Approximation Theorems for Exponentially Bounded a Times Integrated Cosine Function

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http://dx.doi.org/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: [email protected]

Received 26 September 2014; revised 25 October 2014; accepted 31 October 2014

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

In this paper, based on the theories of α-times Integrated Cosine Function, we discuss the ap-proximation 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

[ ]

t∈ 0,∞ , 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,

m∈N. In this paper, we refine the theory by introducing α-times Integrated Cosine Function for positive real numbers α. Moreover, if the semigroups are equicontinuous at each point t∈

[ ]

0,∞ , we give different me-thods toprove the theorem.

Throughout this paper, we will denote by X—a Banach space with norm  , by B X

( )

—the Banach space of all bounded linear operators from X to X; A is a linear operator in X, by

( ) ( ) ( )

, ,

D A R A ρ A ,R

(

λ,A

)

(2)

2. Preliminaries

Definition 2.1. Let α∈R+, then a strongly continuous family

{ }

( )

0

t

S t _≥ in B X

( )

is called an α-times In-tegrated Cosine Function, if the following hold:

1) S

( )

0 =0;

2) For any x∈X , and ∀s t, ≥0,

( ) ( )

_{( ) ( )}

{

(

)

( )

(

)

(

)

( )

(

)

( )

(

)

( )

}

1 0 1 0 0 0

1 1

0 0

1

2 1 d

d

d d .

t s t s t s

t s

S s S t t s r S r x r

t s r S r x r

t s r S r x r t s r S r x r

α α α α α α − − + − − − = − − − Γ + − − + − + − + + − +

∫

∫ ∫ ∫

∫

Definition 2.2. A is a linear operator in X, α∈R+, A is called the generator of an α -times Integrated Cosine Function if there are nonnegative numbers ω,M and a mapping S: 0,

[

∞ →

)

B X

( )

such that

1)

{ }

( )

0

t

S t _≥ is strongly continuous and

( )

0 d e

t _wt

S s s ≤M

∫

for all t≥0; 2)

(

ω,∞

)

is contained in the resolvent set of A;

3)

(

2

)

1

( )

0

, e t d

R λ A =λα−

∫

∞ −λS t t for λ ω> .

Lemma 2.3.[9] For each n∈N let fn∈L1loc

(

[

0,∞

)

,X

)

, with

( )

0 d e , 0

t _t

n

f s s ≤M ω t≥

∫

and let

( )

₀ e t

( )

d ,

n n

F λ =

∫

∞ −λ f t t λ ω>

Assume that

( )

lim _n exists for ,

n

F λ λ ω

→∞ >

and that for a fixed t0∈

(

0,∞

)

, sup n

( )

0 n N

f t

∈ < ∞, and

(

)

( )

(

0 0

)

0 0

1

lim h _n _n d 0

h→ _h

∫

f t + −s f t s=

with uniform concergence for n∈N. Then lim n

( )

0

m→∞f t exists.

Lemma 2.4.[10] If A is a linear operator in X, α ≥0. The following assertions are equivalent: 1) There exist constant ω,M ≥0, such that

(

ω2,∞ ⊂

)

ρ

( )

A , and

(

)

1

(

1

(

2

)

( )

, k !

k

R A Mk

α

λ ω₋ + λ− λ _≤

.

for λ ω> ，k∈N₀=N

{ }

0 .

2) ∀ ∈β

(

α α, +1

]

, A generate a β-times Integrated Cosine Function

{

( )

}

0 t

S_β t

≥ , and exist constant k

such that α +1-times Integrated Cosine Function

{

Sα +1

( )

t

}

_t_≥₀ hold

(

)

( )

(

)

1 1

0 1

lim sup e t 0, 0 .

h _h S t h S t k t h ω

α+ α+

→ + − ≤ ≥ ≥

3. Main Results

Theorem 3.1. If An generates a α-times Integrated Cosine Function

{

Sn

( )

t

}

_t_≥₀, and there is M,ω R

+

(3)

such that

( )

e ,t n

S t ≤M ω then the following statements are equivalent: 1)

(

2

)

(

2

)

0

lim , _n ,

n→∞R λ A x=R λ A x,

,

x X

∀ ∈ for some λ0 >ω, and

{

Sn

( )

t

}

_t_≥₀ is equicontinuous at each

point t∈

[ ]

0,∞ ;

2)

(

2

)

(

2

)

0

lim , _n ,

n→∞R λ A x=R λ A x , ∀ ∈x X, λ ω> , and

{

Sn

( )

t

}

t≥0 is equicontinuous at each point

[ ]

0,

t∈ ∞ ;

3) lim n

( )

0

( )

n→∞S t x=S t x, ∀ ∈x X uniformly on compacts of t≥0.

Proof: 1) ⇒ 2) Consider the set

(

)

(

)

{

2 2

}

0

: lim , n , , ,

n R A x R A x x X

λ λ λ λ ω

→∞

Ω = = ∀ ∈ > ,

which is nonempty by assumption. Let µ∈ Ω, then

(

) (

)

2 2 2 2 2 2 2 2

,

n n n n

A A I R A A

λ − =µ − +λ −µ =_ − µ −λ µ _ µ −

when

(

)

2 2 2 1 n R A µ λ µ − < −

(

₂

) (

₂

) (

₂ ₂

) (

₂

)

1

(

₂ ₂

) (

₂

)

1

0

, n n , n k , n k

k

R λ A R µ A I µ λ R µ A µ λ R µ A

∞

− +

=

 

= − _ − − _ =

∑

−

Obviously R

(

λ2,An

)

converges as n→ ∞. Therefore, the set Ω is open.

On the other hand, taking an accumulation point λ of Ω with λ ω> , we can find µ∈Ω, such that

(

)

2 2 2 1 n R A µ λ µ − <

− . By the above considerations, λ must belong to Ω, i.e., Ω is relatively closed in

{

:

}

S= λ λ ω> , which leads to the conclusion.

2) ⇒ 3) Let

( )

1

(

2

)

( )

0

, e t d ,

n n n

F λ =λ− +α R λ A =

∫

∞ −λS t t

for

(

)

(

)

( )

2 2

0

lim , ,

lim

n n

R A x R A x

S t x

λ λ

→∞

=

,

and

{

}

0 ( ) n _t

S t

≥ is equicontinuous at each point t∈

[ ]

0,∞ ; using Lemma 2.2, it is easy to know that

( )

lim _n

n→∞S t x exists. We now fix b>0, then for each ε >0, ∃ ∈K N; when , ,

[ ]

0,

b

t s t s b

K

− ≤ ∈ , we have

( )

3 m m

S t −S s <ε (1)

Pick t_i ib

[ ]

0,b i, 1, 2,3, ,K k

= ∈ =  , then ∃ ∈N0 N such that

( )

, , 0, 1, 2,3, , .

3 n i l i

S t −S t <ε n l≥N i= k (2)

From (1) (2), we have

( )

3 n l

S t −S t <ε ，n l, ≥N₀，t∈

[ ]

0,b . It shows that 3) is right.

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We have

( )

0

( )

3 n N

S s −S s <ε , s∈

[

0,t+1

]

.

For Sn

( )

t is continuous on

[

0,t+1

]

, then ∃ >δ0 0, s t− <δ0, when s∈

[

0,t+1

]

We have

( )

3 n n

S s −S t <ε ，n=1, 2,3,,N₀

Therefore, if n≥N0，s∈

[

0,t+1

]

, then

( )

₀

( )

₀

( )

₀

( )

₀

( )

n n n N N N N n

S s −S t ≤ S s −S S + S s −S t + S t −S t <ε

In conclusion

{

S_n

( )

t n, ∈N

}

is equicontinuous at t. By using the dominated convergence theorem, we obtain

( )

1

(

2

)

( )

0

0 0

lim , e t d e t d

n n n

n F R A S t t S t t

α λ λ

λ λ− + λ ∞ − ∞ −

→∞ = =

∫

=

∫

So 2) is right.

2) ⇒ 1) the proof is obvious. The proof is completed.

Corollary 3.2. If An is the generator of α-times Integrated Cosine Function

{

Sn

( )

t

}

_t_≥₀ satisfying:

(

)

( )

_e ( )t h _, _{, ,} _0,

₍

_0,1

_]

n n

S t+h −S t ≤M ω + hγ n∈N t h≥ γ∈ (3)

Then (1)-(3) are equivalent: 1)

(

2

)

(

2

)

0

lim , _n ,

n→∞R λ A x=R λ A x

，∀ ∈x X, for some λ0>ω.

2)

(

2

)

(

2

)

0

lim , _n ,

n→∞R λ A x=R λ A x

，∀ ∈x X, λ ω> .

3) lim n

( )

0

( )

n→∞S t x=S t x

，∀ ∈x X, uniformly on compacts of t≥0.

Theorem 3.3. If An is the generator of α-times Integrated Cosine Function

{

Sn

( )

t

}

_t_≥₀, and there is

,

M ω∈R+ such that

( )

e ,t n

S t ≤M ω ∀ ∈x X, λ ω> ,

{

( )

}

0

n _t

S t _≥ is equicontinuous at each point t∈

[ ]

0,∞ .

(

2

)

( )

2

lim , _n

n→∞R λ A x=R λ x exist, for some λ0 ω

> , kerR

( )

λ =02

{ }

0 , then there is a linear operator A—ge-

nerator of α-times Integrated Cosine Function S t

( )

, such that lim _n

( )

n→∞S t x=S t x, ∀ ∈x X, and uniformly

on compacts of t≥0.

Proof: By

(

2

)

( )

2

lim , n

n→∞R λ A x=R λ x, from the resolvent identity, we have

(

2

) (

2

) (

2 2

) (

2

) (

2

)

, n , n , n , n

R λ A −R µ A = µ −λ R λ A R µ A

then R

( ) ( ) (

λ2 −R µ2 = µ2−λ2

) ( ) ( )

R λ2 R µ2 , λ µ ω, > hence kerR

( )

λ2 and RangR

( )

λ2 independent

λ. Since kerR

( )

λ =02

{ }

0 , then there is a linear operator A,

( )

2

D A =RangR λ , R

( ) (

λ2 x= λ2I−A

)

−1x. By Definition 2.2, we know that

(

)

( )

1 2

0

, e t d , , ,

n n

R A x S t x t x X

α λ

λ− λ ₌

_∫

∞ − _{∀ ∈} λ ω_>

(4)

for lim

(

2, _n

)

( )

2

n→∞R λ A x=R λ x exist, by the proof of the Theorem 3.1, we obtain that

( )

lim _n

(5)

hence 1

(

2

)

( )

0

, e t d

R A x S t x t

α λ

λ− λ ₌

_∫

∞ −

, ∀ ∈x X, λ ω> .

then A generates a α-times Integrated Cosine Function

{ }

( )

0

t

S t _≥ , such that lim _n

( )

n→∞S t x=S t x, ∀ ∈x X,

and uniformly on compacts of t≥0.

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[3] Davies, E.B. and Pang, M.M.H. (1987) The Cauchy Problem and a Generalization of the Hill-Yosida Theorem. Pr o-ceedings of the London Mathematical Society, 55, 181-208. http://dx.doi.org/10.1112/plms/s3-55.1.181

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http://dx.doi.org/10.1007/s10092-008-0152-5

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