In this work, we study the existence of periodic solutions for the following neutral partial functional differential equations of the following form
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
In  , Ezzinbi et al. established the existence of periodic solutions for the following partial functional differential equation:
where is a continuous w-periodic function, is a con- tinuous function w-in t, periodic and G is a positive function.
In  , Arendt gave necessary and sufficient conditions for the existence of periodic solutions of the following evolution equation.
where A is a closed linear operator on an UMD-space Y.
In  , C. Lizama established results on the existence of periodic solutions of Equation (1) when namely, for the following partial functional differential equation
where is a linear operator on an UMD-space X.
where and are closed linear operator such that
2. UMD Spaces
Let X be a Banach space. Firstly, we denote By the group defined as the quotient. There is an identification between functions on and 2p-periodic func- tions on. We consider the interval as a model for.
Given, we denote by the space of 2p-periodic locally p-inte- grable functions from into X, with the norm:
For, we denote by, the k-th Fourier coefficient of f that is defined by:
Definition 2.1 Let and. Define the operator by: for all
if exists in Then, is called the Hilbert transform of f on.
Definition 2.2 
A Banach space X is said to be UMD space if the Hilbert transform is bounded on for all.
Example 2.1  1) Any Hilbert space is an UMD space.
2) (0.1) are UMD spaces for every.
3) Any closed subspace of UMD space is an UMD space.
R-Bounded and Lp-Multipliers
Let X and Y be Banach spaces. Then denotes the space of bounded linear ope- rators from X to Y.
Definition 2.3 
A family of operators is called R-bounded (Rademacher bounded or randomized bounded), if there is a constant and such that for each, and for all independent, symmetric, -va- lued random variables on a probability space the inequality
is valid. The smallest C is called R-bounded of and it is denoted by.
Lemma 2.1 (  , Remark 2.2)
1) If is R-bounded then it is uniformly bounded, with
2) The definition of R-boundedness is independent of
Definition 2.4  For, a sequence is said to be an -multiplier if for each, there exists such that
Proposition 2.1 (  , Proposition 1.11) Let X be a Banach space and be an -multiplier, where. Then the set is R-bounded.
Theorem 2.1 (Marcinkiewicz operator-valud multiplier Theorem).
Let X, Y be UMD spaces and. If the sets and
are R-bounded, then is an -multiplier for.
Theorem 2.2  Let. Then
Theorem 2.3 (Neumann Expansion) Let, where X is a Banach space.
If then is invertible, moreover
3. Periodic Solutions for Equation (1)
Lemma 3.1 Let. If and. Then
Proof. Let. Then by applying the Fourier transform, we obtain that
Integration by parts we obtain that
The proof is complete.
Lemma 3.2  Let and. Then the following assertions are equivalent:
1) and there exists such that
2) for any.
By a Lemma 3.2 we obtain that
():such that and there exists with
Definition 3.1  . For, we say that a sequence is an -multiplier, if for each there exists such that
Lemma 3.3  Let and (is the set of all boun- ded linear operators from X to X). Then the following assertions are equivalent:
1) is an -multiplier.
2) is an -multiplier.
3.1. Existence of Strong Solutions for Equation (2)
Then the Equation (1) is equivalent:
Denote by; and for all. We define
We begin by establishing our concept of strong solution for Equation (2).
Definition 3.2 Let. A function is said to be a 2p- periodic strong -solution of Equation (2) if for all and Equation (2) holds almost every where.
Lemma 3.4 Let be a bounded linear operateur. Then
Proof. Let. Then
Since G is bounded, then
Lemma 3.5  Let X be a Banach space, independent, symmetric, -valued random variables on a probability space, and such that, for each. Then
Proposition 3.1 Let A be a closed linear operator defined on an UMD space X. Suppose that. Then the following assertions are equivalent:
1) is an -multiplier for
2) is R-bounded.
Proof. 1) Þ 2) As a consequence of Proposition 2.1
2) Þ 1) We claim first that the set is R-bounded. In fact, for we have:
By Lemma 3.4, we obtain that
We conclude that
Next define, where. By Theorem 2.1 it is su- fficient to prove that the set is R-bounded. Since
Since products and sums of R-bounded sequences is R-bounded [10. Remark 2.2]. Then the proof is complete.
Lemma 3.6 Let. Suppose that and that for every
there exists a 2p-periodic strong -solution x of Equation (2). Then, x is the unique 2p-periodic strong -solution.
Proof. Suppose that and two strong -solution of Equation (2) then
is a strong -solution of Equation (2) corresponding to. Taking Fourier transform in (2), we obtain that
It follows that for every and therefore. Then.
Theorem 3.1 Let X be a Banach space. Suppose that for every there exists a unique strong solution of Equation (2) for. Then
1) for every the operator has bounded inverse
2) is R-bounded.
Before to give the proof of Theorem 3.1, we need the following Lemma.
Lemma 3.7 if for all, then is a 2p-periodic strong -solution of the following equation
Proof of Lemma 3.7.
We have and
Proof of Theorem 3.1: 1) Let and. Then for, there exists such that:
Taking Fourier transform, G and D are bounded. We have
by Lemma 3.2 and Lemma 3.4 , we deduce that:
Consequently, we have
If, then by Lemma 3.7, is a 2p-periodic strong -solution of Equation (2) corresponing to the function Hence and then is injective.
2) Let. By hypothesis, there exists a unique such that the Equation (2) is valid. Taking Fourier transforms, we deduce that
Since then there exists such that
Then is an -multiplier and is R-bounded.
3.2. Periodic Mild Solutions of Equation (2) When A Generates a C0-Semigroup
It is well known that in many important applications the operator A can be the infini- tesimal generator of -semigroup on the space X.
Definition 3.3 Assume that A generates a -semigroup on X. A func- tion x is called a mild solution of Equation (2) if:
Remark 3.1 (  , Remark 4.2) Let be the -semigroup generated by A.
If is a continuous function, then and
Lemma 3.8  Assume that A generates a -semigroup on X, if x is a mild solution then
Theorem 3.2 Assume that A generates a -semigroup on X and
. For some; if x is a mild solution of Equation (2). Then
Proof. Let x be a mild solution of Equation (2). Then by Lemma 3.8, we have
For, we have
which shows that the assertion holds for.
Now, define and by Lemma 3.1 We have:
Corollary 3.1 Assume that A generates a -semigroup on X and let and x be a mild solution of Equation (2). If
has a bounded inverse. Then
Proof. From Theorem (3.2), we have that
Our main result in this work is to establish that the converse of Theorem 3.1 and Corollary 3.1 are true, provided X is an UMD space.
Theorem 3.3 Let X be an UMD space and be an closed linear operator. Then the following assertions are equivalent for.
1) for every there exists a unique 2p-periodic strong -solution of Equation (2).
2) and is R-bounded.
Lemma 3.9  Let. If and for all Then
Proof of Theorem 3.3:
1) Þ 2) see Theorem 3.1
1) Ü 2) Let. Define.
By proposition 3.1, the family is an -multiplier it is equivalent to the family is an -multiplier that maps into, namely
there exists such that
In particular, and there exists such that
By Theorem 2.2, we have
Hence in, we obtain that
Since G is bounded, then
Using now (3) and (4) we have:
Since A is closed, then [Lemma 4.1] and from the uniqueness theorem of Fourier coefficients, that Equation (2) is valid.
Theorem 3.4 Let. Assume that A generates a -semigroup on X. If and is an -multiplier Then there exists a unique mild periodic solution of Equation (2).
Proof. For, we define
By Theorem 2.2 we can assert that as for the norm in.
We have is an -multiplier then there exists such that
Using again Theorem 2.2, we obtain that and is strong - solution of Equation (2) and verified
For, we obtain that
From which we infer that the sequence is convergent to some element y as
. Moreover, y satisfies the following condition
let n go to infinity in (5), we can write
Then, we conclude that x is a 2p-periodic mild solution of Equation (2).
Example 5.1: Let A be a closed linear operator on a Hilbert space H and suppose that
If then for every there exists a unique strong -
solution of Equation (2).
From the identity
it follows that is invertible whenever [Theo-
rem 2.3], we observe that.
Then and by Theorem 2.3 we deduce that
We conclude that there exists a unique strong -solution of Equation (2). Using Corollary 3.8 in  .
Let A be a closed linear operator and X be a Hilbert space such that and. Suppose that. Then using Lemma 2.1
(1), we obtain that
From the identity it follows that is invertible whenever
Then and by Theorem 2.3, we have
This proves that is R-bounded and by Theorem 3.3, we get that there exists a unique strong -solution of (2).
The authors would like to thank the referee for his remarks to improve the original version.
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