- Research Article
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Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-Differential Equation
Advances in Difference Equations volume 2010, Article number: 310951 (2010)
Abstract
The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay is studied.
1. Introduction
In this paper, we study the existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations modelled in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ1_HTML.gif)
where and
,
, are closed linear operators;
is a Banach space; the history
,
, belongs to some abstract phase space
defined axiomatically
are appropriated functions.
The study of abstract neutral equations is motivated by different practical applications in different technical fields. The literature related to ordinary neutral functional differential equations is very extensive and we refer the reader to Chukwu [1], Hale and Lunel [2], Wu [3], and the references therein. As a practical application, we note that the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ2_HTML.gif)
arises in the study of the dynamics of income, employment, value of capital stock, and cumulative balance of payment; see [1] for details. In the above system, is a real number, the state
,
,
are
continuous functions matrices,
is a constant
matrix,
represents the government intervention, and
the private initiative. We note that by assuming the solution
is known on
, we can transform this system into an abstract system with unbounded delay described as (1.1).
Abstract partial neutral differential equations also appear in the theory of heat conduction. In the classic theory of heat conduction, it is assumed that the internal energy and the heat flux depend linearly on the temperature and on its gradient
. Under these conditions, the classic heat equation describes sufficiently well the evolution of the temperature in different types of materials. However, this description is not satisfactory in materials with fading memory. In the theory developed in [4, 5], the internal energy and the heat flux are described as functionals of
and
. The next system, see for instance [6–9], has been frequently used to describe this phenomenon,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ3_HTML.gif)
In this system, is open, bounded, and with smooth boundary;
;
represents the temperature in
at the time
;
is a physical constant
,
are the internal energy and the heat flux relaxation, respectively. By assuming that the solution
is known on
and
, we can transform this system into an abstract system with unbounded delay described in the form (1.1).
Recent contributions on the existence of solutions with some of the previously enumerated properties or another type of almost periodicity to neutral functional differential equations have been made in [10, 11], for the case of neutral ordinary differential equations, and in [12–15] for partial functional differential systems.
The purpose of this work is to study the existence of asymptotically almost periodic mild solutions for the neutral system (1.1). To this end, we study the existence and qualitative properties of an exponentially stable resolvent operator for the integro-differential system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ4_HTML.gif)
There exists an extensive literature related to the existence and qualitative properties of resolvent operator for integro-differential equations. We refer the reader to the book by Gripenberg et al. [16] which contains an overview of the theory for the case where the underlying space has finite dimension. For abstract integro-differential equations described on infinite dimensional spaces, we cite the Prüss book [17] and the papers [18–20], Da Prato et al. [21, 22], and Lunardi [9, 23]. To finish this short description of the related literature, we cite the papers [24–26] where some of the above topics for the case of abstract neutral integro-differential equations with unbounded delay are treated.
To the best of our knowledge, the study of the existence of asymptotically almost periodic solutions of neutral integro-differential equations with unbounded delay described in the abstract form (1.1) is an untreated topic in the literature and this is the main motivation of this article.
To finish this section, we emphasize some notations used in this paper. Let and
be Banach spaces. We denote by
the space of bounded linear operators from
into
endowed with norm of operators, and we write simply
when
. By
, we denote the range of a map
, and for a closed linear operator
, the notation
represents the domain of
endowed with the graph norm,
,
. In the case
, the notation
stands for the resolvent set of
and
is the resolvent operator of
. Furthermore, for appropriate functions
and
, the notation
denotes the Laplace transform of
and
the convolution between
and
, which is defined by
. The notation
stands for the closed ball with center at
and radius
in
. As usual,
represents the subspace of
formed by the functions which vanish at infinity.
2. Preliminaries
In this work, we will employ an axiomatic definition of the phase space similar to that in [27]. More precisely,
will denote a vector space of functions defined from
into
endowed with a seminorm denoted by
and such that the following axioms hold.
-
(A)
If
with
is continuous on
and
, then for each
the following conditions hold:
-
(i)
is in
,
-
(ii)
,
-
(iii)
where
is a constant, and
are functions such that
and
are respectively continuous and locally bounded, and
are independent of
.
-
(i)
(A1) If is a function as in
, then
is a
-valued continuous function on
.
-
(B)
The space
is complete.
(C2) If is a sequence in
formed by functions with compact support such that
uniformly on compact, then
and
as
Remark 2.1.
In the remainder of this paper, is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ5_HTML.gif)
for every continuous and bounded; see [27, Proposition
] for details.
Definition 2.2.
Let be the
-semigroup defined by
on
and
on
. The phase space
is called a fading memory if
as
for each
with
.
Remark 2.3.
In this work, we suppose that there exists a positive such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ6_HTML.gif)
for each . Observe that this condition is verified, for example, if
is a fading memory, see [27, Proposition
].
Example 2.4 (The phase space ).
Let
, and let
be a nonnegative measurable function which satisfies the conditions (g-5) and (g-6) in the terminology of [27]. Briefly, this means that
is locally integrable, and there exists a nonnegative, locally bounded function
on
such that
for all
and
, where
is a set with Lebesgue measure zero. The space
consists of all classes of functions
such that
is continuous on
, Lebesgue-measurable, and
is Lebesgue integrable on
. The seminorm in
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ7_HTML.gif)
The space satisfies axioms (A), (A-1), and (B). Moreover, when
and
, we can take
,
, and
, for
; see [27, Theorem
] for details.
Now, we need to introduce some concepts, definitions, and technicalities on almost periodical functions.
Definition 2.5.
A function is almost periodic (a.p.) if for every
, there exists a relatively dense subset of
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ8_HTML.gif)
Definition 2.6.
A function is asymptotically almost periodic (a.a.p.) if there exists an almost periodic function
and
such that
.
The next lemmas are useful characterizations of a.p and a.a.p functions.
Lemma 2.7 (see [28, Theorem ]).
A function is asymptotically almost periodic if and only if for every
there exist
and a relatively dense subset of
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ9_HTML.gif)
In this paper, and
are the spaces
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ10_HTML.gif)
endowed with the norms and
respectively. We know from the result in [28] that
and
are Banach spaces.
Next, and
are abstract Banach spaces.
Definition 2.8.
Let be an open subset of
-
(a)
A continuous function
(resp.,
) is called pointwise almost periodic (p.a.p.), (resp., pointwise asymptotically almost periodic (p.a.a.p.) if
(resp.,
) for every
.
-
(b)
A continuous function
is called uniformly almost periodic (u.a.p.), if for every
and every compact
there exists a relatively dense subset of
, denoted by
, such that
(2.7)
-
(c)
A continuous function
is called uniformly asymptotically almost periodic (u.a.a.p.), if for every
and every compact
there exists a relatively dense subset of
, denoted by
, and
such that
(2.8)
The next lemma summarizes some properties which are fundamental to obtain our results.
Lemma 2.9 (see [29, Theorem ]).
Let be an open set. Then the following properties hold.
-
(a)
If
is p.a.p. and satisfies a local Lipschitz condition at
, uniformly at
, then
is u.a.p.
-
(b)
If
is p.a.a.p. and satisfies a local Lipschitz condition at
, uniformly at
, then
is u.a.a.p.
-
(c)
If
then
. Moreover, if
is a fading memory space and
is such that
and
, then
.
-
(d)
If
is u.a.p. and
is such that
, then
.
-
(e)
If
is u.a.a.p and
is such that
, then
.
3. Resolvent Operators
In this section, we study the existence and qualitative properties of an exponentially resolvent operator for the integro-differential abstract Cauchy problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ13_HTML.gif)
The results obtained for the resolvent operator in this section are similar to those that can be found, for instance, in the papers [21, 23, 30]. In this paper, we prove the necessary estimates for the proof of an existence theorem of asymptotically almost periodic solutions for (1.1). For the better comprehension of the subject, we will introduce the following definitions, hypothesis, and results.
We introduce the following concept of resolvent operator for integro-differential problem (3.1).
Definition 3.1.
A one-parameter family of bounded linear operators on
is called a resolvent operator of (3.1) if the following conditions are verified.
-
(a)
Function
is strongly continuous and
for all
.
-
(b)
For
,
, and
(3.2)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ15_HTML.gif)
for every ,
-
(c)
There exist constants
such that
for every
Definition 3.2.
A resolvent operator of (3.1) is called exponentially stable if there exist positive constants
such that
In this work, we always assume that the following conditions are verified.
(H1) The operator is the infinitesimal generator of an analytic semigroup
on
, and there are constants
, and
such that
and
for all
.
(H2) For all is a closed linear operator,
, and
is strongly measurable on
for each
. There exists
such that
exists for
and
for all
and
. Moreover, the operator valued function
has an analytical extension (still denoted by
) to
such that
for all
, and
as
.
(H3) There exist a subspace dense in
and positive constants
,
such that
,
, and
for every
and all
.
In the sequel, for,
, and
, set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ16_HTML.gif)
and for,
, the paths
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ17_HTML.gif)
with are oriented counterclockwise. In addition,
is the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ18_HTML.gif)
We next study some preliminary properties needed to establish the existence of a resolvent operator for the problem (3.1).
Lemma 3.3.
There exists such that
and the function
is analytic. Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ19_HTML.gif)
and there exist constants for
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ22_HTML.gif)
for every .
Proof.
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ23_HTML.gif)
fixed there exists a positive number
such that
for
. Consequently, the operator
has a continuous inverse with
. Moreover, for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ24_HTML.gif)
and for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ25_HTML.gif)
which shows (3.7) and that . Now, from (3.7) we obtain
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ26_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ27_HTML.gif)
the functions are analytic, and estimates (3.8), and (3.10) are valid. In addition, for
, we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ28_HTML.gif)
for sufficiently large. This proves (3.9) and completes the proof.
Observation 1.
If is a resolvent operator for (3.1), it follows from (3.3) that
for all
. Applying Lemma 3.3 and the properties of the Laplace transform, we conclude that
is the unique resolvent operator for (3.1).
In the remainder of this section, and
are numbers such that
and
. Moreover, we denote by
a generic constant that represents any of the constants involved in the statements of Lemma 3.3 as well as any other constant that arises in the estimate that follows. We now define the operator family
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ29_HTML.gif)
We will next establish that is a resolvent operator for (3.1).
Lemma 3.4.
The function is exponentially bounded in
.
Proof.
If , from (3.17) and estimate (3.8), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ30_HTML.gif)
On the other hand, using that is analytic on
, for
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ31_HTML.gif)
This complete the proof.
Lemma 3.5.
The operator function is exponentially bounded in
.
Proof.
It follows from (3.9) that the integral in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ32_HTML.gif)
is absolutely convergent in and defines a linear operator
. Using that
is closed, we can affirm that
. From Lemma 3.3,
is analytic and
. If
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ33_HTML.gif)
For and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ34_HTML.gif)
From before and Lemma 3.4, we infer that is exponentially bounded in
. The proof is finished.
Lemma 3.6.
The function is strongly continuous.
Proof.
It is clear from (3.17) that is continuous at
for every
. We next establish the continuity at
. Let
and
be sufficiently large, using that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ35_HTML.gif)
where represent the curve
for
.
For and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ36_HTML.gif)
Furthermore, it follows from (3.8), and assumption (H2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ37_HTML.gif)
where is integrable for
. From the Lebesgue dominated convergence theorem, we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ38_HTML.gif)
Let now be the curve
for
. Turning to apply Cauchy's theorem combining with the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ39_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ40_HTML.gif)
we can affirm that for all
, which completes and the proof since
is dense in
and
is bounded on
.
Notice that the sectors
from Lemma 3.3,
is analytic. Consider the contours
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ41_HTML.gif)
and oriented counterclockwise. By Cauchy theorem for
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ42_HTML.gif)
The following estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ43_HTML.gif)
is the one responsible for the fact that the integral tends to
as
tend to
in a similar way the integral
tend to
as
tend to
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ44_HTML.gif)
For , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ45_HTML.gif)
and proceeding as before, we obtain for all
which ends the proof.
The following result can be proved with an argument similar to that used in the proof of the preceding lemma with changing by
Lemma 3.7.
The function is strongly continuous.
We next set .
Lemma 3.8.
The function has an analytic extension to
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ46_HTML.gif)
Proof.
For and
, we can write
where
,
and
If
, from (3.8) and (3.17), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ47_HTML.gif)
Using that is analytic on
, for
,
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ48_HTML.gif)
This property allows us to define the extension by this integral.
Similarly, the integral on the right hand side of (3.34) is also absolutely convergent in and strong, continuous on
for
. For
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ49_HTML.gif)
where is integrable for
. From the Lebesgue dominated convergence theorem, we obtain that
verifies (3.34). The proof is ended.
Lemma 3.9.
For every with
,
.
Proof.
Using that is analytic on
and that the integrals involved in the calculus are absolutely convergent, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ50_HTML.gif)
Theorem 3.10.
The function is a resolvent operator for the system (3.1).
Proof.
Let . From Lemma 3.9, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ51_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ52_HTML.gif)
Applying [31, Proposition , Corollary
], we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ53_HTML.gif)
which in turn implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ54_HTML.gif)
Arguing as above but using the equality we obtain that (3.2) holds.
On the other hand, by Lemma 3.8 we infer that . Next, we analyze the differentiability on
. Let
and
for all
we can choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ55_HTML.gif)
For and
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ56_HTML.gif)
Consequently, for we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ57_HTML.gif)
which proves the existence of the right derivative of at zero and that
This proves that resolvent equation (3.3) is valid for every
and
for every
. This completes the proof.
Corollary 3.11.
If then the function
is an exponentially stable resolvent operator for the system (3.1).
In the next result, we denote by the fractional power of the operator
(see [32] for details).
Theorem 3.12.
Suppose that the conditions are satisfied. Then there exists a positive number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ58_HTML.gif)
for all
Proof.
Let From [32, Theorem
], there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ59_HTML.gif)
Since is a
valued function, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ60_HTML.gif)
where is independent of
. From (3.48), we get for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ61_HTML.gif)
On the other hand, using that is analytic on
, for
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ62_HTML.gif)
From the previous facts, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ63_HTML.gif)
which ends the proof.
Corollary 3.13.
If and
, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ64_HTML.gif)
In the remainder of this section, we discuss the existence and regularity of solutions of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ65_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ66_HTML.gif)
where . In the sequel,
is the operator function defined by (3.17). We begin by introducing the following concept of classical solution.
Definition 3.14.
A function ,
, is called a classical solution of (3.53)-(3.54) on
if
, the condition (3.54) holds and (3.53) is verified on
.
The next result has been established in [30].
Theorem 3.15 ([30, Theorem ]).
Let . Assume that
and
is a classical solution of (3.53)-(3.54) on
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ67_HTML.gif)
An immediate consequence of the above theorem is the uniqueness of classical solutions.
Corollary 3.16.
If are classical solutions of (3.53)-(3.54) on
, then
on
.
Motivated by (3.55), we introduce the following concept.
Definition 3.17.
A function is called a mild solution of (3.53)-(3.54) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ68_HTML.gif)
4. Existence Result of Asymptotically Almost Periodic Solutions
In this section, we study the existence of asymptotically almost periodic mild solutions for the abstract integro-differential system (1.1). To establish our existence result, motivated by the previous section we introduce the following assumptions.
(P1) There exists a Banach space continuously included in
such that the following conditions are verified.
-
(a)
For every
,
and
. In addition,
for every
.
-
(b)
There are positive constants
such that
(4.1)
-
(c)
There exists
such that
(P2) The continuous function is p.a.a.p, and there exists a continuous function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ70_HTML.gif)
(P3) The continuous function is p.a.a.p, and there exists a continuous function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ71_HTML.gif)
Motivated by the theory of resolvent operator, we introduce the following concept of mild solution for (1.1).
Definition 4.1.
A function is called a mild solution of (1.1) on
, if
the functions
and
are integrable on
for every
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ72_HTML.gif)
Lemma 4.2.
Let condition hold and let
be a function in
. If
is the function defined by
then
.
Proof.
Let . Let
,
be as in Lemma 2.7 and
such that
. For
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ73_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ74_HTML.gif)
Now, from inequality (4.6) and Lemma 2.7, we conclude that is a.a.p. The proof is complete.
Lemma 4.3.
Assume that the condition is fulfilled. Let
and let
be the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ75_HTML.gif)
Then .
Proof.
Let ,
be as in Lemma 2.7 and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ76_HTML.gif)
for where
. For
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ77_HTML.gif)
We obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ78_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ79_HTML.gif)
From inequality (4.11) and Lemma 2.7, we conclude that is a.a.p., which ends the proof.
Now, we can establish our existence result.
Theorem 4.4.
Assume that is a fading memory space and
, and
are held. If
and
for every
, then there exists
such that for each
, there exists a mild solution,
, of (1.1) on
such that
and
.
Proof.
Let and
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ80_HTML.gif)
where is the constant introduced in Remark 2.3. We affirm that the assertion holds for
Let
On the space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ81_HTML.gif)
endowed with the metric , we define the operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ82_HTML.gif)
where is the function defined by the relation
and
on
. From the hypothesis
, and
, we obtain that
is well defined and that
Moreover, from Lemmas 4.2 and 4.3 it follows that
.
Next, we prove that is a contraction from
into
. If
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ83_HTML.gif)
where the inequality has been used and
represent the continuous inclusion of
on
. Thus,
. On the other hand, for
we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ84_HTML.gif)
which shows that is a contraction from
into
The assertion is now a consequence of the contraction mapping principle. The proof is complete.
5. Applications
In this section, we study the existence of asymptotically almost periodic solutions of the partial neutral integro-differential system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ85_HTML.gif)
for ,
, and
Moreover, we have identified
.
To represent this system in the abstract form (1.1), we choose the spaces and
; see Example 2.4 for details. We also consider the operators
,
, given by
,
for
Moreover,
has discrete spectrum, the eigenvalues are
,
with corresponding eigenvectors
, and the set of functions
is an orthonormal basis of
and
for
. For
from [32] we can define the fractional power
of
is given by
where
In the next theorem, we consider
. We observe that
and
for
from [33, Proposition
], we obtain that
is a sectorial operator satisfying
Moreover, it is easy to see that conditions (H2)-(H3) in Section 3 are satisfied with
, and
is the space of infinitely differentiable functions that vanish at
and
. Under the above conditions, we can represent the system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ86_HTML.gif)
in the abstract form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ87_HTML.gif)
We define the functions by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ88_HTML.gif)
where
-
(i)
the functions
are continuous and
;
-
(ii)
the functions
,
are measurable,
for all
and
(5.5)
Moreover, are bounded linear operators,
,
, and a straightforward estimation using (ii) shows that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ90_HTML.gif)
for all . This allows us to rewrite the system (5.1) in the abstract form (1.1) with
Theorem 5.1.
Assume that the previous conditions are verified. Let and
such that
then there exists a mild solution
of (5.1) with
.
Proof.
For from
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F310951/MediaObjects/13662_2009_Article_1276_Equ91_HTML.gif)
since . By using a similar procedure as in the proofs of Lemma 3.3 and Theorem 3.10, we obtain the existence of resolvent operator for (5.2). From the hypothesis, we obtain
by the Lemma 3.3, Corollaries 3.11 and 3.13, the assumption
is satisfied. From Theorem 4.4, the proof is complete.
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Acknowledgment
José Paulo C. dos Santos is partially supported by FAPEMIG/Brazil under Grant CEX-APQ-00476-09.
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dos Santos, J.P.C., Guzzo, S.M. & Rabelo, M.N. Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-Differential Equation. Adv Differ Equ 2010, 310951 (2010). https://doi.org/10.1155/2010/310951
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DOI: https://doi.org/10.1155/2010/310951