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

Authors: Hicham Zoubeir
Journal: International Journal of Differential Equations
 (2020)

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## 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−1,1 of a class of nonlinear fractional functional differential equations.


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## Body

## 1. Introduction

Fractional calculus has evolved from the speculations of early mathematicians of the17th 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 Gl,q1q1,q2 of bound q1 and index l>0 on an interval q1,q2. Our purpose in this paper is to prove, under some regularity conditions on the data, the solvability in a Gevrey class of the form Gk,−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

LetF:E⟶E be a mapping from a nonempty set E into itself. Fnn∈ℕ denotes the iterate of F of order n for the composition of mappings.Forz∈ℂ and h>0,Bz,h is the open ball in ℂ≃ℝ2 with the center z and radius h.LetS1 and S2 be two nonempty subsets of ℂ such that S1⊂S2 and f:S2⟶ℂ a mapping. We denote by fS1 the restriction of the mapping f to the set S1.Forz∈ℂ and S⊂ℂS nonempty) we set(1)ϱz,S:=infζ∈Sz−ζ.Forl, φ, r>0, and n∈ℕ∗, we set for every nontrivial compact interval q1,q2 of ℝ(2)q1,q2r:=x+ζ:x∈q1,q2,ζ∈B0,r,q1,q2l,r,n:=q1,q2rn−1/l,q1,q2φ,r:=q1+seiθ:s∈0,q2−q1+r,θ∈−φ,φ,q1,q2r:=q1,q2r,r,q1,q2l,r,n:=q1,q2rn−1/l.Thus, we have(3)q1,q2r=z∈ℂ:ϱz,q1,q2<r,q1,q2l,r,n=z∈ℂ:ϱz,q1,q2<rn−1/l.Remark 1.
The following inclusions hold for everyd∈q1,q2,r∈0,d−q1 and n∈ℕ∗:(4)d,q2r⊂q1,q2r,d,q2l,r,n⊂q1,q2l,r,n.
Letf:S⟶ℂ be a bounded function. f∞,S denotes the quantity:(5)f∞,S:=supz∈Sfs.
ByC0q1,q2 (resp. C1q1,q2), we denote the complex vector space of all complex valued functions defined and continuous (resp. defined and of class C1) on the interval q1,q2. C0q1,q2 is a Banach space when it is endowed with the uniform norm:(6)⋅∞,q1,q2:f∈C0q1,q2⟼f∞,q1,q2.For every r≥0, we denote by Δ¯∞r the closed ball in C0−1,1) of radius r and center, the null function.
Letξ1,ξ2∈ℂ. We denote by ξ1,ξ2⟶ the linear path joining ξ1 to ξ2:(7)ξ1,ξ2⟶:0,1⟶ℂ,t⟼1−tξ1+tξ2.
In this paper,k>0 and α∈0,1 are fixed numbers.


### 2.2. Fractional Derivatives and Integrals

Definition 1.
Letδ∈0,1 and f be a Lebesgue-integrable function on the nontrivial compact interval q1,q2. The Caputo fractional integral of order δ and lower bound q1 of the function f [19, 23, 25, 26, 28] is the function denoted by Icaδf and defined by(8)Icq1δft:=1Γδ∫q1tt−sδ−1fsds,t∈q1,q2,where Γ denotes the classical gamma function.Remark 2.
If the functionf is continuous on the interval q1,q2, then the function Icq1δf is well defined and continuous on the entire interval q1,q2, and we have(9)Icq1δfq1=0.Definition 2.
Letf : q1,q2⟶ℂ be an absolutely continuous function on q1,q2; then, the Caputo fractional derivative of f of order δ and lower bound q1 [19, 23, 25, 26, 28] is the function denoted by Dcq1δf and defined by(10)Dcq1δft:=1Γ1−δ∫q1tt−s−δf′sds,t∈q1,q2.Remark 3.
Letf∈C1q1,q2. We have for every x∈q1,q2(11)Icq1δ∘Dcq1δfx=fx−fq1.
Iffq1=0, then the Caputo fractional integral of the function f of order δ,Icq1δf, is also of class C1 on the interval q1,q2 and we have [19, 23, 25, 26, 28](12)Dcq1δ∘Icq1δf=f.


### 2.3. Gevrey Classes

Definition 3.
Letl>0. The Gevrey class of index l on q1,q2, denoted by Glq1,q2, is the set of all functions f of class C∞ on q1,q2 such that(13)fn∞,q1,q2≤Bn+1nn1+1/l,n∈ℕ,where B>0 is a constant (with the convention that 00=1).Definition 4.
The Gevrey class of boundq1 and index l on the interval q1,q2, denoted by Gl,q1q1,q2, is the set of all functions f of class C1 on q1,q2 and of class C∞ on q1,q2 such that the restriction fq,q2 of f belongs to the Gevrey class Glq,q2, for every q∈q1,q2.


### 2.4. The PropertySl

Definition 5.
A functionφ defined on the set q1∪q1,q2rr∈0,π is said to satisfy the property Sl on the interval q1,q2 if φq1,q2r is holomorphic on q1,q2r,φq1,q2 is a function of class C1 on q1,q2, and there exists a constant τφ∈0,π such that for all D∈0,τφ there exist Nl,φD∈ℕ∗ depending only on D,l, and φ such that the inclusion(14)φq1,q2l,D,n+1⊂q1,q2l,D,nholds for every integer n≥Nl,φD. The number τφ is then called a Sl-threshold for the function φ.Remark 4.
Letφ be a function verifying the property Sl. Then,(15)φq1,q2⊂q1,q2.
On the other hand, it follows from (14) that we have for every D∈0,τφ(16)φq1,q2l,D,Nl,φDp+1⊂q1,q2l,A,Nl,φDp,p∈ℕ∗.
Thence, we have(17)φq1,q2l,DNl,φD−1/l,p+1⊂q1,q2l,DNl,φD−1/l,p,p∈ℕ∗.
It follows that for everyD∈0,τφ there exists E∈0,D such that(18)φq1,q2l,E,p+1⊂q1,q2l,E,p,p∈ℕ∗.


### 2.5. Statement of the Main Result

Our main result in this paper is the following.Theorem 1.
Letλ∈ℂ and σ>0. Let a,b, and ψ be holomorphic functions on −1,1σ and Φ be an entire function. We assume that the function a is not identically vanishing and that there exist constants α0,β0>0 such that(19)Φz≤α0eβ0z,z∈ℂand that ψ satisfies the property Sk. We also assume that the following conditions are fulfilled:(20)a−1=b−1=0,(21)2ααΓαb∞,−1,1+λ<lnαΓα/eα0β02αa∞,−1,1β0,(22)a∞,−1,1Φ∞,−lnαΓα/α0β0a∞,−1,1/β0,lnαΓα/α0β0a∞,−1,1/β0+a∞,−1,1<1,(23)a∞,−1,1Φ′∞,−lnαΓα/α0β0a∞,−1,1/β0,lnαΓα/α0β0a∞,−1,1/β0<αΓα2α,(24)ψ′∞,−1,1≤1+α.
Then, the FFDE(25)E:Dc−1αyt=atΦyψt+bthas a solution u which belongs to the Gevrey class Gk,−1−1,1 and verifies the initial condition(26)E1:y−1=λ.


## 2.1. Basic Notations

LetF:E⟶E be a mapping from a nonempty set E into itself. Fnn∈ℕ denotes the iterate of F of order n for the composition of mappings.Forz∈ℂ and h>0,Bz,h is the open ball in ℂ≃ℝ2 with the center z and radius h.LetS1 and S2 be two nonempty subsets of ℂ such that S1⊂S2 and f:S2⟶ℂ a mapping. We denote by fS1 the restriction of the mapping f to the set S1.Forz∈ℂ and S⊂ℂS nonempty) we set(1)ϱz,S:=infζ∈Sz−ζ.Forl, φ, r>0, and n∈ℕ∗, we set for every nontrivial compact interval q1,q2 of ℝ(2)q1,q2r:=x+ζ:x∈q1,q2,ζ∈B0,r,q1,q2l,r,n:=q1,q2rn−1/l,q1,q2φ,r:=q1+seiθ:s∈0,q2−q1+r,θ∈−φ,φ,q1,q2r:=q1,q2r,r,q1,q2l,r,n:=q1,q2rn−1/l.Thus, we have(3)q1,q2r=z∈ℂ:ϱz,q1,q2<r,q1,q2l,r,n=z∈ℂ:ϱz,q1,q2<rn−1/l.Remark 1.
The following inclusions hold for everyd∈q1,q2,r∈0,d−q1 and n∈ℕ∗:(4)d,q2r⊂q1,q2r,d,q2l,r,n⊂q1,q2l,r,n.
Letf:S⟶ℂ be a bounded function. f∞,S denotes the quantity:(5)f∞,S:=supz∈Sfs.
ByC0q1,q2 (resp. C1q1,q2), we denote the complex vector space of all complex valued functions defined and continuous (resp. defined and of class C1) on the interval q1,q2. C0q1,q2 is a Banach space when it is endowed with the uniform norm:(6)⋅∞,q1,q2:f∈C0q1,q2⟼f∞,q1,q2.For every r≥0, we denote by Δ¯∞r the closed ball in C0−1,1) of radius r and center, the null function.
Letξ1,ξ2∈ℂ. We denote by ξ1,ξ2⟶ the linear path joining ξ1 to ξ2:(7)ξ1,ξ2⟶:0,1⟶ℂ,t⟼1−tξ1+tξ2.
In this paper,k>0 and α∈0,1 are fixed numbers.


## 2.2. Fractional Derivatives and Integrals

Definition 1.
Letδ∈0,1 and f be a Lebesgue-integrable function on the nontrivial compact interval q1,q2. The Caputo fractional integral of order δ and lower bound q1 of the function f [19, 23, 25, 26, 28] is the function denoted by Icaδf and defined by(8)Icq1δft:=1Γδ∫q1tt−sδ−1fsds,t∈q1,q2,where Γ denotes the classical gamma function.Remark 2.
If the functionf is continuous on the interval q1,q2, then the function Icq1δf is well defined and continuous on the entire interval q1,q2, and we have(9)Icq1δfq1=0.Definition 2.
Letf : q1,q2⟶ℂ be an absolutely continuous function on q1,q2; then, the Caputo fractional derivative of f of order δ and lower bound q1 [19, 23, 25, 26, 28] is the function denoted by Dcq1δf and defined by(10)Dcq1δft:=1Γ1−δ∫q1tt−s−δf′sds,t∈q1,q2.Remark 3.
Letf∈C1q1,q2. We have for every x∈q1,q2(11)Icq1δ∘Dcq1δfx=fx−fq1.
Iffq1=0, then the Caputo fractional integral of the function f of order δ,Icq1δf, is also of class C1 on the interval q1,q2 and we have [19, 23, 25, 26, 28](12)Dcq1δ∘Icq1δf=f.


## 2.3. Gevrey Classes

Definition 3.
Letl>0. The Gevrey class of index l on q1,q2, denoted by Glq1,q2, is the set of all functions f of class C∞ on q1,q2 such that(13)fn∞,q1,q2≤Bn+1nn1+1/l,n∈ℕ,where B>0 is a constant (with the convention that 00=1).Definition 4.
The Gevrey class of boundq1 and index l on the interval q1,q2, denoted by Gl,q1q1,q2, is the set of all functions f of class C1 on q1,q2 and of class C∞ on q1,q2 such that the restriction fq,q2 of f belongs to the Gevrey class Glq,q2, for every q∈q1,q2.


## 2.4. The PropertySl

Definition 5.
A functionφ defined on the set q1∪q1,q2rr∈0,π is said to satisfy the property Sl on the interval q1,q2 if φq1,q2r is holomorphic on q1,q2r,φq1,q2 is a function of class C1 on q1,q2, and there exists a constant τφ∈0,π such that for all D∈0,τφ there exist Nl,φD∈ℕ∗ depending only on D,l, and φ such that the inclusion(14)φq1,q2l,D,n+1⊂q1,q2l,D,nholds for every integer n≥Nl,φD. The number τφ is then called a Sl-threshold for the function φ.Remark 4.
Letφ be a function verifying the property Sl. Then,(15)φq1,q2⊂q1,q2.
On the other hand, it follows from (14) that we have for every D∈0,τφ(16)φq1,q2l,D,Nl,φDp+1⊂q1,q2l,A,Nl,φDp,p∈ℕ∗.
Thence, we have(17)φq1,q2l,DNl,φD−1/l,p+1⊂q1,q2l,DNl,φD−1/l,p,p∈ℕ∗.
It follows that for everyD∈0,τφ there exists E∈0,D such that(18)φq1,q2l,E,p+1⊂q1,q2l,E,p,p∈ℕ∗.


## 2.5. Statement of the Main Result

Our main result in this paper is the following.Theorem 1.
Letλ∈ℂ and σ>0. Let a,b, and ψ be holomorphic functions on −1,1σ and Φ be an entire function. We assume that the function a is not identically vanishing and that there exist constants α0,β0>0 such that(19)Φz≤α0eβ0z,z∈ℂand that ψ satisfies the property Sk. We also assume that the following conditions are fulfilled:(20)a−1=b−1=0,(21)2ααΓαb∞,−1,1+λ<lnαΓα/eα0β02αa∞,−1,1β0,(22)a∞,−1,1Φ∞,−lnαΓα/α0β0a∞,−1,1/β0,lnαΓα/α0β0a∞,−1,1/β0+a∞,−1,1<1,(23)a∞,−1,1Φ′∞,−lnαΓα/α0β0a∞,−1,1/β0,lnαΓα/α0β0a∞,−1,1/β0<αΓα2α,(24)ψ′∞,−1,1≤1+α.
Then, the FFDE(25)E:Dc−1αyt=atΦyψt+bthas a solution u which belongs to the Gevrey class Gk,−1−1,1 and verifies the initial condition(26)E1:y−1=λ.


## 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:(27)ℑ:r=α02ααΓαa∞,−1,1eβ0r+2ααΓαb∞,−1,1+λ.
The study of the variations of the function(28)H:t⟼α02ααΓαa∞,−1,1eβ0t+2ααΓαb∞,−1,1+λ−t,shows, under condition (21), that H is strictly decreasing on 0,lnαΓα/α0β02αa∞,−1,1/β0 and strictly increasing on lnαΓα/α0β02αa∞,−1,1/β0,+∞. But,(29)H0=α02ααΓαa∞,−1,1+2ααΓαb∞,−1,1+λ>0,(30)HlnαΓα/α0β02αa∞,−1,1β0=2ααΓαb∞,−1,1+λ−lnαΓα/eα0β02αa∞,−1,1β0<0.
Therefore, the equationℑ has on ℝ+ exactly two solutions R0<R1 and the following inequalities hold:(31)0<R0<lnαΓα/α0β02αa∞,−1,1β0<R1.Step 2.
Proof of the existence of a solutionu of the FFDE E in C1−1,1 such that the initial condition E1 holds.
Consider the operatorT:C0−1,1⟶C0−1,1 defined by the following formula:(32)Tft=Ic−1αa.Φ∘f∘ψ+bt+λ,t∈−1,1.
We have for allf∈Δ¯∞R0(33)Tf∞,−1,1≤Ic−1αa.Φ∘f∘ψ+b∞,−1,1+λ≤2ααΓαa∞,−1,1Φ∘f∞,−1,1+2ααΓαb∞,−1,1+λ≤2ααΓαa∞,−1,1α0eβ0f∞,−1,1+2ααΓαb∞,−1,1+λ≤α02ααΓαa∞,−1,1eβR0+2ααΓαb∞,−1,1+λ=R0.
Thence, the closed ballΔ¯∞R0 is stable by the operator T. On the other hand, we have for all f, g∈Δ¯∞R0(34)Tf−Tg∞,−1,1≤2ααΓαa∞,−1,1Φ′∞,−R0,R0f−g∞,−1,1.
Since0<R0<lnαΓα/α0β02αa∞,−1,1/β0, it follows from condition (23) that(35)2ααΓαa∞,−1,1Φ′∞,−R0,R0<1.
Thence,T has, in Δ¯∞R0, a unique fixed point u.
Consider the sequence of functionsfnn∈ℕ defined on −1,1 by the following formula:(36)fn:=Tnf0,n∈ℕ,where f0 is the null function. Direct computations show that the functions fn belonging to Δ¯∞R0 are of class C1 on −1,1 and verify the following inequality:(37)fn+1−fn∞,−1,1≤f1∞,−1,1Qn,n∈ℕ,where(38)Q:=2ααΓαa∞,−1,1Φ′∞,−R0,R0.
Let us set for eachn∈ℕ,Fn:=fn+1−fn. Since Q∈0,1, it follows that the function series ∑Fn is uniformly convergent on −1,1 to a function v∈Δ¯∞R0 which is a fixed point of the operator T. It follows that v=u. Consequently, the function series ∑Fn is uniformly convergent on −1,1 to the function u∈C0−1,1.
On the other hand, we have for allx∈−1,1 and n∈ℕ∗:(39)Fn+1′x=αx+1α−1Γα∫011−tα−1a−1+tx+1⋅Φfnψ−1+tx+1−Φfn−1ψ−1+tx+1dt+x+1αΓα∫011−tα−1ta′−1+tx+1⋅Φfnψ−1+tx+1−Φfn−1ψ−1+tx+1dt+x+1αΓα∫011−tα−1ta−1+tx+1ψ′−1+tx+1⋅Φ′fnψ−1+tx+1−Φ′fn−1ψ−1+tx+1⋅fn′ψ−1+tx+1+Φ′fn−1ψ−1+tx+1⋅fn′ψ−1+tx+1−fn−1′ψ−1+tx+1dt.
Sincea−1=0, it follows that(40)Fn+1′∞,−1,1≤2αα+1Γαa′∞,−1,1Φ′∞,−R0,R0Fn∞,−1,1+2ααα+1Γαa′∞,−1,1Φ′∞,−R0,R0Fn∞,−1,1+2ααα+1Γαψ′∞,−1,1a∞,−1,1Φ″∞,−R0,R0fn′∞,−1,1Fn∞,−1,1+2ααα+1Γαψ′∞,−1,1a∞,−1,1Φ′∞,−R0,R0Fn′∞,−1,1≤2ααΓαa′∞,−1,1Φ′∞,−R0,R0f1∞,−1,1Qn+2ααΓαa∞,−1,1Φ″∞,−R0,R0fn′∞,−1,1f1∞,−1,1Qn+2ααΓαa∞,−1,1Φ′∞,−R0,R0Fn′∞,−1,1.
To achieve the proof of this step we need the following result.Proposition 1.
The sequencefn′∞,−1,1n∈ℕ∗ is bounded.Proof.
We have for allx∈−1,1 and n∈ℕ∗(41)fn+1′x=αx+1α−1Γα∫011−sα−1⋅a−1+sx+1⋅Φfnψ−1+sx+1+b−1+sx+1ds+x+1αΓα∫011−sα−1s⋅a′−1+sx+1⋅Φfnψ−1+sx+1+b′−1+sx+1ds+x+1αΓα∫011−sα−1sa−1+sx+1⋅ψ′−1+sx+1Φ′fnψ−1+sx+1⋅fn′ψ−1+sx+1ds.
It follows from assumption (20) that(42)fn+1′∞,−1,1≤2ααΓαa′∞,−1,1Φ∞,−R0,R0+b′∞,−1,1+2ααα+1Γαa∞,−1,1⋅ψ′∞,−1,1Φ′∞,−R0,R0fn′∞,−1,1.
But, according to assumption (24) and (35) we have(43)2ααα+1Γαa∞,−1,1ψ′∞,−1,1Φ′∞,−R0,R0≤Q<1.
Consequently, the following inequality holds for eachn∈ℕ∗(44)fn+1′∞,−1,1≤2ααΓαa′∞,−1,1Φ∞,−R0,R0+b′∞,−1,1+Qfn′∞,−1,1.
SinceQ∈0,1, it follows that the sequence fn′∞,−1,1n∈ℕ∗ is bounded.
The proof of the proposition is complete.
Now, we set(45)θ:=f1∞,−1,12ααΓαa′∞,−1,1Φ′∞,−R0,R0+2ααΓαa∞,1,1Φ″∞,−R0,R0⋅supn∈ℕ∗fn′∞,−1,1.
Then, we can write(46)Fn+1′∞,−1,1≤θQn+QFn′∞,−1,1,n∈ℕ∗.
Direct computations show then that(47)Fn+1′∞,−1,1≤θnQn+F1′∞,−1,1Qn,n∈ℕ∗.
SinceQ∈0,1, it follows that the function series ∑Fn′ is uniformly convergent on −1,1. Thence, the function u is of class C1 on −1,1 and satisfies the following relation:(48)Ic−1αa.Φ∘u∘ψ+b+λ=u.
Consequently, according to assumption (20), we can write for all t∈−1,1(49)Dc−1αut=Dc−1αIc−1αa.Φ∘u∘ψ+bt=atΦuψt+bt.
So,u is a solution of the FFDE E which belongs to C1−1,1 and fulfills the relation u−1=λ.Step 3.
Proof thatu belongs to the Gevrey class Gk,−1−1,1.
Since the functionΛ defined on 0,min1,σ by(50)Λ:0,min1,σ⟶ℝ,0⟼2ααΓαmaxa∞,−1,1Φ′∞,−R0,R0,a∞,−1,1Φ∞,−R0,R0+b∞,−1,1,s>0⟼2+sααΓαmaxa∞,−1,1sΦ′∞,−R0,R0s,a∞,−1,1sΦ∞,−R0,R0s+b∞,−1,1sis continuous on 0,min1,σ and verifies by virtue of assumptions (22) and (23), the inequality Λ0<1. It follows that there exists s1∈0,min1,σ,τψ such that(51)Λ0,s1⊂0,1,where τψ is a Sl-threshold of ψ. Let d be an arbitrary but fixed element of −1,1. Thanks to remark 4, there exists s2∈0,s1 such that the functions a and b are both holomorphic on −1,1s2 and the following condition holds:(52)ψ−1,1k,s2,n+1⊂−1,1k,s2,n,n∈ℕ∗.
Consider the sequence of functionsωn:−1,1k,s2,n⟶ℂn∈ℕ∗, where(53)ω1z=0,for  allz∈−1,1k,s2,1,(54)ωn+1z=z+1αΓα∫011−sα−1a−1+sz+1Φωnψ−1+sz+1++b−1+sz+1ds+λ,for each n∈ℕ∗ and z∈−1,1k,s2,n. Then, direct computations, based on (52), show that the function ωn is for every n∈ℕ∗ holomorphic on −1,1k,s2,n.Proposition 2.
The inclusionωn−1,1k,s2,n⊂−R0,R0k,s2,n holds for every n∈ℕ∗.Proof.
We denote the last inclusion byPn. We denote for every z∈ℂ by z^ the closest point of −1,1 to z. It is obvious that P1 is true. Assume for a certain n∈ℕ∗ that Pp is true for every p∈1,…,n. Since the function ωn+1 is holomorphic on −1,1k,s2,n+1, we have then for each z∈−1,1k,s2,n+1(55)ϱωn+1z,−R0,R0≤ωn+1z−ωn+1z^≤1Γα∫z^,z⟶z−ζα−1aζΦωnψζ.dζ+1Γα∫z^,z⟶z−ζα−1bζ.dζ≤2+s2ααΓαa∞,−1,1k,s2,nΦ∞,−R0,R0k,s2,n+b∞,−1,1k,s2,n⋅ϱz,−1,1≤Λs2s2n+1−1/k<s2n+1−1/k.
Thence, the assertionPn+1 is true. Consequently, Pn is true for all n∈ℕ∗.
The proof of the proposition is then complete.
By virtue of the Proposition 2., we have for alln∈ℕ∗\1 and z∈−1,1k,s2,n+1(56)ωn+1z−ωnz≤z+1αΓα∫011−sα−1a−1+sz+1⋅Φωnψ−1+sz+1−Φωn−1ψ−1+sz+1ds≤2+s2ααΓαa∞,−1,1k,s2,n+1Φ′∞,−R0,R0k,s2,n+1ωn−ωn−1−1,1k,s2,n∞,−1,1k,s2,n≤Λs2ωn−ωn−1−1,1k,s2,n∞,−1,1k,s2,n.
It follows that(57)ωn+1−ωn−1,1k,s2,n+1∞,−1,1k,s2,n+1≤ω2−ω1−1,1k,s2,2∞,−1,1k,s2,2Λs2Λs2n,n∈ℕ∗\1.
Let us setΩ1:=ω1 and denote, for all n∈ℕ∗\1, by Ωn the function(58)Ωn:−1,1k,s2,n+1⟶ℂ,z⟼ωn+1z−ωnz.
Then, the functionΩn is holomorphic on −1,1k,s2,n+1 for each n∈ℕ∗. Furthermore, the following relations hold for every n∈ℕ∗\1:(59)Ωn∞,−1,1k,s2,n+1≤ω2−ω1−1,1k,s2,1∞,−1,1k,s2,1Λs2Λs2n,Ωnd,1=fn−1.
SinceΛs2∈0,1, it follows then from (59) that the function series ∑Ωn−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:(60)d,1k,s2,n⊂−1,1k,s2,n,n∈ℕ∗.
It follows then that(61)Ωn∞,d,1k,s2,n≤ω2−ω1−1,1k,s2,n∞,−1,1k,s2,2Λs2Λs2n,n∈ℕ∗\1,Ωnd,1=fn−1,n∈ℕ∗\1.
The relations (61) entail, thanks to the main result of [41], that ud,1 belongs to Gkd,1, for each d ϵ −1,1. Thence, since u is of class C1 on −1,1, it follows that u belongs to the Gevrey class Gk,−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(62)ℒ:ℂ⟶ℂ,z⟼2ez−1/2−1,satisfies the property Sl for every l∈0,1.Proof.
Letl∈0,1,ε∈0,1 and z∈−1,1ε. We have(63)ℒz=2ez−1/2−1.
It follows that(64)Reℒz+1>0.
We consider then the principal argumentargℒz+1 of ℒz+1 which satisfies the following estimates:(65)argℒz+1=Imz2≤1+ε2tanε.
But, direct computations prove that(66)0<tanε−ε+ε3/3ε3<tan1−43.
Thence, we have(67)1+ε2tanε≤1+ε2ε+ε33+tan1−43ε3≤ε+32tan1−1ε2.
It follows that(68)argℒz+1≤ε+32tan1−1ε2.
On the other hand, we have(69)ℒz+1=2eRez−1/2≤2eε/2.
But, we know that(70)2eε/2=2+ε+ε2∑n=0+∞εnn+2!2n+2≤2+ε+e−32ε2.
It follows that(71)ℒz+1≤2+ε+e−32ε2.
We derive, from the estimates (68) and (71), the following inclusion:(72)ℒ−1,1ε⊂−1,1ε+με2,where(73)μ:=max32tan1−1,e−32=32tan1−1>0.
Letn∈ℕ∗ and A∈0,1/μl. We have (74)An+1−1/l+μA2n+1−2/l−An−1/l=An−1/l1+1n−1/l+μAn1/ln+12/l−1≤An−1/l1+1n−1/l+μAn−1.
But, we have(75)1+1n−1/l+μAn−1∼n⟶+∞μA−1/μln.
It follows that there exists an integerNA,l≥1 such that the following inequality holds for every integer n≥NA,l:(76)An+1−1/l+μA2n+1−2/l≤An−1/l.
Consequently, we have(77)ℒ−1,1An+1−1/l⊂−1,1An−1/l,n≥NA,l,that is,(78)ℒ−1,1l,A,n+1⊂−1,1l,A,n,n≥NA,l.
It follows that the function satisfies the propertySl.
The proof of the proposition is then complete.Recall that the following estimate holds for everyz∈ℂ(79)maxsinz,cosz≤ez.It means that the functionsΦ1:=sin and Φ2:=cos satisfy the estimates:(80)Φjz≤α0eβ0z,z∈ℂ,j=1,2,with α0=β0=1.Example 1.
LetC∈ℂ and γ∈−1,1. We assume that(81)0<C<minαΓα2α+1e2α/αΓαγ+1,1−γ2.
Consider the FFDE(82)E1:Dc−1αfx=Cx+1sinf2ex−1/2−1+γsinx+1with the initial condition(83)E11:f−1=0.
Consider then the following entire functions:(84)Φ1:z⟼sinz,a1:z⟼Cx+1,b1:z⟼γsinz+1.
It is clear thata1 is not identically vanishing and that a1−1=b1−1=0. Furthermore, we have
(85)2ααΓαb1∞,−1,1+0=2ααΓαγ<lnαΓαe2α+1C=lnαΓα/eα0β02αa1∞,−1,1β0.
We also have(86)a1∞,−1,1Φ1′∞,−lnαΓα/α0β0a1∞,−1,1/β0,lnαΓα/α0β0a1∞,−1,1/β0≤2C<αΓα2α,a1∞,−1,1Φ1∞,−lnαΓα/α0β0a1∞,−1,1/β0,lnαΓα/α0β0a1∞,−1,1/β0+b1∞,−1,1≤2C+γ<1.
Consequently, it follows from the main result that the problemE1−E11 has a solution which belongs to the Gevrey class G1,−1−1,1.Example 2.
Letη>0 and λ∈ℂ. We assume that(87)η<minαΓα2αeλ+1,1.
Consider the FFDE(88)E2:Dc−1αfx=ηsinx+1cosf2ex−1/2−1with the initial condition(89)E12:f−1=λ.
Consider then the following functions:(90)Φ2:z⟼cosz,a2:z⟼ηsinz+1,b2:z⟼0.
It is clear thata2 is not identically vanishing and that a2−1=b2−1=0. Furthermore, we have the following inequalities:(91)2ααΓαb2∞,−1,1+λ=lnαΓαe2αη<lnαΓα/eα0β02αa2∞,−1,1β0,a2∞,−1,1Φ2′∞,−lnαΓα/α0β0a1∞,−1,1/β0,lnαΓα/α0β0a1∞,−1,1/β0≤η<αΓα2α,a2∞,−1,1Φ2∞,−lnαΓα/α0β0a1∞,−1,1/β0,lnαΓα/α0β0a1∞,−1,1/β0+b2∞,−1,1=η<1.
Consequently, it follows from the main result that the problemE2−E12 has a solution which belongs to the Gevrey class G1,−1−1,1.



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