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Solvability in Gevrey Classes of Some Nonlinear Fractional Functional Differential Equations.

1. Introduction

Fractional calculus has evolved from the speculations of early mathematicians of the 17th and 18th centuries like G. W. Leibnitz, I. Newton, L. Euler, G. F. de L'Hospital, and J. L. Lagrange [1]. In the 19th 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 20th 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 [mathematical expression not reproducible] ([[q.sub.1], [q.sub.2]]) of bound q1 and index l > 0 on an interval [[q.sub.1], [q.sub.2]]. Our purpose in this paper is to prove, under some regularity conditions on the data, the solvability in a Gevrey class of the form [G.sub.k,-1] ([-1,1]) 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 F : E [right arrow] E be a mapping from a nonempty set E into itself. [F.sup.<n>] (n [member of] N) denotes the iterate of F of order n for the composition of mappings.

For z [member of] C and h > 0, B(z, h) is the open ball in C [equivalent] [R.sup.2] with the center z and radius h.

Let [S.sub.1] and [S.sub.2] be two nonempty subsets of C such that [S.sub.1] [subset] [S.sub.2] and f: [S.sub.2] [right arrow] C a mapping. We denote by [mathematical expression not reproducible] the restriction of the mapping f to the set S1. For z e C and S c C(S nonempty) we set

[mathematical expression not reproducible]. (1)

For l, [phi], r > 0, and n [member of] [N.sup.*], we set for every nontrivial compact interval [[q.sub.1], [q.sub.2]] of R

[mathematical expression not reproducible]. (2)

Thus, we have

[mathematical expression not reproducible]. (3)

Remark 1. The following inclusions hold for every d [member of] ][q.sub.1], [q.sub.2]], r [member of]]0,d - [q.sub.1] [ and n [member of] [N.sup.*]:

[mathematical expression not reproducible]. (4)

Let f : S [right arrow] C be a bounded function. [[parallel]f[parallel].sub.[infinity],S] denotes the quantity:

[mathematical expression not reproducible]. (5)

By [C.sup.0]([[q.sub.1], [q.sub.2]]) (resp. [C.sup.1] ([[q.sub.1], [q.sub.2]])), we denote the complex vector space of all complex valued functions defined and continuous (resp. defined and of class [C.sup.1]) on the interval [[q.sub.1], [q.sub.2]]. [C.sup.0]([[q.sub.1], [q.sub.2]]) is a Banach space when it is endowed with the uniform norm:

[mathematical expression not reproducible]. (6)

For every r [greater than or equal to] 0, we denote by [[DELTA].sub.[infinity]] (r) the closed ball in [C.sup.0] ([-1,1])) of radius r and center, the null function.

Let [[xi].sub.1], [[xi].sub.2] [member of] C. We denote by [mathematical expression not reproducible] the linear path joining

[mathematical expression not reproducible]. (7)

In this paper, k > 0 and [alpha] [member of] ]0,1 [ are fixed numbers.

2.2. Fractional Derivatives and Integrals

Definition 1. Let o e ]0,1[ and f be a Lebesgue-integrable function on the nontrivial compact interval [[q.sub.1], [q.sub.2]]. The Caputo fractional integral of order 8 and lower bound [q.sub.1] of the function f [19, 23, 25, 26, 28] is the function denoted by [sup.C][I.sup.[delta].sub.[alpha]] and defined by

[mathematical expression not reproducible], (8)

where [GAMMA] denotes the classical gamma function.

Remark 2. If the function f is continuous on the interval [[q.sub.1], [q.sub.2]], then the function [mathematical expression not reproducible] is well defined and continuous on the entire interval [[q.sub.1], [q.sub.2]], and we have

[mathematical expression not reproducible]. (9)

Definition 2. Let f: [[q.sub.1], [q.sub.2]] [right arrow] C be an absolutely continuous function on [[q.sub.1], [q.sub.2]]; then, the Caputo fractional derivative of f of order [delta] and lower bound [q.sub.1] [19, 23, 25, 26, 28] is the function denoted by [mathematical expression not reproducible] and defined by

[mathematical expression not reproducible]. (10)

Remark 3. Let f [member of] [C.sup.1]([[q.sub.1], [q.sub.2]]). We have for every x [member of] [[q.sub.1], [q.sub.2]]

[mathematical expression not reproducible]. (11)

If f ([q.sub.1]) = 0, then the Caputo fractional integral of the function f of order [mathematical expression not reproducible] is also of class [C.sup.1] on the interval [[q.sub.1], [q.sub.2]] and we have [19, 23, 25, 26, 28]

[mathematical expression not reproducible]. (12)

2.3. Gevrey Classes

Definition 3. Let l > 0. The Gevrey class of index l on [[q.sub.1], [q.sub.2]], denoted by [G.sub.l] ([[q.sub.1], [q.sub.2]]), is the set of all functions f of class [C.sup.[infinity]] on [[q.sub.1], [q.sub.2]] such that

[mathematical expression not reproducible], (13)

where B > 0 is a constant (with the convention that [0.sup.0] = 1).

Definition 4. The Gevrey class of bound [q.sub.1] and index l on the interval [[q.sub.1], [q.sub.2]], denoted by [mathematical expression not reproducible], is the set of all functions f of class [C.sup.1] on [[q.sub.1], [q.sub.2]] and of class [C.sup.[infinity]] on ][q.sub.1], [q.sub.2]] such that the restriction [mathematical expression not reproducible] of f belongs to the Gevrey class [G.sub.l] ([q, [q.sub.2]]), for every q [member of] ][q.sub.1], [q.sub.2][.

2.4. The Property S(l)

Definition 5. A function [phi] defined on the set {[q.sub.1]} [union] [[[q.sub.1], [q.sub.2]].sup.r] (r [member of] ]0, [pi][) is said to satisfy the property S(l) on the interval [[q.sub.1], [q.sub.2]] if [mathematical expression not reproducible] is holomorphic on [mathematical expression not reproducible] is a function of class [C.sup.1] on [[q.sub.1], [q.sub.2]], and there exists a constant [[tau].sub.[phi]] [member of] ]0, [pi][ such that for all D [member of] ]0, [[tau].sub.[phi]]] there exist [N.sub.l,[phi]] (D) [member of] [N.sup.*] depending only on D, l, and [phi] such that the inclusion

[mathematical expression not reproducible] (14)

holds for every integer n [greater than or equal to] [N.sub.l,[phi]] (D). The number [[tau].sub.[phi]] is then called a S(l)-threshold for the function [phi].

Remark 4. Let [phi] be a function verifying the property S(l). Then,

[phi]([[q.sub.1], [q.sub.2]]) [subset] [[q.sub.1], [q.sub.2]]. (15)

On the other hand, it follows from (14) that we have for every D [member of] ]0, [[tau].sub.[phi]][

[mathematical expression not reproducible]. (16)

Thence, we have

[mathematical expression not reproducible]. (17)

It follows that for every D [member of] ] 0, [[tau].sub.[phi]][ there exists E [member of]] 0, D [ such that

[mathematical expression not reproducible]. (18)

2.5. Statement of the Main Result. Our main result in this paper is the following.

Theorem 1. Let [lambda] [member of] C and [sigma] > 0. Let a, b, and [psi] be holomorphic functions on [[-1,1].sub.[sigma]] and [PHI] be an entire function. We assume that the function a is not identically vanishing and that there exist constants [[alpha].sub.0], [[beta].sub.0] > 0 such that

[mathematical expression not reproducible] (19)

and that [psi] satisfies the property S (k). We also assume that the following conditions are fulfilled:

a(-1) = b(-1) = 0, (20)

[mathematical expression not reproducible], (21)

[mathematical expression not reproducible], (21)

[mathematical expression not reproducible], (22)

[mathematical expression not reproducible]. (24)

Then, the FFDE

(E): [sup.c][D.sup.[alpha].sub.-1]y(t) = a(t)[phi](y ([psi](t))) + b(t) (25)

has a solution u which belongs to the Gevrey class [G.sub.k,-1] ([- 1, 1]) and verifies the initial condition

([E.sub.1]) : y (-1) = [lambda] (26)

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:

[mathematical expression not reproducible]. (27)

The study of the variations of the function

[mathematical expression not reproducible], (28)

shows, under condition (21), that H is strictly decreasing on [mathematical expression not reproducible] and strictly increasing on [mathematical expression not reproducible]. But,

[mathematical expression not reproducible], (29)

[mathematical expression not reproducible]. (30)

Therefore, the equation (I) has on [R.sup.+] exactly two solutions [R.sub.0] < [R.sub.1] and the following inequalities hold:

[mathematical expression not reproducible]. (31)

Step 2. Proof of the existence of a solution u of the FFDE (E) in [C.sup.1] ([-1,1]) such that the initial condition ([E.sub.1]) holds.

Consider the operator T : [C.sup.0]([-1,1]) [right arrow] [C.sup.0] ([-1,1]) defined by the following formula:

[mathematical expression not reproducible]. (32)

We have for all f [member of] [[bar.[DELTA]].sub.[infinity]]([R.sub.0])

[mathematical expression not reproducible]. (33)

Thence, the closed ball [[bar.[DELTA]].sub.[infinity]] ([R.sub.0]) is stable by the operator T. On the other hand, we have for all f, g [member of] [[bar.[DELTA]].sub.[infinity]] ([R.sub.0])

[mathematical expression not reproducible]. (34)

Since [mathematical expression not reproducible], it follows from condition (23) that

[mathematical expression not reproducible]. (35)

Thence, T has, in [[bar.[DELTA]].sub.[infinity]] ([R.sup.0]), a unique fixed point u.

Consider the sequence of functions [(fn).sub.n[member of]N] defined on [-1,1] by the following formula:

[f.sub.n] := [T.sup.<n>] ([f.sub.0]), n [member of] N, (36)

where [f.sub.0] is the null function. Direct computations show that the functions [f.sub.n] belonging to [[bar.[DELTA]].sub.[infinity]] ([R.sub.0]) are of class [C.sup.1] on [-1,1] and verify the following inequality:

[mathematical expression not reproducible], (37)

where

[mathematical expression not reproducible]. (38)

Let us set for each n [member of] N, [F.sub.n] := [f.sub.n+1] - [f.sub.n]. Since Q [member of] [0,1 [, it follows that the function series [summation] [F.sub.n] is uniformly convergent on [-1,1] to a function v [member of] [[bar.[DELTA]].sub.[infinity]] ([R.sub.0]) which is a fixed point of the operator T. It follows that v = u. Consequently, the function series [summation] [F.sub.n] is uniformly convergent on [-1,1] to the function u [member of] [C.sup.0] ([ -1,1]).

On the other hand, we have for all x [member of] ] 1, 1] and n [member of] [N.sup.*] :

[mathematical expression not reproducible]. (39)

Since a(- 1) = 0, it follows that

[mathematical expression not reproducible]. (40)

To achieve the proof of this step we need the following result.

Proposition 1. The sequence [mathematical expression not reproducible] is bounded.

Proof. We have for all x [member of]] - 1,1] and n [member of] [N.sup.*]

[mathematical expression not reproducible]. (41)

It follows from assumption (20) that

[mathematical expression not reproducible]. (42)

But, according to assumption (24) and (35) we have

[mathematical expression not reproducible]. (43)

Consequently, the following inequality holds for each n [member of] [N.sup.*]

[mathematical expression not reproducible]. (44)

Since Q [member of] [0,1 [, it follows that the sequence [mathematical expression not reproducible] is bounded.

The proof of the proposition is complete.

Now, we set

[mathematical expression not reproducible]. (45)

Then, we can write

[mathematical expression not reproducible]. (46)

Direct computations show then that

[mathematical expression not reproducible]. (47)

Since Q [member of] [0,1 [, it follows that the function series [summation] [F'.sub.n] is uniformly convergent on [-1, 1]. Thence, the function u is of class [C.sup.1] on [- 1, 1] and satisfies the following relation:

[mathematical expression not reproducible]. (48)

Consequently, according to assumption (20), we can write for all t [member of] [- 1, 1]

[mathematical expression not reproducible]. (49)

So, u is a solution of the FFDE (E) which belongs to [C.sup.1] ([-1,1]) and fulfills the relation u(-1) = [lambda].

Step 3. Proof that u belongs to the Gevrey class [G.sub.k,-1] ([-1,1]).

Since the function A defined on [0, min(1, [sigma])[ by [LAMBDA] : [0, min(1, [sigma])[ [right arrow] R,

[mathematical expression not reproducible], (50)

is continuous on [0, min(1, [sigma])[ and verifies by virtue of assumptions (22) and (23), the inequality [LAMBDA](0) < 1. It follows that there exists [s.sub.1] [member of] ]0, min(1, [sigma], [[tau].sub.[psi]])[ such that

[LAMBDA]([0, [s.sub.1]]) [subset] [0,1[, (51)

where [[tau].sub.[psi]] is a S(l)-threshold of [psi]. Let d be an arbitrary but fixed element of ]-1,1 [. Thanks to remark 4, there exists [s.sub.2] [member of] ]0, [s.sub.1][ such that the functions a and b are both holomorphic on [mathematical expression not reproducible] and the following condition holds:

[mathematical expression not reproducible]. (52)

Consider the sequence of functions ([mathematical expression not reproducible], where

[mathematical expression not reproducible], (53)

[mathematical expression not reproducible], (54)

for each n [member of] [N.sup.*] and [mathematical expression not reproducible]. Then, direct computations, based on (52), show that the function [w.sub.n] is for every n [member of] [N.sup.*] holomorphic on [mathematical expression not reproducible].

Proposition 2. The inclusion [mathematical expression not reproducible] holds for every n [member of] [N.sup.*].

Proof. We denote the last inclusion by P (n). We denote for every z [member of] C by [??] the closest point of [-1,1] to z. It is obvious that P(1) is true. Assume for a certain n [member of] [N.sup.*] that P(p) is true for every p e {1,... ,n}. Since the function [[omega].sub.n+1] is holomorphic on [mathematical expression not reproducible], we have then for each [mathematical expression not reproducible]

[mathematical expression not reproducible]. (55)

Thence, the assertion P(n + 1) is true. Consequently, P(n) is true for all n [member of] [N.sup.*].

The proof of the proposition is then complete.

By virtue of the Proposition 2, we have for all n [member of] [N.sup.*]\{1} and [mathematical expression not reproducible]

[mathematical expression not reproducible]. (56)

It follows that

[mathematical expression not reproducible]. (57)

Let us set [[OMEGA].sub.1] := [[omega].sub.1] and denote, for all n [member of] [N.sup.*]\{1}, by [[OMEGA].sub.n] the function

[mathematical expression not reproducible]. (58)

Then, the function [[OMEGA].sub.n] is holomorphic on [mathematical expression not reproducible] for each n [member of] [N.sup.*]. Furthermore, the following relations hold for every n [member of] [N.sup.*]\{1}:

[mathematical expression not reproducible]. (59)

Since [LAMBDA]([s.sup.2]) [member of] [0, 1 [, it follows then from (59) that the function series [summation] [[OMEGA].sub.n|[-1,1]] is uniformly convergent on [-1,1] to the function u However, we know, according to relation (4) of Remark 1, that the following inclusion hold:

[mathematical expression not reproducible]. (60)

It follows then that

[mathematical expression not reproducible]. (61)

The relations (61) entail, thanks to the main result of [41], that [u.sub.|[d,1]] belongs to [G.sub.k]([d, 1]), for each d e ] - 1,1 [. Thence, since u is of class [C.sup.1] on [-1,1], it follows that u belongs to the Gevrey class [G.sub.k-1] ([-1,1]).

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 function

L: C [right arrow] C,

z [??] 2[e.sup.(z-1)1/2] - 1, (62)

satisfies the property S(l) for every l [member of] ]0,1].

Proof. Let l [member of] ]0,1], [epsilon] [member of] ]0,1] and z [member of] [[-1,1].sup.[epsilon]]. We have

L(z) = 2[e.sup.(z-1)1/2] - 1. (63)

It follows that

Re(L(z) + 1) > 0. (64)

We consider then the principal argument arg(L(z) + 1) of L(z) + 1 which satisfies the following estimates:

[mathematical expression not reproducible] (65)

But, direct computations prove that

0 < tan [epsilon] - ([epsilon] + ([[epsilon].sup.3]/3))/[[epsilon].sup.3] < 1 - 4/3

Thence, we have

[mathematical expression not reproducible]. (67)

It follows that

arg(L(z)+ 1) [less than or equal to] e + (3/2 tan1 - 1)[[epsilon].sup.2]. (68)

On the other hand, we have

[absolute value of (L(z)+ 1)] = 2[e.sup.(Re(z)-1)/2] [less than or equal to] 2[e.sup.[epsilon]/2].

But, we know that

[mathematical expression not reproducible]. (70)

It follows that

[absolute value of (L(z) + 1)] [less than or equal to] 2 + [epsilon] + ([square root of e] - 3/2)[[epsilon].sup.2]. (71)

We derive, from the estimates (68) and (71), the following inclusion:

[mathematical expression not reproducible], (72)

where

[mu]:= max (3/2 tan 1 - 1, [square root of e] - 3/2) = 3/2 tan 1 - 1 > 0. (73)

Let n [member of] [N.sup.*] and A [member of]]0, 1/[mu]l[. We have

[mathematical expression not reproducible]. (74)

But, we have

[mathematical expression not reproducible]. (75)

It follows that there exists an integer [N.sub.A,l] [greater than or equal to] 1 such that the following inequality holds for every integer n [greater than or equal to] [N.sub.A,l]:

A[(n + 1).sup.-1/l] + [mu][A.sup.2] [(n + 1).sup.-2/l] [less than or equal to] [An.sup.-1/l]. (76)

Consequently, we have

[mathematical expression not reproducible], (77)

that is,

[mathematical expression not reproducible]. (78)

It follows that the function satisfies the property S(l). The proof of the proposition is then complete.

Recall that the following estimate holds for every z [member of] C

max([absolute value of (sin(z))], [absolute value of (cos(z))]) [less than or equal to] [e.sup.[absolute value of (z)]]. (79)

It means that the functions [[PHI].sub.1] := sin and [[PHI].sub.2] := cos satisfy the estimates:

[mathematical expression not reproducible], (80)

with [[alpha].sub.0] = [[beta].sub.0] = 1.

Example 1. Let C [member of] C and [gamma] [member of]]-1,1 [. We assume that

[mathematical expression not reproducible]. (81)

Consider the FFDE

[mathematical expression not reproducible] (82)

with the initial condition

([E.sup.1.sub.1]) : f(-1) = 0. (83)

Consider then the following entire functions:

[mathematical expression not reproducible]. (84)

It is clear that [a.sub.1] is not identically vanishing and that [a.sub.1] (-1) = [b.sub.1] (-1) = 0. Furthermore, we have

[mathematical expression not reproducible]. (85)

We also have

[mathematical expression not reproducible]. (86)

Consequently, it follows from the main result that the problem ([E.sup.1]) - ([E.sup.1.sub.1]) has a solution which belongs to the Gevrey class [G.sub.1,-1] ([-1,1]).

Example 2. Let [eta] > 0 and [lambda] [member of] C We assume that

[mathematical expression not reproducible]. (87)

Consider the FFDE

[mathematical expression not reproducible] (88)

with the initial condition

([E.sup.2.sub.1]) : f(-1)= [lambda]. (89)

Consider then the following functions:

[mathematical expression not reproducible]. (90)

It is clear that [a.sub.2] is not identically vanishing and that [a.sub.2] (-1) = [b.sub.2](-1) = 0. Furthermore, we have the following inequalities:

[mathematical expression not reproducible] (91)

Consequently, it follows from the main result that the problem ([E.sup.2]) - ([E.sup.2.sub.1]) has a solution which belongs to the Gevrey class [G.sub.1,=1] ([-1,1]).

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

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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Hicham Zoubeir [ID]

Ibn Tofail University, Department of Mathematics, Faculty of Sciences, P.O. Box 133, Kenitra, Morocco

Correspondence should be addressed to Hicham Zoubeir; hzoubeir2014@gmail.com

Received 11 August 2019; Accepted 28 October 2019; Published 29 June 2020

Academic Editor: Julio D. Rossi
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Title Annotation:Research Article
Author:Zoubeir, Hicham
Publication:International Journal of Differential Equations
Geographic Code:4EUGE
Date:Jul 31, 2020
Words:4974
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