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.