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Almost Automorphic and Pseudo-Almost Automorphic Solutions to Semilinear Evolution Equations with Nondense Domain
Journal of Inequalities and Applications volume 2009, Article number: 298207 (2009)
Abstract
We study the existence and uniqueness of almost automorphic (resp., pseudo-almost automorphic) solutions to a first-order differential equation with linear part dominated by a Hille-Yosida type operator with nondense domain.
1. Introduction
In recent years, the theory of almost automorphic functions has been developed extensively (see, e.g., Bugajewski and N'guérékata [1], Cuevas and Lizama [2], and N'guérékata [3] and the references therein). However, literature concerning pseudo-almost automorphic functions is very new (cf. [4]). It is well known that the study of composition of two functions with special properties is important and basic for deep investigations. Recently an interesting article has appeared by Liang et al. [5] concerning the composition of pseudo-almost automorphic functions. The same authors in [6] have applied the results to obtain pseudo-almost automorphic solutions to semilinear differentail equations (see also [7]). On the other hand, in article by Blot et al. [8], the authors have obtained existence and uniqueness of pseudo-almost automorphic solutions to some classes of partial evolutions equations.
In this work, we study the existence and uniqueness of almost automorphic and pseudo-almost automorphic solutions for a class of abstract differential equations described in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ1_HTML.gif)
where is an unbounded linear operator, assumed to be Hille-Yosida (see Definition 2.5) of negative type, having the domain
, not necessarily dense, on some Banach space
is a continuous function, where
. The regularity of solutions for (1.1) in the space of pseudo-almost periodic solutions was considered in Cuevas and Pinto [9] (see [10–12]). We note that pseudo-almost automorphic functions are more general and complicated than pseudo-almost periodic functions (cf. [5]).
The existence of almost automorphic and pseudo-almost automorphic solutions for evolution equations with linear part dominated by a Hille-Yosida type operator constitutes an untreated topic and this fact is the main motivation of this paper.
2. Preliminaries
Let be Banach spaces. The notations
and
stand for the collection of all continuous functions from
into
and the Banach space of all bounded continuous functions from
into
endowed with the uniform convergence topology. Similar definitions as above apply for both
and
We recall the following definition (cf. [7]).
Definition 2.1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_IEq14_HTML.gif)
A continuous function is called almost automorphic if for every sequence of real numbers
there exists a subsequence
such that
is well defined for each
and
, for each
Since the range of an almost automorphic function is relatively compact, then it is bounded. Almost automorphic functions constitute a Banach space,
when it is endowed with the supremum norm.
A continuous function is called almost automorphic if
is almost automorphic in
uniformly for all
in any bounded subset of
.
is the collection of those functions.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_IEq29_HTML.gif)
A continuous function (resp.,
) is called pseudo-almost automorphic if it can be decomposed as
where
(resp.,
) and
is a bounded continuous function with vanishing mean value, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ2_HTML.gif)
(resp., is a bounded continuous function with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ3_HTML.gif)
uniformly for in any bounded subset of
). Denote by
(resp.,
) the set of all such functions. In both cases above,
and
are called, respectively, the principal and the ergodic terms of
.
We define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ4_HTML.gif)
Remark 2.2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_IEq44_HTML.gif)
is a Banach space, where is the supremum norm (see [6]).
Lemma 2.3 (see [13]).
Let be an almost automorphic function in
for each
and assume that
satisfies a Lipschitz condition in
uniformly in
. Let
be an almost automorphic function. Then the function
defined by
is almost automorphic.
Let and assume that
is uniformly continuous in any bounded subset
uniformly in
. If
, then the function
belongs to
.
We recall some basic properties of extrapolation spaces for Hille-Yosida operators which are a natural tool in our setting. The abstract extrapolation spaces have been used from various purposes, for example, to study Volterra integro differential equations and retarded differential equations (see [14]).
Definition 2.5.
Let be a Banach space, and let
be a linear operator with domain
. One says that
is a Hille-Yosida operator on
if there exist
and a positive constant
such that
and
The infinimum of such
is called the type of
. If the constant
can be chosen smaller than zero,
is called of negative type.
Let be a Hille-Yosida operator on
, and let
;
, and let
be the operator defined by
. The following result is well known.
Lemma 2.6 (see [12]).
The operator is the infinitesimal generator of a
-semigroup
on
with
for
. Moreover,
and
, for
.
For the rest of paper we assume that is a Hille-Yosida operator of negative type on
. This implies that
, that is,
. We note that the expression
defines a norm on
. The completion of
, denoted by
, is called the extrapolation space of
associated with
. We note that
is an intermediary space between
and
and that
(see [12]). Since
, we have that
which implies that
has a unique bounded linear extension
to
. The operator family
is a
-semigroup on
, called the extrapolated semigroup of
. In the sequel,
is the generator of
.
Lemma 2.7 (see [12]).
Under the previous conditions, the following properties are verified.
(i) and
for every
.
(ii)The operator is the unique continuous extension of
, and
is an isometry from
into
.
(iii)If , then (
exists and
. In particular,
and
.
(iv)The space is dense in
. Thus, the extrapolation space
is also the completion of
and
. Moreover,
is an extension of
to
. In particular, if
, then
and
.
Lemma 2.8 (see [12]).
Let . Then the following properties are valid.
(i), for every
.
(ii) where
is independent of
and
.
(iii)The linear operator defined by
is continuous.
(iv), for every
.
(v) is the unique bounded mild solution in
of
3. Existence Results
3.1. Almost Automorphic Solutions
The following property of convolution is needed to establish our result.
Lemma 3.1.
If is an almost automorphic function and
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ5_HTML.gif)
then .
Proof.
Let be a sequence of real numbers. There exist a subsequence
and a continuous functions
such that
converges to
and
converges to
for each
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ6_HTML.gif)
Using the Lebesgue dominated convergence theorem, it follows that converges to
for each
. Proceeding as previously, one can prove that
converges to
for each
. This completes the proof.
Theorem 3.2.
Assume that is an almost automorphic function in
for each
and assume that satisfies a
Lipschitz condition in
uniformly in
. If
where
is the constant in Lemma 2.8, then (1.1) has a unique almost automorphic mild solution which is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ7_HTML.gif)
Proof.
Let be a function in
, from Lemma 2.3 the function
is in
From Lemma 2.8 and taking into account Lemma 3.1, the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ8_HTML.gif)
has a unique solution in
, which is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ9_HTML.gif)
It suffices now to show that the operator has a unique fixed point in
For this, let
and
be in
and we can infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ10_HTML.gif)
This proves that is a contraction, so by the Banach fixed point theorem there exists a unique
such that
This completes the proof of the theorem.
3.2. Pseudo-Almost Automorphic Solutions
To prove our next result, we need the following result.
Lemma 3.3.
Let , and let
be the function defined in Lemma 3.1. Then
.
Proof.
It is clear that . If
, where
and
. From Lemma 3.1
To complete the proof, we show that
. For
we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ11_HTML.gif)
The preceding estimates imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ12_HTML.gif)
The proof is now completed.
Now, we are ready to state and prove the following result.
Theorem 3.4.
Assume that is a pseudo-almost automorphic function and that there exists a bounded integrable function
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ13_HTML.gif)
Then (1.1) has a unique pseudo-almost automorphic (mild) solution.
Proof.
Let be a function in
, from Lemma 2.4 the function
belongs to
. From Lemmas 2.8 and 3.3, (3.4) has a unique solution in
which is given by (3.5). Let
and
be in
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ14_HTML.gif)
hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ15_HTML.gif)
In general, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ16_HTML.gif)
Hence, since for
sufficiently large, by the contraction principle
has a unique fixed point
This completes the proof.
A different Lipschitz condition is considered in the following result.
Theorem 3.5.
Let be a pseudo-almost automorphic function. Assume that
verifies the Lipschitz condition (3.9) with
a bounded continuous function. Let
If there is a constant
such that
for all
where
is the constant in Lemma 2.8, then (1.1) has a unique pseudo-almost automorphic (mild) solution.
Proof.
We define the map on
by (3.5). By Lemmas 2.4 and 3.3,
is well defined. On the other hand, we can estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ17_HTML.gif)
Therefore is a contraction.
The following consequence is now immediate.
Corollary 3.6.
Let be a pseudo-almost automorphic function. Assume that
verifies the uniform Lipschitz condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ18_HTML.gif)
If where
is the constant in Lemma 2.8, then (1.1) has a unique pseudo-almost automorphic (mild) solution.
3.3. Application
In this section, we consider a simple application of our abstract results. We study the existence and uniqueness of pseudo-almost automorphic solutions for the following partial differential equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ19_HTML.gif)
with boundary initial conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ20_HTML.gif)
Let , and let the operator
be defined on
by
with domain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ21_HTML.gif)
It is well known that is a Hille-Yosida operator of type-1 with domain nondense (cf. [15]). Equation (3.15) can be rewritten as an abstract system of the form (1.1), where
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F298207/MediaObjects/13660_2009_Article_1934_Equ22_HTML.gif)
for all and
. By [5, Example 2.5],
is a pseudo-almost automorphic function. If we assume that
then, by Corollary 3.6, (3.15) has a unique pseudo-almost automorphic mild solution.
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Acknowledgment
Claudio Cuevas is partially supported by CNPQ/Brazil under Grant 300365/2008-0.
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de Andrade, B., Cuevas, C. Almost Automorphic and Pseudo-Almost Automorphic Solutions to Semilinear Evolution Equations with Nondense Domain. J Inequal Appl 2009, 298207 (2009). https://doi.org/10.1155/2009/298207
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DOI: https://doi.org/10.1155/2009/298207