Approximations of Laplace transforms and integrated semigroups

Approximations of Laplace transforms and
integrated semigroups Ti-Jun Xiao and Jin Liang
Department of Mathematics, University of Science and Technology of China
Hefei 230026, People's Republic of China
e-mail:[email protected]; [email protected]
Let ffm; m 2 N g be a sequence of functions from [0; 1) to a Banach space E . We
give a new and essential condition on fm , which is weaker than the usual \local Lipschitz continuity" condition, ensuring that the convergence of fm is equivalent to the
convergence of their Laplace transforms. This enables us to establish new approximation theorems for r-times integrated semigroups on E , for all r 0. As a consequence,
an open problem for the convergence of integrated semigroups on the whole space E ,
is solved in essence. Moreover, we present an application to nonhomogeneous Cauchy
problems.
1. INTRODUCTION
One of the fundamental theorem in the theory of operator semigroups and abstract
Cauchy problems is the Trotter-Kato theorem, which tells us that for a sequence of C0
semigroups fSm (); m 2 N g on a Banach space E satisfying the stability condition
kSm (t)k Me!t ; m 2 N; t 0;
(1:1)
(M , ! are constants), it converges (pointwise on E and uniformly on bounded intervals
of t 0) if and only if the sequence of resolvents R(; Am ) converges (pointwise on E and
for > !), where Am is the generator of Sm (t). Since the Trotter-Kato theorem came
out in the late fties, we have seen many interesting variants (cf., e.g., [8, 11, 13, 14, 17,
19]). Several years ago, a Laplace transform version of the Trotter-Kato theorem (LTV
Trotter-Kato theorem in short) appeared in [3, 9, 15]. It says that the convergence of a
sequence of functions fm : [0; 1) ! E is equivalent to the convergence of their Laplace
This work was supported partly by the National NSF of China, the Key Project Foundation of the
Chinese Academy of Sciences and the K. C. Wong Education Foundation
R
transforms 01 e?t fm (t)dt, whenever fm satisfy the following local Lipschitz continuity
condition:
fm(0) = 0; kfm(t + h) ? fm(t)k Mhe!(t+h) ; m 2 N; t; h 0:
(1:2)
We emphasize that the constants M , ! in (1.2) are independent of m as well as t and h.
Moreover, it is clear that condition (1.2) implies
kfm (t)k Me !
t;
( +1)
m 2 N; t 0:
(1:3)
The LTV Trotter-Kato theorem was used in [15] to obtain the extensions of the TrotterKato theorem to r-times integrated semigroups for r 2 N . Such extensions were also
given in [4, 16] with proofs of operator theoretical nature. However, there exists a weakpoint among these theorems (by the reason that the generators of integrated semigroups,
unlike C0 semigroups, may fail to possess dense domains). That is, the convergence of
integrated semigroups Sm () on the whole space E (pointwise) is obtained only under the
additional condition that Sm () are locally Lipschitz continuous (with constants independent of m) besides (1.1) and the convergence of the resolvents. \Without this condition,
it is an open problem, which seems to be dicult to solve" ([16, p. 314]).
The present paper aims at the open problem. First in Section 2, we consider a
similar problem in a more general setting based on Laplace transforms. One advantage
of the consideration is that a theorem for Laplace transforms produces results for many
types of operator families and linear evolution equations in the same time. We develop
completely the LTV Trotter-Kato theorem mentioned above, for a sequence of Banach
space valued functions ffm ; m 2 N g fullling (1.3), by replacing (1.2) with the much
weaker one that ffm ; m 2 N g is equicontinuous at each point t 2 [0; 1) and showing
that the equicontinuity is necessary for ffm ; m 2 N g to converge uniformly on bounded
t-intervals. Moreover, we give an example to illustrate that without the equicontinuity,
the convergence of fm (t) can not be guaranteed even for a single t and even in the
scalar case. A typical example for the equicontinuity is the local Holder continuity
(with the related constants independent of m), which is weaker than the local Lipschitz
continuity. In the proof of our theorem, we use the idea of reducing the approximation to
a stationary problem, which originates from [12] and has been propagated by Goldstein
(see [6, 7]). It was also used in [15] to obtain the previous LTV Trotter-Kato theorem, by
invoking the integrated version (see [1]) of the classical Widder representation theorem
for Laplace transforms. It can be seen that our approach is much more straightforward
and concise, with the aid of the classical Post-Widder inversion formula for Laplace
transforms. This work in terms of Laplace transforms enables us to establish, in Section
3, new approximation theorems for r-times integrated semigroups for all r 0, which
cover the corresponding results in [4, 15, 16] (see Remark 3.12). As a consequence,
we solve the above open problem in essence (see Theorem 3.6). Finally, an interesting
application to nonhomogeneous Cauchy problems is given in Section 4.
Throughout this paper, E is a Banach space, A a linear operator in E , and M , !, r are
nonnegative constants. We will write L(E ) for the space of all bounded linear operators
from E to E . By D(A), R(A), N (A), (A) and R(; A), we denote respectively the
domain, the range, the kernel, the resolvent set and the resolvent of A. N will be the set
of positive integers and N0 := N [ f0g. [r] denotes the least integer > r ? 1. Finally
(t ? s) g(s)ds if > ?1;
0 ?( + 1)
g(t)
if = ?1:
8 Z
>
<
(j g)(t) := >
:
t
2. APPROXIMATION OF LAPLACE TRANSFORMS
Lemma 2.1. For each m 2 N let fm 2 L1loc ([0; 1); E ) with
Z
t
fm(s)ds Me!t ; t 0;
0
and let
Fm () =
Assume that
Z
0
1 ?t
e fm (t)dt;
mlim
!1 Fm ()
> !:
exists for > !;
(2:1)
(2:2)
(2:3)
and that for a xed t0 2 (0; 1),
Z
1
lim
h!0 h
h
0
(fm (t0 + s) ? fm (t0 )) ds = 0
(2:4)
with uniform convergence for m 2 N . Then limm!1 fm (t0 ) exists.
Proof. Set
E = fu := (um )m2N E ; sup kum k < 1g:
m2N
It is clear that E is a Banach space under the norm kukE := supm2N kum k. Let
E = fu := (um )m2N E ; mlim
!1 um existsg:
0
Then E0 is a closed subspace of E. we dene the function f : [0; 1) ! E by f (t) =
(fm (t))m2N ; t 2 [0; 1): It follows from (2.1) and (2.4) that
Z
t
0
lim 1
h!0 h
f (s)ds Me!t ; t 0;
E
Z
h
0
(f (t0 + s) ? f (t0 )) ds = 0:
(2:5)
(2:6)
R
Write F() = 01 e?t f (t)dt; > !: Then F() = (Fm ())m2N ( > !). By (2.3) we
have that
F() 2 E0; > !:
(2:7)
Applying the vector valued version of the Post-Widder inversion formula for Laplace
transforms (cf. [3, 9] or [21, Ch.1]), we obtain
k
k k :
k1 k
F
f (t ) = klim
(
?
1)
!1
k! t
t
+1
0
( )
0
0
So, (2.7) implies that f (t0 ) 2 E0 , since E0 is closed. This nishes the proof.
Theorem 2.2. For each m 2 N , let fm 2 C ([0; 1); E ) satisfying (2:1) and let Fm be
as in (2:2). Then the following assertions are equivalent.
(i) ffm ; m 2 N g is equicontinuous at each point t 2 [0; 1), and limm!1 Fm () exists
for > !.
(ii) limm!1 fm (t) exists for t 0 and the convergence is uniform on bounded tintervals.
Proof. (i) ) (ii). An application of Lemma 2.1 yields immediately that for each
t 2 (0; 1), limm!1 fm (t) exists. We now x b > 0. Then for each " > 0, there exists
k" 2 N such that
kfm (t) ? fm(s)k < 3" ; m 2 N; t; s 2 [0; b] with jt ? sj bk"? ;
1
(2:8)
since ffm ; m 2 N g is equicontinuous on [0; b]. Pick ti = ki" b 2 [0; b], i = 1; : : : ; k" .
Then there is m" 2 N such that
kfm (ti) ? fl (ti)k < 3" ; m; l m"; i = 1; : : : ; k" :
(2:9)
Combining (2.8) and (2.9) gives
kfm (t) ? fl (t)k < "; m; l m"; t 2 [0; b]:
Consequently, fm (t) converges as m ! 1 uniformly on [0,b].
(ii) ) (i). Fix t 2 [0; 1). Then for each " > 0, there exists m0 2 N such that
kfm(s) ? fm0 (s)k < 3" ; m m ; s 2 [0; t + 1]:
0
On the other hand, there exists 0 > 0 such that
kfm (s) ? fm(t)k < 3" ; m = 1; 2; : : : ; m ; s 2 [0; t + 1] with js ? tj < ;
0
0
(2:10)
due to fm being continuous on [0; t + 1]. Hence for m > m0 and s as in (2.10),
kfm (s) ? fm(t)k kfm (s) ? fm0 (s)k + kfm0 (s) ? fm0 (t)k + kfm0 (t) ? fm(t)k
< 3" + 3" + 3" = ":
In conclusion, ffm ; m 2 N g is equicontinuous at t. Clearly,
Z t
Z 1
?
1
?
t
Fm () = e
fm(s)ds dt;
0
0
> !; m 2 N:
Thus we obtain the existence of limm!1 Fm () for > !, by (2.1) and the dominated
convergence theorem. The proof is then complete.
The following example shows that without the equicontinuity of fm in Theorem 2.2
(i), the existence of limm!1 Fm () alone does not imply the convergence of fm (t), even
for a single t and even in the scalar case.
Example
2.3. Let E = R and fm (t) = cos(mt + m), m 2 N . Then fm 2 C ([0; 1); E )
R
and 0t fm (s)ds = m1 j sin(mt + m) ? sin mj 2, t 2 [0; 1): We see that for > 0,
Z 1
Fm () = e?t cos(mt + m)dt = 2 +1 m2 ( cos m ? m sin m)
0
?! 0; as m ! 1:
But for any t 2 [0; 1), fm (t) does not converge.
2 (0; 1]. For each m 2 N , let fm : [0; 1) ! E satisfy (2:1)
and kfm (t + h) ? fm (t)k Me! t h h (t; h 0), and let Fm be as in (2:2). Then the
Corollary 2.4. Let ( + )
following statements are equivalent.
(i) limm!1 Fm () exists for > !.
(ii) There exists f : [0; 1) ! E with kf (t + h) ? f (t)k Me!(t+h) h (t; h 0) such
R
that limm!1 Fm () = 01 e?t f (t)dt uniformly on [! + ; 1) for any > 0.
(iii) limm!1 fm (t) exists for all t 0.
(iv) There exists f : [0; 1) ! E with kf (t + h) ? f (t)k Me!(t+h) h (t; h 0) such
that limm!1 fm (t) = f (t) uniformly on bounded intervals of t 0.
Proof. From the hypothesis we know that ffm ; m 2 N g is equicontinuous at each
point t 2 [0; 1). Accordingly, the equivalence of (i), (iii) and (iv) follows from Theorem
2.2.
(iv) ) (ii). It is easy to be seen by observing that for each xed > 0
Fm () ?
Z
0
Z 1
1 ?t
e f (t)dt e?(!+)t kfm (t) ? f (t)kdt;
0
2 [! + ; 1):
(ii) ) (i). Obviously.
The proof is then complete.
3. APPROXIMATION OF INTEGRATED SEMIGROUPS
Definition 3.1. Let r 0. If (!; 1) (A) and there exists a strongly continuous
family S () : [0; 1) ! L(E ) with kS (t)k Me!t for all t 0 such that
R(; A)u = r
Z
0
1 ?t
e S (t)udt;
> !; u 2 E;
then we say that A is the generator of an r-times integrated semigroup S (), and denote
by A 2 Gr (M; !).
It is known that 0-times integrated semigroups coincide with C0 semigroups, and that
Z t
r
t
S (0) = 0; S (t)u = ?(r + 1) u + S (s)Auds; t 0; u 2 D(A):
0
(3:1)
Definition 3.2. We say that A 2 Hr (M; !) if (!; 1) (A) and
?
?r R(; A)
(m) Mm!( ? !)?m? ; > !; m 2 N :
1
0
Proposition 3.3 (Real characterization). Let r 2 (r; r + 1]: Then A is the generator
of an r-times integrated semigroup S () satisfying
k(S jr?r ) (t + h) ? (S jr?r ) (t)k Me! t h h; t; h 0;
( + )
if and only if A 2 Hr (M; !). In this case, we also have
M
!(t+h) hr?r ; t; h 0:
kS (t + h) ? S (t)k (r ? r2)?(
r ? r) e
For the proofs of Proposition 3.3, please see [10] or [21, Ch. 1].
Proposition 3.4 (A complex condition). If (A) f 2 C; Re > !g (! > 0) and
kR(; A)k M jjr ; Re > !;
then for each > 1, there exists M0 depending only on M ,r, such that A 2 Gr+(M0 ; !),
and so A 2 Hr+ (M0 ; !).
For the proofs of Proposition 3.4, please see [2] or [21, Ch. 1].
Lemma 3.5. Let Am 2 Gr (M; !) (m 2 N ). Assume that limm!1 R(0 ; Am )u exists
for some 0 > ! and for all u 2 E . Then limm!1 R(; Am )u exists for all > ! and
u 2 E.
Proof. Using the same type of arguments as in [4, Theorem 3.1], [5, Proposition 3.4.4]
or [7, Theorem 1.7.8], we can prove this lemma.
Theorem 3.6. Let A; Am
2 Gr (M; !), m 2 N . Let S () and Sm () be the r-times
integrated semigroups generated by A and Am respectively. Then the following statements
are equivalent.
(i) For all u 2 E , limm!1 R(; Am )u = R(; A)u for some / all > !; and fSm (); m 2
N g is equicontinuous at each point t 2 [0; 1).
(ii) For all u 2 E , limm!1 Sm (t)u = S (t)u uniformly on compacts of t 0.
Proof. Apply Theorem 2.2 with fm (t) = Sm (t)u (m 2 N ) and Fm () = ?r R(; Am )u.
Corollary 3.7. Suppose that S () and Sm () are r-times integrated semigroups on
E with generators A and Am respectively, satisfying
kSm(t + h) ? Sm(t)k Me! t h h ; m 2 N; t; h 0;
( + )
(3:2)
for some 2 (0; 1]. Then the following assertions are equivalent.
(i) limm!1 R(; Am )u = R(; A)u for all u 2 E and for some / all > !:
(ii) limm!1 Sm (t)u = S (t)u for all u 2 E and uniformly on compacts of t 0.
Corollary 3.8. Let A, Am , S () and Sm () be as in Theorem 3.6. Let D be a core
for A. Then the implications
(i) () (ii) ) (iii)
hold among the following statements.
(i) limm!1 R(; Am )u = R(; A)u for all u 2 E and for some / all > !:
(ii) For each u 2 D, there exists um 2 D(Am ) (m 2 N ) such that limm!1 um = u and
limm!1 Am um = Au.
(iii) limm!1 Sm (t)u = S (t)u for all u 2 D(A) and uniformly on compacts of t 0.
When D(A) is dense in E , the statements (i), (ii) and (iii) are equivalent.
Proof. The equivalence of (i) and (ii) is known (cf. [5, Theorem 3.4.8], [15, Theorem
2.3]).
(i))(iii). Fix u 2 D(A) and put v = ( ? A)u. Observe that for t 0, m 2 N ,
Sm(t)u = Sm(t)R(; A)v
r
= Sm (t)(R(; A)v ? R(; Am )v) + ?(rt+ 1) R(; Am )v
?
Z
0
t
Sm(s)vds + Z
t
0
Sm(s)R(; Am )vds
by (3.1). Hence we deduce easily, by (i) and
kSm (t)k Me!t (t 0; m 2 N );
(3:3)
that fSm ()u; m 2 N g is equicontinuous at each point t 2 [0; 1). With the help of
Theorem 2.2 we obtain that (iii) holds true for all u 2 D(A), and so for all u 2 D(A)
owing to (3.3).
When D(A) is dense in E , D(A) = E: Therefore (iii) implies (i) in view of Theorem
3.6.
2 N , let Am 2 Gr (M; !) and Sm() be the r-times
integrated semigroup generated by Am , such that for every u 2 E and t 0, fSm ()u; m 2
N g is equicontinuous at t. Let > !: Assume that for every u 2 E , the limit
Theorem 3.9. For each m
0
R(0 )u := mlim
!1 R(0 ; Am )u exists ; with N (R(0 )) = 0:
(3:4)
Then there exists A 2 Gr (M; !) generating an r-times integrated semigroup S () such
that limm!1 Sm (t)u = S (t)u for all u 2 E and uniformly on compacts of t 0.
Proof. By Denition 3.1 we have
Z 1
?
r
R(; Am )u = e?t Sm(t)udt;
0
> !; u 2 E; m 2 N:
(3:5)
Moreover, we know by Lemma 3.5 that for any u 2 E , R()u := limm!1 R(; Am )u
exists for all > !: So an appeal to Theorem 2.2 yields that for each u 2 E , S (t)u :=
limm!1 Sm (t)u exists uniformly on compacts of t 0.
It is clear that R() is a psendo resolvent on > !. Dene
A = 0 I ? R(0 )?1 :
It can be seen that R() = R(; A), > !. Consequently, we obtain by (3.5) that for
> ! and u 2 E ,
Z 1
?r R(; A)u = e?t S (t)udt:
(3:6)
This justies the required conclusion.
0
Corollary 3.10. Suppose that Sm (), m 2 N , are r-times integrated semigroups on
E with generators Am , satisfying (3:2). If (3:4) holds, then the conclusion of Theorem
3.9 holds.
Corollary 3.11. For each m 2 N , let Am
2 Gr (M; !) with (3:4) holding, and let
Sm() be the r-times integrated semigroup generated by Am. Then there is a closed linear
operator A such that the part of A in D(A) generates an r-times integrated semigroup
S () on D(A) and
mlim
!1 Sm (t)u = S (t)u; u 2 D(A)
uniformly on compacts of t 0.
If R(R(0 )) is dense in E , in addition, then the conclusion of Theorem 3:9 holds.
Proof. We know, from the proofs of Corollary 3.8 and Theorem 3.9, that there exists
A with limm!1 R(; Am )u = R(; A)u, > !; and that for each u 2 D(A) and t 0,
fSm ()u; m 2 N g is equicontinuous at t. Accordingly, the limit
S (t)u := mlim
!1 Sm (t)u (u 2 D(A))
exists uniformly on compacts of t 0, in view of Theorem 2.2. Therefore (3.6) holds for
> ! and u 2 D(A), which implies the rst conclusion.
The density of R(R(0 )) implies the density of A. Thus the second conclusion follows
from the rst one.
Remark 3.12. For = 1 and r = 1, Corollary 3.7 and Corollary 3.10 coincide with
Theorem 3.3 and Corollary 3.5 in [4]. In the case of r 2 N0 , Corollary 3.8 and Corollary
3.11 can be found in [15, 16].
4. APPROXIMATION OF SOLUTIONS OF NONHOMOGENEOUS
CAUCHY PROBLEMS
Let x 2 E and f 2 C ([0; b]; E ) (where b > 0). We consider the nonhomogeneous
Cauchy problem
8
< u0 (t) = Au(t) + f (t); t 2 [0; b];
(4:1)
: u(0) = x:
By a mild solution of (4.1), we understand a function u() 2 C ([0; b]; E ) satisfying that
Rt
u(s)ds 2 D(A) for all t 2 [0; b] and
0
u(t) = x + A
Z
0
t
u(s)ds +
Z
0
t
f (s)ds; t 2 [0; b]:
Lemma 4.1. Assume that A 2 Hr (M; !), f (0) 2 D A[r] and
f (t) = f (0) + (jr?1 g) (t); t 2 [0; b]
for some
8
<
g2:
Let x 2 D A[r] with A[r]x 2
solution, given by
8
<
C ([0; b]; E ) if r 2 [0; 1);
L1 ([0; b]; E ) otherwise :
D(A) if r 2 N Then (4:1) admits a unique mild
: D (A) if r 62 N :
0
0
u(t) = dtd S (t)A[r] x + S (t)A[r] f (0) + dtd S jr?[r]?1 g (t) + h(t); t 2 [0; b];
where S () is the ([r] + 1)-times integrated semigroup generated by A, and
8
>
>
>
<
h(t) := >
>
>
:
0
rX
?1
[ ]
j =0
!
tj+1 Aj f (0) + tj Aj x
(j + 1)!
j!
if r 2 [0; 1);
otherwise :
Proof. Proceed analogously as in the proof of [10, Theorem 4.6].
Theorem 4.2. Let the hypotheses of Lemma 4:1 hold for A, f , g, x, and also for
Am , fm , gm , xm (m 2 N ) in place of A, f , g, x respectively. Suppose
(i) limm!1 R(; Am )v = R(; A)v for all v 2 E and for some > !:
(ii) limm!1 Aim fm (0) = Ai f (0); 0 i [r], and limm!1 kgm ? gkL1 ([0;b];E ) = 0.
(iii) limm!1 Aim xm = Ai x (0 i r) and D(Am ) D(A) (m 2 N ), if r 2 N0 ;
limm!1 Aim xm = Ai x (0 i [r] + 1), if r 62 N0 .
Then the mild solution um (t) of
8
<
u0m(t) = Am um(t) + fm(t); t 2 [0; b]
: u (0) = x
m
m
converges to the mild solution u(t) of (4:1) uniformly for t 2 [0; b].
(4:2)
Proof. We denote by Sm (); m 2 N , the ([r] + 1)-times integrated semigroups generated by Am . For convenience, write
A0 = A; S0 = S; f0 = f; g0 = g; x0 = x:
We have from Proposition 3.3 that for any m 2 N0 and t; h 0,
kSm (t + h) ? Sm(t)k M e! t h h r ?r ;
1
( + )
[ ]
+1
(4:3)
kSem(t + h) ? Sem(t)k Me! t h h;
(4:4)
where M is a constant and Sem := Sm jr? r ? . This, together with condition (i), yields
by Corollary 3.7 that for each v 2 E ,
( + )
1
[ ]
1
mlim
!1 Sm (t)v = S0 (t)v
uniformly on compacts of t 0;
(4:5)
mlim
!1 Sm (t)v = S0 (t)v
uniformly on compacts of t 0:
(4:6)
e
e
Moreover, it follows from (4.6) that, given 2 C ([0; b]; E ),
e
e
mlim
!1 Sm (t) = (S0 )(t); t 2 [0; b]:
(4:7)
We observe by (4.4) that fSem ; m 2 N g is equicontinuous on [0; b], and so conclude
(cf. the proof of Theorem 2.2) that the convergence in (4.7) is uniform for all t in [0; b].
Lemma 4.1 says that for every m 2 N0 and t 0,
um (t) = dtd Sm (t)A[mr] xm + dtd Sem gm (t)
+Sm (t)A[mr] fm (0) + hm (t);
where
8
>
>
>
<
hm (t) := >
>
>
:
0
rX
?1
[ ]
j =0
tj+1 Aj f (0) + tj Aj x
(j + 1)! m m
j! m m
!
(4:8)
if r 2 [0; 1);
otherwise:
It is immediate, by (4.5), conditions (ii) and (iii), that
[r ]
[r ]
mlim
!1 Sm (t)Am fm (0) = S0 (t)A0 f0 (0);
mlim
!1 hm (t) = h0 (t);
(4:9)
d Se g (t) = d Se g (t); uniformly for t 2 [0; b]:
lim
m!1 dt m m
dt 0 0
(4:10)
uniformly on the t-interval [0; b].
Next, we show that
To this end, we dene the linear mappings ?m (m 2 N0 ): L1 ([0; b]; E ) ! C ([0; b]; E ) by
(?m ) (t) = dtd Sem (t); t 2 [0; b]; 2 L1 ([0; b]; E ):
?m are well dened by (4.4) and the density of C 1 ([0; b]; E ) in L1 ([0; b]; E ). Observing
by (4.4) that for any m 2 N0 ,
k?m kC
;b ;E )
([0 ]
we see that
= max
t2[0;b] limh!0+
Me!b kkL1
Z
0
t ?1 e
e
h Sm(t ? s + h) ? Sm (t ? s) (s)ds
2 L1 ([0; b]; E );
([0;b];E ) ;
k?mkL1 !C Me!b ; m 2 N :
0
(4:11)
Using (4.7) and the statement below it yields that if 2 C 1 ([0; b]; E ),
k?m ? ? kC
0
h
i
= maxt2[0;b] Sem ? Se0 0 (t)
([0;b];E )
?! 0 as m ! 1:
In combination with (4.11), this implies that for all 2 L1 ([0; b]; E ),
k?m ? ? kC
0
;b ;E ) ?! 0
([0 ]
as m ! 1;
since C 1 ([0; b]; E ) is dense in L1 ([0; b]; E ). Accordingly,
k ? m gm ? ? g k C
0 0
;b ;E )
([0 ]
k?m kL1 !C kgm ? g kL1
?! 0 as m ! 1;
0
;b ;E ) + k?m g0 ? ?0 g0 kC ([0;b];E )
([0 ]
by condition (ii). Therefore (4.10) is true.
It remains to be shown that
d
d
[r ]
[r ]
mlim
!1 dt Sm (t)Am xm = dt S0 (t)A0 x0 ; uniformly for t 2 [0; b]:
(4:12)
If r 62 N0 , then xm 2 D(A[mr]+1) (m 2 N0 ) by hypothesis, so that (4.12) follows from (4.5)
and condition (iii) immediately, because
d S (t)A[r] x = t[r] A[r]x + S (t)A[r]+1 x ; t 2 [0; b]; m 2 N ;
0
dt m m m [r]! m m m m m
by (3.1). Let now r 2 N0 . we have by hypothesis that for any m 2 N0 ,
Arm xm 2 D(Am );
r
r
mlim
!1 Am xm = A0 x0 ;
and
D(Am ) D(A ):
Consider the linear mappings m (m 2 N ): D(Am ) ! C ([0; b]; E ), given by
0
(4:13)
(4:14)
0
(m v) (t) = dtd (Sm (t)v) ; t 2 [0; b]; v 2 D(Am ):
From (4.3) and noting [r] ? r + 1 = 1, we obtain that for m 2 N0 and v 2 D(Am ),
km vkC
hence
;b ;E )
([0 ]
= maxt2[0;b] limh!0+ h?1 (Sm (t + h) ? Sm (t)) v M1e!b kvk;
km k M e!t ; m 2 N :
1
0
(4:15)
We x v 2 D(A0 ). The condition (i) gives, by Corollary 3.8, that there exist vm 2 D(Am )
such that vm ! v and A0 vm ! A0 v: On the other hand, we have by (3.1) that
r
(m vm ) (t) = tr! vm + Sm (t)Am vm ; t 2 [0; b]; m 2 N0 :
So using (4.5) gives
mlim
!1 km vm ? 0 vkC ([0;b];E ) = 0:
It follows from (4.14) and (4.15) that
k m v ? v k C
0
;b ;E )
([0 ]
k m k k v m ? v k + k m vm ? v k C
?! 0 as m ! 1;
0
;b ;E )
([0 ]
which implies that for all v 2 D(A0 ),
mlim
!1 km v ? 0 vkC ([0;b];E ) = 0:
Consequently
km (Arm xm ) ? (Ar x )kC
0
0
0
;b ;E )
([0 ]
kmk kArm xm ? Ar x k
+ km (Ar x ) ? (Ar x )kC
?! 0 as m ! 1;
0
0
0
0
0
0
0
;b ;E )
([0 ]
by (4.13) and (4.15). This ends the proof.
Corollary 4.3. Let 0 < r < 1. For each m 2 N0 , let Am 2 Hr (M; !), xm 2 D(Am )
and
fm(t) = fm (0) + (jr?1 gm ) (t); t 2 [0; b];
where gm 2 C ([0; b]; E ), A0 := A, x0 := x, f0 := f , and g0 := g: Suppose
(i) limm!1 R(; Am )v = R(; A)v for all v 2 E and for some > !:
(ii) limm!1 fm (0) = f (0) and limm!1 kgm ? gkL1 ([0;b];E ) = 0.
(iii) limm!1 xm = x and limm!1 Am xm = Ax.
Then the conclusion of Theorem 4:2 holds.
Corollary 4.4. For each m
2 N , let Am 2 H (M; !), xm 2 D(Am ) and fm 2
0
0
C ([0; b]; E ), where A0 := A, x0 := x, and f0 := f . Suppose
(i) limm!1 R(; Am )v = R(; A)v for all v 2 E and for some > !:
(ii) limm!1 kfm ? f kL1 [0;b];E ) = 0.
(iii) limm!1 xm = x and D(Am ) D(A) (m 2 N ):
Then the conclusion of Theorem 4:2 holds.
Proof. Note that, in this case, the term dtd Sem gm (t) + Sm (t)A[mr] fm (0) in (4.8)
can be rewritten as dtd Sem fm (t).
Remark 3.5. Specialized to the case of fm = f and xm = x for all m 2 N , Corollary
4.4 improves Theorem 4.3 in [4], by replacing the condition that D(Am ) D(A) and
Sm(t) : D(A) ! D(A); with D(Am ) D(A):
REFERENCES
1. W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J.Math. 59
(1987), 327-352.
2. W. Arendt and H. Kellermann, Integrated solutions of Volterra integro-dierential equations and applications, in \Volterra Integro-dierential Equations in Banach Spaces and
Applications"(G. Da Prato and M. Iannelli, Eds.), pp. 21-51, Proc. Conf. Trento, Pitman,
1989.
3. B. Baumer and F. Neubrander, Laplace transform methods for evolution equations, Conference del Seminario di Matematica dell' Universita di Bari 259 (1994), 27-60.
4. S. Busenberg and B. H. Wu, Convergence theorems for integrated semigroups, Dierential
and Integral Equations 5 (1992), 509-520.
5. K. -J. Engel and R. Nagel, \One-parameter Semigroups and Linear Evolution Equations,"
GMT, Springer, Berlin, New York, to appear.
6. J. A. Goldstein, On the convergence and approximation of cosine functions, Aequationes
Math. 10 (1974), 201-205.
7. J. A. Goldstein, \Semigroups of Linear Operators and Applications," Oxford Univ. Press,
New York, 1985.
8. M. Hasegawa, On the convergence of resolvents of operators, Pacic J. Math. 21 (1967),
35-47.
9. B. Hennig and F. Neubrander, On representations, inversion and approximations of Laplace
transforms in Banach spaces, Applicable Analysis 49 (1993), 151-170.
10. M. Hieber, Laplace transforms and -times integrated semigroups, Forum Math. 3 (1991),
595-612.
11. T. Kato, \Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1966.
12. J. Kisynski, A proof of the Trotter-Kato theorem on approximation of semi-groups, Colloq.
Math. 18 (1967), 181-184.
13. T. Kurtz, Extensions of Trotter's operator semigroups approximation theorems, J. Funct.
Anal. 3 (1969), 354-375.
14. T. Kurtz, A general theorem of the convergence of operator semigroups, Trans. Amer.
Math. Soc. 148 (1970), 23-32.
15. C. Lizama, On the convergence and approximation of integrated semigroups, J. Math. Anal.
Appl. 181 (1994), 89-103.
16. S. Nicaise, The Hille-Yosida and Trotter-Kato theorems for integrated semigroups, J. Math.
Anal. Appl. 180 (1993), 303-316.
17. S. Oharu and H. Sunouchi, On the convergence of semigroups of linear operators, J. Funct.
Anal. 6 (1970), 292-304.
18. A. Pazy, \Semigroups of Linear Operators and Applications to Partial Dierential Equations," Springer-Verlag, New York, 1983.
19. T. Seidmann, Approximation of operator semi-groups, J. Funct. Anal. 5 (1970), 160-166.
20. D. V. Widder,\The Laplace Transform," Princeton University Press, New Jersey, 1946.
21. T. J. Xiao and J. Liang, \The Cauchy problem for higher order abstract dierential equations," Lect. Notes in Math. 1701, Springer, Berlin, New York, (1998).

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