1. Introduction
Integrated semigroups were introduced by Arent [1] [2] and Davies and Pang [3] in 1987. The approximation theorem is one of the fundamental theorems in the theory of operater semigroups. There have been many results on approximation [4] - [7] . Cao [8] obtained the approximation theorem for m-times Integrated Cosine Function,
. In this paper, we refine the theory by introducing α-times Integrated Cosine Function for positive real numbers
. Moreover, if the semigroups are equicontinuous at each point
, we give different methods to prove the theorem.
Throughout this paper, we will denote by
—a Banach space with norm
, by
—the Banach space of all bounded linear operators from
to
;
is a linear operator in
, by
,
respectively the domain, the range, the resolvent set, and the resolvent of
.
2. Preliminaries
Definition 2.1. Let
, then a strongly continuous family
in
is called an
-times Integrated Cosine Function, if the following hold:
1)
;
2) For any
, and
,
![]()
Definition 2.2.
is a linear operator in
,
,
is called the generator of an
-times Integrated Cosine Function if there are nonnegative numbers
and a mapping
such that
1)
is strongly continuous and
for all
;
2)
is contained in the resolvent set of
;
3)
for
.
Lemma 2.3. [9] For each
let
, with
![]()
and let
![]()
Assume that
![]()
and that for a fixed
,
, and
![]()
with uniform concergence for
. Then
exists.
Lemma 2.4. [10] If
is a linear operator in
,
. The following assertions are equivalent:
1) There exist constant
, such that
, and
.
for
,
.
2)
,
generate a
-times Integrated Cosine Function
, and exist constant
such that
-times Integrated Cosine Function
hold
![]()
3. Main Results
Theorem 3.1. If
generates a
-times Integrated Cosine Function
, and there is
such that
then the following statements are equivalent:
1)
,
for some
, and
is equicontinuous at each point
;
2)
, ![]()
, and
is equicontinuous at each point
;
3)
,
uniformly on compacts of
.
Proof: 1) Þ 2) Consider the set
,
which is nonempty by assumption.
Let
, then
![]()
when ![]()
![]()
Obviously
converges as
. Therefore, the set
is open.
On the other hand, taking an accumulation point
of
with
, we can find
, such that
. By the above considerations,
must belong to
, i.e.,
is relatively closed in
, which leads to the conclusion.
2) Þ 3) Let ![]()
for
,
and
is equicontinuous at each point
; using Lemma 2.2, it is easy to know that
exists. We now fix
, then for each
,
; when
, we have
(1)
Pick
, then
such that
(2)
From (1) (2), we have
,
,
.
It shows that 3) is right.
3) Þ 2) fix
, for each
,
, when
.
We have
,.
For
is continuous on
, then
,
, when ![]()
We have
,
Therefore, if
,
, then
![]()
In conclusion
is equicontinuous at
.
By using the dominated convergence theorem, we obtain
![]()
So 2) is right.
2) Þ 1) the proof is obvious.
The proof is completed.
Corollary 3.2. If
is the generator of
-times Integrated Cosine Function
satisfying:
(3)
Then (1)-(3) are equivalent:
1)
,
for some
.
2)
,![]()
.
3)
,
uniformly on compacts of
.
Theorem 3.3. If
is the generator of
-times Integrated Cosine Function
, and there is
such that
,
is equicontinuous at each point
.
exist, for some
,
, then there is a linear operator
—ge- nerator of
-times Integrated Cosine Function
, such that
,
and uniformly on compacts of
.
Proof: By
, from the resolvent identity, we have
![]()
then
hence
and
independent
. Since
, then there is a linear operator
,
,
.
By Definition 2.2, we know that
(4)
for
exist, by the proof of the Theorem 3.1, we obtain that
exist,
hence
,![]()
.
then
generates a
-times Integrated Cosine Function
, such that
,
and uniformly on compacts of
.