Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2008, Article ID 405092, 13 pages
doi:10.1155/2008/405092
Research Article
Existence and Uniqueness of Pseudo Almost
Automorphic Mild Solutions to Some Classes of
Partial Hyperbolic Evolution Equations
Zhanrong Hu1, 2 and Zhen Jin1, 2
1
2
Department of Mathematics, North University of China, Taiyuan 030051, China
School of Mechatronic Engineering, North University of China, Taiyuan 030051, China
Correspondence should be addressed to Zhanrong Hu, huzhanrong22@126.com
Received 1 June 2008; Revised 4 July 2008; Accepted 10 September 2008
Recommended by Leonid Shaikhet
We will establish an existence and uniqueness theorem of pseudo almost automorphic mild
solutions to the following partial hyperbolic evolution equation d/dtut ft, But Aut gt, Cut, t ∈ R, under some assumptions. To illustrate our abstract result, a concrete
example is given.
Copyright q 2008 Z. Hu and Z. Jin. 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.
1. Introduction
Let X, · be a Banach space, and let Xα , ·α for 0 < α < 1 be an arbitrary abstract
intermediate Banach space between DA domain of the operator A and X. In this paper,
we deal with sufficient conditions for the existence and uniqueness of pseudo almost
automorphic mild solutions to equation
d
ut ft, But Aut gt, Cut,
dt
t ∈ R,
1.1
where A is a sectorial linear operator on a Banach space X and σA ∩ iR ∅, B, C are
bounded linear operators on Xα , f : R × X → Xβ , g : R × X → X are jointly continuous
functions. This turns out to be a nontrivial problem due to the complexity and importance of
pseudo almost automorphic functions e.g., see 1–8 and the references therein.
Upon making some additional assumptions, it will be shown that 1.1 admits a
unique Xα -valued pseudo almost automorphic mild solution. Applications include the study
2
Discrete Dynamics in Nature and Society
of pseudo almost automorphic mild solutions to some nonlinear heat equation:
ut t, x uxx t, x aut, x gt, qxut, x,
ut, 0 ut, π 0,
t ∈ R, x ∈ 0, π,
t ∈ R,
1.2
where a ∈ R is a constant, and g, q are continuous functions.
As a natural and important generalization of almost automorphy as well as pseudo
almost periodicity, pseudo almost automorphy has recently been investigated for more
details, see 1–3. Moreover, Liang et al. in 2 established a composition theorem about
pseudo almost automorphic functions; Xiao et al. in 3 obtained sufficient conditions for
the existence and uniqueness of pseudo almost automorphic mild solutions to semilinear
differential equations in Banach spaces. In particular, they proved in 3 that the space
PAAR, X, ·∞ of pseudo almost automorphic functions, endowed with sup norm, is a
Banach space, thus pushed the door open to study pseudo almost automorphic mild solutions
to various differential equations.
The method used here was firstly developed in 9, in which Diagana established
an existence and uniqueness theorem of pseudo almost periodic mild solutions to 1.1
under some similar assumptions. Combining bounded invariance theorem of pseudo
almost automorphic functions established in 6 with composition theorem of pseudo
almost automorphic functions and completeness of the space PAAR, X, ·∞ , this√paper
extends the results in 9, since such function as gt, u u sin1/2 cos t cos 2t 2 2
maxk∈Z {e−t±k } cos u, t ∈ R, u ∈ X is not pseudo almost periodic, while they are pseudo
almost automorphic.
It should be pointed out that one can also make use of 1.1 to study several types
of evolution equations including partial functional differential equations, integro-differential
equations, and reaction diffusion equations.
2. Preliminaries
In this section, we collect some preliminary facts from 9–11 that will be used later.
Throughout this paper, N, Z, R, and C stand for the sets of positive integer, integer, real and
complex numbers; X, ·, Y, ·Y stand for Banach spaces. If A is a linear operator on X,
then ρA, σA, DA, kerA, RA stand for the resolvent set, spectrum, domain, kernel,
and range of A. The space BX, Y denotes the Banach space of all bounded linear operators
from X into Y equipped with natural norm ABX,Y supx∈X, x / 0 AxY /x. If Y X, it
is simply denoted by BX with ABX supx∈X, x / 0 Ax/x.
2.1. Sectorial linear operators and analytic semigroups
Definition 2.1. A linear operator A : DA ⊂ X → X not necessarily densely defined is said
to be sectorial if the following hold. There exist constants ω ∈ R, θ ∈ π/2, π, and M > 0
such that
ρA ⊃ Sθ,ω : {λ ∈ C : λ /
ω, | argλ − ω| < θ},
M
Rλ, A ≤
, λ ∈ Sθ,ω ,
|λ − ω|
where Rλ, A λI − A−1 for each λ ∈ ρA.
2.1
Z. Hu and Z. Jin
3
Remark 2.2. If A is sectorial, then it generates an analytic semigroup T tt≥0 , which maps
0, ∞ into BX such that there exist M0 , M1 > 0 with
T t ≤ M0 eωt ,
t > 0,
tA − ωIT t ≤ M1 eωt ,
t > 0.
2.2
Definition 2.3. A semigroup T tt≥0 is said to be hyperbolic, if there exist a projection P and
constants M, δ > 0 such that each T t commutes with P , kerP is invariant with respect to
T t, T t : RQ → RQ is invertible and
T tP x ≤ Me−δt x
for t ≥ 0,
T tQx ≤ Meδt x for t ≤ 0,
2.3
where Q : I − P and T t : T −t−1 for t ≤ 0.
Recall that if a semigroup T tt≥0 is analytic, then T tt≥0 is hyperbolic if and only
if σA ∩ iR ∅ see, e.g., 11, Definition 1.14 and Proposition 1.15, page 305.
2.2. Intermediate Banach spaces
Definition 2.4. Let α ∈ 0, 1. A Banach space Xα , ·α is said to be an intermediate space
between DA and X, if DA ⊂ Xα ⊂ X and there is a constant C > 0 such that
xα ≤ Cx1−α xαA ,
x ∈ DA,
2.4
where ·A is the graph norm of A.
Concrete examples of Xα include DAα for α ∈ 0, 1, the domains of the fractional
powers of A, the real interpolation spaces DA α, ∞, α ∈ 0, 1, defined as follows:
DA α, ∞ :
x ∈ X : xα sup t1−α A − ωIe−ωt T tx < ∞ ,
0<t≤1
2.5
xα x xα ,
·α
and the abstract Hölder spaces DA α : DA
.
Lemma 2.5. For the hyperbolic analytic semigroup T tt≥0 , there exist constants Cα > 0, δ >
0, Mα > 0, and γ > 0 such that
T tQxα ≤ Cαeδt x for t ≤ 0,
T tP xα ≤ Mαt−α e−γt x
for t > 0.
2.6
4
Discrete Dynamics in Nature and Society
Lemma 2.6 see 9. Let 0 < α < β < 1. For the hyperbolic analytic semigroup T tt≥0 , there exist
constants c > 0, δ > 0, and γ > 0 such that
AT tQxα ≤ ceδt xβ
for t ≤ 0,
AT tP xα ≤ ctβ−α−1 e−γt xβ
2.7
for t > 0.
3. Pseudo almost automorphic functions
In this section, we recall some recent results on almost automorphic functions and pseudo
almost automorphic functions. Let CR, X denote the collection of continuous functions from
R into X. Let BCR, X denote the Banach space of all X-valued bounded continuous functions
equipped with the sup norm u∞ : supt∈R ut for each u ∈ BCR, X. Similarly, CR ×
X, X denotes the collection of continuous functions from R × X into X, BCR × X, X denotes
the collection of all bounded continuous functions f : R × X → X.
Definition 3.1 see 8. A function f ∈ CR, X is said to be almost automorphic if for every
sequence of real numbers σn n∈N , there exists a subsequence sn n∈N such that
lim
lim ft sn − sm ft
m → ∞ n → ∞
for each t ∈ R.
3.1
This limit means that
gt lim ft sn 3.2
ft lim gt − sn 3.3
n → ∞
is well defined for each t ∈ R, and
n → ∞
for each t ∈ R. The collection of all such functions will be denoted by AAR, X.
Theorem 3.2 see 7. Assume f, g : R → X are almost automorphic, and λ is any scalar. Then the
following holds true:
: f−t are almost automorphic;
1 f g, λf, fτ t : ft τ and ft
2 the range Rf of f is precompact, so f is bounded;
3 if {fn } is a sequence of almost automorphic functions and fn → f uniformly on R, then f
is almost automorphic.
Theorem 3.3 see 7. If one equips AAR, X with the sup norm u∞ supt∈R ut, then
AAR, X turns out to be a Banach space.
One sets
1 r
AA0 R, X : f ∈ BCR, X : lim
fsds 0 .
r → ∞ 2r −r
3.4
Z. Hu and Z. Jin
5
Definition 3.4 see 2. A function f ∈ BCR, X is said to be pseudo almost automorphic if
it can be decomposed as f g φ, where g ∈ AAR, X and φ ∈ AA0 R, X. The collection of
such functions will be denoted by PAAR, X.
Theorem 3.5 see 3. If one equips PAAR, X with the sup norm u∞ supt∈R ut, then
PAAR, X turns out to be a Banach space.
Definition 3.6 see 2. A function f ∈ CR × X, X is said to be almost automorphic if
f ∈ CR × X, X is almost automorphic in t ∈ R uniformly for all x ∈ K, where K is any
bounded subset of X. That is to say, for every sequence of real numbers σn n∈N there exists a
subsequence sn n∈N such that
gt, x lim ft sn , x
n → ∞
3.5
is well defined in t ∈ R, for all x ∈ K, and
ft, x lim gt − sn , x
n → ∞
3.6
for each t ∈ R and x ∈ K. Denote by AAR × X, X the collection of all such functions.
One also defines AA0 R × X, X as the collection of functions f ∈ BCR × X, X such
that
1
lim
r → ∞ 2r
r
−r
ft, xdt 0
3.7
uniformly for any bounded subset of X.
Definition 3.7 see 2. A function f ∈ BCR × X, X is said to be pseudo almost automorphic
if it can be decomposed as f g φ, where g ∈ AAR × X, X and φ ∈ AA0 R × X, X. The
collection of such functions will be denoted by PAAR × X, X.
Theorem 3.8 see 2. Assume f g φ ∈ PAAR × X, X with gt, x ∈ AAR × X, X,
φt, x ∈ AA0 R × X, X, satisfying the following conditions:
1 gt, x is uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R;
2 ft, x is uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R.
If x· ∈ PAAR, X, then f·, x· ∈ PAAR, X.
Remark 3.9. If Xα for α ∈ 0, 1 is an intermediate space between DA and X, BCR, Xα ,
AAR, Xα , PAAR, Xα are equipped with the α-sup norm: u∞,α supt∈R utα , then they
all constitute Banach spaces in view of Definition 2.4, Theorems 3.3 and 3.5.
Theorem 3.10 see 6. Let u ∈ PAAR, Y, B ∈ BY, X. If vt : But for each t ∈ R, then
v ∈ PAAR, X.
6
Discrete Dynamics in Nature and Society
4. Pseudo almost automorphic mild solutions
In this section, combining Theorem 3.10 with composition theorem of pseudo almost
automorphic functions Theorem 3.8 and completeness of the space PAAR, X, ·∞ Theorem 3.5, we will establish the existence and uniqueness theorem of pseudo almost
automorphic mild solutions to 1.1 under the following assumptions.
H1 For 0 < α < β < 1, Xβ → Xα → X are continuously embedded and there exist
k0 > 0, kα > 0 such that u ≤ k0 uβ , uα ≤ kαuβ for each u ∈ Xβ .
H2 A is a sectorial linear operator on a Banach space X and σA ∩ iR ∅.
H3 f M ϕ ∈ PAAR × X, Xβ with M ∈ AAR × X, Xβ , ϕ ∈ AA0 R × X, Xβ and
g N ψ ∈ PAAR × X, X with N ∈ AAR × X, X, ψ ∈ AA0 R × X, X.
H4 The functions f, g are uniformly Lipschitz with respect to the second argument in
the sense that: there exists L > 0 such that
ft, x − ft, yβ ≤ Lx − y,
gt, x − gt, y ≤ Lx − y,
4.1
for all t ∈ R and x, y ∈ X.
H5 M, N are uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R.
H6 The operators B, C ∈ BXα , X and ω maxBBXα ,X , CBXα ,X .
Definition 4.1 see 9. A function u ∈ BCR, Xα is said to be a mild solution to 1.1 if
s → AT t − sP fs, Bus is integrable on −∞, t, s → AT t − sQfs, Bus is integrable on
t, ∞ and
ut −ft, But −
t
−∞
t
−∞
AT t − sP fs, Busds T t − sP gs, Cusds −
∞
∞
AT t − sQfs, Busds
t
4.2
T t − sQgs, Cusds
t
for each t ∈ R.
Lemma 4.2. Let assumptions (H1)–(H6) hold. Consider the nonlinear operator Λ1 defined by
Λ1 ut t
−∞
AT t − sP fs, Busds −
Then, Λ1 maps PAAR, Xα into itself.
∞
t
AT t − sQfs, Busds.
4.3
Z. Hu and Z. Jin
7
Proof. From H1, H3, H4, H5, we deduce that f M ϕ ∈ PAAR × X, X with M ∈
AAR × X, X, ϕ ∈ AA0 R × X, X, satisfying the following conditions.
1 Mt, x is uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R.
2 ft, x is uniformly Lipschitz with respect to the second argument in the sense that
there exists k0 L > 0 such that
ft, x − ft, y ≤ k0 Lx − y,
4.4
for all t ∈ R and x, y ∈ X.
Let u ∈ PAAR, Xα . Since B ∈ BXα , X, it follows from Theorem 3.10 that Bu· ∈
PAAR, X. Setting h· f·, Bu· and applying Theorem 3.8, we get that h ∈ PAAR, X.
Using H1 and Theorem 3.10, we obtain that h ∈ PAAR, Xβ . Now, write h ξ η where
ξ ∈ AAR, Xβ and η ∈ AA0 R, Xβ , then
Λ1 ut t
−∞
AT t − sP ξsds t
∞
t
−∞
AT t − sQξsds AT t − sP ηsds
t
∞
4.5
AT t − sQηsds.
Set
Ξt Πt t
−∞
t
−∞
AT t − sP ξsds AT t − sP ηsds t
∞
AT t − sQξsds,
t
∞
4.6
AT t − sQηsds.
Next, we show that Ξ ∈ AAR, Xα and Π ∈ AA0 R, Xα .
Now, to prove that Ξ ∈ AAR, Xα . Let us take a sequence σn n∈N and show that there
exists a subsequence sn n∈N such that
lim
lim Ξt sn − sm − Ξtα 0 for each t ∈ R.
m → ∞ n → ∞
4.7
Since ξ ∈ AAR, Xβ , there exists a subsequence sn n∈N such that
lim
lim ξt sn − sm − ξtβ 0
m → ∞ n → ∞
for each t ∈ R.
4.8
8
Discrete Dynamics in Nature and Society
On the other hand, we have
Ξt sn − sm − Ξt tsn −sm −∞
AT t sn − sm − sP ξsds −
tsn −sm
∞
∞
∞
0
−
∞
−∞
−∞
AT t sn − sm − sQξsds −
AT sP ξt sn − sm − sds −
0
t
0
AT sQξt sn − sm − sds −
AT t − sP ξsds
t
∞
AT t − sQξsds
AT sP ξt − sds
0
−∞
AT sQξt − sds
4.9
AT sP ξt sn − sm − s − ξt − sds
0
−
0
−∞
AT sQξt sn − sm − s − ξt − sds.
Passing to the norm ·α , 0 < α < 1, remembering the triangle inequalities, 2.7, then we
obtain that
Ξt sn − sm − Ξtα ≤
∞
AT sP ξt sn − sm − s − ξt − sα ds
0
≤c
0
−∞
∞
AT sQξt sn − sm − s − ξt − sα ds
4.10
β−α−1 −γs
s
e
ξt sn − sm − s − ξt − sβ ds
0
c
0
−∞
eδs ξt sn − sm − s − ξt − sβ ds.
Thus, 4.8 and Lebesgue dominated convergence theorem lead to 4.7, therefore, to Ξ ∈
AAR, Xα . To finish the proof, we will prove that Π ∈ AA0 R, Xα . It is obvious that Π ∈
BCR, Xα , the left task is to show that
1
r → ∞ 2r
lim
r
−r
Πtα dt 0.
4.11
Πtα dt ≤ I J,
4.12
Using 2.7, we have
1
0 ≤ lim
r → ∞ 2r
r
−r
Z. Hu and Z. Jin
9
where
t
c r
dt
t − sβ−α−1 e−γt−s ηsβ ds
I : lim
r → ∞ 2r −r
−∞
∞
c r
lim
dt
σ β−α−1 e−γσ ηt − σβ dσ
r → ∞ 2r −r
0
∞
r
1
lim c
σ β−α−1 e−γσ
ηt − σβ dt dσ,
r →∞ 0
2r −r
r ∞
c
J : lim
dt
eδt−s ηsβ ds
r → ∞ 2r −r
t
r 0
c
dt
eδσ ηt − σβ dσ
lim
r → ∞ 2r −r
−∞
0
r
δσ 1
e
ηt − σβ dt dσ.
lim c
r → ∞ −∞
2r −r
4.13
Then, by the Lebesgue dominated convergence theorem and the fact that η ∈ AA0 R, Xβ ,
one has I J 0. Hence, Π ∈ AA0 R, Xα and we end the proof.
Lemma 4.3. Let assumptions (H2)–(H6) hold. Define the nonlinear operator Λ2 by
Λ2 ut t
−∞
T t − sP gs, Cusds −
∞
T t − sQgs, Cusds.
4.14
t
Then, Λ2 maps PAAR, Xα into itself.
Proof. The proof is similar to that of Lemma 4.2, so we omit it.
Theorem 4.4. Under the assumptions (H1)–(H6), partial hyperbolic evolution equation 1.1 admits
a unique pseudo almost automorphic mild solution if
c Cα cΓβ − α MαΓ1 − α
< 1.
Θ Lω kα δ
δ
γ 1−α
γ β−α
4.15
Proof. Firstly, define the nonlinear operator Λ on BCR, Xα by
Λut −ft, But −
t
−∞
t
−∞
AT t − sP fs, Busds T t − sP gs, Cusds −
∞
t
−ft, But − Λ1 ut Λ2 ut.
∞
AT t − sQfs, Busds
t
T t − sQgs, Cusds
4.16
10
Discrete Dynamics in Nature and Society
Let u ∈ PAAR, Xα , then f·, Bu· ∈ PAAR, Xβ ⊂ PAAR, Xα , as proved in Lemma 4.2.
Together with Lemmas 4.2 and 4.3, it follows that the operator Λ maps PAAR, Xα into itself.
Secondly, we will show that Λ admits a unique fixed point in PAAR, Xα .
Let v, w ∈ PAAR, Xα , then the triangle inequality reads
Λvt − Λwtα ≤ ft, Bvt − ft, Bwtα Λ1 vt − Λ1 wtα
Λ2 vt − Λ2 wtα .
4.17
By H1, H5, and H6, we obtain
ft, Bvt − ft, Bwtα ≤ kαLBvt − Bwt ≤ kαLωv − w∞,α .
4.18
By H2, Lemma 2.6, H5, and H6, we obtain
Λ1 vt − Λ1 wtα ≤
t
−∞
AT t − sP fs, Bvs − fs, Bwsα ds
∞
AT t − sQfs, Bvs − fs, Bwsα ds
t
≤ cLωv − w∞,α
t
−∞
cLωv − w∞,α
≤ cLωv − w∞,α
t − sβ−α−1 e−γt−s ds
∞
4.19
eδt−s ds
t
Γβ − α 1
.
δ
γ β−α
Similarly, by H2, Lemma 2.5, H5, and H6, we obtain
Λ2 vt − Λ2 wtα ≤
t
−∞
T t − sP gs, Cvs − gs, Cwsα ds
∞
T t − sQgs, Cvs − gs, Cwsα ds
t
≤ MαLωv − w∞,α
t
CαLωv − w∞,α
≤ Lωv − w∞,α
−∞
t − s−α e−γt−s ds
∞
t
eδt−s ds
MαΓ1 − α Cα
.
δ
γ 1−α
4.20
Z. Hu and Z. Jin
11
Combining the above inequality together, we obtain Λv − Λw∞,α ≤ Θv − w∞,α , where
c Cα cΓβ − α MαΓ1 − α
Θ Lω kα .
δ
δ
γ 1−α
γ β−α
4.21
Clearly, if Θ < 1, then the operator Λ becomes a strict contraction on PAAR, Xα .
Remembering that PAAR, Xα equipped with the α-sup norm: u∞,α supt∈R utα is a
Banach space by Remark 3.9, the classical Banach fixed-point theorem leads to the desired
conclusion.
5. Application
Example 5.1. Take X : C0, π equipped with the sup norm. Define the operator A by
Aϕξ : ϕ ξ aϕξ,
∀ξ ∈ 0, π, ϕ ∈ DA,
5.1
where DA : {ϕ ∈ C2 0, π, ϕ0 ϕπ 0} ⊂ C0, π and a ∈ R is a constant.
Clearly, A is sectorial, and hence is the generator of an analytic semigroup. If a /
n2 ,
then A generates a hyperbolic analytic semigroup T tt≥0 on X.
Let
Xα DA α, ∞ ⎧
⎪
⎨C2α 0, π,
⎪
⎩C2α 0, π,
ν
if 0 < α <
1
,
2
5.2
1
if < α < 1,
2
where Cν2α 0, π {ϕ ∈ C2α 0, π, ϕ0 ϕπ 0} see 10 for more details.
Define the operator C by
Cϕξ : qξϕξ,
∀ξ ∈ 0, π, ϕ ∈ DC,
5.3
where DC : {ϕ ∈ C0, π, ϕ0 ϕπ 0} ⊂ C0, π, then Xα ⊂ DC and CBXα ,X q∞ supξ∈0,π qξ.
Let
gt, u u sin
1
2 cos t cos
√
2t
2 2
max e−t±k cos u,
k∈Z
t ∈ R, u ∈ X,
5.4
then g ∈ PAAR × X, X, g is Lipschitz
with respect to the second argument with L √
2, Nt, u u sin1/2 cos t cos 2t is uniformly continuous in any bounded subset
K ⊂ X uniformly for t ∈ R.
12
Discrete Dynamics in Nature and Society
Now the above discussion can be formulated as follows.
√
2 2
Proposition 5.2. Let gt, u u sin1/2cos tcos 2tmaxk∈Z {e−t±k } cos u, t ∈ R, u ∈
X. Suppose that the constant a /
n2 for n ∈ N. Then, nonlinear heat equation 1.2 admits a unique
Xα -valued pseudo almost automorphic mild solution if
2q∞
Cα MαΓ1 − α
< 1.
δ
γ 1−α
5.5
Appendix
Proof of Theorem 3.10. Write u φ ψ where φ ∈ AAR, Y and ψ ∈ AA0 R, Y, then
vt Bφt Bψt.
A.1
Set
Φt Bφt,
Ψt Bψt.
A.2
We are going to prove that Φ ∈ AAR, X and Ψ ∈ AA0 R, X.
Let us take a sequence σn n∈N and show that there exists a subsequence sn n∈N such
that
lim
lim Φt sn − sm − Φt 0
for each t ∈ R.
m → ∞ n → ∞
A.3
Since φ ∈ AAR, Y, there exists a subsequence sn n∈N such that
lim
lim φt sn − sm − φtY 0
for each t ∈ R.
m → ∞ n → ∞
A.4
On the other hand, we have
0 ≤ Φt sn − sm − Φt Bφt sn − sm − Bφt ≤ BBY,X φt sn − sm − φtY .
A.5
Thus, A.4 leads to A.3, therefore, to Φ ∈ AAR, X.
To complete the proof, we will prove that Ψ ∈ AA0 R, X. In fact, it is easy to check
that Ψ ∈ BCR, X and
1
r → ∞ 2r
lim
r
−r
1
r → ∞ 2r
Ψtdt ≤ BBY,X lim
Hence, Ψ ∈ AA0 R, X and we end the proof.
r
−r
ψtY dt 0.
A.6
Z. Hu and Z. Jin
13
Acknowledgments
The authors would like to thank referees for their helpful suggestions. This work is supported
by the National Science Foundation of China 60771026, the Special Scientific Research
Foundation for the Subjects of Doctors in University 20060110005, and the Program for
New Century Excellent Talents in University NCET050271.
References
1 T. Diagana, “Existence of p-almost automorphic mild solution to some abstract differential
equations,” International Journal of Evolution Equations, vol. 1, no. 1, pp. 57–67, 2005.
2 J. Liang, J. Zhang, and T.-J. Xiao, “Composition of pseudo almost automorphic and asymptotically
almost automorphic functions,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp.
1493–1499, 2008.
3 T.-J. Xiao, J. Liang, and J. Zhang, “Pseudo almost automorphic solutions to semilinear differential
equations in Banach spaces,” Semigroup Forum, vol. 76, no. 3, pp. 518–524, 2008.
4 T. Diagana, G. M. N’Guérékata, and N. Van Minh, “Almost automorphic solutions of evolution
equations,” Proceedings of the American Mathematical Society, vol. 132, no. 11, pp. 3289–3298, 2004.
5 R. A. Johnson, “A linear, almost periodic equation with an almost automorphic solution,” Proceedings
of the American Mathematical Society, vol. 82, no. 2, pp. 199–205, 1981.
6 Z. R. Hu and Z. Jin, “Pseudo almost automorphic mild solutions to some classes of partial evolution
equations,” submitted to Nonlinear Analysis: Real World Applications.
7 G. M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer
Academic/Plenum, New York, NY, USA, 2001.
8 G. M. N’Guérékata, Topics in Almost Automorphy, Springer, New York, NY, USA, 2005.
9 T. Diagana, “Existence of pseudo almost periodic solutions to some classes of partial hyperbolic
evolution equations,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–12,
2007.
10 A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16 of Progress in
Nonlinear Differential Equations and Their Applications, Birkhäuser, Basel, Switzerland, 1995.
11 K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate
Texts in Mathematics, Springer, New York, NY, USA, 2000.