Solvability in Gevrey Classes of Some Nonlinear Fractional Functional Differential Equations

Zoubeir, Hicham

doi:https://doi.org/10.1155/2020/3739249

International Journal of Differential Equations

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Research Article | Open Access

Volume 2020 | Article ID 3739249 | https://doi.org/10.1155/2020/3739249

Solvability in Gevrey Classes of Some Nonlinear Fractional Functional Differential Equations

Hicham Zoubeir¹

Academic Editor: Julio D. Rossi

Received11 Aug 2019

Accepted28 Oct 2019

Published29 Jun 2020

Abstract

Our purpose in this paper is to prove, under some regularity conditions on the data, the solvability in a Gevrey class of bound −1 on the interval of a class of nonlinear fractional functional differential equations.

1. Introduction

Fractional calculus has evolved from the speculations of early mathematicians of the and centuries like G. W. Leibnitz, I. Newton, L. Euler, G. F. de L’Hospital, and J. L. Lagrange [1]. In the century, other eminent mathematicians like P. S. Laplace, J. Liouville, B. Riemann, E. A. Holmgren, O. Heaviside, A. Grunwald, A. Letnikov, J. B. J. Fourier, and N. H. Abel have used the ideas of fractional calculus to solve some physical or mathematical problems [1]. In the century, several mathematicians (S. Pincherle, O. Heaviside, G. H. Hardy, H. Weyl, E. Post, T. J. Fa Bromwich, A. Zygmund, A. Erdelyi, R. G. Buschman, M. Caputo, etc.) have made considerable progress in their quest for rigor and generality to build fractional calculus and its applications on rigorous and solid mathematical foundations [1]. Actually, fractional calculus allows mathematical modeling of social and natural phenomena in a more powerful way than the classical calculus. Indeed fractional calculus has a lot of applications in different areas of pure and applied sciences like mathematics, physics, engineering, fractal phenomena, biology, social sciences, finance, economy, chemistry, anomalous diffusion, and rheology [1–22]. It is then of capital importance to develop for fractional calculus the mathematical tools analogous to those of classical calculus [1, 3, 4, 19, 23]. The fractional differential equations [23–28] are a particularly important case of such fundamental tools. An important type of fractional differential equations is that of fractional functional differential equations (FFDEs) [10, 29–31] which are the fractional analogues to functional differential equations [17, 32–34], enable the study of some physical, biological, social, and economical processes (automatic control, financial dynamics, economical planning, population dynamics, blood cell dynamics, infectious disease dynamics, etc.) with fractal memory and nonlocality effects, where the rate of change of the state of the systems depends not only on the present time but on other different times which are functions of the present time [11, 35, 36]. The question then arises of the choice of a suitable framework for the study of the solvability of these equations. But, since the functional Gevrey spaces play an important role in various branches of partial and ordinary differential equations [37– 40], we think that these functional spaces can play the role of such convenient framework. However, let us point out that in order to make these spaces adequate to our specific setting, it is necessary to make a modification to their definition. This leads us to the definition of new Gevrey classes, namely, the Gevrey classes of bound and index on an interval . Our purpose in this paper is to prove, under some regularity conditions on the data, the solvability in a Gevrey class of the form of a class of nonlinear FFDE. Our approach is mainly based on a theorem that we have proved in [41]. The notion of fractional calculus we are interested in is the Caputo fractional calculus. Some examples are given to illustrate our main results.

2. Preliminary Notes and Statement of the Main Result

2.1. Basic Notations

Let be a mapping from a nonempty set E into itself. denotes the iterate of F of order n for the composition of mappings.

For and is the open ball in with the center z and radius .

Let and be two nonempty subsets of such that and a mapping. We denote by the restriction of the mapping f to the set .

For and nonempty) we set

For , , , and , we set for every nontrivial compact interval of

Thus, we have

Remark 1. The following inclusions hold for every and :Let be a bounded function. denotes the quantity:By (resp. ), we denote the complex vector space of all complex valued functions defined and continuous (resp. defined and of class ) on the interval . is a Banach space when it is endowed with the uniform norm:For every , we denote by the closed ball in ) of radius and center, the null function.
Let . We denote by the linear path joining to :In this paper, and are fixed numbers.

2.2. Fractional Derivatives and Integrals

Definition 1. Let and f be a Lebesgue-integrable function on the nontrivial compact interval . The Caputo fractional integral of order δ and lower bound of the function f [19, 23, 25, 26, 28] is the function denoted by and defined bywhere denotes the classical gamma function.

Remark 2. If the function f is continuous on the interval , then the function is well defined and continuous on the entire interval , and we have

Definition 2. Let f : be an absolutely continuous function on ; then, the Caputo fractional derivative of f of order δ and lower bound [19, 23, 25, 26, 28] is the function denoted by and defined by

Remark 3. Let . We have for every If then the Caputo fractional integral of the function f of order , is also of class on the interval and we have [19, 23, 25, 26, 28]

2.3. Gevrey Classes

Definition 3. Let . The Gevrey class of index l on , denoted by is the set of all functions f of class on such thatwhere is a constant (with the convention that ).

Definition 4. The Gevrey class of bound and index l on the interval , denoted by is the set of all functions f of class on and of class on such that the restriction of f belongs to the Gevrey class , for every .

2.4. The Property

Definition 5. A function φ defined on the set is said to satisfy the property on the interval if is holomorphic on is a function of class on , and there exists a constant such that for all there exist depending only on , and φ such that the inclusionholds for every integer . The number is then called a -threshold for the function .

Remark 4. Let φ be a function verifying the property . Then,On the other hand, it follows from (14) that we have for every Thence, we haveIt follows that for every there exists such that

2.5. Statement of the Main Result

Our main result in this paper is the following.

Theorem 1. Let and . Let b, and ψ be holomorphic functions on and be an entire function. We assume that the function a is not identically vanishing and that there exist constants such thatand that ψ satisfies the property . We also assume that the following conditions are fulfilled:Then, the FFDEhas a solution u which belongs to the Gevrey class and verifies the initial condition

3. Proof of the Main Result

The proof of the theorem is subdivided in three steps.

Step 1. The localisation of the solutions of the equation:The study of the variations of the functionshows, under condition (21), that H is strictly decreasing on and strictly increasing on . But,Therefore, the equation has on exactly two solutions and the following inequalities hold:

Step 2. Proof of the existence of a solution u of the FFDE in such that the initial condition holds.
Consider the operator defined by the following formula:We have for all Thence, the closed ball is stable by the operator . On the other hand, we have for all f, Since , it follows from condition (23) thatThence, T has, in a unique fixed point .
Consider the sequence of functions defined on by the following formula:where is the null function. Direct computations show that the functions belonging to are of class on and verify the following inequality:whereLet us set for each Since it follows that the function series is uniformly convergent on to a function which is a fixed point of the operator It follows that Consequently, the function series is uniformly convergent on to the function
On the other hand, we have for all and :Since , it follows thatTo achieve the proof of this step we need the following result.

Proposition 1. The sequence is bounded.

Proof. We have for all and It follows from assumption (20) thatBut, according to assumption (24) and (35) we haveConsequently, the following inequality holds for each Since it follows that the sequence is bounded.
The proof of the proposition is complete.
Now, we setThen, we can writeDirect computations show then thatSince it follows that the function series is uniformly convergent on Thence, the function u is of class on and satisfies the following relation:Consequently, according to assumption (20), we can write for all So, u is a solution of the FFDE which belongs to and fulfills the relation .

Step 3. Proof that u belongs to the Gevrey class .
Since the function defined on byis continuous on and verifies by virtue of assumptions (22) and (23), the inequality It follows that there exists such thatwhere is a -threshold of Let d be an arbitrary but fixed element of Thanks to remark 4, there exists such that the functions a and b are both holomorphic on and the following condition holds:Consider the sequence of functions , wherefor each and Then, direct computations, based on (52), show that the function is for every holomorphic on .

Proposition 2. The inclusion holds for every .

Proof. We denote the last inclusion by We denote for every by the closest point of to It is obvious that is true. Assume for a certain that is true for every Since the function is holomorphic on we have then for each Thence, the assertion is true. Consequently, is true for all .
The proof of the proposition is then complete.
By virtue of the Proposition 2., we have for all and It follows thatLet us set and denote, for all by the functionThen, the function is holomorphic on for each Furthermore, the following relations hold for every Since it follows then from (59) that the function series is uniformly convergent on to the function However, we know, according to relation (4) of Remark 1, that the following inclusion hold:It follows then thatThe relations (61) entail, thanks to the main result of [41], that belongs to , for each d ϵ . Thence, since u is of class on it follows that u belongs to the Gevrey class .
The proof of the main result is then complete.

4. Examples

To obtain examples illustrating our main result, we need first to prove the following proposition.

Proposition 3. The functionsatisfies the property for every .

Proof. Let and We haveIt follows thatWe consider then the principal argument of which satisfies the following estimates:But, direct computations prove thatThence, we haveIt follows thatOn the other hand, we haveBut, we know thatIt follows thatWe derive, from the estimates (68) and (71), the following inclusion:whereLet and We have But, we haveIt follows that there exists an integer such that the following inequality holds for every integer Consequently, we havethat is,It follows that the function satisfies the property S.
The proof of the proposition is then complete.

Recall that the following estimate holds for every

It means that the functions and satisfy the estimates:with .

Example 1. Let and We assume thatConsider the FFDEwith the initial conditionConsider then the following entire functions:It is clear that is not identically vanishing and that Furthermore, we have
We also haveConsequently, it follows from the main result that the problem has a solution which belongs to the Gevrey class .

Example 2. Let and We assume thatConsider the FFDEwith the initial conditionConsider then the following functions:It is clear that is not identically vanishing and that Furthermore, we have the following inequalities:Consequently, it follows from the main result that the problem has a solution which belongs to the Gevrey class .

Data Availability

No data were used to support this study.

Disclosure

This modest work is dedicated to the memories of two great men: our beloved master Ahmed Intissar (1951–2017), a brilliant mathematician (PhD at M.I.T, Cambridge), a distinguished professor, a man with a golden heart; our brother and indeed friend Mohamed Saber Bensaid (1965–2019), the man who belongs to the time of jasmine and sincere love, the comrade who devoted his whole life to the fight for socialism, democracy, and human rights.

Conflicts of Interest

The author declares that there are no conflicts of interest.

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Copyright © 2020 Hicham Zoubeir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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