Advances in Pure Mathematics
Vol.04 No.11(2014), Article ID:51410,4 pages
10.4236/apm.2014.411066
Approximation Theorems for Exponentially Bounded α-Times Integrated Cosine Function
Lufeng Ling
School of Mathematics and Information Science, Shangqiu Teachers College, Shangqiu, Henan, China
Email: sqsxlfl@126.com
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 26 September 2014; revised 25 October 2014; accepted 31 October 2014
ABSTRACT
In this paper, based on the theories of α-times Integrated Cosine Function, we discuss the approximation theorem for α-times Integrated Cosine Function and conclude the approximation theorem of exponentially bounded α-times Integrated Cosine Function by the approximation theorem of n-times integrated semigroups. If the semigroups are equicontinuous at each point, we give different methods to prove the theorem.
Keywords:
α-Times Integrated Cosine Function, Exponentially Bounded, Approximation
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
.
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