New Results on the Solvability of Abstract Sequential Caputo Fractional Differential Equations with a Resolvent-Operator Approach and Applications

: This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative.


Introduction
Fractional calculus is a more advanced version of traditional calculus and has a wider range of applications.It has been particularly useful in areas such as signal processing, chemistry, biology, control theory, physics, economic systems and mechanics [7,8,16,21,24,25,35].
In the field of biology, the authors of [2] utilized the Caputo fractional derivative as a mathematical technique to develop a model for the transmission of a coronavirus (specifically, MERS-CoV) between humans and camels.Camels are suspected to be the primary source of the infection.The paper investigates how the transmission of MERS-CoV disease changes over time by employing a nonlinear fractional order based on the Caputo operator.In the realm of physics, the nonlinear space-time fractional partial differential symmetric regularized long-wave equation is a useful tool for summarizing various physical phenomena.For example, it can describe ion-acoustic waves in plasma, as well as solitary waves and shallow-water waves.In [23], the authors used this novel approach to obtain the traveling wave solutions of two equations: the space-time fractional Cahn-Hilliard equation and the space-time fractional symmetric regularized long-wave equation.
In [19], Hernandez et al. investigated the existence and uniqueness of a specific problem, defined as follows: D α χ(κ) = Aχ(κ) + ς(κ, Bχ(κ), χ(κ)), κ ∈ [0, a], χ(0) = χ 0 + g(χ), where D α represents the Caputo fractional derivative, A is a closed linear operator with a domain contained in a Banach space X, and ς and g are continuous functions.The researchers employed various techniques, including the use of the resolvent operator and other properties of fractional differential equations, to study this problem.
In [10], Aqlan et al. investigated the following sequential fractional equation: with boundary conditions of the form and with the nonlocal integral boundary conditions where D α is the Liouville-Caputo fractional derivative, and ς is a continuous function.Salem and Almaghamsi [37] studied the existence of the solution of the following sequential fractional differential equation: with the boundary conditions where 1 < α ≤ 2, D α represents the Caputo derivative, and D denotes the first-order derivative.
In this paper, we extend Equations ( 2), ( 3) and ( 5) by considering the case where λ represents a closed linear operator A. We investigate the following problem with an abstract sequential fractional differential equation of the form where D α , D β are two Caputo fractional derivatives, and I σ is the Riemann-Liouville fractional integral, with 0 A is a closed linear unbounded operator, with the domain D(A) contained in a Banach space X, and H ω depends on a parameter ω ≥ 0, with H ω : [0, T] × X 2 → X, g : C(J, X) → X, J ⊂ R being continuous functions.Equation ( 7) can be interpreted as an abstract form of the following partial fractional differential equation: with Aχ = χ ττ and D The main objective of this paper is to examine the existence and uniqueness of mild solutions of (1).In our study, we use the Caputo fractional derivative and Riemann-Liouville fractional integral operators, with a specific emphasis on the significance of resolvent operators.To demonstrate the uniqueness, we use the Banach contraction principle, and for the existence of solutions, we apply the Krasnoseskii fixed-point theorem.

Preliminaries
In this section, we will present definitions and preliminary concepts that will serve as building blocks for the next sections.These essential definitions and preliminary explanations will be referred to frequently in the upcoming sections.
In the following two definitions, we will present some results that can be found in [36].
For the proof of this lemma, refer to [36].The following example shows how operators and their properties are used.
Example 1.Let us consider the following abstract fractional problem: where D δ is the Caputo fractional derivative, 0 < δ < 1, B is a closed linear operator, ς is a continuous function, and χ 0 ∈ X. Problem ( 13) is equivalent to The equation mentioned above can be expressed as an integral equation in thefollowing form: where f (κ Assuming the existence of a differentiable resolvent operator S(κ), κ ≥ 0, for Problem (13), then by using Point (2) of Lemma 3, we can write We present two fixed-point theorems that allow us to establish the uniqueness and existence results, as mentioned in references [15] and [38].
Theorem 1 (Banach's fixed-point theorem).Let Ω be a nonempty closed subset of a Banach space X; then, any contraction mapping Ψ of Ω onto itself has a unique fixed point.

Theorem 2 (Krasnoselskii fixed-point theorem).
Let Ω be a closed convex and nonempty subset of a Banach space X.Let Ψ 1 and Ψ 2 be two operators such that 1.
Ψ 2 is compact and continuous.
Then, there exists z This paper is organized as follows.
In Section 3, we examine the existence of mild solutions and establish the theorems regarding the existence and uniqueness of the mild solution to Problem (1).Section 4 presents the results concerning the existence in the specific case of A ≡ λ, λ ∈ R. In Section 5, we investigate an example of partial differential equations with the Caputo fractional derivative.

Main Results
In this section, we investigate the existence of mild solutions to Problem (1).We make the following assumptions throughout this study: Hypothesis 2 (H2).The resolvent operator R(t), t ≥ 0, is differentiable, and there exists a function , for all t > 0.
Then, Problem ( 16) is equivalent to By applying the Riemann-Liouville fractional integral of order β toEquation (17), we obtain By once again applying the Riemann-Liouville fractional integral of orderα to Equation (18), we obtain the following result: where c 0 , c 1 are constants.
Using the first boundary condition χ(0) = g(χ), we obtain c 1 = g(χ).The second boundary condition, Consequently, Problem ( 16) is equivalent to the following: Let us denote the solution of Problem ( 16) as follows: Remark 1. Equation ( 19) can be alternatively represented as an integral equation in the following scientific form: where Using Lemma 4, we can establish the equivalence of Problem (1) to the following integral equation: where κ ∈ [0, T].
In the subsequent definition, we present a conceptually similar definition for the mild solution of Problem (1).
Then, we have This leads to the conclusion that Λχ ∈ C([0, T], X), and as a result, Λ is well defined.Moreover, for χ, ψ ∈ C([0, T], X), we have and by using hypotheses (H2), (H3) and (H4), we obtain Since we have we finally obtain Due to assumption (H5), there exists ω * ≥ 0 such that for all ω ≥ ω * , the operator Λ is a contraction.By applying Banach's fixed-point theorem, we conclude that there exists a unique mild solution to Problem (1).Thus, the proof is complete.
Proof.We convert the existence of a solution to Problem (1) into a fixed-point problem.We introduce a map denoted by Λ : C([0, T], X) → C([0, T], X), which is defined according to Equation (22), stated in the proof of the previous theorem.
We decompose Λ into two parts, denoted by Λ 1 and Λ 2 , on the closed ball B r (0, E), where B r (0, E) represents the closed ball centered at 0 with the radius r in the space E = C([0, T], X), where and Obviously, due to hypothesis (H3), we have Then, for χ, ψ ∈ B r (0, E) and ω ≥ 0, we have and then, since ω ≥ 0, we obtain Using hypotheses (H3) and (H4), we can deduce that . Consequently, .

Particular Case A ≡ λ
We consider the following problem: where D α , D β are Caputo fractional derivatives, and I σ is the Riemann-Liouville fractional integral, where 0 Here, H ω depends on a parameter ω ≥ 0, where H ω : [0, T] × X 2 → X, g : C(J, X) → X are continuous functions, and X is a Banach space.Problem ( 24) is equivalent to the following integral equation: Corollary 1.Under assumptions (H2), (H3), (H4) and (H5), we can conclude the existence and uniqueness of a mild solution to Problem (24).

Application
In this section, our focus is on investigating the existence and uniqueness of a mild solution for a differential system that involves Caputo derivatives.

Conclusions
In this study, we have extended the concept of sequential fractional differential equations by introducing an operator coefficient, thus creating what we refer to as an abstract sequential fractional differential equation.We have examined the uniqueness and existence of mild solutions to such abstract sequential fractional differential equations with nonlocal boundary conditions.Our investigation utilizes the Caputo fractional derivative and Riemann-Liouville fractional integral operators, with a particular focus on the role of resolvent operators.To establish uniqueness, we apply the Banach contraction principle, while, for existence, we utilize the Krasnoseskii fixed-point theorem.We also provide an application of our results to a partial differential equation to demonstrate their applicability to practical problems.In our future research, we will concentrate on investigating the Ulam-Hyers and Ulam-Hyers-Rassias stability of similar problems using the approach of resolvent operators.