Periodic Solutions of Integro-Differential Equations

The aim of this work is to study the existence of periodic solutions of integro-differential equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d d d t t t t x t L x A x t L x G x a t s x s s f t t −∞     − = − + + − +     ∫ , ( 0 2 t ≤ ≤ π ) with the periodic condition ( ) ( ) 0 2 x x = π , where ( ) 1 a L + ∈  . Our approach is based on the M-boundedness of linear operators , s p q B -multipliers and some results in Besov space.

Our approach is based on the M-boundedness of linear operators

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 or-DOI: 10.4236/apm.2022.123011 126 Advances in Pure Mathematics dinary 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 p L -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: and Aϕ ϕ′′ = .
Then we have: In [8], the author investigated the existence of solutions of the following fractional integrodifferential equation: In [9], S. Koumla, Kh. Ezzinbi and R. Bahloul., study the existence of mild solutions for some partial functional integrodifferential equations with finite delay 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 asser- ist a unique strong , s p q B -solution of (1). In section 4, we give the conclusion.

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 2π   . There is an identification between functions on  and 2π -periodic functions on  . We consider the interval [ ) 0, 2π as a model for  .
the space of 2π -periodic locally p-integrable functions from  into X, with the norm: be a bounded linear operateur. Then: Next we give some preliminaries. Given tended by periodicity to  ), we define:

Preliminary
In this section, we consider the periodic solutions of Equation (1) . In order to define Besov spaces, we consider the dyadic-like subsets of  : ; For more information about the standard definitions and properties, see [7].
therefore u is twicely differentiable a.e. and ( ) ( ) We recall the following operator-valued Fourier multiplier theorem on Besov spaces.

Main Result
For convenience, we introduce the following notations: In order to give our result, the following hypotheses are fundamental.
For convenience, we introduce the following result: Then we have:   a q a a r a p On the other hand, we have: Then by (9) we have: Then by (9) we have:  ( ) On the other hand, we have: Then by (10)     for k ∈  . Since by (7) and (8)