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J. Nonlinear Sci. Appl. 5 (2012), 14–26
Research Article
Pseudo almost automorphic and weighted pseudo
almost automorphic mild solutions to a partial
functional differential equation in Banach spaces
Yong-Kui Changa,∗, Zhi-Han Zhaob , Juan J. Nietoc , Zhi-Wei Liud
a
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China.
Institute of Mathematics and Information Engineering, Sanming University, Sanming,365004, Fujian, China.
c
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de
Compostela, 15782, Spain.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.
d
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China.
b
This paper is dedicated to Professor Ljubomir Ćirić
Communicated by Professor V. Berinde
Abstract
In this paper we prove some existence and uniqueness results for pseudo almost automorphic and weighted
pseudo almost automorphic mild solutions to a class of partial functional differential equation in Banach
spaces. The main technique is based upon some appropriate composition theorems combined with the
c
Banach contraction mapping principle and fractional powers of operators.2012
NGA. All
Keywords: Pseudo almost automorphic, Weighted pseudo almost automorphic, Partial functional
differential equations, Fractional powers of operators.
2010 MSC: 44A35, 42A85, 42A75.
1. Introduction
In this paper, we are mainly concerned with the existence and uniqueness of pseudo almost automorphic
and weighted pseudo almost automorphic mild solutions for a class of partial neutral functional differential
∗
Corresponding author
Email addresses: lzchangyk@163.com (Yong-Kui Chang), zhaozhihan841110@126.com (Zhi-Han Zhao),
juanjose.nieto.roig@usc.es (Juan J. Nieto), lzw9981@yeah.net (Zhi-Wei Liu)
Received 2011-5-14
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
15
equations in the abstract form
d
[u(t) + f (t, u(h1 (t)))] = Au(t) + g (t, u(h2 (t))) , t ∈ R,
dt
(1.1)
where A : D(A) ⊂ X → X is the infinitesimal generator of an analytic semigroup of linear operators
{T (t)}t≥0 on a Banach space (X, | · |) and there exist positive numbers M , δ such that kT (t)k ≤ M e−δt for
t ≥ 0, and f (·), g(·), hi (·), i = 1, 2, are appropriate functions specified later.
The concept of pseudo almost automorphic function, which was initiated by Xiao et al. in [1], is more
general than that of almost automorphic function created by Bochner in [2, 3]. Since then, those functions
has been studied and developed extensively. For more details on those functions we refer the reader to
[4, 5, 6, 7, 8, 9] and the references therein. Very recently, J. Blot et al [10] have introduced the concept
of weighted pseudo almost automorphic, which generalizes the concept of weighted pseudo almost periodic
[11, 12, 13, 14]. In [10], the authors have proved some useful properties of the space of weighted pseudo
almost automorphic functions and established a general existence and uniqueness theorem for weighted
pseudo almost automorphic mild solutions to some semi-linear differential equations. The existence of
almost automorphic, pseudo almost automorphic and weighted pseudo almost automorphic solutions are
among the most attractive topics in qualitative theory of differential equations due to their significance
and applications in physics, mechanics and mathematical biology [15]. In recent years, the existence of
almost automorphic and pseudo almost automorphic solutions on different kinds of differential equations
have been considered in many publications such as [6, 16, 17, 18, 19, 20, 21, 22, 23, 24] and references
therein. Especially, authors in [23] have proved the existence of almost automorphic and weighted pseudo
almost automorphic solutions to a semi-linear evolution equation in a Banach space X such as
x0 (t) = A(t)x(t) + f (t, x(t)), t ∈ R,
where the family {A(t), t ∈ R} of operators in X generates an exponentially stable evolution family
{U (t, s), t ≥ s} and f : R × X → X an almost automorphic function(resp. a weighted pseudo almost
automorphic function).
And in [24], authors have investigated the existence of almost automorphic and pseudo-almost automorphic mild solutions to the following equation
du (t)
d
= Au(t) + F1 (t, u(h1 (t))) + F2 (t, u(h2 (t))),
dt
dt
where A : D(A) ⊂ X → X is the infinitesimal generator of a C0 -semigroup {T (t), t ≥ 0} on a Banach space
X.
Motivated by the above mentioned works [10, 23, 24], the main purpose of this paper is to deal with
the existence and uniqueness of pseudo almost automorphic and weighted pseudo almost automorphic mild
solutions to the problem (1.1). We obtain the new results by using fractional powers of linear operators and
the Banach contraction mapping principle.
The rest of this paper is organized as follows: In section 2 we recall some basic definitions, lemmas and
preliminary facts which will be need in the sequel. Our main results and their proofs are arranged in Section
3.
2. Preliminaries
This section is concerned with some notations, definitions, lemmas and preliminary facts which are used
in what follows.
Throughout the paper, let (X, | · |) be a Banach space and C (R, X) stand for the collection of continuous
functions from R into X. We denote by BC(R, X) the Banach space of all bounded continuous functions
from R into X endowed with the supremum norm defined by kxkBC(R,X) := supt∈R {|x(t)|}. Furthermore,
BC(R × X, X) is the space of all bounded continuous functions F : R × X → X.
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
16
Let U denote the set of all functions (weights) ρ : R → (0, ∞), which are locally integrable over R such
that ρ > 0 almost everywhere. For a given r > 0 and for each ρ ∈ U, we set
Z r
m(r, ρ) =
ρ(x)dx.
−r
We denote by U∞ the set of all ρ ∈ U with limr→∞ m(r, ρ) = ∞ and Ub the set of all ρ ∈ U∞ such that ρ is
bounded and inf x∈R ρ(x) > 0.
It is clear that Ub ⊂ U∞ ⊂ U, with strict inclusions.
Let 0 ∈ ρ(A), then it is possible to define the fractional power (−A)α , for 0 < α ≤ 1, as a closed linear
operator on its domain D((−A)α ). Furthermore, the subspace D((−A)α ) is dense in X and the expression
|x|α = |(−A)α x|, x ∈ D((−A)α ),
defines a norm on D((−A)α ). Hereafter we denote by Xα the Banach space D((−A)α ) with norm |x|α .
The following properties hold by [25, Lemma 2.1] and [26].
Lemma 2.1. Let 0 < γ ≤ µ ≤ 1. Then the following properties hold:
(i) Xµ is a Banach space and Xµ ,→ Xγ is continuous.
(ii) The function s → (−A)µ T (s) is continuous in the uniform operator topology on (0, ∞) and there
exists Mµ > 0 such that k(−A)µ T (t)k ≤ Mµ e−δt t−µ for each t > 0.
(iii) For each x ∈ D((−A)µ ) and t ≥ 0, (−A)µ T (t)x = T (t)(−A)µ x.
(iv) (−A)−µ is a bounded linear operator in X with D((−A)µ ) = Im((−A)−µ ).
Definition 2.2. A continuous function F : R → X is said to be almost automorphic if for every sequence
of real numbers (sn 0 )n∈N , there exists a subsequence (sn )n∈N such that
G(t) := lim F(t + sn )
n→∞
is well defined for each t ∈ R and
lim G(t − sn ) = F(t)
n→∞
for each t ∈ R. The collection of such functions will be denote by AA(X).
We recall that every almost periodic function is almost automorphic, but the class of almost automorphic
functions is larger than the class of almost periodic solutions. For example
1
√
f (t) = cos
, t∈R
2 + sin 2t + sin t
is almost automorphic but not almost periodic.
Definition 2.3. A continuous function F : R × X → X is said to be almost automorphic if F(t, x) is almost
automorphic for each t ∈ R uniformly for all x ∈ B, where B is any bounded subset of X. The collection
of such functions will be denote by AA(R × X, X).
Lemma 2.4. ([7]) AA(X), k · kAA(X) is a Banach space endowed with the supremum norm given by
kFkAA(X) = sup |F(t)|.
t∈R
The notation P AA0 (X) stands for for the spaces of functions
Z r
1
P AA0 (X) = F ∈ BC(R, X) : lim
|F(t)|dt = 0 .
r→∞ 2r −r
Similarly, the notation P AA0 (R × X, X) stands for for the spaces of functions
Rr
1
F ∈ BC(R × X, X) : limr→∞ 2r
|F(t, x)|dt = 0,
−r
P AA0 (R × X, X) =
.
uniformly in x in any bounded subset of X
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
17
Definition 2.5. A continuous function F : R → X(resp. R × X → X) is called pseudo almost automorphic
if it can be decomposed as F = G + φ, where G ∈ AA(X)(resp. AA(R × X, X)) and φ ∈ P AA0 (X)(resp.
P AA0 (R × X, X)). The class of all such functions will be denote by P AA(X)(resp. P AA(R × X, X)).
Lemma 2.6. ([27, Theorem 2.2]) P AA(X), k · kP AA(X) is a Banach space endowed with the supremum
norm given by
kFkP AA(X) = sup |F(t)|.
t∈R
Lemma 2.7. ([27, Lemma 2.4]) Assume F = G + φ ∈ P AA(R × X, X), where G (t, x) ∈ AA(R × X, X) and
φ(t, x) ∈ P AA0 (R × X, X), and suppose that F(t, x) is uniformly continuous in any bounded subset K ⊂ X
uniformly for t ∈ R. If x(t) ∈ P AA(R, X), then F (·, x(·)) ∈ P AA(R, X).
Now for ρ ∈ U∞ , we define
P AA0 (R, ρ) =
P AA0 (R × X, ρ) =
1
F ∈ BC(R, X) : lim
r→∞ m(r, ρ)
Z
r
|F(t)|ρ(t)dt = 0 ;
−r
Rr
1
F ∈ BC(R × X, X) : limr→∞ m(r,ρ)
−r |F(t, x)|ρ(t)dt = 0
.
uniformly in x ∈ X
Definition 2.8. ([10]) A bounded continuous function F : R → X(resp. R × X → X) is called weighted
pseudo almost automorphic if it can be decomposed as F = G + φ, where G ∈ AA(X)(resp. AA(R ×
X, X)) and φ ∈ P AA0 (R, ρ)(resp. P AA0 (R × X, ρ)). The class of all such functions will be denote by
W P AA(R, ρ)(resp. W P AA(R × X, ρ)).
Remark 2.9. ([10, Remark 2.2.]) When ρ = 1, we obtain the standard spaces P AA(R, X) and P AA(R ×
X, X).
Lemma 2.10. [5, Theorem 3.4] Let ρ ∈ U∞ . Suppose that P AA0 (R, ρ) is translation invariant. Then the
decomposition of weighted pseudo almost automorphic functions is unique.
Lemma 2.11. [28, Theorem 2.15]Let ρ ∈ U∞ . If P AA0 (R, ρ) is translation invariant,
then W P AA(R, ρ), k · kW P AA(R,ρ) is a Banach space endowed with the supremum norm given by
kFkW P AA(R,ρ) = sup |F(t)|.
t∈R
Lemma 2.12. ([10, Corollary 2.11.]) Let F = G +φ ∈ W P AA(R×X, ρ) where ρ ∈ U∞ , G ∈ AA(R×X, X)
and φ ∈ P AA0 (R × X, ρ). Assume both F and G are Lipschitzian in x ∈ X uniformly in t ∈ R. If
x(t) ∈ W P AA(R, ρ) then the function F(·, x(·)) ∈ W P AA(R, ρ).
Lemma 2.13. ([10, Lemma 3.1.]) Let F = RG + φ ∈ W P AA(R, ρ) where ρ ∈ U∞ and {T (t)}t≥0 is an
t
exponentially stable semigroup. Then F (t) := −∞ T (t − s)F(s)ds ∈ W P AA(R, ρ).
The next result is a straightforward consequence of Lemma 2.13 when ρ = 1.
Lemma 2.14. Let F = G +φ ∈ P AA(X) and {T (t)}t≥0 is an exponentially stable semigroup. If F : R → X
be the function defined by
Z
t
T (t − s)F(s)ds, t ≥ s ∈ R,
F (t) =
−∞
then F (·) ∈ P AA(X).
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
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Definition 2.15. A function u ∈ BC(R, X) is called a pseudo almost automorphic mild solution of Eq.
(1.1) on R if u ∈ P AA(X) and the function s → AT (t − s)f (s, u(h1 (s))) is integrable on (−∞, t) for each
t ∈ R, and u(t) satisfies
Z t
AT (t − s)f (s, u(h1 (s))) ds
u (t) = T (t − a) [u(a) + f (a, u(h1 (a)))] − f (t, u(h1 (t))) −
a
Z t
T (t − s)g (s, u(h2 (s))) ds
+
a
for all t ≥ a and all a ∈ R.
Definition 2.16. A function u ∈ BC(R, X) is called a weighted pseudo almost automorphic mild solution
of Eq. (1.1) on R if u ∈ W P AA(R, ρ) and the function s → AT (t − s)f (s, u(h1 (s))) is integrable on (−∞, t)
for each t ∈ R, and u(t) satisfies
Z t
AT (t − s)f (s, u(h1 (s))) ds
u (t) = T (t − a) [u(a) + f (a, u(h1 (a)))] − f (t, u(h1 (t))) −
a
Z t
+
T (t − s)g (s, u(h2 (s))) ds
a
for all t ≥ a and all a ∈ R.
Now we list the following basic assumptions of this paper:
(H1) (I) There exists a positive number α ∈ (0, 1) such that f : R × X → Xα is continuous and (−A)α f ∈
(1)
P AA(R × X, X). Let Lf > 0 be such that for each (t, x), (t, y) ∈ R × X
(1)
|(−A)α f (t, x) − (−A)α f (t, y)| ≤ Lf |x − y|.
(II) There exists a positive number α ∈ (0, 1) such that f : R × X → Xα is continuous and (−A)α f =
(2)
ϕ1 +ψ1 ∈ W P AA(R×X, ρ), and there exist positive numbers Lf , Lϕ1 such that for each (t, x), (t, y) ∈ R×X
(2)
|(−A)α f (t, x) − (−A)α f (t, y)| ≤ Lf |x − y|, |ϕ1 (t, x) − ϕ2 (t, y)| ≤ Lϕ1 |x − y|.
(1)
(H2) (I) g ∈ P AA(R × X, X) and there exists a positive number Lg such that for each (t, x), (t, y) ∈ R × X
|g(t, x) − g(t, y)| ≤ L(1)
g |x − y|.
(2)
(II) g = ϕ2 + ψ2 ∈ W P AA(R × X, ρ), and there exist positive numbers Lg , Lϕ2 such that for each
(t, x), (t, y) ∈ R × X
|g(t, x) − g(t, y)| ≤ L(2)
g |x − y|, |ϕ2 (t, x) − ϕ2 (t, y)| ≤ Lϕ2 |x − y|.
(H3) ([24]) The functions hi : R → R, hi (R) = R are continuously differentiable on R, and for u(·) ∈
0
AA(X), u(hi (·)) ∈ AA(X), hi (t) > 0, i = 1, 2, are nondecreasing with
|hi (−r)| + |hi (r)|
lim sup
< ∞.
0
rhi (−r)
r→∞
(H4) The functions hi : R → R, hi (R) = R are continuously differentiable on R, and for u(·) ∈ AA(X), u(hi (·)) ∈
0
AA(X), hi (t) > 0, i = 1, 2, are nondecreasing with
m(ri∗ , ρ)
ρ (t)
lim sup
< ∞, and 0 < sup
< ∞,
0
m(r, ρ)hi (−r)
r→∞
t∈R ρ (hi (t))
where ri∗ = |hi (−r)| + |hi (r)| for i = 1, 2.
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
19
3. Main results
In this section, we present and prove our main results. In order to establish our main results, we need
the following auxiliary results.
Lemma 3.1. Let α ∈ (0, 1] and (−A)α v ∈ W P AA(R, ρ). If u(·) : R → X be the function defined by
Z
t
AT (t − s)v(s)ds, t ≥ s,
u(t) =
−∞
then u(·) ∈ W P AA(R, ρ).
Proof. . First we observe that u(·) is well defined. Since (−A)α v ∈ W P AA(R, ρ), then (−A)α v is bounded,
we assume that there exists M1 > 0, such that k(−A)α vkW P P A(R,ρ) ≤ M1 . So
Z t
|AT (t − s)v(s)|ds
|u(t)| ≤
−∞
t
Z
k(−A)1−α T (t − s)k|(−A)α v(s)|ds
≤
−∞
Z
t
M1−α e−δ(t−s) (t − s)α−1 ds
≤ M1
−∞
+∞
Z
≤ M1 M1−α
σ α−1 e−δσ dσ
0
≤ M1 M1−α δ −α Γ(α),
where Γ(·) is the gamma function. Thus s → AT (t − s)v(s) is integrable on (−∞, t) for each t ∈ R
and so that u(t) is a bounded continuous functions. Now we prove that u(·) ∈ W P AA(R, ρ). We let
(−A)α v(t) = m(t) + n(t), where m(·) ∈ AA(X) and n(·) ∈ P AA0 (R, ρ).
Then
Z t
u(t) = −
(−A)1−α T (t − s)(−A)α v(s)ds
−∞
t
Z
= −
(−A)
1−α
−∞
Z
t
T (t − s)m(s)ds −
(−A)1−α T (t − s)n(s)ds.
−∞
Rt
Rt
Let G(t) = − −∞ (−A)1−α T (t−s)m(s)ds, H(t) = − −∞ (−A)1−α T (t−s)n(s)ds. Clearly, u(t) = G(t)+H(t).
Now we show that G(t) ∈ AA(X). Let (sn 0 )n∈N be an arbitrary sequence of real numbers. Since
m(t) ∈ AA(X), there exists a subsequence (sn )n∈N of (sn 0 )n∈N such that ϕ(t) := limn→∞ m(t + sn ) is well
defined for each t ∈ R, and m(t) = limn→∞ ϕ(t − sn ) for each t ∈ R.
Now, we consider
Z t+sn
G(t + sn ) = −
(−A)1−α T (t + sn − s)m(s)ds
−∞
t
Z
= −
−∞
Z t
= −
−∞
(−A)1−α T (t − s)m(s + sn )ds
(−A)1−α T (t − s)mn (s)ds,
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
20
where mn (s) = m(s + sn ), n = 1, 2, · · · . Also, we have
Z t
|(−A)1−α T (t − s)mn (s)|ds
|G(t + sn )| ≤
−∞
t
Z
≤
M1−α e−α(t−s) (t − s)α−1 |mn (s)|ds
−∞
≤ M1−α δ −α Γ(α)kmkAA(X) ,
for n = 1, 2, · · · . By the property (ii) of Lemma 2.1 it follows that
(−A)1−α T (t − s)mn (s) → (−A)1−α T (t − s)ϕ(s), as n → ∞,
for each s ∈ R fixed and any t ≥ s, and we get
t
Z
(−A)1−α T (t − s)ϕ(s)ds,
lim G(t + sn ) = −
n→∞
−∞
by the Lebesgue’s dominated convergence theorem. Analogously to the above proof, it can be shown that
Z t−sn
1−α
lim −
(−A)
T (t − sn − s)ϕ(s)ds = G(t).
n→∞
−∞
This shows that G(t) ∈ AA(X).
Next we prove that H(t) ∈ P AA0 (R, ρ), that is we need to prove that
Z r
1
lim
|H(t)|ρ(t)dt = 0.
r→∞ m(r, ρ) −r
We have
1
lim
r→∞ m(r, ρ)
Z rZ t
1
|H(t)|ρ(t)dt ≤ lim
|(−A)1−α T (t − s)n(s)|ρ(s)dsdt
r→∞ m(r, ρ) −r −∞
−r
Z r Z −r
1
≤ lim
dt
|(−A)1−α T (t − s)n(s)|ρ(s)ds
r→∞ m(r, ρ) −r
−∞
Z r Z t
1
|(−A)1−α T (t − s)n(s)|ρ(s)ds
dt
+ lim
r→∞ m(r, ρ) −r
−r
= I1 + I2 ,
Z
r
where
r
1
r→∞ m(r, ρ)
Z
1
I2 := lim
r→∞ m(r, ρ)
Z
I1 := lim
−r
Z
|(−A)1−α T (t − s)n(s)|ρ(s)ds
dt
−r
and
−∞
r
Z
t
dt
−r
−r
|(−A)1−α T (t − s)n(s)|ρ(s)ds.
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
21
We get
I1 =
≤
≤
≤
≤
≤
≤
Z r Z −r
1
|(−A)1−α T (t − s)n(s)|ρ(s)ds
lim
dt
r→∞ m(r, ρ) −r
−∞
Z r Z −r
1
lim
dt
M1−α e−δ(t−s) (t − s)α−1 |n(s)|ρ(s)ds
r→∞ m(r, ρ) −r
−∞
Z −r
Z
knkBC(R,X) r
M1−α e−δ(t−s) (t − s)α−1 ds
ρ(t)dt
lim
r→∞
m(r, ρ)
−∞
−r
Z +∞
Z
M1−α knkBC(R,X) r
lim
σ α−1 e−δσ dσ
ρ(t)dt
r→∞
m(r, ρ)
t+r
−r
Z
Z +∞
M1−α knkBC(R,X) r
lim
ρ(t)dt
σ α−1 e−δσ dσ
r→∞
m(r, ρ)
−r
2r
Z +∞
lim M1−α knkBC(R,X)
(2r)α−1 e−δσ dσ
r→∞
2r
M1−α knkBC(R,X)
,
(2r)1−α e2δr δ
where converges to zero as r → ∞.
Z r Z t
1
dt
|(−A)1−α T (t − s)n(s)|ρ(s)ds
I2 = lim
r→∞ m(r, ρ) −r
−r
Z r Z t
M1−α
≤ lim
dt
e−δ(t−s) (t − s)α−1 |n(s)|ρ(s)ds
r→∞ m(r, ρ) −r
−r
Z r
Z t+r
M1−α
≤ lim
|n(t)|ρ(t)dt
σ α−1 e−δσ dσ
r→∞ m(r, ρ) −r
0
Z r
Z +∞
M1−α
|n(t)|ρ(t)dt
σ α−1 e−δσ dσ
≤ lim
r→∞ m(r, ρ) −r
0
Z r
1
−α
≤ lim M1−α δ Γ(α)
|n(t)|ρ(t)dt.
r→∞
m(r, ρ) −r
Rr
1
Since n(·) ∈ P AA0 (R, ρ), then limr→∞ m(r,ρ)
−r |n(t)|ρ(t)dt = 0. Therefore limr→∞ I2 = 0. In view of the
above it is clear that u ∈ W P AA(R, ρ). The proof is achieved.
The next result is a straightforward consequence of Lemma 3.1 when ρ = 1, and so we omit its proof.
Lemma 3.2. Let α ∈ (0, 1] and (−A)α v ∈ P AA(X). If u(·) : R → X be the function defined by
Z
t
AT (t − s)v(s)ds, t ≥ s,
u(t) =
−∞
then u(·) ∈ P AA(X).
Lemma 3.3. ([24, Lemma 4.1]) Assume that both h1 (·) and h2 (·) satisfy (H3). If u ∈ P AA(X), then
u (hi (·)) ∈ P AA(X) for i = 1, 2.
The proof of the following lemma is similar to that of Lemma 3.2 in [29]. For the completeness, we give
the detailed proof here.
Lemma 3.4. Assume that both h1 (·) and h2 (·) satisfy (H4).
W P AA(R, ρ) for i = 1, 2.
If u ∈ W P AA(R, ρ), then u (hi (·)) ∈
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
22
Proof. . Let u(·) = v(·) + w(·), where v(·) ∈ AA(X) and w(·) ∈ P AA0 (R, ρ). By condition (H4) and the
definition of almost automorphic functions, we can easily conclude that v (hi (t)) ∈ AA(X).
On the other hand, we need to prove that w (hi (t)) ∈ P AA0 (R, ρ). We have
Z r
1
0 ≤
|w (hi (t)) |ρ (t) dt
m(r, ρ) −r
Z r
0
1
ρ(t) hi (t)
=
dt
|w (hi (t)) |ρ (hi (t))
m(r, ρ) −r
ρ (hi (t)) h0i (t)
Z r
ρ(t)
1
0
|w (hi (t)) |ρ (hi (t)) hi (t)dt
≤
sup
0
m(r, ρ)hi (−r) t∈R ρ (hi (t)) −r
Z |hi (r)|
1
ρ(t)
≤
|w(t)|ρ(t)dt
sup
0
m(r, ρ)hi (−r) t∈R ρ (hi (t)) −|hi (−r)|
Z |hi (−r)|+|hi (r)|
1
ρ(t)
≤
|w(t)|ρ(t)dt
sup
0
m(r, ρ)hi (−r) t∈R ρ (hi (t)) −|hi (−r)|−|hi (r)|
Z r∗
i
ρ(t)
m(ri∗ , ρ)
1
sup
=
|w(t)|ρ(t)dt,
0
∗
m(r, ρ)hi (−r) t∈R ρ (hi (t)) m(ri , ρ) −ri∗
where ri∗ = |hi (−r)| + |hi (r)|, i = 1, 2. Since (H4) and w(·) ∈ P AA0 (R, ρ), then the last inequality converges
to zero as r → ∞. Thus w (hi (t)) ∈ P AA0 (R, ρ). As a consequence of the above proof, we can see that
u (hi (t)) ∈ W P AA(R, ρ) for i = 1, 2. The proof is finished.
Remark 3.5. Let ρ = 1, then the condition (H4) is reduced to the condition (H3). It is consistent with the
fact that the space W P AA(R × X, ρ) is turned into the space P AA(R × X, X) when ρ = 1.
Now, we state and prove our main results.
Theorem 3.6. Assume the conditions (H1)(I), (H2)(I) and (H3) hold, then the problem (1.1) has a unique
pseudo almost automorphic mild solution on R provide that
(1)
(1)
k(−A)−α kLf + M1−α δ −α Γ(α)Lf +
M (1)
L < 1,
δ g
(3.1)
where Γ(·) is the gamma function.
Proof. Let Λ : P AA(X) → C(R, X) be the operator defined by
Z t
−α
α
Λu(t) = −(−A) (−A) f (t, u(h1 (t))) +
(−A)1−α T (t − s)(−A)α f (s, u(h1 (s))) ds
−∞
Z
t
T (t − s)g (s, u(h2 (s))) ds
+
−∞
Z
t
= −f (t, u(h1 (t))) −
AT (t − s)f (s, u(h1 (s))) ds
−∞
Z
t
T (t − s)g (s, u(h2 (s))) ds, t ∈ R.
+
−∞
First we prove that Λu(t) is well defined. From the continuity of s → AT (t − s) and s → T (t − s) in the
uniform operator topology on (−∞, t) for each t ∈ R and the estimate
|AT (t − s)f (s, u(h1 (s))) | = |(−A)1−α T (t − s)(−A)α f (s, u(h1 (s))) |
≤ M1−α e−δ(t−s) (t − s)α−1 k(−A)α f (s, u(h1 (s))) kBC(R×X,X) ,
it follows that s → AT (t − s)f (s, u(h1 (s))) and s → T (t − s)g (s, u(h2 (s))) are integrable on (−∞, t) for
every t ∈ R and so that Λu is well defined and continuous. Moreover, from Lemmas 3.3, 2.7, 3.2 and 2.14
we infer that Λu(t) ∈ P AA(X), that is, Λ maps P AA(X) into itself.
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
23
Next, we show that Λ is a contraction on P AA(X). Indeed, for each t ∈ R, u, v ∈ P AA(X), we have
|Λu(t) − Λv(t)|
≤ |f (t, u(h1 (t))) − f (t, v(h1 (t))) |
Z t
|AT (t − s)f (s, u(h1 (s))) − AT (t − s)f (s, v(h1 (s))) |ds
+
−∞
t
Z
|T (t − s)g (s, u(h2 (s))) − T (t − s)g (s, v(h2 (s))) |ds
+
−∞
−α
≤ k(−A) k|(−A)α f (t, u(h1 (t))) − (−A)α f (t, v(h1 (t))) |
Z t
k(−A)1−α T (t − s)k|(−A)α f (s, u(h1 (s))) − (−A)α f (s, v(h1 (s))) |ds
+
−∞
t
Z
+
M e−δ(t−s) |g (s, u(h2 (s))) − g (s, v(h2 (s))) |ds
−∞
(1)
≤ k(−A)−α kLf |u (h1 (t)) − v (h1 (t)) |
Z t
(1)
+Lf
M1−α e−δ(t−s) (t − s)α−1 |u (h1 (s)) − v (h1 (s)) |ds
−∞
t
+L(1)
g
Z
−∞
−α
≤ k(−A)
+L(1)
g
M e−δ(t−s) |u (h2 (s)) − v (h2 (s)) |ds
(1)
(1)
kLf ku − vkP AA(X) + Lf M1−α δ −α Γ(α)ku − vkP AA(X)
M
ku − vkP AA(X)
δ
(1)
(1)
= [k(−A)−α kLf + M1−α δ −α Γ(α)Lf +
M (1)
L ]ku − vkP AA(X) .
δ g
Thus
M (1)
L ]ku − vkP AA(X) ,
δ g
which implies that Λ is a contraction by (3.1). By the contraction principle, we conclude that there exists
a unique fixed point u(·) for Λ in P AA(X), such that Λu = u, that is
(1)
(1)
kΛu − ΛvkP AA(X) ≤ [k(−A)−α kLf + M1−α δ −α Γ(α)Lf +
Z
t
u(t) = −f (t, u(h1 (t))) −
Z
t
AT (t − s)f (s, u(h1 (s))) ds +
−∞
T (t − s)g (s, u(h2 (s))) ds,
−∞
Ra
Ra
for all t ∈ R. If we let u(a) = −f (a, u(h1 (a)))− −∞ AT (a−s)f (s, u(h1 (s))) ds+ −∞ T (a−s)g (s, u(h2 (s))) ds,
then
Z a
AT (t − s)f (s, u(h1 (s))) ds
T (t − a)u(a) = −T (t − a)f (a, u(h1 (a))) −
−∞
Z a
+
T (t − s)g (s, u(h2 (s))) ds.
−∞
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
24
But for t ≥ a,
Z t
T (t − s)g (s, u(h2 (s))) ds
a
t
Z
Z
a
T (t − s)g (s, u(h2 (s))) ds
T (t − s)g (s, u(h2 (s))) ds −
=
−∞
−∞
t
Z
AT (t − s)f (s, u(h1 (s))) ds
= u(t) + f (t, u(h1 (t))) +
−∞
−T (t − a)[u(a) + f (a, u(h1 (a)))]
Z a
−
AT (t − s)f (s, u(h1 (s))) ds
−∞
t
Z
AT (t − s)f (s, u(h1 (s))) ds
= u(t) + f (t, u(h1 (t))) +
a
−T (t − a)[u(a) + f (a, u(h1 (a)))].
In conclusion,
u (t) = T (t − a) [u(a) + f (a, u(h1 (a)))] − f (t, u(h1 (t)))
Z t
Z t
AT (t − s)f (s, u(h1 (s))) ds +
T (t − s)g (s, u(h2 (s))) ds
−
a
a
is a mild solution of equation (1.1) and u ∈ P AA(X). This finishes the proof.
Theorem 3.7. Assume the conditions (H1)(II), (H2)(II) and (H4) are satisfied, then the problem (1.1) has
a unique weighted pseudo almost automorphic mild solution on R provide that
(2)
(2)
k(−A)−α kLf + M1−α δ −α Γ(α)Lf +
M (2)
L < 1,
δ g
(3.2)
where Γ(·) is the gamma function.
Proof. We define the operator Λ : W P AA(R, ρ) → C(R, X) as
Z t
−α
α
(−A)1−α T (t − s)(−A)α f (s, u(h1 (s))) ds
Λu(t) = −(−A) (−A) f (t, u(h1 (t))) +
−∞
Z
t
T (t − s)g (s, u(h2 (s))) ds
+
−∞
Z
t
= −f (t, u(h1 (t))) −
AT (t − s)f (s, u(h1 (s))) ds
−∞
Z
t
T (t − s)g (s, u(h2 (s))) ds, t ∈ R.
+
−∞
The same arguments used in the proof of Theorem 3.6, we can prove that Λu(t) is well defined and continuous.
Moreover, from Lemmas 3.4, 2.12, 3.1 and 2.13, we infer that Λu(t) ∈ W P AA(R, ρ), that is, Λ maps
W P AA(R, ρ) into itself.
Now we prove that Λ is a contraction on W P AA(R, ρ). Indeed, for each t ∈ R, u, v ∈ W P AA(R, ρ), we
Y. -K. Chang, Z. -H. Zhao, J. J. Nieto, Z. -W. Liu, J. Nonlinear Sci. Appl. 5 (2012), 14–26
25
have
|Λu(t) − Λv(t)|
≤ |f (t, u(h1 (t))) − f (t, v(h1 (t))) |
Z t
|AT (t − s)f (s, u(h1 (s))) − AT (t − s)f (s, v(h1 (s))) |ds
+
−∞
t
Z
|T (t − s)g (s, u(h2 (s))) − T (t − s)g (s, v(h2 (s))) |ds
+
−∞
−α
≤ k(−A) k|(−A)α f (t, u(h1 (t))) − (−A)α f (t, v(h1 (t))) |
Z t
k(−A)1−α T (t − s)k|(−A)α f (s, u(h1 (s))) − (−A)α f (s, v(h1 (s))) |ds
+
−∞
t
Z
+
M e−δ(t−s) |g (s, u(h2 (s))) − g (s, v(h2 (s))) |ds
−∞
(2)
≤ k(−A)−α kLf |u (h1 (t)) − v (h1 (t)) |
Z t
(2)
+Lf
M1−α e−δ(t−s) (t − s)α−1 |u (h1 (s)) − v (h1 (s)) |ds
−∞
t
+L(2)
g
Z
−∞
−α
≤ k(−A)
+L(2)
g
M e−δ(t−s) |u (h2 (s)) − v (h2 (s)) |ds
(2)
(2)
kLf ku − vkW P AA(R,ρ) + Lf M1−α δ −α Γ(α)ku − vkW P AA(R,ρ)
M
ku − vkW P AA(R,ρ)
δ
(2)
(2)
= [k(−A)−α kLf + M1−α δ −α Γ(α)Lf +
M (2)
L ]ku − vkW P AA(R,ρ) .
δ g
Thus
(2)
(2)
kΛu − ΛvkW P AA(R,ρ) ≤ [k(−A)−α kLf + M1−α δ −α Γ(α)Lf +
M (2)
L ]ku − vkW P AA(R,ρ) .
δ g
It follows that Λ is a contraction from (3.2). By the contraction principle, we draw a conclusion that there
exists a unique fixed point u(·) for Λ in W P AA(R, ρ), such that Λu = u. Moreover, using the same
R t proof
as in Theorem 3.6, we can see that u (t) = T (t − a) [u(a) + f (a, u(h1 (a)))] − f (t, u(h1 (t))) − a AT (t −
Rt
s)f (s, u(h1 (s))) ds + a T (t − s)g (s, u(h2 (s))) ds is a mild solution of equation (1.1) and u ∈ W P AA(R, ρ).
This completes the proof.
Acknowledgements:
The first author was supported by NNSF of China (10901075), Program for New Century Excellent
Talents in University (NCET-10-0022), the Key Project of Chinese Ministry of Education (210226), and
NSF of Gansu Province of China (1107RJZA091). The third author’s work was supported by Ministerio de
Ciencia e Innovación and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project
PGIDIT06PXIB207023PR.
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