Solvability in Gevrey classes of some nonlinear fractional functional differential equations

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


Introduction
The fractional calculus has grown up from the speculations of early mathematicians of the 17 th and 18 th centuries like G. W. Leibnitz, I. Newton, L. Euler, G. F. de L'Hospital, J. L. Lagrange ( [33]).In the 19 th 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, N. H. Abel have used the ideas of fractional calculus to solve some physical or mathematical problems ( [33]).In the 20 th 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 the fractional calculus and its applications on rigorous and solid mathematical foundations ( [33]).Actually the fractional calculus allows the 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, engeneering, fractal phenomena, biology, social sciences, finance, economy, chemistry, anomalous diffusion, rheology ([4]- [7], [14], [18], [23]- [34], [38], [40], [41], [44]).It is then of capital importance to develop for fractional calculus, the mathematical tools analogous to those of classical calculus ( [5], [6], [33], [38], [39]).The fractional differential equations ( [1], [15], [16], [17], [36], [39]) are a particularly important case of such fundamental tools.An important type of fractional differential equations is that of fractional functional differential equations (FFDE) ( [9], [13], [25], [42]) which are the fractional analogue to functional differential equations ( [3], [19], [21], [32]).FFDE enable the study of some physical, biological, social, economical processes (automatic control, financial dynamics, economical planning, population dynamics, blood cell dynamics, infectious disease dynamics.etc) with fractal memory and non-locality effects and 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 ( [8], [26], [35]).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 ( [2], [11], [22], [43]), we think that these functional spaces can play the role of such convenient framework.However let us pointwise 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 a new Gevrey classes namely the Gevrey classes G l,q1 ([q 1 , q 2 ]) of bound q 1 and index l > 0 on an interval [q 1 , q 2 ] .Our purpose in this paper is to prove, under some regularity conditions on the datas, the solvability in a Gevrey class of the form G k,−1 ([−1, 1]) of a class of nonlinear FFDE.Our approach is mainly based on a theorem that we have proved in ( [10]).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 −→ E be a mapping from a nonempty set E into itself.F n (n ∈ N) denotes iterate of F of order n for the composition of mappings.
For z ∈ C and h > 0, B(z, h) is the open ball in C ≃ R 2 with the center z and radius h.
Let S 1 and S 2 be two nonempty subsets of C such that S 1 ⊂ S 2 and f : S 2 → C a mapping.We denote by f |S1 the restriction of the mapping f to the set S 1 .
For z ∈ C and S ⊂ C (S nonempty) we set : For l, ϕ, r > 0 and n ∈ N * we set for every nontrivial compact interval [q 1 , q 2 ] of R : Thus we have : becomes a Banach space.For every r ≥ 0, ∆ ∞ (r) denotes then the closed ball, in this Banach space, of center the null function and radius r.
Definition 2.7.The Gevrey class of bound q 1 and index l on the interval [q 1 , q 2 ], denoted by G l,q1 ([q 1 , q 2 ]), is the set of all functions f of class C 1 on [q 1 , q 2 ] and of class C ∞ on ]q 1 , q 2 ] such that the restriction f |[q,q2] of f belongs to the Gevrey class G l ([q, q 2 ]), for every q ∈ ]q 1 , q 2 [ .

The property S (l).
Definition 2.8.A function ϕ defined on the set {q and there exists a constant τ ϕ ∈]0, π[ such that for all D ∈]0, τ ϕ ] there exist N l,ϕ (D) ∈ N * depending only on D, l and ϕ such that the following inclusion : holds for every integer n ≥ N l,ϕ (D).The number τ ϕ is then called a S(l)-threshold for the function ϕ.
Remark 2.9.Let ϕ be a function verifying the property S(l).Then : On the other hand, it follows from (2.2) that we have for every D ∈]0, τ ϕ [ : Thence we have : Statement of the main result .Our main result in this paper is the following.
Let be given λ ∈ C and σ > 0. Let a, b and ψ be a holomorphic functions on [−1, 1] σ and Φ an entire function.We assume that the function a is not identically vanishing, that there exist a constants α 0 , β 0 > 0 such that : and that ψ satisfies the property S(k).We assume also that the following conditions are fullfiled : Then the FFDE : ) and verifies the initial condition : (E 1 ) : y(−1 ) = λ

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 function : shows, under the condition (2.4), that H is strictly decreasing on Therefore the equation (ℑ) has on R + exactly two solutions R 0 < R 1 and the following inequalities hold : ) such that the initial condition (E 1 ) holds.Consider the operator T : ) defined by the formula : We have for all f ∈ ∆ ∞ (R 0 ) : Thence the closed ball ∆ ∞ (R 0 ) is stable by the operator T. On the other hand, we have for all f , g ∈ ∆ ∞ (R 0 ) : β 0 it follows from the condition (2.6) that : Thence T has, in ∆ ∞ (R 0 ), a unique fixed point u.
Consider the sequence of functions (f n ) n∈N defined on [−1, 1] by the formula : where f 0 is the null function.Direct computations show that the functions f n belong to ∆ ∞ (R 0 ), are of class C 1 on [−1, 1] and verify the inequality : where : Let us set for each n ∈ N, ).On the other hand we have for all x ∈] − 1, 1] and n ∈ N * : Since a(−1) = 0 it follows that : To achieve the proof of this step we need the following result.
Proposition 3.1.The sequence ) n∈N * is bounded.Proof.We have for all x ∈] − 1, 1] and n ∈ N * : ))ds It follows from the assumption (2.3) that : But we have according to the assumption (2.7) that : Consequently the following inequality holds for each n ∈ N * : ) n∈N * is bounded.The proof of the proposition is complete.Now we set : Then we can write : , n ∈ N * Direct computations show then that : 1] and satisfies the relation : So u is a solution of the FFDE (E) which belongs to C 1 ([−1, 1]) and fullfiles the relation u(−1) = λ.
By virtue of the proposition 3.2., we have for all n ∈ N * and z ∈ [−1, 1] k,s2,n+1 : It follows that : Let us set Ω 1 := ω 1 and denote, for all n ∈ N * \ {1} , by Ω n the function : Then the function Ω n is holomorphic on [−1, 1] k,s2,n +1 for each n ∈ N * .Furthermore the following relations hold for every n ∈ N * : 1] to the function u.But we know, according to the relation (2.1) of remark 2.1., that the following inclusions hold : The relations (3.4) entail, thanks to the main result of ( The proof of the main result is then complete.

Examples
To obtain examples illustrating our main result, we need first to prove the following proposition.
Proof.Let l ∈]0, 1], ε ∈]0, 1] and z ∈ [−1, 1] ε .We have : It follows that : Re (L(z) + 1) > 0 We consider then the principal argument arg(L(z) + 1) of L(z) + 1 which satisfies the following estimates : It follows that : On the other hand we have : But we know that : It follows that : We derive, from the estimates (4.1) and (4.2), the following inclusion : where : But we have : It follows that there exists an integer N A,l ≥ 1 such that the following inequality holds for every integer n ≥ N A,l : Consequently we have :

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