1. Introduction
The aim of this paper is to study the existence and solutions for some neutral functional integro-differential equations with delay by using methods of maximal regularity in spaces of vector-valued functions and Besov space. Motivated by the fact that neutral functional integro-differential equations with finite delay arise in many areas of applied mathematics, this type of equation has received much attention in recent years. In particular, the problem of the existence of periodic solutions has been considered by several authors. We refer the readers to papers [1] [2] [3] and the references listed therein for information on this subject. One of the most important tools to prove maximal regularity is the theory of Fourier multipliers. They play an important role in the analysis of parabolic problems. In recent years, it has become apparent that one needs not only the classical theorems but also vector-valued extensions with operator-valued multiplier functions or symbols. These extensions allow treating certain problems for evolution equations with partial differential operators in an elegant and efficient manner in analogy to ordinary differential equations. For some recent papers on the subjet, we refer to Lizama et al. [4], Hino [5], Hale [6] and Pazy [7].
We characterize the existence of periodic solutions for the following integro-differential equations in vector-valued spaces and Besov. In the case of vector-valued space, our results involve UMD spaces, the concept of R-boundedness and a condition on the resolvent operator. We remark that many of the most powerful modern theorems are valid in UMD spaces, i.e., Banach space in which martingale is unconditional differences. The probabilistic definition of UMD spaces turns out to be equivalent to the
-boundedness of the Hilbert transform, a transformation, which is, in a sense, the typical representative example of a multiplier operator. On the other hand, the notion of R-boundedness has played an important role in the functional analytic approach to partial differential equations.
In the case of, Besov space, our results involve only boundedness of the resolvent.
In this work, we study the existence of periodic solutions for the following integro-differential equations:
(1)
where
is a linear closed operator on Banach space
and
for all
. For
(some
) L and G are in
is the space of all bounded linear operators and
is an element of
which is defined as follows:
For example:
Put
,
,
,
and
.
Then we have:
In [8], the author investigated the existence of solutions of the following fractional integrodifferential equation:
(2)
In [9], S. Koumla, Kh. Ezzinbi and R. Bahloul., study the existence of mild solutions for some partial functional integrodifferential equations with finite delay in a Fréchet space for equation:
In [10], Ezzinbi et al. gave necessary and sufficient conditions for the existence of periodic solutions of Equation (1) for
.
This work is organized as follows: after preliminaries in the second section, we are able to characterize in Section 3 the existence and uniqueness of the strong solution of the Equation (1) in Besov space, we obtain that the following assertion are equivalent If
and
then for every
there exist a unique strong
-solution of (1). In section 4, we give the conclusion.
2. Preliminaries
In this section we introduce some of the concepts to be used thereafter. We also review the classical results that provide material for a better understanding of the paper. We study the notion of M-boundedness. We present the notion of multipliers. Fourier multiplier theorems are of crucial importance in the study of maximal regularity of evolution equations. Let X be a Banach Space. Firstly, we denote By
the group defined as the quotient
. There is an identification between functions on
and
-periodic functions on
. We consider the interval
as a model for
.
Given
, we denote by
the space of
-periodic locally p-integrable functions from
into X, with the norm:
For
, we denote by
,
the k-th Fourier coefficient of f that is defined by:
For
, the periodic vector-valued space is defined by
(3)
Lemma 2.1 [7]:
Let
be a bounded linear operateur. Then:
Next we give some preliminaries. Given
and
(extended by periodicity to
), we define:
Let
be the Laplace transform of a. An easy computation shows that:
(4)
3. Periodic Strong Solutions in Besov Spaces
3.1. Preliminary
In this section, we consider the periodic solutions of Equation (1) in periodic Besov spaces
. Firstly, we briefly recall the definition of periodic Besov spaces. Let
be the Schwartz space of all rapidly decreasing smooth functions on
. Let
be the space of all infinitely differentiable functions on
equipped with the locally convex topology given by the seminorms
for
. Let
. In order to define Besov spaces, we consider the dyadic-like subsets of
:
for
. Let
be the set of all systems
satisfying
, for each
,
.
Let
,
and
the X-valued periodic Besov space is defined by:
For more information about the standard definitions and properties, see [7].
Proposition 3.1 1)
is a Banach space;
2) Let
. Then
in and only if f is differentiale and
Definition 3.1 For
, a sequence
is a
-multiplier if for each
, there exists
such that
for all
.
Remark 3.1 1) When
, then
.
2) when
is a strong
-solution of (1), then
, therefore u is twicely differentiable a.e. and
.
Definition 3.2 We say that
is M-bounded if:
(5)
(6)
We recall the following operator-valued Fourier multiplier theorem on Besov spaces.
Theorem 3.1 [7]
Let X, Y be Banach spaces and let
be a M-bounded sequence. Then for
,
,
is an
-multiplier:
3.2. Main Result
For convenience, we introduce the following notations:
In order to give our result, the following hypotheses are fundamental.
Definition 3.3: Let
and
. We say that a function
is a strong
-solution of (1) if
,
and Equation (1) holds for all
.
We prove the following result.
Theorem 3.2: Let A be a linear closed operator. Suppose that
is invertible for all
. If
and
then
is an
-multiplier for
and
.
Proof. Let
,
,
,
and
.
For convenience, we introduce the following result:
Then we have:
Now, we are going to show that:
(7)
and:
(8)
Put
,
and
. We have:
We obtain:
(9)
On the other hand, we have:
Then:
Finally we have:
Then by (9) we have:
and:
Then by (9) we have:
proving (7). To verify (8), put
,
we conclude that,
(10)
So,
is M-bounded and therefore, by Theorem 3.1 is an
-multiplier.
Furthermore:
Then:
(11)
On the other hand, we have:
Then by (10) and (11), we have:
Finally we have:
Then by (10) we have:
and:
Then by hypotheses and (10) we have
So,
and
are M-bounded and therefore, by Theorem 3.1 are an
-multiplier.
Theorem 3.3 Let
and
. Let X be a Banach space. Suppose that
is invertible for all
.
If
and
then for every
there exist a unique strong
-solution of (1).
Proof. Define
,
,
and
for
. Since by (7) and (8),
and
are M-bounded, we have by Theorem 3.1 that
and
are an
-multipliers. Since
(because (
), we deduce
is also an
-multiplicateur.
Now let
. Then there exist
, such that
,
,
,
and
for all
. So, We have
and
for all
, we deduce that
. On the other hand
such that
. By Lemma 2.2 we obtain
a.e. Since
.
We have
,
,
and
for all
, It follows from the identity
that:
For the uniqueness we suppose two solutions
and
, then
is strong
-solution of equation (1) corresponding to the function
, taking Fourier transform, we get
, which implies that
for all
and
. Then
.
On the other hand, we have
and by Remark 3.1 we deduce
. The proof completed.
4. Conclusion
We are obtained necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equation
in terms
of either the R-boundedness of the modified resolvent operator determined by the equation. Our results are obtained in the vector-valued space and Besov space.
Acknowledgements
The authors would like to thank the referee for his remarks to improve the original version.