Approximation of regularized evolution operators

Arch. Math. 81 (2003) 38–490003–889X/03/010038–12DOI 10.1007/s00013-003-0511-z© Birkhauser Verlag, Basel, 2003 Archiv der Mathematik

Approximation of regularized evolution operators

By

Naoki Tanaka

Abstract. This paper is devoted to the approximation of a regularized evolution operator by asequence of “regularized” discrete parameter evolution operators, and the results obtained here areapplied to some degenerate partial differential equation.

Introduction. Throughout this paper let X and Y be real Banach spaces such that Y iscontinuously and densely embedded in X. Let C be an injective operator in B(X) withdense range R(C), and let {A(t) : t ∈ [0, T ]} be a family of closed linear operators in X

satisfying the following conditions:

(A1) For t ∈ [0, T ], D(A(t)) ⊃ Y . A(t) is strongly continuous in B(Y, X) on [0, T ].(A2) For t ∈ [0, T ], A(t) ⊂ C−1A(t)C.

We discuss the approximation of a regularized evolution operator {U(t, s)} in B(X),defined on the triangle � = {(t, s) : 0 � s � t � T }, satisfying the four properties listedbelow.

(E1) U(t, t) = C for t ∈ [0, T ], and U(t, r)U(r, s) = U(t, s)C for (t, r), (r, s) ∈ �.(E2) U(t, s) is strongly continuous in B(X) on �.(E3) For (t, s) ∈ �, U(t, s)(Y ) ⊂ Y , and U(t, s) is strongly continuous in B(Y ) on �.(E4) For (s, y) ∈ [0, T ) × Y , U(t, s)y is differentiable in X with respect to t ∈ [s, T ],

and (∂/∂t)U(t, s)y = A(t)U(t, s)y for y ∈ Y and (t, s) ∈ �.

A family {U(t, s) : (t, s) ∈ �} satisfying (E1) through (E4) is unique for each family{A(t) : t ∈ [0, T ]}, and such a family is called a regularized evolution operator on X, witha regularizing operator C, generated by {A(t) : t ∈ [0, T ]} in this paper. In the specialcase where C is the identity operator on X, the notion of regularized evolution operatorscoincides with that of evolution operators due to Kato [5]. The generation of regularizedevolution operators was studied in [8] and applied to some degenerate partial differentialequation [9]. For the related information we refer to deLaubenfels [3, Section XXXI].

Mathematics Subject Classification (2000): Primary 47D60, 47D06; Secondary 65J10.

Vol. 81, 2003 Approximation of regularized evolution operators 39

According to Kurtz [6], we assume that a sequence {Xn}of Banach spaces approximates X

in the following sense (see also [4]): There exist Pn ∈ B(X, Xn) such that limn→∞ ‖Pnx‖n =

‖x‖ for x ∈ X.Under this assumption there exists L > 0 such that

‖Pnx‖n � L‖x‖ for x ∈ X and n � 1.(1)

Let {U(t, s) : (t, s) ∈ �} be a regularized evolution operator on X, with a regularizingoperator C, generated by {A(t) : t ∈ [0, T ]}. It is necessary to consider an approximationof the operator C. In this paper we assume that there exists a sequence {Cn}, where Cn isinjective in B(Xn), satisfying the condition

(H1) if limn→∞ ‖xn − Pnx‖n = 0, then lim

n→∞ ‖Cnxn − PnCx‖n = 0.

For each n � 1, let {Fn(t) : t ∈ [0, T ]} be a family in B(Xn) such that

(H2) Fn(t)Cn = CnFn(t) for t ∈ [0, T ].

Let {hn} be a null sequence of positive numbers as n → ∞, and set

An(t) = (Fn(t) − I )/hn for t ∈ [0, T ] and n � 1.

Analogously to Kato’s stability condition [5], we can introduce the stability condition that

sup ‖k∏

i=l+1Fn(ti)Cn‖n < ∞, where the supremum is taken over all integers n, partitions

{ti}Ni=1 of [0, T ] and 0 � l � k � N . If this stability condition is assumed, then the mainresult (Theorem 2) in this paper may be proved by defining a family of norms so that eachFn(t) is quasi-contractive on Xn and using a nonlinear semigroup theoretical technique dueto Miyadera and Kobayashi [7]. However, we do not know whether this type of stabilitycondition can be checked or not, in an application to some degenerate equation discussedin Section 3.

If one considers an approximation of the solution of u′(t) = A(t)u(t) with u(0) = u0 ∈R(C) by solutions of some finite difference approximation with un(0) = un,0 ∈ R(Cn)

such as (un(t+hn)−un(t))/hn = An(t+hn)un(t) or un(t+hn) = Fn(t+hn)un(t), then it

is natural to assume that if u0 is approximated by un,0 in some sense, then[t/hn]∏i=1

Fn(ihn)un,0

(= un([t/hn]hn)) is bounded for 0 � t � T in some sense, as n → ∞. We emphasizethat in this assumption, the special partition {ihn : 1 � i � [T/hn]} is used as partitions of[0, T ], differently from the first-introduced stability condition. Under such an assumption,we are interested in studying the approximation of a regularized evolution operator by asequence of “regularized” discrete parameter evolution operators and applying the resultsto the problem of approximation which arises when the solution of some degenerate partialdifferential equation is numerically computed.

40 Naoki Tanaka arch. math.

1. Discrete parameter evolution operators. In this section we give a key estimate onthe difference between the solution given in terms of a discrete parameter evolution operatorand that of difference equation associated with its “generator”.

Let T > 0 and h > 0, and consider a discrete parameter evolution operator on Xk∏

i=l+1

F(ih) : 0 � l � k � [T/h]

,

where {F(t) : t ∈ [0, T ]} is a family in B(X) satisfying the following condition: Thereexist M � 1 and an injective operator R ∈ B(X) such that∥∥∥∥∥∥

k∏i=l+1

F(ih)R

∥∥∥∥∥∥ � M for 0 � l � k � [T/h],(2)

F(t)R = RF(t) for t ∈ [0, T ].(3)

Let s ∈ [0, T ) and let {s = t0 < t1 < . . . < tN = T } be a partition of [s, T ]. Let {ui}Ni=0be a finite sequence in R(R) satisfying the difference equation

(ui − ui−1)/(ti − ti−1) = B(ti)ui + zi for 1 � i � N,

where B(t) = (F (t) − I )/h for t ∈ [0, T ], and {zi}Ni=1 is a finite sequence in R(R).Notice that the set R(R) is invariant under each operator B(t), by condition (3). Letu : [s, T ] → R(R) be the step function defined by u(t) = u0 for t = s, and ui fort ∈ (ti−1, ti] and 1 � i � N .

The following is a key lemma to prove Theorem 2 in Section 2.

Lemma 1. If the number h > 0 satisfies s + 2h � T and the condition

h � min1�i�N

(ti − ti−1),(4)

then we have, for (t, s) ∈ �,∥∥∥∥∥∥[t/h]∏

i=[s/h]+1

F(ih)u0 − u(t)

∥∥∥∥∥∥ � h max1�i�N

( supt∈[0,T ]

|||B(t)ui |||)

+ 3M max1�i�N

|||ui − ui−1||| + MT (α + 2β + 2γ ),(5)

where ||| · ||| is the norm defined by |||u||| = ‖u‖ + ‖R−1u‖ for u ∈ R(R), and the symbolsα, β and γ are defined by

α = max1�i�N

|||zi |||,β = max

1�i�N( supt∈[ti−1,ti ]

|||(B(t) − B(ti))ui |||),γ = max

1�i�N( supt∈[0,T ]

|||B(t)(ui − ui−1)|||).


P r o o f. For t ∈ [s, ([s/h] + 1)h) we have u(t) = u0 or u1 by condition (4), and[t/h]∏

i=[s/h]+1F(ih)u0 = u0; hence (5) is true for t ∈ [s, ([s/h] + 1)h).

Now, let us consider the function v : [s, T ] → R(R) defined by

v(t) = ui−1 + (t − ti−1)(ui − ui−1)/(ti − ti−1)

for t ∈ [ti−1, ti] and 1 � i � N. Then we have

|||u(t) − v(t)||| � max1�i�N

|||ui − ui−1||| for t ∈ [s, T ], and by condition (4) we have

|||u(t) − u([t/h]h)||| � max1�i�N

|||ui − ui−1||| for t ∈ [([s/h] + 1)h, T ]; hence

|||u(t) − v([t/h]h)||| � 2 max1�i�N

|||ui − ui−1||| for t ∈ [([s/h] + 1)h, T ].(6)

Now, we set

fl = (v(lh) − v((l − 1)h))/h − B((l − 1)h)v((l − 1)h)

for l = [s/h] + 2, . . . , [T/h]. Notice that fl ∈ R(R) for l = [s/h] + 2, . . . , [T/h]. SinceF(t) = I + hB(t) for t ∈ [0, T ], we have v(lh) = F((l − 1)h)v((l − 1)h) + hfl forl = [s/h] + 2, . . . , [T/h]. By using this relation it is shown inductively that

v(kh) = k−1∏

i=[s/h]+1

F(ih)

v(([s/h] + 1)h) + h

k∑l=[s/h]+2

(k−1∏i=l

F (ih)

)fl

for k = [s/h] + 1, . . . , [T/h]. It follows from (2) that∥∥∥∥∥∥F([t/h]h)v([t/h]h) − [t/h]∏

i=[s/h]+1

F(ih)

v(([s/h] + 1)h)

∥∥∥∥∥∥� MT max

[s/h]+2�l�[T/h]|||fl |||

for t ∈ [([s/h] + 1)h, T ]. Since t0 = s < ([s/h] + 1)h � s + h � t1 by condi-tion (4), we deduce from the definition of function v that |||v(([s/h] + 1)h) − u0||| �|||u1 − u0|||. Since F(t) = I + hB(t) for t ∈ [0, T ], F([t/h]h)v([t/h]h) − u(t) is writtenas F([t/h]h)(v([t/h]h) − u(t)) + hB([t/h]h)u(t); hence for t ∈ [([s/h] + 1)h, T ],

‖F([t/h]h)v([t/h]h) − u(t)‖� 2M max

1�i�N|||ui − ui−1||| + h max

1�i�N( supt∈[0,T ]

|||B(t)ui |||)

by condition (2) and (6). It follows that∥∥∥∥∥∥u(t) −[t/h]∏

i=[s/h]+1

F(ih)u0

∥∥∥∥∥∥ � 3M max1�i�N

|||ui − ui−1|||

+ h max1�i�N

(sup

t∈[0,T ]|||B(t)ui |||

)+ MT max

[s/h]+2�l�[T/h]|||fl |||(7)


for t ∈ [([s/h] + 1)h, T ]. We want to estimate |||fl ||| for [s/h] + 2 � l � [T/h]. For thispurpose, let (l − 1)h ∈ [ti−1, ti] for some i. Since

(ui − ui−1)/(ti − ti−1) − B((l − 1)h)v((l − 1)h)

= zi + (B(ti) − B((l − 1)h))ui + B((l − 1)h)(ui − v((l − 1)h))

and

ui − v((l − 1)h) = (ti − (l − 1)h)(ui − ui−1)/(ti − ti−1),(8)

we have

|||(ui − ui−1)/(ti − ti−1) − B((l − 1)h)v((l − 1)h)||| � α + β + γ.(9)

To estimate |||fl |||, it suffices to consider the following cases, by condition (4): (i) lh ∈[ti , ti+1], and (ii) lh ∈ [ti−1, ti]. We begin by considering the case of (i). Since fl is writtenas ((v(lh)−ui)+ (ui −v((l −1)h))/h−B((l −1)h)v((l −1)h) we have, by the definitionof function v,

fl = {(lh − ti )((ui+1 − ui)/(ti+1 − ti ) − B((l − 1)h)v((l − 1)h))

+ (ti − (l − 1)h)((ui − ui−1)/(ti − ti−1) − B((l − 1)h)v((l − 1)h))}/h.

Since (ui+1 − ui)/(ti+1 − ti ) − B((l − 1)h)v((l − 1)h) = zi+1 + (B(ti+1) − B(ti))ui+1 +B(ti)(ui+1 − ui) + (B(ti) − B((l − 1)h))ui + B((l − 1)h)(ui − v((l − 1)h)), we have|||(ui+1 − ui)/(ti+1 − ti ) − B((l − 1)h)v((l − 1)h)||| � α + 2β + 2γ by inequality (8). Itfollows from (9) and this estimate that

|||fl ||| � α + 2β + 2γ.(10)

By using (9) it is easily seen that the estimate (10) is also valid in the case of (ii). Substituting(10) into (7) we obtain the desired estimate (5). �

2. Approximation of regularized evolution operators. The main result in this paper isgiven by the following theorem, which is a generalization of the Chernoff product formula([1] and [2, Theorem 3.1]) and new even if C = Cn = I (the identity).

Theorem 2. Assume that the operator A(t) is approximated by a sequence {An(t)} inthe following sense:

(H3) For w ∈ Y , there exist wn ∈ Xn such that limn→∞ ‖wn − Pnw‖n = 0 and

limn→∞ ‖An(t)wn − PnA(t)w‖n = 0 uniformly for t ∈ [0, T ].

Then the following statements are equivalent:

(i) There exists M � 1 such that ‖k∏

i=l+1Fn(ihn)Cn‖n � M for 0 � l � k � [T/hn]

and n � 1.


(ii) For every x ∈ X, xn ∈ Xn with limn→∞ ‖xn − Pnx‖n = 0, we have

limn→∞(sup{‖

[t/hn]∏i=[s/hn]+1

Fn(ihn)Cnxn − PnU(t, s)x‖n : (t, s) ∈ �}) = 0.

P r o o f. To prove the implication “(ii) ⇒ (i)", let X be the set of all sequences (xn) suchthat xn ∈ Xn for n � 1 and that there exists x ∈ X satisfying lim

n→∞ ‖xn − Pnx‖n = 0.

Clearly, the set X becomes a Banach space under the norm ‖(xn)‖X = supn�1

‖xn‖n. For

(t, s) ∈ �, let us consider the operator V (t, s) on X defined by V (t, s)x = {[t/hn]∏

i=[s/hn]+1

Fn(ihn)Cnxn} for x = (xn) ∈ X. It is seen that each V (t, s) is a bounded linear operatoron X by the closed graph theorem. Statement (ii) asserts that sup{‖V (t, s)x‖X: (t, s) ∈ �}is finite for all x ∈ X, by virtue of (1) and the strong continuity of U(t, s) in X on �. Wetherefore deduce from the principle of uniform boundedness that sup{‖V (t, s)‖X: (t, s) ∈�} is finite, which implies the desired assertion (i), by considering a special element x suchthat the n-th component is x where x ∈ Xn and all the other components are zero.

Conversely, we shall prove that (i) implies (ii). To do this, let x ∈ X, xn ∈ Xn andlim

n→∞ ‖xn − Pnx‖n = 0. Let (s, y) ∈ [0, T ) × C2(Y ) and ε > 0. By (E1) with t = r we

have C−1U(τ, s)y = U(τ, s)C−1y for (τ, s) ∈ �. Since C(Y ) ⊂ Y by (E3) with t = s,conditions (A1), (A2) and (E3) together imply thatC−1A(σ)U(τ, s)y = A(σ)U(τ, s)C−1y

for σ ∈ [0, T ] and (τ, s) ∈ �. By conditions (E2) and (E3) there exists a partitionPε = {s = tε0 < tε1 < . . . < tεNε

= T } of [s, T ] such that

|Pε| := max1�i�Nε

(tεi − tεi−1) � ε,(11)

sup{N(U(τ, s)y − U(τ , s)y) : |τ − τ | � |Pε|} � ε,(12)

sup{N(A(σ)U(τ, s)y − A(σ )U(τ , s)y) : |τ − τ | + |σ − σ | � |Pε|} � ε,(13)

where N(·) is the norm defined by N(u) = ‖u‖ + ‖C−1u‖ for u ∈ R(C). Now, weset uε

i = U(tεi , s)C−1y for 0 � i � Nε, and define a step function uε : [s, T ] → X byuε(s) = uε

0 = y, and uε(t) = uεi for t ∈ (tεi−1, t

εi ] and 1 � i � Nε. Then we have, by (11)

and (12),

‖U(t, s)x − uε(t)‖ � K‖x − C−1y‖ + ε(14)

for t ∈ [s, T ], where K = sup{‖U(τ, s)‖ : s � τ � T }. For 0 � i � Nε, since uεi ∈

C(Y ) by condition (E3), hypothesis (H3) asserts that there exist vεi,n ∈ Xn satisfying

limn→∞ ‖vε

i,n − PnC−1uε

i ‖n = 0 and limn→∞ ‖An(t)v

εi,n − PnA(t)C−1uε

i ‖n = 0 uniformly

on [0, T ]. Set uεi,n = Cnv

εi,n for 0 � i � Nε and n � 1. Then we have lim

n→∞ ‖uεi,n −

Pnuεi ‖n = 0 by condition (H1). In the following arguments, let us consder integers n such

that hn � min1�i�Nε

(tεi − tεi−1) and s + 2hn � T . Notice by hypothesis (H2) that Fn(t) and


Cn commute, and so do An(t) and Cn. If we define a step function uεn : [s, T ] → R(Cn)

by uεn(s) = uε

0,n, and uεn(t) = uε

i,n for t ∈ (tεi−1, tεi ] and 1 � i � Nε, then we have by

Lemma 1 ∥∥∥∥∥∥[t/hn]∏

i=[s/hn]+1

Fn(ihn)uε0,n − uε

n(t)

∥∥∥∥∥∥n

� hn max1�i�Nε

(sup

t∈[0,T ]|||An(t)u

εi,n|||n

)+ 3M max

1�i�Nε

|||uεi,n − uε

i−1,n|||n+ MT (αn + 2βn + 2γn)(15)

for t ∈ [s, T ], where |||u|||n = ‖u‖n + ‖C−1n u‖n for u ∈ R(Cn), and

αn = max1�i�Nε

|||(uεi,n − uε

i−1,n)/(tεi − tεi−1) − An(t

εi )uε

i,n|||n,

βn = max1�i�Nε

(sup

t∈[tεi−1,tεi ]

|||(An(t) − An(tεi ))uε

i,n|||n)

,

γn = max1�i�Nε

(sup

t∈[0,T ]|||An(t)(u

εi,n − uε

i−1,n)|||n)

.

Since An(t)uεi,n − PnA(t)uε

i = Cn(An(t)vεi,n − PnA(t)C−1uε

i ) + (CnPnA(t)C−1uεi −

PnCA(t)C−1uεi ) we have, by condition (H1), lim

n→∞ ‖An(t)uεi,n − PnA(t)uε

i ‖n = 0 uni-

formly on [0, T ]. The fact that the convergence is uniform is proved by using the con-tinuity of A(t)C−1uε

i on [0, T ], (1) and the estimate that supn�1

‖Cn‖n < ∞. Note that if

f ∈ C([a, b]; X), then limn→∞ ‖Pnf (t)‖n = ‖f (t)‖ uniformly on [a, b], by virtue of (1).

Since the convergence of {An(t)uεi,n} and {An(t)v

εi,n} is both uniform on [0, T ], we deduce

from (11) and (13) that limn→∞(αn + 2βn + 2γn) � 5ε. It follows that

lim supn→∞

sup

∥∥∥∥∥∥

[t/hn]∏i=[s/hn]+1

Fn(ihn)Cnxn − PnU(t, s)x

∥∥∥∥∥∥n

: t ∈ [s, T ]

� (M + LK)‖x − C−1y‖ + (L + 3M + 5MT )ε,

by combining (14) and (15), and taking the lim sup as n → ∞. Since C(Y ) is dense in X,it is concluded that for s ∈ [0, T ), we have

limn→∞

∥∥∥∥∥∥[t/hn]∏

i=[s/hn]+1

Fn(ihn)Cnxn − PnU(t, s)x

∥∥∥∥∥∥n

= 0(16)


uniformly for t ∈ [s, T ]. But, we see that for each t ∈ [s, T ], the sequence {[t/hn]∏

i=[s/hn]+1Fn

(ihn)Cnxn}n�1 of functions in s ∈ [0, t] is pseudo-equicontinuous in the following sense:∥∥∥∥∥∥[t/hn]∏

i=[s/hn]+1

Fn(ihn)Cnxn −[t/hn]∏

i=[s/hn]+1

Fn(ihn)Cnxn

∥∥∥∥∥∥n

� M(|s − s| + hn) sup{‖An(τ)vε0,n‖n : τ ∈ [0, T ]} + 2M‖xn − vε

0,n‖n

for s, s ∈ [0, t]. In view of this fact, the uniform convergence of (16) on � is shown by anindirect proof. �

3. Application to degenerate partial differential equations. This section is devotedto an approximation problem of the solution of the following degenerate equation by a finitedifference scheme of Lax-Friedrichs type.{

utt (t, x) = a(t)uxx(t, x) for (t, x) ∈ [0, T ] × R

u(0, x) = u0(x), ut (0, x) = v0(x) for x ∈ R,(17)

where a is a nonnegative, nondecreasing, Lipschitz continuous function on [0, T ].Let X = H 1(R)×L2(R) and Y = H 2(R)×H 1(R). Then it is shown [9] that the family

{A(t) : t ∈ [0, T ]} of closed linear operators in X defined by{(A(t)(u, v))(x) = (v(x), (a(t)u(t, x))xx) for u ∈ D(A(t)),

D(A(t)) = {(u, v) ∈ H 1(R) × H 1(R) : a(t)u ∈ H 2(R)}generates a regularized evolution operator {U(t, s) : (t, s) ∈ �} on X with a regularizingoperator C = ((1 + ∂x)

−1, (1 + ∂x)−1), from which it follows that for each (u0, v0) ∈

H 3(R) × H 2(R), the Cauchy problem (17) has a unique solution u in the class C([0, T ];H 2(R)) ∩ C1([0, T ]; H 1(R)) ∩ C2([0, T ]; L2(R)).

Let {hn} and {kn} be two null sequences of positive numbers such that hn/kn = r , wherer is a positive constant such that (sup{a(t) : t ∈ [0, T ]} + 1)r2 � 1, and consider thedifference scheme of the form

(ui+1,l − (ui,l+1 + ui,l−1)/2)/hn = vi,l

(vi+1,l − (vi,l+1 + vi,l−1)/2)/hn

= a((i + 1)hn)(ui,l+2 − 2ui,l + ui,l−2)/(4k2n)

(18)

for i = 0, 1, . . . , [T/hn] − 1 and l = 0, ±1, ±2, . . . , where

u0,l = 1

kn

(l+1/2)kn∫(l−1/2)kn

u0(x) dx, v0,l = 1

kn

(l+1/2)kn∫(l−1/2)kn

v0(x) dx

for l = 0, ± 1, ±2, . . . . For each n � 1, let Xn be the Banach space l2 × l2 equippedwith the norm ‖(u, v)‖n = (‖u‖2

n + ‖δnu‖2n + ‖v‖2

n)1/2 for (u, v) ∈ l2 × l2, where


δnu = {(ul+1 − ul−1)/(2kn)} and ‖u‖n = (∞∑

l=−∞|ul |2kn)

1/2 for u = {ul} ∈ l2. Then

it is known [4,10] that the sequences {Xn} approximates X by considering the operatorPn ∈ B(X, Xn) defined by Pn(u, v) = ({ul}, {vl}) where

ul = 1

kn

(l+1/2)kn∫(l−1/2)kn

u(x) dx, vl = 1

kn

(l+1/2)kn∫(l−1/2)kn

v(x) dx

for l = 0, ±1, ±2, . . . . The main result in this section is given by

Theorem 3. For (u0, v0) ∈ H 3(R) × H 2(R), the solution (u(t, ·), ut (t, ·)) of (17) isapproximated by a sequence of solutions {({ui,l}, {vi,l})} of (18) in the sense that

limn→∞(‖Pn(u(t, ·), ut (t, ·)) − ({u[t/hn],l}, {v[t/hn],l})‖n = 0

uniformly on [0, T ].

To prove the theorem, for each n � 1 let us define Fn(t) ∈ B(Xn) by Fn(t)({ul}, {vl}) =({wl}, {zl}) where

wl = (ui,l+1 + ui,l−1)/2 + hnvi,l,

zl = (vi,l+1 + vi,l−1)/2 + hna(t)(ui,l+2 − 2ui,l + ui,l−2)/(4k2n),

and an injective operator Cn ∈ B(Xn) by

Cn = ((1 + δn)−1, (1 + δn)

−1).

Then it is shown by Appendix that the operator Cn is well-defined and that the two operatorsFn(t) and Cn satisfy hypotheses (H1) through (H3). The proof is now complete because it

will be shown that the regularized discrete parameter evolution operator {k∏

i=l+1Fn(ihn)Cn :

0 � l � k � [T/hn]} satisfies the regularized stability condition (i) of Theorem 2, byProposition 4 below.

Let h > 0 and {ai}Ni=1 be a sequence satisfying the following conditions:

(i) There exists M > 0 such that 0 � ai � M for 1 � i � N .(ii) There exists L > 0 such that 0 � ai+1 − ai � Lh for 1 � i � N − 1.

Let k > 0 and {(u0,l , v0,l)}∞l=−∞ ∈ l2×l2. Then we define inductively {(ui,l , vi,l)}∞l=−∞ ∈l2 × l2 by

ui+1,l = (ui,l+1 + ui,l−1)/2 + hvi,l(19)

vi+1,l = (vi,l+1 + vi,l−1)/2 + hai+1(ui,l+2 − 2ui,l + ui,l−2)/(4k2)(20)


for 0 � i � N − 1. We use the following three operators on l2 in later arguments.

δu = {(ul+1 − ul−1)/(2k)}∞l=−∞, τ+u = {ul+1}∞l=−∞, τ−u = {ul−1}∞l=−∞

for u = {ul}∞l=−∞ ∈ l2.

Proposition 4. Assume that (M + 1)(h/k)2 � 1 and Nh � T . Then there existsC(T , M, L) > 0 such that

‖ui‖2 + ‖δui‖2 + ‖vi‖2

� C(T , M, L)(‖u0‖2 + ‖δu0‖2 + ‖v0‖2 + ‖δv0‖2)

for 1 � i � N , where ‖u‖ = (∞∑

l=−∞|ul |2k)1/2 for u = {ul}∞l=−∞ ∈ l2.

P r o o f. Set zi = δui for 0 � i � N , where ui = {ui,l}∞l=−∞ for 0 � i � N . Then wehave by (19) and (20)

zi+1 = (τ+zi + τ−zi)/2 + (h/k)(τ+vi − τ−vi)/2(21)

vi+1 = (τ+vi + τ−vi)/2 + ai+1(h/k)(τ+zi − τ−zi)/2(22)

for 0 � i � N − 1, respectively. Now, let 1 � j � N , and set wj = 0 and define induc-tively

wi = (τ+wi+1 + τ−wi+1)/2 − (h/k)(τ+zi − τ−zi)/2(23)

for 0 � i � j − 1. By (19) we have ‖ui+1‖ � ‖ui‖ + h‖vi‖ for 0 � i � j − 1. Addingthese inequalities for 0 � i � j − 1, and using Schwarz’s inequality we find

‖uj‖2 �(

‖u0‖ +j−1∑i=0

h‖vi‖)2

� 2‖u0‖2 + 2jh

j−1∑i=0

h‖vi‖2.(24)

By (21) through (23) we find

(ai+1 + 1)(zi+1,l)2 + (vi+1,l)

2 + 2vi+1,l(wi+1,l+1 + wi+1,l−1)/2

− ai+1((wi+1,l+1 + wi+1,l−1)/2)2

= (ai+1 + 1){(zi,l+1 + zi,l−1)2/4 + ai+1(h/k)2(zi,l+1 − zi,l−1)

2/4}+ 2(ai+1 + 1)(h/k)(zi,l+1vi,l+1 − zi,l−1vi,l−1)/2

+ (vi,l+1 + vi,l−1)2/4 + (ai+1 + 1)(h/k)2(vi,l+1 − vi,l−1)

2/4

+ 2(vi,l+1 + vi,l−1)/2 · wi,l − ai+1(wi,l)2

for 0 � i � j − 1 and l = 0, ±1, ±2, . . . . Let us define

Ei = (ai + 1)‖zi‖2 + ‖vi‖2 + 2〈vi, (τ+wi + τ−wi)/2〉 − ai‖wi‖2


for 0 � i � j , where 〈·, ·〉 is the associated inner product. Since (ai+1 + 1)(h/k)2 � 1 for0 � i � N − 1 and ‖(τ+wi+1 + τ−wi+1)/2‖ � ‖wi+1‖, the above identity implies

Ei+1 − Ei � (ai+1 − ai)‖zi‖2 − (ai+1 − ai)‖wi‖2 � Lh‖zi‖2

for 0 � i � j , where we have used condition (ii) to obtain the last inequality. Summingup the above inequalities from i = 0 to i = j − 1 we have

‖zj‖2 + ‖vj‖2 � (a0 + 1)‖z0‖2 + ‖v0‖2

+ 2〈v0, (τ+w0 + τ−w0)/2〉 + Lh

j−1∑i=0

‖zi‖2.(25)

We shall estimate the third term on the right-hand side. By (23) we have wi = (τ+wi+1 +τ−wi+1)/2 − hδzi for 0 � i � j − 1; hence

〈v0, ((τ+ + τ−)/2)i+1wi〉 − 〈v0, ((τ+ + τ−)/2)i+2wi+1〉= h〈((τ+ + τ−)/2)i+1(δv0), zi〉 � h‖δv0‖‖zi‖ � h(‖δv0‖2 + ‖zi‖2)/2

for 0 � i � j − 1. Adding these inequalities for 0 � i � j − 1 we have

〈v0, ((τ+ + τ−)/2)w0〉 � (T /2)‖δv0‖2 + (1/2)h

j−1∑i=0

‖zi‖2.

We add (24) to the inequality obtained by substituting the above estimate into (25), andthen apply a discrete version of Gronwall’s inequality to the resultant inequality, so that thedesired inequality is proved. �

4. Appendix. Let {kn} be a null sequence of positive numbers as n → ∞, and let En be

the Banach space l2 equipped with the norm ‖u‖n = (∞∑

l=−∞|ul |2kn)

1/2 for u = {ul} ∈ l2.

If we define pn : L2(R) → En by pnu = { 1kn

(l+1/2)kn∫(l−1/2)kn

u(x) dx} then it is known [4, 10] that

‖pnu‖n � ‖u‖ for u ∈ L2(R).

Proposition 5. The following assertions hold for i � 1.

(i) limn→∞ ‖pn∂

ixu − δi

n(pnu)‖n = 0 for u ∈ Hi(R), where the operator δn on En is

defined by δnu = {(ul+1 − ul−1)/(2kn)} for u = {ul} ∈ En.(ii) (1 + δn)

−1 ∈ B(En) and ‖(1 + δn)−1‖n � 1.

(iii) If un ∈ En and u ∈ L2(R) satisfy limn→∞ ‖un − pnu‖n = 0 then we have

limn→∞ ‖(1 + δn)

−iun − pn(1 + ∂x)−iu‖n = 0.


P r o o f. For u ∈ L2(R), let us define (Dnu)(x) = (u(x + kn) − u(x − kn))/(2kn) for

x ∈ R. Then (Dnu)(x) =1∫

0(∂xu)(x + (2θ − 1)kn) dθ , by which we have (Di

nu)(x) =1∫

0· · ·

1∫0(∂i

xu)(x+(2θi −1)kn +· · ·+(2θ1 −1)kn) dθ1 · · · θi . Since δin(pnu) = pn(D

inu) the

desired claim (i) follows from the Lebesgue theorem. The Lax-Milgram theorem assertsthat for u ∈ En, there exists a unique w ∈ En such that (1 + δn)w = u and ‖w‖n � ‖u‖n,which means that assertion (ii) is true. To prove (iii) it suffices to demonstrate the case ofi = 1. For this purpose, let un ∈ En, u ∈ L2(R) and lim

n→∞ ‖un − pnu‖n = 0, and then

set wn = (1 + δn)−1un and w = (1 + ∂x)

−1u. Then we have (1 + δn)(wn − pnw) =(un − pnu) − (pn(Dnw) − pn∂xw), and the second term on the right-hand side vanishesas n → ∞. The desired claim is obtained by using assertion (ii). �

References

[1] P. Chernoff, Product formulas, nonlinear semigroups and addition of unbounded operators. Mem. Amer.Math. Soc. 140 (1974).

[2] A. Chorin, T. Hughes, M. McCracken and J. Marsden, Product formulas and numerical algorithms.Comm. Pure Appl. Math. 31, 205–256 (1978).

[3] R. de Laubenfels, Existence Families, Functional Calculi and Evolution Equations. LNM 1570, Berlin-Heidelberg-New York 1994.

[4] H. O. Fattorini, The Cauchy Problem. Reading, Mass. 1983.[5] T. Kato, Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo 17, 241–258 (1970).[6] T. Kurtz, Extension of Trotter’s operator semigroup approximation theorems. J. Func. Anal. 3, 354–375

(1969).[7] I. Miyadera and Y. Kobayashi, Convergence and approximation of nonlinear semigroups. In: Proc.

Japan-France Seminar on Functional Analysis and Numerical Analysis, 277–295. Japan Society for thePromotion of Science, Tokyo 1978.

[8] N. Tanaka, Linear evolution equations in Banach spaces. Proc. London Math. Soc. 63, 657–672 (1991).[9] N. Tanaka, A class of abstract quasi-linear evolution equations of second order. J. London Math. Soc. 62,

198–212 (2000).[10] H. Trotter, Approximation of semi-groups of operators. Pacific J. Math. 8, 887–919 (1958).

Received: 4 December 2001

Naoki TanakaDepartment of MathematicsFaculty of ScienceOkayama UniversityOkayama [email protected]

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Approximation of regularized evolution operators