Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 895079, 23 pages
doi:10.1155/2011/895079
Research Article
Existence of Pseudo-Almost Automorphic Mild
Solutions to Some Nonautonomous Partial
Evolution Equations
Toka Diagana
Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, DC 20059, USA
Correspondence should be addressed to Toka Diagana, [email protected]
Received 15 September 2010; Accepted 29 October 2010
Academic Editor: Jin Liang
Copyright q 2011 Toka Diagana. 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 use the Krasnoselskii fixed point principle to obtain the existence of pseudo almost
automorphic mild solutions to some classes of nonautonomous partial evolutions equations in
a Banach space.
1. Introduction
Let X be a Banach space. In the recent paper by Diagana 1, the existence of almost automorphic
mild solutions to the nonautonomous abstract differential equations
u
t
A
t
A
t
u
t
F
t, u
t
,t∈ R, 1.2
2 Advances in Difference Equations
where At for t ∈ R is a family of linear operators satisfying Acquistpace-Terreni conditions
and F, G are pseudo-almost automorphic functions. For that, we make use of exponential
dichotomy tools as well as the well-known Krasnoselskii fixed point principle to obtain
some reasonable sufficient conditions, which do guarantee the existence of pseudo-almost
automorphic mild solutions to 1.2.
The concept of pseudo-almost automorphy is a powerful generalization of both the
notion of almost automorphy due to Bochner 3 and that of pseudo-almost periodicity
due to Zhang see 4, which has recently been introduced in the literature by Liang et
al. 5–7. Such a concept, since its introduction in the literature, has recently generated
several developments; see, for example, 8–12. The question which consists of the existence
of pseudo-almost automorphic solutions to abstract partial evolution equations has been
made; see for instance 10, 11, 13. However, the use of Krasnoselskii fixed point principle
to establish the existence of pseudo-almost automorphic solutions to nonautonomous partial
A
t
− ω
λ,
R
λ, A
t
− ω
≤
K
1
|
λ
|
,
A
t
λ
|
−ν
,
2.1
for t, s ∈ R, λ ∈ S
θ
: {λ ∈ C \{0} : | arg λ|≤θ}.
Advances in Difference Equations 3
It should mentioned that H.1 was introduced in the literature by Acquistapace et
al. in 14, 15 for ω 0. Among other things, it ensures that there exists a unique evolution
family
U
{
U
t, s
: t, s ∈ R such that t ≥ s
}
, 2.2
on X associated with At such that Ut, sX ⊂ DAt for all t, s ∈ R with t ≥ s,and
a Ut, sUs, rUt, r for t, s, r ∈ R such that t ≥ s ≥ r;
b Ut, t
I for t ∈ R where I is the identity operator of X;
ct, s → Ut, s ∈ BX is continuous for t>s;
d U·,s ∈ C
1
s, ∞,BX, ∂U/∂tt, sAtUt, s and
continuous in t and constants δ>0andN ≥ 1 such that
f Ut, sPsPtUt, s;
g the restriction U
Q
t, s : QsX → QtX of Ut, s is invertible we then set
U
Q
s, t : U
Q
t, s
−1
;
h Ut, sPs≤Ne
−δt−s
and
U
Q
s, tQt≤Ne
−δt−s
for t ≥ s and t, s ∈ R.
Under Acquistpace-Terreni conditions, the family of operators defined by
Γ
t, s
⎧
⎨
A
α
:
x ∈ X :
x
A
α
: sup
r>0
r
α
A − ω
R
r, A − ω
x
< ∞
, 2.5
4 Advances in Difference Equations
which, by the way, is a Banach space when endowed with the norm ·
A
ω − A
x
.
2.6
Moreover, let
X
A
: DA of X. In particular, we have the following continuous embedding:
D
A
→ X
A
β
→ D
ω − A
α
→ X
A
α
,
X
t
:
X
At
2.9
for 0 ≤ α ≤ 1andt ∈ R, with the corresponding norms.
Now the embedding in 2.7 holds with constants independent of t ∈ R. These
interpolation spaces are of class J
α
32, Definition 1.1.1, and hence there is a constant cα
such that
y
t
α
≤ c
α
y
t
α
≤ c
α
e
−δ/2t−s
t − s
−α
x
. 2.11
ii There is a constant mα, such that
U
Q
s, t
Q
t
n
n∈N
, there exists a subsequence s
n
n∈N
such that
g
t
: lim
n →∞
f
t s
n
2.13
is well defined for each t ∈ R,and
lim
n →∞
g
t − s
n
f
: {ft : t ∈ R} is relatively compact in X, thus f is bounded in norm,
v if f
n
→ f uniformly on R, where each f
n
∈ AAX,thenf ∈ AAX too.
Let Y, ·
Y
be another Banach space.
Definition 2.5. A jointly continuous function F : R × Y → X is said to be almost automorphic
in t ∈ R if t → Ft, x is almost automorphic for all x ∈ K K ⊂ Y being any bounded subset.
Equivalently, for every sequence of real numbers s
n
n∈N
, there exists a subsequence s
n
n∈N
such that
G
t, x
: lim
n →∞
F
t s
f ∈ BC
R, X
: lim
r →∞
1
2r
r
−r
f
s
ds 0
. 2.17
Similarly, PAP
0
Y, X will denote the collection of all bounded continuous functions
F : R × Y → X such that
lim
T →∞
1
uniformly continuous on any bounded subset K of Y uniformly in t ∈ R. Furthermore, one supposes
that there exists L>0 such that
f
t, x
− f
t, y
≤ L
x − y
Y
2.19
for all x, y ∈ Y and t ∈ R.
Then the function defined by htft, ϕt belongs to PAAX provided ϕ ∈ PAAY.
Advances in Difference Equations 7
We also have the following.
Theorem 2.10 see 6. If f : R × Y → X belongs to PAAY, X and if x → ft, x is uniformly
continuous on any bounded subset K of Y for each t ∈ R, then the function defined by htft, ϕt
belongs to PAAX provided that ϕ ∈ PAAY.
3. Main Results
− Γ
t τ
n
,s τ
n
BX,X
α
≤ εH
0
t − s
3.2
whenever n>N
0
for t, s ∈ R,and
Γt, s − R
t − τ
n
,s− τ
n
BX,X
F
t, u
∞
≤M
u
α,∞
, 3.4
where u
α,∞
sup
t∈R
ut
α
and M : R
→ R
is a continuous, monotone
increasing function satisfying
lim
r →∞
M
> 0 such that
sup
t,s∈R
AsA
t
−1
BX,X
β
<c
0
. 3.7
To study the existence and uniqueness of pseudo-almost automorphic solutions to
1.2 we first introduce the notion of a mild solution, which has been adapted to the one
given in the studies of Diagana et al. 35, Definition 3.1.
Definition 3.1. A continuous function u : R → X
α
is said to be a mild solution to 1.2
provided that the function s → AsUt, sPsGs, us is integrable on s, t, the function
s → AsU
Q
t, sQsGs, us is integrable on t, s and
s
U
t, s
P
s
G
s, u
s
ds
s
t
A
s
U
Q
t, s
Q
s
t
U
Q
t, s
Q
s
F
s, u
s
ds,
3.8
for t ≥ s and for all t, s ∈ R.
Under assumptions H.1, H.2,andH.5, it can be readily shown that 1.2 has a
mild solution given by
u
t
−G
t, u
s
U
Q
t, s
Q
s
G
s, u
s
ds
t
−∞
U
t, s
P
s
We denote by S and T the nonlinear integral operators defined by
Su
t
t
−∞
U
t, s
P
s
F
s, u
s
ds −
∞
t
U
Q
s
U
t, s
P
s
G
s, u
s
ds
∞
t
A
s
U
Q
t, s
t
U
t, s
P
s
x
β
≤ r
α, β
e
−δ/4t−s
t − s
−β
x
,t>s,
3.11
x
,t≤ s.
3.12
Proof. Let x ∈ X. First of all, note that AtUt, s
BX,X
β
≤ Kt − s
−1−β
for all t, s such that
0 <t− s ≤ 1andβ ∈ 0, 1.
Letting t − s ≥ 1andusingH.2 and the above-mentioned approximate, we obtain
A
t
U
t, s
x
β
A
t
x
≤ MKe
δ
e
−δt−s
x
K
1
e
−δt−s
x
K
1
e
−3δ/4t−s
t − s
β
t − s
−β
e
β
t − s
−β
e
−δ/4t−s
x
. 3.14
Now, let 0 <t− s ≤ 1. Using 2.11 and the fact 2β>α 1, we obtain
A
t
U
t, s
x
β
U
t,
t s
2
BX,X
β
U
t s
2
,s
x
≤ k
,s
x
α
≤ k
1
K
t − s
2
β−1
c
α
t − s
2
−α
e
−δ/4t−s
x
In summary, there exists r
β, α > 0 such that
AtUt, sx
β
≤ r
α, β
t − s
−β
e
−δ/4t−s
x
, 3.16
for all t, s ∈ R with t>s.
Let x ∈ X. Since the restriction of As to RQs is a bounded linear operator it
follows that
A
U
Q
t, s
Q
s
x
β
≤
A
t
A
s
−1
A
t
A
s
−1
BX,X
β
A
s
U
Q
t, s
Q
β
≤ c
U
Q
t, s
Q
s
x
β
≤ cm
β
e
−δs−t
x
x
β
≤ r
α, β
e
−δ/4t−s
t − s
−β
x
,t>s,
3.18
As
U
Q
t, sQsx
A
s
A
−1
t
A
t
U
t, s
P
s
x
β
≤
x
≤ c
0
k
A
t
U
t, s
P
s
x
β
≤ c
0
k
r
BCR, X
α
is well defined and continuous.
Proof. We first show that SBCR, X
α
⊂ BCR, X
α
. For that, let S
1
and S
2
be the integral
operators defined, respectively, by
S
1
u
t
t
−∞
U
t, s
P
s, u
s
ds.
3.21
12 Advances in Difference Equations
Now, using 2.11 it follows that for all v ∈ BCR, X
α
,
S
1
v
t
∞
t
−∞
−α
e
−δ/2t−s
F
s, v
s
ds
≤
t
−∞
c
α
t − s
−α
e
−δ/2t−s
M
v
u
α,∞
≤ s
α
M
v
α,∞
, 3.23
where sαcα2δ
−1
1−α
Γ1 − α.
It remains to prove that S
1
is continuous. For that consider an arbitrary sequence
of functions u
n
∈ BCR, X
α
which converges uniformly to some u ∈ BCR, X
α
,thatis,
s, u
s
ds
α
≤ c
α
t
−∞
t − s
−α
e
−δ/2t−s
F
s, u
n
F
s, u
n
s
− F
s, u
s
ds
α
−→ 0asn −→ ∞ , 3.25
and hence S
1
u
n
− S
1
u
α,∞
→ 0asn →∞.
1
∈ AAX
α
. Indeed, since u
1
∈ AAX, for every sequence of real numbers τ
n
n∈N
there exists a subsequence τ
n
n∈N
such that
v
1
t
: lim
n →∞
u
1
t τ
n
3.26
is well defined for each t ∈ R and
t τ
n
− N
t
tτ
n
−∞
U
t τ
n
,s
P
s
u
1
s
ds −
u
1
s τ
n
ds −
t
−∞
U
t, s
P
s
v
1
s
ds
t
−∞
U
n
,s τ
n
P
s τ
n
− U
t, s
P
s
v
1
s
ds.
3.28
Using 2.11 and the Lebesgue Dominated Convergence Theorem, one can easily see
that
α
−→ 0asn →∞,t∈ R. 3.29
Similarly, using H.3 and 40 it follows that
t
−∞
U
t τ
n
,s τ
n
P
s τ
n
− U
t, s
,t∈ R. 3.31
Using similar ideas as the previous ones, one can easily see that
M
t
lim
n →∞
N
t − τ
n
,t∈ R. 3.32
14 Advances in Difference Equations
Again using 2.11 it follows that
lim
r →∞
1
2r
r
−r
S
1
u
2
r →∞
c
α
∞
0
s
−α
e
−δ/2s
1
2r
r
−r
u
2
t − s
dt ds.
3.33
Set
Γ
s
−r
u
2
t − s
dt 0 3.35
for each s ∈ R.
One completes the proof by using the well-known Lebesgue dominated convergence
theorem and the fact Γ
s
r → 0asr →∞for each s ∈ R.
The proof for S
2
is similar to that of S
1
and hence omitted. For S
2
, one makes use of
2.12 rather than 2.11.
Let γ ∈ 0, 1,andletBC
γ
R, X
α
{u ∈ BCR, X
α
: u
α,γ
|
γ
. 3.36
Clearly, the space BC
γ
R, X
α
equipped with the norm ·
α,γ
is a Banach space, which is
the Banach space of all bounded continuous H
¨
older functions from R to X
α
whose H
¨
older
exponent is γ.
Lemma 3.7. Under assumptions (H.1), (H.2), (H.3), (H.4), and (H.5), V S
1
− S
2
maps bounded
sets of BCR, X
α
into bounded sets of BC
γ
R, X
α
for some 0 <γ<1,whereS
β
≤ k
α
t
−∞
U
t, s
P
s
g
s
β
ds
≤ k
α
k
α
c
β
∞
0
e
−σ
2σ
δ
−β
2dσ
δ
≤M
u
α,∞
k
α
c
β
2
−1
δ
1−β
Γ
1 − β
M
u
α,∞
. 3.38
Similarly,
S
2
t, s
Q
s
g
s
β
ds
≤ k
α
m
β
∞
t
e
−δs−t
α,∞
≤ p
α, β, δ
M
u
α,∞
. 3.40
16 Advances in Difference Equations
Let t
1
<t
2
. Clearly,
S
1
u
t
2
− S
1
u
ds
t
1
−∞
Ut
2
,s − Ut
1
,sPsgsds
α
t
2
t
1
U
P
s
g
s
ds
α
≤
t
2
t
1
U
t
2
2
t
1
A
τ
U
τ,s
P
s
g
s
dτ
ds
α
N
1
α
ds
≤ c
α
t
2
t
1
t
2
− s
−α
e
−δ/2t
2
−s
g
s
α
M
u
α,∞
t
2
t
1
t
2
− s
−α
ds
≤
1 − α
−1
c
α
2
t
1
A
τ
U
τ,s
P
s
g
s
β
dτ
ds
≤ k
α
ds
≤ k
α
r
α, β
M
u
α,∞
t
2
t
1
t
1
−∞
τ − s
−β
∞
τ−t
1
e
−δ/4r
dr
dτ
≤ 4δ
−1
k
α
r
α, β
M
u
α,∞
t
t
2
t
1
e
−δs−t
1
g
s
ds
m
α
∞
t
2
t
2
,
3.44
where Nα, δ is a positive constant.
Consequently, letting γ 1 − β it follows that
Vu
t
2
− Vu
t
1
α
≤ s
α, β, δ
M
u
α,∞
|
α
≤ R
3.47
for all t ∈ R, where R
depends on R.
The proof of the next lemma follows along the same lines as that of Lemma 3.6 and
hence omitted.
Lemma 3.8. The integral operator V S
1
− S
2
maps bounded sets of AAX
α
into bounded sets of
BC
1−β
R, X
α
∩ AAX
α
.
Similarly, the next lemma is a consequence of 2, Proposition 3.3.
Lemma 3.9. The set BC
1−β
R, X
α
is compactly contained in BCR, X, that is, the canonical
injection id : BC
Proof. The proof follows along the same lines as that of 2, Proposition 3.4. Recalling that in
view of Lemma 3.7, we have
Vu
α,∞
≤ p
α, β, δ
M
u
α,∞
,
Vut
2
− V
t
1
α
≤ s
α
and u
α,∞
<Ryield Vu∈ BC
1−β
R, X
α
and
Vu
α
<R
1
, 3.50
where R
1
cα, β, δMR.
Therefore, there exists r>0 such that for all R ≥ r, the following hold:
V
B
AAX
α
0,R
⊂ B
t
−∞
A
s
U
t, s
P
s
G
s, u
s
ds,
W
2
u
t
α
into itself.
Proof. Let u ∈ PAAX
α
. Again, using the composition of pseudo-almost automorphic
functions Theorem 2.10 it follows that ψ·G·,u· is in PAAX
β
whenever u ∈
PAAX
α
. In particular,
ψ
β,∞
sup
t∈R
G
t, u
t
β
< ∞. 3.53
Now write ψ φ z, where φ ∈ AAX
s
ds,
Ξz
t
:
t
−∞
A
s
U
t, s
P
s
z
s
ds.
3.54
n →∞
ψ
t − τ
n
φ
t
3.56
for each t ∈ R.
Set Jt
t
−∞
AsUt, sPsφsds and Kt
t
−∞
AsUt, sPsψsds for all t ∈
R.
Now
J
t τ
n
− K
s
U
t, s
P
s
ψ
s
ds
t
−∞
A
s τ
n
U
t τ, s τ
n
ds
t
−∞
A
s τ
n
U
t τ
n
,s τ
n
P
s τ
n
φ
s τ
n
− ψ
U
t, s
P
s
ψ
s
ds.
3.57
Using 3.18 and the Lebesgue Dominated Convergence Theorem, one can easily see
that
t
−∞
A
s τ
n
−→ 0asn −→ ∞ ,t∈ R.
3.58
Similarly, using H.3 it follows that
t
−∞
A
s τ
n
U
t τ
n
,s τ
n
P
s τ
n
− A
n →∞
J
t τ
n
,t∈ R. 3.60
20 Advances in Difference Equations
Using similar ideas as the previous ones, one can easily see that
J
t
lim
n →∞
K
t − τ
n
,t∈ R. 3.61
Now, let r>0. Again from 3.18, we have
1
2r
r
−r
Ξzt
β
ds dt
≤
k
α
r
α, β
2r
r
−r
t
−∞
e
−δ/4t−s
t − s
−β
z
t − s
r →∞
1
2r
r
−r
zt − s
β
dt 0, 3.63
as t → zt − s ∈ PAP
0
X
β
for every s ∈ R. One completes the proof by using the Lebesgue’s
dominated convergence theorem.
The proof for W
2
u· is similar to that of W
1
u· except that one makes use of 3.19
instead of 3.18 .
Theorem 3.12. Under assumptions (H.1), (H.2), (H.3), (H.4), and (H.5) and if L is small enough,
then 1.2 has at least one pseudo-almost automorphic solution.
Proof. We have seen in the proof of Theorem 3.10 that S : D → D is continuous and compact,
where D is the ball in PAAX
α
of radius R with R ≥ r.
Now, if we set a
d
β
δ
3.64
for all u ∈ D.
Choose R
such that
k
α
kLR a
G
1 r
α, β
4
δ
1−β
and hence S TD
⊂ D
.
Advances in Difference Equations 21
To complete the proof we have to show that T is a strict contraction. Indeed, for all
u, v ∈ X
α
Tu− Tv
α,∞
≤ Lk
α
1 r
α, β
4
δ
1−β
Γ
1 − β
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