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Volume 2011, Article ID 414906, 12 pagesdoi:10.1155/2011/414906 Research Article Strong Stability and Asymptotical Almost Periodicity of Volterra Equations in Banach Spaces Jian-Hua Chen

Volume 2011, Article ID 414906, 12 pages

doi:10.1155/2011/414906

Research Article

Strong Stability and Asymptotical Almost

Periodicity of Volterra Equations in Banach Spaces

Jian-Hua Chen1 and Ti-Jun Xiao2

1 School of Mathematical and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China

2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Correspondence should be addressed to Ti-Jun Xiao,xiaotj@ustc.edu.cn

Received 1 January 2011; Accepted 1 March 2011

Academic Editor: Toka Diagana

Copyrightq 2011 J.-H Chen and T.-J Xiao This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We study strong stability and asymptotical almost periodicity of solutions to abstract Volterra equations in Banach spaces Relevant criteria are established, and examples are given to illustrate our results

1 Introduction

Owing to the memory behavior cf., e.g., 1, 2 of materials, many practical problems in

engineering related to viscoelasticity or thermoviscoelasticity can be reduced to the following

Volterra equation:

ut  Aut 

t

0

at − sAusds, t ≥ 0, u0  x

1.1

in a Banach space X, with A being the infinitesimal generator of a C0-semigroup Tt defined

on X, and a· ∈ L pR, C a scalar function R : 0, ∞ and 1 ≤ p < ∞, which is often

called kernel function or memory kernelcf., e.g., 1 It is known that the above equation is

well-posed This implies the existence of the resolvent operator St, and the mild solution is

then given by

which is actually a classical solution if x ∈ DA In the present paper, we investigate strong

stability and asymptotical almost periodicity of the solutions For more information and related topics about the two concepts, we refer to the monographs3,4 In particular, their connections with the vector-valued Laplace transform and theorems of Widder type can be found in4 6 Recall the following

Definition 1.1 Let X be a Banach space and f :R → X a bounded uniformly continuous

function

i f is called almost periodic if it can be uniformly approximated by linear combinations

of e ibt x b ∈ R, x ∈ X Denote by APR, X the space of all almost periodic

functions onR

ii f is called asymptotically almost periodic if f  f1 f2with limt → ∞ f1t  0 and f2∈

APR, X Denote by APPR, X the space of all asymptotically almost periodic

functions onR

iii We call 1.1 or St strongly stable if, for each x ∈ DA, limt → ∞ Stx  0 We

call 1.1 or St asymptotically almost periodic if for each x ∈ DA, S·x ∈

APPR, X.

The following two results on C0-semigroup will be used in our investigation, among which the first is due to Inghamsee, e.g., 7, Section 1 and the second is known as Countable

Spectrum Theorem3, Theorem 5.5.6 As usual, the letter i denotes the imaginary unit and iR the imaginary axis

Lemma 1.2 Suppose that A generates a bounded C0-semigroup Tt on a Banach space X If σA ∩

iR  ∅, then



TtA−1 −→ 0, t −→ 0. 1.3

Lemma 1.3 Let Tt be a bounded C0-semigroup on a reflexive Banach space X with generator A If σA ∩ iR is countable, then Tt is asymptotically almost periodic.

2 Results and Proofs

Asymptotic behaviors of solutions to the special case of at ≡ 0 have been studied

systematically, see, for example,3, Chapter 4 and 8, Chapter V The following example shows that asymptotic behaviors of solutions to1.1 are more complicated even in the finite-dimensional case

Example 2.1 Let X  C, A  −2I, at  −e −tin1.1 Then taking Laplace transform we can calculate

ut  131 2e −3t

It is clear that the following assertions hold.

a The corresponding semigroup Tt  e −2tis exponentially stable

b Each solution with initial value x ∈ DA, x / 0 is not strongly stable and hence not

exponentially stable

c Each solution with x ∈ DA is asymptotically almost periodic.

It is well known that the semigroup approach is useful in the study of 1.1 More information can be found in the book8, Chapter VI.7 or the papers 9 11

LetX : X × L pR, X be the product Banach space with the norm









x f







2

: x2f2

L pR ,X 2.2

for each x ∈ X and f ∈ L pR, X Then the operator matrix

A :

⎛

⎜A δ0

ds

⎞

⎟

⎠, DA : DA × W 1,pR, X 2.3

generates a C0-semigroup onX Here, W 1,pR, X is the vector-valued Sobolev space and δ0

the Dirac distribution, that is, δ0f  f0 for each f ∈ W 1,pR, X; the operator B is given

by

Denote bySt the C0-semigroup generated byA It follows that, for each x ∈ DA, the first

coordinate of



ut

Ft, ·

 : St



x

0



2.5

is the unique solution of1.1

Theorem 2.2 Let A be the generator of a C0-semigroup Tt on the Banach space X and a· ∈

L pR, C with 1 ≤ p < ∞ Assume that

i M is a left-shift invariant closed subspace of L pR, X such that a·Ax ∈ M for all

x ∈ DA;

ii C⊂ ρA|D and

Rλ, A|D ≤ |λ| K , λ ∈ C 2.6

for some constant K > 0 Here, D : DA × {f ∈ W 1,pR, X ∩ M : f ∈ M} :

DA × M1.

Then

a 1.1 is strongly stable if iR ⊂ ρA|D;

b if X is reflexive and 1 < p < ∞, then every solution to 1.1 is asymptotically almost

periodic provided σA|D ∩ iR is countable.

Proof Since the first coordinate of2.5 is the unique solution of 1.1, it is easy to see that the strong stability and asymptotic almost periodicity of1.1 follows from the strong stability and asymptotic almost periodicity ofSt, respectively.

Moreover, from 9, Proposition 2.8 we know that if M is a closed subspace of

L pR, X such that M is S l t-invariant and a·Ax ∈ M for all x ∈ DA, then A|D the restriction ofA to D generates the C0-semigroup



St : St|M, 2.7 which is defined on the Banach space

Thus, by assumptionsi, ii and the well-known Hille-Yosida theorem for C0-semigroups,

we know that St is bounded Hence, in view ofLemma 1.2, we get



 StA|−1

D −→ 0, t −→ ∞. 2.9 Clearly

A|−1 D



Ax a·Ax







x

0



2.10

for each x ∈ DA So, combining 2.5 with 2.9, we have

ut2≤







ut

Ft, ·







2





St



x

0







2





StA| −1

D



Ax a·Ax







2

≤ StA|−1

D2

·1 a·2

L p



· Ax2

−→ 0, t −→ ∞.

2.11

This means thata holds

On the other hand, we note that, to get b, it is sufficient to show that St is asymptotically almost periodic Actually, if X is reflexive and 1 < p <∞, then it is not hard

to verify that L pR, X is reflexive Hence, X × L pR, X is reflexive By assumption i, M

is a closed subspace of X × L pR, X Thus, Pettis’s theorem shows that M is also reflexive.

Hence, in view ofLemma 1.3, we getb This completes the proof

Corollary 2.3 Let A be the generator of a C0-semigroup Tt on the Banach space X and at 

αe −βt β > 0, α / 0 Assume that

i for each λ ∈ C, λ, λλ  β/λ  α  β ∈ ρA,

ii there exists a constant C > 0 satisfying

Hλ2 |α|2

β · λ  α  β2 · I − λHλ2≤ C

with

Hλ : λ  α  β λ  β



λ

λ  β

λ  α  β − A

−1

Then

a if Reα  β / 0 and

λ

−1  iβ

for each λ ∈ R, then 1.1 is strongly stable;

b if X is reflexive and 1 < p < ∞, then 1.1 is asymptotically almost periodic provided



λ ∈ R : iλiλ  βiλ  α  β−1∈ σA 2.15

is countable.

Proof As in9, Section 3, we take

M :e −βs x : x ∈ X⊂ L pR, X. 2.16

In view of the discussion in8, Lemma VI.7.23, we can infer that

C⊂ ρA|D, if C⊂ ρA,

λ

λ  β

Moreover, we have

Rλ, A 

⎛

⎜ I − aλRλ, AA

R



λ, d

ds



BI − aλRλ, AA−1 I

⎞

⎟

⎛

⎜

⎜

Rλ, A Rλ, Aδ0R

λ, d

ds





λ, d

ds



⎞

⎟

⎟



⎛

⎜

⎜



λ, d

ds



R

λ, d

ds



BHλ Rλ, d

ds



BHλδ0R

λ, d

ds



 Rλ, d

ds



⎞

⎟

⎟.

2.18

Hence,

Rλ, A|D2≤



Hλ2 |α|2

β · λ  α  β2 · I − λHλ2



×



2β

λ  β2  1



λ  β1 2

2.19

with Hλ being defined as in 2.13 Thus, it is clear that  St is bounded if 2.12 is satisfied Next, for λ∈ R, we consider the eigenequation

iλ − A|D



x f







y g



Writing f  e −βs f0and g  e −βs g0, we see easily that2.20 is equivalent to

iλ − Ax − f0  y,

−αAx iλ  βf0 g0. 2.21 Thus, if Reα  β / 0 and

λ

−1  iβ

then by2.21 we obtain

x α  β  iλ−1



λ

−1  iβ

α  β  iλ − A

−1



iλ  βy  g0

,

f0α  β  iλ−1iλ − A



λ

−1  iβ

α  β  iλ − A

−1



iλ  βy  g0

− y.

2.23

By the closed graph theorem, the operator

iλ − A



λ

−1  iβ

α  β  iλ − A

−1

2.24

in the second equality of2.23 is bounded Hence, noting that

iλ  βy  g02≤ 2iλ  βy2g02

 2iλ  β2·y2 2βg2

,

2.25

we have

Consequently, in view ofa ofTheorem 2.2, we know that1.1 is strongly stable if 2.14 holds

Furthermore, by9, Lemma 3.3, we have

σA|D ⊂λ ∈ C : λλ  βλ  α  β−1∈ σA∪−α  β. 2.27

Combining this with b of Theorem 2.2, we conclude that 1.1 is asymptotically almost

periodic if X is reflexive, 1 < p <∞, and the set in 2.15 is countable

Theorem 2.4 Let A be the generator of a C0-semigroup Tt on the Banach space X and a· ∈

L pR, C with 1 ≤ p < ∞ Assume that

i for all λ ∈ C,

aλ / − 1, λ1  aλ−1∈ ρA,

sup

λ>0, n0,1,2,







λ n1 λHλ − 1 n λ

n!





< ∞,

2.28

λHλ, λ2Hλ is bounded on C, where Hλ : λ − 1  aλA−1,

ii qλ is analytic on Cand λqλ, λ2qλ are bounded on C, where

for each iη ∈ iE iE is the set of half-line spectrum of Hλ and α > 0,

iii for each x ∈ X and iη ∈ iE, the limit

lim

α → 0 αe αiηt H

exists uniformly for t ≥ 0.

Then every solution to1.1 is asymptotically almost periodic Moreover, if for each x ∈ X and

iη ∈ iE the limit in 2.30 equals 0 uniformly for t ≥ 0, then 1.1 is strongly stable

Proof Take x ∈ DA Then the solution Stx to 1.1 is Lipschitz continuous and hence

uniformly continuous Actually, by assumptioni, we know that

∞

0

e −λt Stxdt  λ − 1  aλA−1x, Re λ > 0, 2.31 and that

is analytic onC Thus, r ∈ C∞0, ∞, X and

Hλx − 1λ x  rλ

∞

0

e −λt Stx − xdt, λ > 0. 2.33

On the other hand, if2.28 holds, then there exists K > 0 such that

sup

λ>0







λ n1 r n λ

n!





 ≤supλ>0







λ n1 λHλ − 1 n λ

n!





x

≤ Kx, n  0, 1, 2,

2.34

Hence, from4, Chapter 1 or 5 and the uniqueness of the Laplace transform, it follows that

satisfies

rλ

∞

0

and that

Ft  h − Ft  St  hx − Stx ≤ Khx, t ≥ 0, h ≥ 0. 2.37

Moreover, by 3, Corollary 2.5.2, the assumption i implies the boundedness of St Therefore,

is bounded and uniformly continuous on0, ∞ In addition, the half-line spectrum set of ft

is just the following set:

iη ∈ iR : Hλ cannot be analytically extened to an eighborhood of iη. 2.39

Write τ  α  iη Then

∞

0

e −τs ft  sds  e τt∞

t e −τs fsds

 e τt





fτ −

t

0

e −τs fsds



 e τt Hτx −e τ· ∗ ft,

2.40

qλ  λ − α − iη Hλx  e τ· ∗ fλ. 2.41

From assumptionii and 3, Corollary 2.5.2, it follows that eτ· ∗ ft is bounded, which

implies

lim

α → 0 α

e τ· ∗ ft  0 2.42

uniformly for t ≥ 0 Finally, combining 2.40 with Theorem 7, Theorem 4.1, we complete the proof

3 Applications

In this section, we give some examples to illustrate our results

First, we applyCorollary 2.3toExample 2.1 As one will see, the previous result will

be obtained by a different point of view

Example 3.1 We reconsiderExample 2.1 First, we note that

This implies that condition Reα  β / 0 is not satisfied Therefore, part a ofCorollary 2.3

is not applicable, and this explains partially why the corresponding Volterra equation is not strongly stable However, it is easy to check that conditionsi and ii inCorollary 2.3are satisfied In particular, we have accordingly

and hence the estimate

Hλ2 |α|2

β · λ  α  β2 · I − λHλ2 |λ  1|2 4

|λλ  3|2 ≤ 13/9

|λ|2 , λ ∈ C. 3.3

Note σA  {−2} and



λ ∈ R : iλiλ  βiλ  α  β−1∈ σA {λ ∈ R : iλ  1  −2}  ∅. 3.4

Applying part b of Corollary 2.3, we know that the corresponding Volterra equation is asymptotically almost periodic

Example 3.2 Consider the Volterra equation

∂u

∂t  t, x  ∂ ∂x2u2t, x  α

t

0

e −βt−s ∂2u

∂x2s, xds, t > 0, 0 ≤ x ≤ π,

u0, t  uπ, t  0, t ≥ 0, u0, x  u0x, 0 ≤ x ≤ π,

3.5

where the constants satisfy

Let H  L20, π, and define

A : d2

dx2, DA f ∈ H20, π : f0  fπ  0. 3.7

Then3.5 can be formulated into the abstract form 1.1 It is well known that A is self-adjoint

see, e.g., 12, page 280,b of Example 3 and that A generates an analytic C0-semigroup

The self-adjointness of A implies



λ − A−1  1

distλ, σA, λ ∈ ρA. 3.8

On the other hand, we can compute

σA  σ p A −n2: n  1, 2, . 3.9

It follows immediately that conditioni inCorollary 2.3holds Moreover, corresponding to

2.13, we have

Combining this with3.6, 3.8, and 3.9, we estimate

Hλ2 |α|2

β · λ  α  β2 · I − λHλ2

 1

|λ|2



λ  β2·

λ  β − A−12

 βI −

λ  βλ  β − A−12

≤ 1

|λ|2

 λ  β2

λ  β  12  β



1 λ  β2

λ  β  12



≤ 1 2β

|λ|2 , λ ∈ C.

3.11

Note that2.15 becomes



λ ∈ R : iλ  β  −n2 for some n∈ N ∅. 3.12

Applying partb ofCorollary 2.3, by3.11, we conclude that 3.5 is asymptotically almost periodiccf 9, Remark 3.6

Acknowledgments

This work was supported partially by the NSF of China11071042 and the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics08DZ2271900

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