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.

1. Introduction

Fractional calculus has evolved 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, and J. L. Lagrange [1]. 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, and N. H. Abel have used the ideas of fractional calculus to solve some physical or mathematical problems [1]. 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 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 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 G_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⟶E be a mapping from a nonempty set E into itself. F^〈n〉(n ∈ ℕ) denotes the 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≃ℝ² with the center z and radius h.

Let S₁ and S₂ be two nonempty subsets of ℂ such that S₁ ⊂ S₂ and f : S₂⟶ℂ a mapping. We denote by the restriction of the mapping f to the set S₁.

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 [q₁, q₂] of ℝ

()

Thus, we have

()

Remark 1. The following inclusions hold for every d ∈ ]q₁, q₂], r ∈ ]0, d − q₁[ and n ∈ ℕ^∗:

()

Let f : S⟶ℂ be a bounded function. ‖f‖_∞,S denotes the quantity:

()

By C⁰([q₁, q₂]) (resp. C¹([q₁, 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]. C⁰([q₁, q₂]) is a Banach space when it is endowed with the uniform norm:

()

For every r ≥ 0, we denote by

the closed ball in C⁰([−1,1])) of radius r and center, the null function.

Let ξ₁, ξ₂ ∈ ℂ. We denote by the linear path joining ξ₁ to ξ₂:

()

In this paper, k > 0 and α ∈ ]0,1[ are fixed numbers.

2.2. Fractional Derivatives and Integrals

Definition 1. Let δ ∈ ]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 and defined by

()

where Γ denotes the classical gamma function.

Remark 2. If the function f is continuous on the interval [q₁, q₂], then the function is well defined and continuous on the entire interval [q₁, q₂], and we have

()

Definition 2. Let f : [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 and defined by

()

Remark 3. Let f ∈ C¹([q₁, q₂]). We have for every x ∈ [q₁, q₂]

()

If f(q₁) = 0, then the Caputo fractional integral of the function f of order δ, , is also of class C¹ on the interval [q₁, q₂] and we have [19, 23, 25, 26, 28]

()

2.3. Gevrey Classes

Definition 3. Let l > 0. The Gevrey class of index l on [q₁, q₂], denoted by G_l([q₁, q₂]), is the set of all functions f of class C^∞ on [q₁, q₂] such that

()

where B > 0 is a constant (with the convention that 0⁰ = 1).

Definition 4. The Gevrey class of bound q₁ and index l on the interval [q₁, q₂], denoted by is the set of all functions f of class C¹ on [q₁, q₂] and of class C^∞ on ]q₁, q₂] such that the restriction of f belongs to the Gevrey class G_l([q, q₂]), for every q ∈ ]q₁, q₂[ .

2.4. The Property S(l)

Definition 5. A function φ defined on the set is said to satisfy the property S(l) on the interval [q₁, q₂] if is holomorphic on is a function of class C¹ on [q₁, q₂], 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 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 4. Let φ be a function verifying the property S(l). Then,

()

On the other hand, it follows from (14) that we have for every D ∈ ]0, τ_φ[

()

Thence, we have

()

It follows that for every D ∈ ]0, τ_φ[ there exists E ∈ ]0, D[ such that

()

2.5. Statement of the Main Result

Our main result in this paper is the following.

Theorem 1. Let λ ∈ ℂ and σ > 0. Let a, b, and ψ be holomorphic functions on [−1,1]_σ 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

()

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

()

Then, the FFDE

()

has a solution u which belongs to the Gevrey class G_k,−1([−1,1]) 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 function

()

shows, under condition (21), that H is strictly decreasing on [0, ln(αΓ(α)/α₀β₀2^α‖a‖_∞,[−1,1])/β₀] and strictly increasing on [ln(αΓ(α)/α₀β₀2^α‖a‖_∞,[−1,1])/β₀ + ∞[. But,

()

Therefore, the equation (ℑ) has on ℝ⁺ exactly two solutions R₀ < R₁ and the following inequalities hold:

()

Step 2. Proof of the existence of a solution u of the FFDE (E) in C¹([−1,1]) such that the initial condition (E₁) holds.

Consider the operator T : C⁰([−1,1])⟶C⁰([−1,1]) defined by the following formula:

()

We have for all

()

Thence, the closed ball is stable by the operator T. On the other hand, we have for all f,

()

Since 0 < R₀ < (ln (αΓ(α)/α₀β₀2^α‖a‖_∞,[−1,1])/β₀), it follows from condition (23) that

()

Thence, T has, in a unique fixed point u.

Consider the sequence of functions defined on [−1,1] by the following formula:

()

where f₀ is the null function. Direct computations show that the functions f_n belonging to

are of class C¹ on [−1,1] and verify the following inequality:

()

where

()

Let us set for each n ∈ ℕ, F_n = f_n+1 − f_n. Since Q ∈ [0,1[, it follows that the function series ∑F_n is uniformly convergent on [−1,1] to a function which is a fixed point of the operator T. It follows that v = u. Consequently, the function series ∑F_n is uniformly convergent on [−1,1] to the function u∈C⁰([−1,1]).

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 1. The sequence is bounded.

Proof. We have for all x ∈ ]−1,1] and n ∈ ℕ^∗

()

It follows from assumption (20) that

()

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

()

Consequently, the following inequality holds for each n ∈ ℕ^∗

()

Since Q ∈ [0,1[, it follows that the sequence is bounded.

The proof of the proposition is complete.

Now, we set

()

Then, we can write

()

Direct computations show then that

()

Since Q ∈ [0,1[, it follows that the function series is uniformly convergent on [−1,1]. Thence, the function u is of class C¹ on [−1,1] and satisfies the following relation:

()

Consequently, according to assumption (20), we can write for all t ∈ [−1,1]

()

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 that u belongs to the Gevrey class G_k,−1([−1,1]).

Since the function Λ defined on [0, min(1, σ)[ by

()

is continuous on [0, min(1, σ)[ and verifies by virtue of assumptions (22) and (23), the inequality Λ(0) < 1. It follows that there exists s₁ ∈ ]0, min(1, σ, τ_ψ)[ such that

()

where τ_ψ is a S(l)-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

and the following condition holds:

()

Consider the sequence of functions , where

()

for each n ∈ ℕ^∗ and

Then, direct computations, based on (52), show that the function ω_n is for every n ∈ ℕ^∗ holomorphic on

Proposition 2. The inclusion holds for every n ∈ ℕ^∗.

Proof. We denote the last inclusion by P(n). We denote for every z ∈ ℂ by the closest point of [−1,1] to z. It is obvious that P(1) is true. Assume for a certain n ∈ ℕ^∗ that P(p) is true for every p ∈ {1 … n}. Since the function ω_n+1 is holomorphic on we have then for each

()

Thence, the assertion P(n + 1) is true. Consequently, P(n) is true for all n ∈ ℕ^∗.

The proof of the proposition is then complete.

By virtue of the Proposition 2., we have for all n ∈ ℕ^∗\{1} and

()

It follows that

()

Let us set Ω₁ = ω₁ and denote, for all n ∈ ℕ^∗\{1}, by Ω_n the function

()

Then, the function Ω_n is holomorphic on for each n ∈ ℕ^∗. Furthermore, the following relations hold for every n ∈ ℕ^∗\{1}:

()

Since Λ(s₂) ∈ [0,1[, it follows then from (59) that the function series ∑Ω_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:

()

It follows then that

()

The relations (61) entail, thanks to the main result of [41], that u_{|[d, 1]} belongs to G_k([d, 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 G_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

()

satisfies the property S(l) for every l ∈ ]0,1].

Proof. Let l ∈ ]0,1], ε ∈ ]0,1] and z ∈ [−1,1]^ε. We have

()

It follows that

()

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

()

But, direct computations prove that

()

Thence, we have

()

It follows that

()

On the other hand, we have

()

But, we know that

()

It follows that

()

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

()

where

()

Let n ∈ ℕ^∗ and A ∈ ]0, 1/μl[ . We have

()

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

()

that is,

()

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 ∈ ℂ

()

It means that the functions Φ₁ : = sin and Φ₂ : = cos satisfy the estimates:

()

with α₀ = β₀ = 1.

Example 1. Let C ∈ ℂ and γ ∈ ] − 1,1[ . We assume that

()

Consider the FFDE

()

with the initial condition

()

Consider then the following entire functions:

()

It is clear that a₁ is not identically vanishing and that a₁(−1) = b₁(−1) = 0. Furthermore, we have

()

We also have

()

Consequently, it follows from the main result that the problem has a solution which belongs to the Gevrey class G_1,−1([−1,1]).

Example 2. Let η > 0 and λ ∈ ℂ. We assume that

()

Consider the FFDE

()

with the initial condition

()

Consider then the following functions:

()

It is clear that a₂ is not identically vanishing and that a₂(−1) = b₂(−1) = 0. 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 G_1,−1([−1,1]).

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.

Open Research

Data Availability

No data were used to support this study.

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Solvability in Gevrey Classes of Some Nonlinear Fractional Functional Differential Equations

Abstract

1. Introduction

2. Preliminary Notes and Statement of the Main Result

2.1. Basic Notations

2.2. Fractional Derivatives and Integrals

2.3. Gevrey Classes

2.4. The Property S(l)

2.5. Statement of the Main Result

3. Proof of the Main Result

4. Examples

Disclosure

Conflicts of Interest

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References

References

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Solvability in Gevrey Classes of Some Nonlinear Fractional Functional Differential Equations

Abstract

1. Introduction

2. Preliminary Notes and Statement of the Main Result

2.1. Basic Notations

2.2. Fractional Derivatives and Integrals

2.3. Gevrey Classes

2.4. The Property S(l)

2.5. Statement of the Main Result

3. Proof of the Main Result

4. Examples

Disclosure

Conflicts of Interest

Open Research

Data Availability

References

References

Related

Information