and p fixed,

there exists n = n(e,p) > 0 such that

(26) h e ]0,n] => Vj = 1,. ..,NP, JA hUP - AuPI < C

Now, we will assume 101 < n and, since p is fixed we will drop the

indexation by p.

Let us denote ai,) = !xi - u.I and let us establish (24) in four

steps. We fix y e D(A).

i) estimate of a0 j: since A is accretive and by (19)

juj - YI < ju. - y + kj(Auj - Ay)j < I - Yj + 3wj j

By induction, for all j ) 1

Jlu. - y1 4 u0 - yl + X kwk + s Ay •

k=1

Hence, for all j > 0

3

(27) a0 j < x 0 - AI + lu0 - Y1 + sjlAyl + k k 'j k= 1

which implies (24) for i = 0 and j =0,...,NP

ii) estimate of aiO:

Ix, - y x - U(h )xi-ll + IU(hi)xi-1 - U(hi)y + UL(hi)Y- yj

< h ilz i I + Ixi_ - yI + hiMO(y)

By induction, for all i > 1

-6-

* , , ,, , -. , _ - , , .. .. . . i L - - u - - , . , . ,

. . .. 7,

Ixi - x o- yj + h hk1zk + tMo(y)k=1 1"

Hence for all i 0 0

i(28) a, 0 J 1u0 - yl + Ixo -yj + > hkjz + tiM (Y)i' ~k=1 ia

iii) estimate of ai, j in terms of ai1 ,11 and ai,j_: for all

i, j ) 1

(29) (i + 2- h-1)ai = Ix- uj + £h 1(x, - U(h )u. + U(h )u - u.)!

Using (14), we have

Ixi - U(hi)ul • a il, j + hilzl

Plugging this inequality in (29) together with (26) applied with h = hi < nf1Jleads to

(30) (1 + x h-1 ai a 2 h a.1 . + .I zii + £C + Ix - u -Au 4 INow by (19)

(31) lxi - Uj - £jAuj ( ai,j_ + jiwjI

Finally, (30) and (31) yield

2.. h. 2.h.(33) a. h ++ ,j + a ,j_ +h- - - (Izi I+ 1wj I + C ) "

1 1 iX

iv) induction procedure: we assume that (24) holds with (i,j) replaced

by (i-1,j) and (i,j). Plugging it into (33) gives (24) after a easy computa-

tion. Thanks to the two first steps, we deduce that (24) holds for all

(iIJ).

nProof of Theorem 1. Let V = v be a sequence of functions definedn n n0 ,Z ,Xby (12) - (15) where

-7-

. .. .i

lrn oaI lrn f~' tzn(t)Idt nlim Ix n- x 1 0flim fol-*li f

n +0 n +0 n+o "

Let u be a mild solution of (16) and let up be an approximated sequence as

in (17) - (20). Let us denote

I0PI = max £P.

1(j4Np 3

By lemma 1, for all p > 1 and E > 0, there exists N(p) such that

Vn > N(p) W 14i4N V 14j4N p Vy e D(A)

n- up x3 -Y + jup - YI + {(t, - P)2 + IonI t n + IOPJsP}1/2M (Y)

+ h(lzn + E)+ p I~k=1 k=1

where

M.(y) = sup h- 1 (y - U(h)y).

he(0,1)Using

IVn(t) - u(t)j I Ivnt(t) _ u(t), + Iup(t)- u(t),

we deduce that for all p > 0 and e > 0:

lim sup (sup IVn(t) - u(t)I) 4 sup IuP(t) - u(t)I + Ix - yl + ju0 -Y1n * 0e t<T 0 tc-T

NP

+ I(P12 + TIOPI}1/2M (y) + ET + P Xp " +

k 1

Now letting p tend to 0, then E tend to 0 yield (according to (20))

lim sup (sup Iv (t) - u(t)l) < 21x 0 - Yl

Since x0 e D(A) and y is arbitrary in D(A), the conclusion of the theorem

follows.

-8-"

: + V + ::+ . .+ + + .+ . + . .

-~~~2 -.T.- .-

3. The counterexample.

Let X = C0 (R) the space of continuous functions u on R such that

lim u(x) = 0, equipped with the norm

ullu. = sup ju(x)jxeR

we define

D(A)= {u e C0 (R); u' e c 0 (R)}

Vu e D(A), Au u'

Then, A is the infinitesimal generator of the semigroup S(t) on X

defined by

vu e c0 (R), S(t)u(x) = u(x - t)

which means that one has

Vu e D(A), lim u(x) - S(t)u(x) = u'(x) uniformly on R.t+0 t

Now, let u0 e X and a: (0,1) + R such that

34) lim a(h) = 0h- 0

We set

(35) Vh e (0,1), Yu e X, U(h)u S(h)u + a(h)(u 0 - S(h)u0 )

Then we have

Theorem 2.

i) Under assumption (34)

Vh e (0,1), Yuu IIU(h)u - U(h)ull < 'lu - ull

A ~ -1 -1

VA > 0, Vu e x iim(I + -(I-U(h ))J u = (I + XA) uh+0' h

-9-

- - .? . :.<. -" . ii " i~ i_ > > :i-i i i. i< i. iiii i . -i .: . . . . . >- . - .. I

ii) One can shoose u0 e X, a satisfying (34) and a sequence

{h, .. .,hn such that" ' nN

nN

n hn'h n T, lir max h. 0

i=1 n- o l(iNn 1

lim UU(h h U(hn U(h n),•n+w Nn N_

Proof. The first statement of i) is immediate from the definition (35) and

the fact that S(h) is nonexpansive in X.

Let u e X, and uh = [I + Xh 1 (I - U(h))] -u, that is

(36) uh + Xh-I 1 - U(h)uh ) = u

or, by (35)

uh + Xh (uh - s(h)uh - a(h)(u 0 - S(h)u0 )) = u

which can be written as

* (37) uh - (h)u0 + Xh- ( - a(h)u0 - S(h)(uh - a(h)uo)) u -(h)u

hu 0 +0 -

" Setting J [I + Xh-(I - S(h))] (36) is equivalent to

(38) uh = a(h)u0 + J(u - a(h)u0 )

hSince S(h) is a linear semigroup of contractions, J is also nonexpansive

and satisfies (see for instance [9])

(39) lim Jhu = (I + XA)- Uh+0

%h

But now, since J is nonexpansive, by (38)

-10-

N

-- %

. .*.* ** * ~ . * . . .. : :... . **i.A.

1Uh - Ju u04 2a(h)Iu0I .

This together with (39) and (34) proves the assertion i).

Now we choose a sequence of numbers (hi) in (0,1) satisfying :1(i (N

Nn

(40) " h" --T > 0i=1I

4

(41) Vn, i jA j => hn 0h

(42)T < min hn < max hn < T(4 )n+-- 14i4N n i 14i*N n i n "

The conditions (41) - (42) insure that all the numbers

{hn, 1 < i . Nn , n > 1} are different. This will allow us to assign thei

values a(hn) as we wish. We do it as follows.i i

Set tI = Z hk, tn = 0, and choose u0 e c0(R) such thatk=0

Nn

(43) n IUo(tn) - uo(t 1 ),

tend to w when n tends to m. Actually any u0 e CO(R) whose total

variation on [0,T] is infinite will be convenient. Indeed n is the totaln

variation of the step function u0 defined by

in ], u (t) = u0 (tn), un(0) u0(0),

and uO converges uniformly on [0,T] to u0 when n tends to .

n1Therefore the total variation of u0 cannot stay bounded if the variation

of u0 is infinite.

Now, for reasons that will become clear later we choose a: (0,1) + R as

follows:

-11-

!S

if h 0 h, a(h) - i2 sign (u (t) - ,t(44) I n

if h f0{h; 1 4 i 4 N', n > 1}, cz(h) =0

Note that a satisfies (34).

We claim that with this choice, if we denote

ff n U(hnj U(h n ) ... UhnO

N N-

then fn (T)1 tends to when n tends to and therefore

definition (35) we have

nf =~ f n- c() (u0 - n~) u)fk S(h k k 1)fkn ~ hk 0

* By induction, we check that

kfn = nh)(S(tn n t u~S(tn -tn ufk 1~ k t Vu0 k i_-i) 0

so that

N n

f = a(h .)ES(T t ~ )u -S( -

i 0 i-i 0

But S(T -t )uo(x) =uo(x -T + ti ), so that according to (44),

N Nn

(46) ff (T) -nl2~ u(~- 0 t 1 I=~/k= 11

From (43), (46) we obtain:

lrn ff (Tr) li 6rn

-12-

Remarks.

i) The same negative conclusion could be reached by choosing

U(h)u = (I + hA) -1 + (.,)(u 0 - (I + hA)- u0 )

which is nothing but the resolvents of the operator u + A(u - a(h)u0 ). Note

that U(h) as defined by (35) is the value at h of the semigroup generated

by the operator u + A(u - a(h)u 0).

ii) According to this counterexample, the conclusion (23) dramatically

fails as soon one switches from a regular step-size to a very slightly

irregular path. In view of (41), (42) we need all the h to be different

but they can be as close to each other as we pleased (see (41), (42)).

iii) It would be interesting to know whether there exists a condition

stronger than (4) but weaker than (2) such that the expected conclusion (8)

holds. Let us mention some positive results with variable step-size obtained

in [10) under stability assumptions on the mappings U(h).

iv) If B, C and B + C are infinitesimal generators of linear

semigroups of contractions and if one sets

A=B+C

U(h) (I+h B) 1 (I+h C) - or U(h) SB(h) SC(h),

B+Cone easily verifies that assumption (11) holds. Since S (t) u0 is a mild

dusolution of -+ Au = 0, u(0) = u0 for all u0 (see [1]), for both choices

one has, as a consequence of Theorem 2.1

lim U(hn) uthn ) ..n U(hl u0 = sB+C(t) U0n n0 N N-i

v) The same conclusion holds if A = B + C where B is the

infinitesimal gnerator of a linear semigroup of contractions and C is a

continuous accretive operator defined on the whole space.

-13-

" 'i.. . ... . " " " "'" .. ..... .. ...... " "" ' ' '"t./ ' '" " '... .. "'

References

[1; Benllan Ph., M. G. Crandall, A. Pazy, Evolution equations governed bym-accretive operators, to appear.

[2) Brezis H., A. Pazy, Convergence and approximation of semigroups ofnonlinear operators in Banach Spaces, J. Funct. Anal. 9 (1972), 63-74.

(3[ Chernoff P. R., Note on product formulas for operator semigroups, J.

Funct. Anal. 2 (1968), 238-242.

[4) Chernoff P. R., Product formulas, nonlinear semi-groups, and addition ofunbounded operators, Mem. Amer. Math. Soc. 140 (1974).

(5) Crandall M. G., T. M. Liggett, Generation of semigroups of nonlineartransformation on general Banach spaces, Amer. J. Math. 93 (1971).

[6] Kobayashi Y., Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semi-groups, J. Math.Soc. Japan, 27 (1975), 640-665.

[71 Kurtz T. G., Extensions of Trotter's operator semigroup approximation

theorem, J. of Functional Ana., 3 (1969), 354-375.

[8) Miyadera I., Kobayashi Y., Convergence and approximation of nonlinear

semi-groups, in Functional Analysis and Numerical Analysis, Japan-FranceSeminar, Tokyo and Kyoto, (1976); H. Fujila (Ed.): Japan Society for thePromotion of Science (1978).

[9] Pazy A., Semigroups of linear operators and applications to partialdifferential equations, Applied Mathematical Sciences 44, Springer-

Verlag, New York (1983).

(10] Schechter E., Stability conditions for nonlinear products and semi-groups, Pacific S. of Math. 85 (1979), 179-200.

[11] Trotter, H. F., Approximation of semigroups of operators, Pac. J. Math.,8 (1958), 887-919.

L

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!! ; :!:: i~i ! i~i. . .>c :ii: .i . : * * * ; K*: . - -:... ... i : .i ii~~i --ii . . i~.::i

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6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(a) 9. CONTRACT OR GRANT NUMBER(&)

Michel Pierre and Mounir Rihani DAAGZ9-80-C-00419, PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

AREA & WORK UNIT NUMBERSMathematics Research Center, University of Work Unit Number 1 -

610 Walnut Street Wisconsin Applied AnalysisMadison, Wisconsin 53706

I I. CONTROLLMIG OFFICE NAME AND ADDRESS 12. REPORT DATE

U. S. Army Research Office January 1985P.O. Box 12211 1s. NUMBER OF PAGESResearch Triangle Park, North Carolina 27709 1414. MONITORING aGENCY NAME & ADDRESS(If different from Controlling Office) IS. SECURITY CLASS. (of this report)

UNCLASSIFIEDIS.. DECLASSIFICATION/OOWNGRADING

SCHEDULE

IS. DISTRIBUTION STATEMENT (of title Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different he Report)

IIS. SUPPLEMENTARY NOTES

12. KEY WORDS (Continue on reveres side it neceseary mid identify by block nhmber)

-Product formulas, nonlinear semigroups, accretive operators,approximation scheme, stability and convergence ,

. 20. ABSTRACT (Continue on revere side if necessary and Identify by block nionbor)

A continuous semigroup of linear contractions (S(t)) t > 0 on a Banach

space X can be generated through the product formula

(1) S(t)x - lim U(.-)nx Vx e X Vt ) 0nn+w I

'FORM 1473 EDITION OF I NOV8 IS OBSOLETE-. '; O , ,N ,UNCLASSIFIED "

SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)

. 7.

20. Abstract (cont.)

where (U(h)) h>o is any family of contractions on X whose right-derivativeat h = 0 coincides with the infinitesimal generator A of S(t), that is

(2) Yx e D(A) lim x - U(h)x . Ax (= lim x - S(t)x)h th4O t+O

Actually formula (1) extends to the case when S(t) is a semigroup ofnonlinear contractions "generated" by an m-accretive operator A andcondition (2) may also be weakened to

(3) I - U(h) h+ A in the sense of graphs.hWe look here at the same formula (1) when the regular step-size t/n is

replaced by a variable step-size, namely

(4) lim U(hnn) U(hn J .. " U(h)x = S(t)x Vx e D(A)n-- N N-1

-where h hn = t and lim ( max hn) =0.

r 1i(N n+oo 1fi4N

Surprisingly, it turns out that (4) fails under assumption (3); we

exhibit a counterexample. However, we prove that (4) holds true under thestonger assumption (2).

S%

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6-85

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