WELL-POSEDNESS OF RIEMANN-LIOUVILLE FRACTIONAL DEGENERATE EQUATIONS WITH FINITE DELAY IN BANACH SPACES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. We study the Existence and uniqueness of solutions of the Riemann-Liouville fractional integrodifferential degenerate equations d dt (B 1 Γ(1−α) ∫ t −∞(t− s)−α x(s)ds) = Ax(t)+ ∫ t −∞ a(t− s)x(s)ds+L(xt)+ 1 Γ(β ) ∫ t −∞(t− s)β−1x(s)ds+ f (t). where A and B are a linear closed operators in a Banach space.


INTRODUCTION
Differential equations play an important role in describing many real-world processes. For many years the models are successfully used to study a number of physical, biological. A particular interest is in differential equations with many variables such as partial differential equations and/or integral differential equations in the case when one of the variables is times.
In this work, we study the existence of periodic solutions for the following Riemann-Liouville fractional integro-differential degenerate equations.
where Γ(.) is the Euler gamma function, α, β ∈ R + , 0 ≤ β ≤ α and for all p ≥ 1 and r 2π := 2πN ( some N ∈ N), a ∈ L 1 (R + ), L is a linear operator and x t is an element of L p ([−r 2π , 0], X) which is defined as follows The operator-valued Fourier multiplier Theorems 2.8 have been used by Keyantuo and Lizama in [19] to establish maximal regularity results for an integro-differential equation in Banach space. The authors consider the following problem Maximal regularity for the evolution problem in L p was treated earlier by Weis [30,31] (see also [12] for a different proof of the operator-valued Mikhlin multiplier theorem using a transference principle). The study in the L p framework (when 1 < p < ∞) was made possible thanks to the introduction of the concept of randomized boundedness (hereafter R-boundedness, also known as Riesz-boundedness or Rademacher-boundedness). With this, necessary conditions for operator-valued Fourier multipliers were found in this context. In addition, the space X must have the UMD property. This was done initially by L. Weis [30,31] for the evolutionary problem and then by Arendt-Bu [2] for periodic boundary conditions. For non-degenerate integrodifferential equations both in the periodic and non periodic cases, operator-valued Fourier multipliers have been used by various authors to obtain well-posedness in various scales of function spaces: [7,9,10,19,25,20,21,27] and the corresponding references. The well-posedness or maximal regularity results are important in that they allow for the treatment of nonlinear problems. Earlier results on the application of operator-valued Fourier multiplier theorems to evolutionary integral equations can be found in [12].
In [22], S.Koumla, Kh.Ezzinbi, R.Bahloul established mild solutions for some partial functional integrodifferential equations with finite delay This work is organized as follows : In Section 2 we collect some preliminary results and definitions. In section 3, we study the existence and uniqueness of strong L p -solution of the Eq. (1.1) solely in terms of a property of R-boundedness for the sequence of operators We optain that the following assertion are equivalent in UMD space :

PRELIMINARIES
In this section, we collect some results and definitions that will be used in the sequel. Let X be a complex Banach space. We denote as usual by L 1 (0, 2π, X) the space of Bochner integrable functions with values in X. For a function f ∈ L 1 (0, 2π; X), we denote byf (k), k ∈ Z the kth where e k (t) = e ikt ,t ∈ R.

Lemma 2.1. [24]
Let L : L p (T, X) → X be a bounded linear operateur. Then Let a ∈ L 1 (R + ). We consider the the function Now using Fubini's theorem and (2.1) we obtain, for k ∈ Z, that Let X,Y be Banach spaces. We denote by L (X,Y ) the set of all bounded linear operators from X to Y . When X = Y , we write simply L (X).
R-boundedness-UMD space,L p -multiplier and Riemann-Liouville fractional integral. For j ∈ N, denote by r j the j-th Rademacher function on [0, 1], i.e. r j (t) = sgn(sin(2 j πt)). For x ∈ X we denote by r j ⊗ x the vector valued function t → r j (t)x.
The important concept of R-bounded for a given family of bounded linear operators is defined as follows.
Definition 2.10. The Riemann-Liouville fractional integral derivative operator of order α > 0 is defined by Those familiar with the Fourier transform know that the Fourier transform of a derivative can be expressed by the following: dx dt (k) = ikx(k), ∀k ∈ Z and more generally, d n x dt n (k) = (ik) nx (k), ∀k ∈ Z A similar identity holds for anti-derivatives

PERIODIC SOLUTIONS IN UMD SPACE
For a ∈ L 1 (R + ), we denote by a * x the function a(t − s)x(s)ds with this notation we may rewrite Eq. (1.1) in the following was: we have a * x(k) =ã(ik)x(k). We define   ators from X to X). Then the following assertions are equivalent: We begin by establishing our concept of strong solution for Eq.
Proof. (i) ⇒ (ii) As a consequence of Proposition (2.7) Observe that for α > 0 we have that |(i(k + 1)) α − (ik) α | can be estimated by (ik) α−1 uniformly in k according to the definition of |(ik) α | and the mean value theorem. This implies that k(a α,k+1 −a α,k ) a α,k is bounded sequence. Since ka α,k also is bounded for α > 0. Since products and sums of R-bounded sequences is R-bounded [24, Remark 2.2]. Then the proof is complete. Then It follows thatx(k) = 0 for every k ∈ Z and therefore x = 0. Then x 1 = x 2 .
Theorem 3.6. Let X be a Banach space. Suppose that for every f ∈ L p (T; X) there exists a unique strong solution of Eq. (3.1) for 1 ≤ p < ∞. Then Before to give the proof of Theorem 3.6, we need the following Lemma.
We have u(t) = e ikt x. In fact, since u t (θ ) = e ikθ u(t) we obtain u t = e k u(t). By Remark 2.11 (2), Proof of Theorem 3.6: 1) Let k ∈ Z and y ∈ X. Then for f (t) = e ikt y , there exists x ∈ H α,p (T; X) such that: Taking Fourier transform. We have D α −∞ Bx(k) = (ik) α Bx(k) and I β −∞ x(k) = (ik) −βx (k) Consequently, we have 2) Let f ∈ L p (T; X). By hypothesis, there exists a unique x ∈ H α,p (T, X) such that the Eq. (3.1) is valid. Taking Fourier transforms, we deduce that

MAIN RESULT
Our main result in this work is to establish that the converse of Theorem 3.6, are true, provided X is an UMD space. Then the following assertions are equivalent for 1 < p < ∞.
(1): for every f ∈ L p (T; X) there exists a unique 2π-periodic strong L p -solution of Eq.    In particular, x ∈ L p (T; X) and there exists v ∈ L p (T; X) such thatv(k) = (ik) αx (k) Using now (4.1) and (4.2) we have: for all k ∈ Z. Since A is closed, then x(t) ∈ D(A) [ Lemma 4.2 ] and from the uniqueness theorem of Fourier coefficients, that Eq. (3.1) is valid .