Faedo-Galerkin approximation of mild solutions of fractional functional differential equations


 In the paper, we discuss the existence and uniqueness of mild solutions of a class of fractional functional differential equations in Hilbert space separable using the Banach fixed point theorem technique. In this sense, Faedo-Galerkin approximation to the solution is studied and demonstrated some convergence results.


Introduction
From the beginning of the fractional calculus, namely, on September 30, 1695, in a letter written by l'Hospital to his friend Leibniz, in which the meaning of a middle order derivative is proposed and discussed [30][31][32]. Leibniz's response to his friend, coupled with the contribution of countless brilliant mathematicians such as Lagrange, Laplace, Fourier, Liouville, among others, led to the rst de nitions of non integer orders fractional derivatives and integrals, that at the end of the nineteenth century, due primarily to the de nitions proposed by Riemann-Liouville and Grünwald-Letnikov, seemed complete [25,29,51]. From then on, innumerable definitions of fractional derivatives and integrals were introduced by numerous researchers and scientists, each one with its own importance and relevance. Thus, countless incredible applications in various elds, such as mechanics, population dynamics, medicine, physics, engineering, among others, have been gaining strength over the years, making the theory well-established [34,36,49,50]. But, an important question arise how do you know, what is the best fractional derivative to look at data for a given problem? One way to overcome this problem is to propose more general fractional derivatives and integrals, where the existing ones are particular cases. Then, in 2018, Sousa and Oliveira [41], introduced the so-called ψ-Hilfer fractional derivative, which contains as a particular case a wide class of fractional derivatives. To complete the ψ-Hilfer fractional derivative theory, in 2019, the same authors [42] introduced the two-part Leibniz-type rule, which, depending on the chosen parameter, gives the Leibniz rule and the Leibniz-type rule for their particular cases.
Also, another question, why study fractional di erential equations? What are the advantages of the results obtained from them? In recent years, investigating fractional di erential equations has attracted a great deal of attention from several researchers, for better describing physical phenomena and providing results more consistent with the reality compared to integer order di erential equations [25, 27-29, 36, 43, 46, 47, 49]. On the other hand, investigating the existence, uniqueness, Ulam-Hyers stability, attractivity, continuous dependence on data, among others, of fractional di erential equations has been a very attractive eld for researchers from various elds, speci cally for mathematicians. To study these numerous solution properties, useful tools are needed, namely: xed point theorem, Gronwall inequality, Arzelà-Ascoli theorem, Laplace transform, Fourier transform, measure of non compactness and others [2-4, 6, 12, 15, 16, 48, 52, 53].
The Faedo-Galerkin approach has been used by many researchers to investigate more regular solutions in fractional di erential equations [7,11,23,37,38]. This approach can be used within a variational formulation to provide solutions of possibly weaker equations [17]. In this regard, in 2010 Muslim [40] did important work on the global existence and uniqueness of mild solutions of the fractional order integral equation in Banach space and also discussed these same properties in Hilbert separable space. In addition, through the Faedo-Galerkin approach, the approximate solution convergence was investigated. In 2013, Lizama and N'Guérékata [26] approached the existence of mild solutions for the fractional di erential equation with nonlocal conditions and investigated the asymptotic behavior of mild solutions for abstract fractional relaxation equations towards the Caputo fractional derivative. On the other hand, we suggest other work on the existence and uniqueness of mild solutions for semilinear nonlocal fractional Cauchy problem, as discussed by Ghour and Omari [1]. In the literature there are numerous works on interesting properties of solutions of fractional differential equations, we refer some papers for a more detailed reading [14,18,19,35,44,45].
On the other hand, the theme Faedo-Galerkin approximation, in fact, continues to be the subject of study by a class of researchers [6,[20][21][22]24]. In 2016, Chaddha et al. [8] using the semigroup theory and the Banach xed point theorem considered an impulsive fractional di erential equation structured over a separable Hilbert space, and investigated the existence and uniqueness of solutions for each approximate integral equation. Also, using Faedo-Galerkin approximation the solution was investigated. In the same year, Chadha and Pandey [10], devoted a work on the Faedo-Galerkin approximation of the solution to a nonlocal neutral fractional di erential equation with into separable Hilbert space.
Finally, in 2019, an interesting and important work on Faedo-Galerkin approximate solutions of a neutral stochastic nite delay fractional di erential equation, performed by Chadha et al. [9], comes to highlight the importance of the theme in the academic community. In this paper, using Banach's xed point theorem and semigroup theory, the authors investigated the existence and uniqueness of mild solutions of a class of neutral stochastic fractional di erential equations. Also, they showed the convergence of solutions using Faedo-Galerkin approximation. Other works on Faedo-Galerkin approximation can be found at [11,13,14,37,40]. Although there is a range of relevant and important work published so far, there are still many ways to go when it comes to mild solutions of fractional di erential equations. We note that, the investigation of a mild solution to a fractional di erential equation towards the ψ-Hilfer fractional derivative, as some properties and tools are still under discussion. Thus, through the work commented above, we were motivated to propose an investigation of the existence, uniqueness and convergence for a class of solutions of the nonlocal fractional functional di erential equations, in order to contribute with new results that can be useful for future research.
In this paper, we consider a class of abstract fractional functional di erential equation with condition in a separable Hilbert space H , given by where H D µ,ν + (·) is the Hilfer fractional derivative of order < µ ≤ and type ≤ ν ≤ , Motivated by the works above and by the numerous questions restricted in di erent directions of the theory of fractional di erential equations, we present here the main results obtained in this paper in two stages. In the rst step of the paper, we discuss approximate solutions and convergence (see Theorem 3.1-3.3). In the second step of the paper, we discuss the Faedo-Galerkin approximation of a solution and show the convergence results for such an approximation (see .
In that sense, we have a class of an abstract fractional functional di erential equations in the sense of Hilfer fractional derivative with condition in a separable Hilbert space H and its respective class of mild solutions. In this sense, we have that from the choice of the limits ν → and ν → , we have the problems with their respective solutions, for the Caputo and Riemann-Liouville fractional derivatives, respectively. The special case is the integer case when we choose µ = .
In the rest, the article is organized as follows: In section 2, we present the idea of some function spaces with their respective norms, fundamental in the course of the work. In this sense, concepts of Riemann-Liouville fractional integral with respect to another function, ψ-Hilfer fractional derivative, the one and two parameter Mittag-Le er functions, and Gronwall inequality, are presented. To nish the section, some conditions about the Mittag-Le er function, and the f function are discussed, and we show that the investigated problem is well-de ned. In section 3, we will investigate the main results of the paper, approximation of solutions and convergence, i.e., we present results on existence and uniqueness of mild solutions for a class of abstract fractional functional di erential equations. Finally, in section 4, we will use Galerkin approach to ensure the uniqueness of solutions.

Preliminaries
In this section, we present the spaces and their respective norms that will be very important for the elaboration of this paper. In this sense, we introduce concepts of Riemann-Liouville fractional integral with respect to another function and the ψ-Hilfer fractional derivative. We discuss the mild solution of the nonlocal functional fractional di erential equation with respect to the Mittag-Le er functions. Let with ≤ γ ≤ and the norm given by be a nite or in nite interval of the real line R and µ > . Also let ψ(t) be an increasing and positive monotone function on (a, b], having a continuous derivative ψ ′ (t) on (a, b). The left-sided and right-sided fractional integrals of a function f with respect to another function ψ on [a, b] are de ned by [29,41] and respectively.
On the other hand, let n − < µ < n with n ∈ N, I = [a, b] be the interval such that −∞ < a < b < ∞ and f , ψ ∈ C n ([a, b], R) two functions such that ψ is increasing and ψ ′ (x) ≠ , for all x ∈ I. The left-sided and right-sided ψ-Hilfer fractional derivative of order µ and type ≤ ν ≤ of a function, denoted by H D µ,ν;ψ a+ (·) are de ned by [41,42] respectively.
Choosing ψ(x) = x and replacing in Eq.(4) and Eq. (5), we obtain left-sided and right-sided Hilfer fractional derivative, which we use in the formulation of the nonlinear functional fractional di erential equation according to Eq.(1), given by [41] respectively.
In what follows, let us state some properties of the special function M ξ also called Mainardi function. This function is a particular case of the Wright type function, introduced by Mainardi. More precisely, for ξ ∈ ( , ), the entire function M ξ : C → C is given by [5] M ξ (z) := ∞ n= z n n!Γ( − ξ ( + n)) · Proposition 2.1. [5] For ξ ∈ ( , ) and − < r < ∞, when we restrict M ξ to the positive real line, it holds that In the sequence, we introduce the Mittag-Le er operators. Then, for each ξ ∈ ( , ), we de ne the Mittag- respectively. The functions E ξ (·) and E ξ ,ξ (·), are the one and two parameters Mittag-Le er functions, respectively.
To this end, let H be a Hilbert space and −A : D(A) ⊂ H → H be the in nitesimal generators of a semigroup S(t), t ≥ .
We consider the following assumptions on the operator A, T and the function f , namely: (H1) A is a closed, positive de nite, self-adjoint linear operator A : D(A) ⊂ H → H such that D(A) is dense in H and A has the pure point spectrum and a corresponding complete orthonormal system of eigenfunctions {ϕ i }, i.e., Throughout the paper we assume that there exists an operator B on D(B) = H given by the formula hence the operator B exists. It follows that for ≤ δ ≤ , A δ can be de ned as a closed linear invertible operator with domain D(A δ ) being dense in H . We have H θ → H δ for < δ < θ and the embedding is continuous.
(H3) Let < T , R < ∞ be arbitrarily xed and M, C > constants with < tp ≤ T such that and where C δµ is a positive constant depending on δ and µ satisfying is the domain of A δ , is the Banach space of all weighted space of continuous functions with the norm For any Banach space Z and r > we de ne Br(Z) = {x ∈ Z, x Z ≤ r}. We say that the function u ∈ C T −γ is called a mild solution of Eq.(1) on [ , T ] if it satis es the equation and Hµ(t k , s; A) fs,u b(s)ds. (10) Hence, from Eq.(9) and Eq.(10) and the de nition of operator B, we get

Main results
In this section, our main results, namely, the existence, uniqueness, and approximation solutions and convergence of a class of solutions of the nonlinear abstract fractional di erential equation in the Hilbert space H, are investigated.

. Approximate solutions and convergence
Let  u(b(s))). So, next the rst main result of this paper, that is, the solution un ∈ S satisfying the approximate integral equation Eq.(11), is presented as a theorem.  un(b(s))).
Proof. Our goal here is to establish the uniqueness of solution of approximate integral equation, Eq.(11), on [ , T ]. Two points are necessary and su cient for the proof of this theorem, namely: 1. Fn is a mapping from S into S.

Fn is a contraction mapping on S.
Then for u ∈ S, we have , for all t ∈ [ , T ] and h > . So we get, On the other hand, for any u ∈ S and t ∈ [ , T ], we get Therefore, from inequality (12), it follows that where R is given by Eq. (6). Hence, we conclude that Fn(S) ⊂ S. Through inequality (7), we have (13) Using the inequalities (7) and (13), we have Then, we have Therefore, it implies that the operator Fn is a contraction operator and has a unique xed point that is Fn un = un, for un ∈ S given by Hµ(t k , s; A) fn,s,u n b(s)ds with t ∈ [ , T ]. Hence, the proof of the theorem is completed.
Using the inequality (16) in inequality (15), we have where and Considering t ′ such that < t ′ < t < T , we have Integrating and introducing the notation N R = L R (T )M we can write Taking the following change t = t +θ in inequality (18), whereθ ∈ [t ′ − t, ], we obtain Introducing s − θ = γ in inequality (19), we get Thus, we have For t + θ ≤ , we have un(t + θ) = K(t + θ) for all n ≥ n . Thus, we get sup −t≤ θ≤ Then, for each t ∈ ( , t ′ ], we have Then, we can write Now, using the Gronwall inequality, we have where Eµ(·) is an one-parameter Mittag-Le er function. Since t ′ is arbitrary and taking m → ∞, therefore the right hand side can be made as small as desired by taking t ′ su ciently small. This complete the proof.  Proof. In fact, using Eq.(1) and Eq. (8), we obtain Thus, we get Through the Theorem 3.4, we conclude the result