A Unified Approach to Implicit Fractional Differential Equations with Anti-Periodic Boundary Conditions

Abstract

This paper develops a unified analytical framework for implicit fractional differential equations subject to anti-periodic boundary conditions. The study considers two main cases: fractional derivatives of order $\alpha \in (0,1)$ and $\alpha \in (1,2)$, both defined with respect to a general kernel function. The existence and uniqueness of solutions are established using Banach’s and Schaefer’s fixed-point theorems under suitable Lipschitz conditions. Furthermore, Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated for each problem. Examples are provided to illustrate the applicability of the main results.

1. Introduction

Fractional differential equations (FDEs) are a generalization of classical integer-order calculus, obtained by considering derivatives and integrals of a non-integer order [1]. This approach leads to a more comprehensive theory, capable of modeling complex natural phenomena that exhibit a certain degree of unpredictability often observed in real-world systems. Over the past decades, fractional calculus has proven effective in describing situations where classical derivatives fall short.

In the field of FDEs, recent studies have explored a variety of fractional operators and related topics. These include investigations into the existence and uniqueness of solutions, stability analyses (e.g., Ulam–Hyers, Mittag–Leffler stability), qualitative theories (such as the positivity of solutions), and the development of numerical methods and approximation techniques. Some of the works in this area include [2], which presents results on the existence and uniqueness of solutions for nonlinear FDEs, and [3], which establishes existence and uniqueness for fractional boundary value problems involving the p-Laplacian operator. Additionally, Ref. [4] discusses existence and uniqueness results for uncertain FDEs with various fractional operators, while [5] addresses coupled systems of sequential FDEs. Stability issues are explored in [6] for the Riemann–Liouville derivative, particularly in the context of linear systems, perturbed systems, and time-delayed systems. Furthermore, Ref. [7] introduces new stability concepts for FDEs, including Hyers–Ulam–Rassias and Hyers–Ulam stability. The positivity of solutions for various FDEs, including boundary value problems with different differential operators or initial conditions, is studied in [8,9,10]. Finally, numerical methods for approximating solutions are extensively discussed in the literature, with a comprehensive overview of various approaches presented in [11,12]. For some references on the stability of ordinary or fractional differential equations, we refer to the works [13,14,15,16,17,18,19].

Notable applications include problems in physics, such as viscoelasticity [20,21] and anomalous diffusion [22], in engineering (e.g., PID controllers [23] and the modeling of electrical and mechanical systems [24,25]), biology (including epidemiological models [26,27]), medicine [28], contact problems [29,30], and also in economics and finance, particularly in the modeling of financial markets [31,32]. By retaining memory (unlike integer-order derivatives, which are a local concept), fractional derivatives can capture long-lasting effects, such as in biological processes or viscoelastic materials, depending on the system’s behavior throughout its entire history. Additionally, because of the inclusion of more parameters in their formulation, such as the order of the derivative or different types of fractional operators, they provide a greater degree of freedom and flexibility in describing the behavior of complex dynamic systems.

In this work, we study implicit FDEs subject to anti-periodic boundary conditions. The literature contains numerous contributions in this topic. For example, in [33], the authors considered the case of the Riemann—Liouville derivative with generalized boundary conditions, while in [34], the study focused on the Hadamard fractional derivative. In [35], the analysis was carried out using nonlocal Erdélyi–Kober q-fractional integral conditions. Research involving the Caputo fractional derivative is presented in [36,37], where stability conditions of the system are also investigated. In [38], the authors examined nonlinear fractional relaxation impulsive implicit delay differential equations. In [39], problems involving the Riemann–Liouville fractional derivative with respect to an increasing function were studied. In [40], the Hilfer fractional derivative was considered in the presence of impulses acting on the system. We also refer to [41], which addresses a problem with two-point anti-periodic boundary conditions. In [42], existence and uniqueness results with periodic and anti-periodic boundary conditions for a nonlinear multi-order fractional differential equation were considered. In [43], a coupled system of implicit fractional boundary value problem was studied. Anti-periodic boundary conditions are useful in contexts such as the Bogoliubov—de Gennes equations, optical fibers where they model modes with phase inversion across repeating units, toroidal domains to represent fields with half-integer winding numbers, vibrating beams where they describe out-of-phase oscillations of coupled segments, and electrorheological fluids [44]. For results dealing with discrete nabla and delta problems, we mention [45,46]. For a numerical approach with this type of derivative, we suggest [47,48].

The aim of this work is to generalize some of the aforementioned studies by introducing a more generalized form of the fractional derivative. The derivative depends on a kernel $\mathcal{K}$, and different choices of this kernel lead to important special cases. For instance, if $\mathcal{K}\left(t\right)=t$, we recover the Riemann—Liouville derivative; if $\mathcal{K}\left(t\right)=ln(t+1)$, we obtain the Hadamard derivative; and if $\mathcal{K}\left(t\right)=t^{\sigma }$, we obtain the Erdélyi—Kober derivative. As a result, several previously studied cases appear as particular instances of our framework. We consider two types of implicit fractional differential equations: the first involves fractional derivatives of an order between 0 and 1, subject to anti-periodic boundary conditions; the second involves derivatives of an order in the interval $(1,2)$, with two-point anti-periodic boundary conditions.

The structure of this paper is as follows. Section 2 introduces the fundamental concepts and definitions needed for our analysis. In Section 3, we formulate the two main problems under consideration. The existence and uniqueness of solutions for the first problem are established in Section 3.1, while Section 3.2 deals with the second problem. Section 4 is devoted to the study of Ulam–Hyers stability, and in Section 5, we present illustrative examples to support our theoretical findings. Finally, we conclude the paper with a summary of results and potential directions for future work.

2. Preliminaries

In this section, we provide a brief overview of the concepts that will be needed throughout this work. For a more detailed study, we refer the reader to [49,50]. We present the concepts of the Riemann—Liouville fractional integral and the Caputo fractional derivative, both depending on a kernel. This approach allows for a broader and more general study, extending previous results and opening new directions for future research. Indeed, by choosing specific forms of the kernel, we recover, for instance, the classical Caputo derivative, as well as the Caputo—Hadamard and Caputo—Katugampola derivatives, among others. For the sake of notation, we consider $\alpha \in {\mathbb{R}}^{+}$ and the interval $J:=[a,b]$. The kernel function is denoted by $\mathcal{K}:J\to \mathbb{R}$, where $\mathcal{K}$ is a smooth function with a positive derivative.

Definition 1

([50]). Let $x:J\to \mathbb{R}$ be an integrable function. The Riemann–Liouville fractional integral of x of order α with respect to the kernel function $\mathcal{K}$ is defined by

$${\mathbb{I}}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right):={\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}x\left(\tau \right)d\tau .$$

The corresponding Riemann–Liouville fractional derivative of order α with respect to $\mathcal{K}$ is given by

$${\mathbb{D}}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right):={\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{\mathbb{I}}_{a+}^{n-\alpha ,\mathcal{K}}x\left(t\right),$$

where $n=\left[\alpha \right]+1$ and the kernel function satisfies $\mathcal{K}\in C^n(J,\mathbb{R})$.

Definition 2

([50]). Let $\mathcal{K}\in C^n(J,\mathbb{R})$ and $x\in C^{n-1}(J,\mathbb{R})$, the Caputo fractional derivative of x, with respect to the kernel $\mathcal{K}$ and order α, is defined as

$${}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right):={\mathbb{D}}_{a+}^{\alpha ,\mathcal{K}}\left[x\left(t\right)-\sum _{i=0}^{n-1}{\displaystyle \frac{{(\mathcal{K}\left(t\right)-\mathcal{K}\left(a\right))}^i}{i!}}{D}_{\mathcal{K}}^{i}x\left(a\right)\right]$$

where

$$n=\left[\alpha \right]+1for\alpha \notin \mathbb{N},n=\alpha for\alpha \in \mathbb{N},$$

and

$$D_{\mathcal{K}}^{i}x\left(t\right):={\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{i}x\left(t\right).$$

It is worth mentioning that when $\alpha$ is an integer, say $\alpha =m$, then

$${}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)={\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{m}x\left(t\right),t\in J.$$

Thus, when $\mathcal{K}\left(t\right)=t$, we obtain the usual derivative of order m of the function x. For a reference containing some exact expressions of Caputo derivatives, we refer to [49].

Define the set

$$A{C}_{\mathcal{K}}^n(J,\mathbb{R}):=\left\{x:J\to \mathbb{R}|{\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n-1}x\in AC(J,\mathbb{R})\right\},$$

where $AC(J,\mathbb{R})$ denotes the set of absolutely continuous functions on J. Throughout this work, we consider two norms, depending on the regularity of the function x. If $x\in AC(J,\mathbb{R})$, we use the supremum norm:

$$\parallel x\parallel =\underset{t\in J}{sup}\left|x\left(t\right)\right|.$$

(1)

If $x\in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$, we employ

$$\parallel x\parallel =\underset{t\in J}{sup}\left|x\left(t\right)\right|+\underset{t\in J}{sup}\left|x^{\prime }\left(t\right)\right|.$$

(2)

When dealing with fractional differential equations, an important property is that the concepts of integration and differentiation are inverses of each other. This property holds for the operators mentioned above, and the result is expressed as follows:

Theorem 1

(cf. [50]). Let $\alpha >0$ be the fractional order and $t\in J$. For any $x\in AC(J,\mathbb{R})$, the following holds:

$${}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{\mathbb{I}}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)=x\left(t\right).$$

Moreover, if $x\in A{C}_{\mathcal{K}}^n(J,\mathbb{R})$, then

$${\mathbb{I}}_{a+}^{\alpha ,\mathcal{K}}{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)=x\left(t\right)-\sum _{i=0}^{n-1}{\displaystyle \frac{{(\mathcal{K}\left(t\right)-\mathcal{K}\left(a\right))}^i}{i!}}{D}_{\mathcal{K}}^{i}x\left(a\right).$$

To end, we recall two important theorems on the existence of fixed points.

Theorem 2

(Banach fixed-point theorem). Suppose $(X,d)$ is a complete metric space, and let $F:X\to X$ be a contraction mapping with contraction constant $0\le L<1$. Then, F admits a unique point $x\in X$. Moreover, for any $y\in X$, the following holds:

$$d(x,y)\le {\displaystyle \frac{d(y,F[y\left]\right)}{1-L}}.$$

Theorem 3

(Schaefer fixed-point theorem). Let $F:B\to B$ be a completely continuous operator on a Banach space B. Assume that the set

$$\{u\in B:u=\lambda F[u]forsome\lambda \in [0,1\left]\right\}$$

is bounded. Then, F possesses at least one fixed point in B.

3. Formulation of the Implicit Fractional Differential Problem

This work addresses two distinct problems, each subject to anti-periodic boundary conditions. In the first problem, the fractional order is between 0 and 1, while in the second, it lies between 1 and 2. In both cases, the boundary conditions play a crucial role in the formulation of the problems. Let $f:J\times {\mathbb{R}}^2\to \mathbb{R}$ be a continuous function.

(FDE1): Given $\alpha \in (0,1)$, consider the following system:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)=f\left(t,x\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)\right),t\in J,\hfill \\ x\left(a\right)+x\left(b\right)=0,\hfill \end{array}\right.$$

(3)

where $x\in AC(J,\mathbb{R})$, equipped with the norm (1).

(FDE2): Given $\alpha \in (1,2)$, consider the following system:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)=f\left(t,x\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)\right),t\in J,\hfill \\ x\left(a\right)+x\left(b\right)=0,\hfill \\ x^{\prime }\left(a\right)+x^{\prime }\left(b\right)=0,\hfill \end{array}\right.$$

(4)

where $x\in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$, equipped with the norm (2).

3.1. Existence and Uniqueness of the Solution for the Problem (FDE1)

First, we rewrite system (3) in its integral form, and to do so, we prove the following lemma:

Lemma 1.

A function is a solution to system (3) if and only if it satisfies the following integral equation:

$$\begin{array}{c}x\left(t\right)=-{\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \hfill \\ +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau .\hfill \end{array}$$

Proof. 

(⇒): Applying the fractional integral to both sides of the FDE in (3) and utilizing Theorem 1, we obtain

$$x\left(t\right)=x\left(a\right)+{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau .$$

At $t=b$, and applying the boundary condition, we get

$$x\left(a\right)=-{\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau ,$$

which gives the system in its integral form.

(⇐): For the converse, applying the Caputo fractional derivative to both sides of the integral equation, and again using Theorem 1, we recover the FDE as presented in system (3). Additionally, it is evident that

$$x\left(a\right)+x\left(b\right)=0,$$

since

$${\mathbb{I}}_{a+}^{\alpha ,\mathcal{K}}f\left(a,x\left(a\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(a\right)\right)=0$$

(see [51] Theorem 5). □

Motivated by Lemma 1, we define the following operator. Given a curve x in a suitable space (to be specified later), the operator F is defined as follows:

$$\begin{array}{c}F\left[x\right]\left(t\right)=-{\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \\ +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau .\end{array}$$

(5)

The next result is an immediate consequence of Banach’s fixed-point theorem.

Theorem 4.

Suppose that the function f in (3) is Lipschitz with respect to the second and third variables, with a Lipschitz constant $L\in (0,1)$:

$$|f(t,x_1,v_1)-f(t,x_2,v_2)|\le L\left(|x_1-x_2|+|v_1-v_2|\right),\forall t\in J,\forall x_1,x_2,v_1,v_2\in \mathbb{R}.$$

Additionally, the constant L satisfies the following condition:

$${\displaystyle \frac{3L{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L)\Gamma (\alpha +1)}}<1.$$

Then, system (3) admits a unique solution.

Proof. 

Define the constants $C_1$ and $C_2$ as

$$C_1:={\displaystyle \frac{3L{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L)\Gamma (\alpha +1)}}\mathrm{and}{C}_2:={\displaystyle \frac{3M{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L)\Gamma (\alpha +1)}},$$

where

$$M:=\underset{t\in J}{max}\left|f(t,0,0)\right|.$$

Let $R>0$ be such that

$$R>{\displaystyle \frac{C_2}{1-C_1}},$$

and consider the set

$$B_R:=\{x\in AC(J,\mathbb{R}):\parallel x\parallel \le R\}.$$

We now prove that $F\left(B_R\right)\subseteq B_R$, where F is defined in (5). Let $x\in B_R$. For all $\tau \in J$, observe that

$$\begin{array}{c}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|\le \left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)-f\left(\tau ,0,0\right)\right|+\left|f\left(\tau ,0,0\right)\right|\hfill \\ \le L\left|x\left(\tau \right)\right|+L\left|{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right|+M\le LR+L\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|+M.\hfill \end{array}$$

Solving for $\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|$, we obtain

$$\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|\le {\displaystyle \frac{LR+M}{1-L}}.$$

Therefore, for any $t\in J$,

$$\begin{array}{cc}\hfill \left|F\right[x\left]\right(t\left)\right|& \le {\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{3(LR+M){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L)\Gamma (\alpha +1)}}=C_{1}R+C_2\le R.\hfill \end{array}$$

Thus, the operator $F:B_R\to B_R$ is well defined. To complete the proof, we now show that F is a contraction mapping. Let $x_1,x_2\in B_R$ and $t\in J$. Then,

$$\begin{array}{c}|F\left[x_1\right]\left(t\right)-F\left[x_2\right]\left(t\right)|\hfill \\ \le {\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|f\left(\tau ,x_1\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_1\left(\tau \right)\right)-f\left(\tau ,x_2\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_2\left(\tau \right)\right)\right|d\tau \\ +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|f\left(\tau ,x_1\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_1\left(\tau \right)\right)-f\left(\tau ,x_2\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_2\left(\tau \right)\right)\right|d\tau .\hfill \end{array}$$

Since

$$\begin{array}{c}\left|f\left(\tau ,x_1\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_1\left(\tau \right)\right)-f\left(\tau ,x_2\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_2\left(\tau \right)\right)\right|\hfill \\ \le L|x_1\left(\tau \right)-x_2\left(\tau \right)|+L\left|{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_1\left(\tau \right)-{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_2\left(\tau \right)\right|\\ \hfill =L|x_1\left(\tau \right)-x_2\left(\tau \right)|+L\left|f\left(\tau ,x_1\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_1\left(\tau \right)\right)-f\left(\tau ,x_2\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_2\left(\tau \right)\right)\right|,\end{array}$$

it follows that

$$\left|f\left(\tau ,x_1\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_1\left(\tau \right)\right)-f\left(\tau ,x_2\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_2\left(\tau \right)\right)\right|\le {\displaystyle \frac{L}{1-L}}\parallel x_1-x_2\parallel .$$

Thus,

$$\parallel F\left[x_1\right]-F\left[x_2\right]\parallel \le {\displaystyle \frac{3L{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L)\Gamma (\alpha +1)}}\parallel x_1-x_2\parallel .$$

The desired conclusion follows from Banach’s fixed-point theorem. □

Our next result is a consequence of Schaefer’s fixed-point theorem.

Theorem 5.

Suppose that there exist two constants $L_0\in {\mathbb{R}}^{+}$ and $L_1\in (0,1)$ such that

$$\left|f(t,x,v)\right|\le L_0+L_1\left(\left|x\right|+\left|v\right|\right),\forall t\in J,\forall x,v\in \mathbb{R}.$$

If

$${\displaystyle \frac{3L_1{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L_1)\Gamma (\alpha +1)}}<1,$$

then system (3) admits at least one solution.

Proof. 

Let F be the functional as defined in (5). The proof is divided in several steps.

First, we prove that F maps bounded sets into bounded sets. Let $R>0$ and define

$$B_R=\left\{x\in AC(J,\mathbb{R}):\parallel x\parallel \le R\right\}.$$

We show that for all $x\in B_R$, there exists a constant $\overline{R}>0$ such that $\parallel F\left[x\right]\parallel \le \overline{R}$. Given that

$$\left|f\left(t,x\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)\right)\right|\le L_0+L_1\left(\parallel x\parallel +∥{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x∥\right),t\in J,$$

it follows that

$$∥{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x∥\le {\displaystyle \frac{L_0+L_1\parallel x\parallel }{1-L_1}},$$

and thus,

$$∥f\left(·,x,{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\right)∥\le {\displaystyle \frac{L_0+L_1\parallel x\parallel }{1-L_1}}.$$

Therefore,

$$∥F\left[x\right]∥\le {\displaystyle \frac{3(L_0+L_{1}R)}{2(1-L_1)\Gamma (\alpha +1)}}{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha },$$

And it suffices to take

$$\overline{R}={\displaystyle \frac{3(L_0+L_{1}R)}{2(1-L_1)\Gamma (\alpha +1)}}{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }.$$

Next, we prove that F is continuous. For that, consider a sequence ${\left(x_n\right)}_n$ in $AC(J,\mathbb{R})$ convergent to $x\in AC(J,\mathbb{R})$. Then,

$$\begin{array}{cc}\hfill \parallel F\left[x_n\right]-F\left[x\right]\parallel & \le {\displaystyle \frac{1}{2}}{\mathbb{I}}_{a+}^{\alpha ,\mathcal{K}}\left|f\left(b,x_n\left(b\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_n\left(b\right)\right)-f\left(b,x\left(b\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(b\right)\right)\right|\hfill \\ \hfill & +\underset{t\in J}{sup}{\mathbb{I}}_{a+}^{\alpha ,\mathcal{K}}\left|f\left(t,x_n\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_n\left(t\right)\right)-f\left(t,x\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{3{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2\Gamma (\alpha +1)}}∥f\left(·,x_n,{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_n\right)-f\left(·,x,{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\right)∥.\hfill \end{array}$$

By the continuity of f, we conclude that

$$f\left(·,x_n,{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_n\right)\underset{n\to \infty }{\to }f\left(·,x,{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\right),$$

which implies that $F\left[x_n\right]\to F\left[x\right]$ as n goes to infinity, proving the desired.

Next, we show that F maps bounded sets into equicontinuous sets. Let $x\in B_R$ and $t_1,t_2\in J$ with $t_1<t_2$. Then,

$$\begin{array}{cc}\hfill |F\left[x\right]\left(t_2\right)& -F\left[x\right]\left(t_1\right)|\hfill \\ \hfill & \le {\int }_a^{t_1}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right)[{(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}-{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}]}{\Gamma \left(\alpha \right)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{L_0+L_{1}R}{(1-L_1)\Gamma (\alpha +1)}}\left[2{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(t_1\right))}^{\alpha }+{(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(a\right))}^{\alpha }-{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(a\right))}^{\alpha }\right],\hfill \end{array}$$

which tends to zero as $t_2\to t_1$. By the Arzelà—Ascoli theorem, we conclude that F is completely continuous.

To conclude the proof, define the set

$$\Lambda =\{x\in AC(J,\mathbb{R}):x=\lambda F[x],\mathrm{for}\mathrm{some}\lambda \in (0,1\left)\right\}.$$

We show that $\Lambda$ is a bounded set. Let $x\in \Lambda$ and $\lambda \in (0,1)$ such that $x=\lambda F\left[x\right]$. Since

$$\parallel F\left[x\right]\parallel \le {\displaystyle \frac{3(L_0+L_1\parallel x\parallel )}{2(1-L_1)\Gamma (\alpha +1)}}{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha },$$

it follows that

$$\parallel x\parallel \le \parallel F\left[x\right]\parallel \le {\displaystyle \frac{3(L_0+L_1\parallel x\parallel )}{2(1-L_1)\Gamma (\alpha +1)}}{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }.$$

Rearranging this inequality gives

$$\parallel x\parallel \le {\displaystyle \frac{\frac{3L_0{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L_1)\Gamma (\alpha +1)}}{1-\frac{3L_1{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L_1)\Gamma (\alpha +1)}}},$$

which shows that $\Lambda$ is bounded. As a consequence of Schaefer’s fixed-point theorem, we conclude that F admits a fixed point. □

3.2. Existence and Uniqueness of the Solution for the Problem (FDE2)

This section follows the same approach as the previous one; therefore, some technical details are omitted for brevity. We begin by rewriting problem (4) in the form of a Volterra integral equation.

Lemma 2.

A function is a solution to system (4) if and only if it satisfies the integral equation

$$\begin{array}{c}x\left(t\right)=-{\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \hfill \\ +{\displaystyle \frac{\left(\mathcal{K}\left(a\right)+\mathcal{K}\left(b\right)-2\mathcal{K}\left(t\right)\right){\mathcal{K}}^{\prime }\left(b\right)}{2({\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right))}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \\ \hfill +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau .\end{array}$$

Proof. 

(⇒): Applying the fractional integral operator to both sides of the equation, we obtain

$$x\left(t\right)=x\left(a\right)+(\mathcal{K}\left(t\right)-\mathcal{K}\left(a\right)){\displaystyle \frac{x^{\prime }\left(a\right)}{{\mathcal{K}}^{\prime }\left(a\right)}}+{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau .$$

Using the boundary condition $x\left(a\right)+x\left(b\right)=0$, we obtain

$$x\left(a\right)=-{\displaystyle \frac{1}{2}}\left[(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right)){\displaystyle \frac{x^{\prime }\left(a\right)}{{\mathcal{K}}^{\prime }\left(a\right)}}+{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \right].$$

Since

$$x^{\prime }\left(t\right)={\displaystyle \frac{{\mathcal{K}}^{\prime }\left(t\right)}{{\mathcal{K}}^{\prime }\left(a\right)}}{x}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(t\right){\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau ,$$

and applying the second boundary condition $x^{\prime }\left(a\right)+x^{\prime }\left(b\right)=0$, we obtain

$${\displaystyle \frac{x^{\prime }\left(a\right)}{{\mathcal{K}}^{\prime }\left(a\right)}}=-{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(b\right)}{{\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right)}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau .$$

After simplifying, it yields the desired Volterra integral equation.

(⇐): Applying the Caputo-type derivative to both sides of the equation and using the linearity of ${}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}$, we recover the original differential equation in system (4). In particular, we use the fact that

$${}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}(\mathcal{K}\left(t\right)-\mathcal{K}\left(a\right))=0,\mathrm{for}t\in J,$$

when $\alpha \in (1,2)$. The boundary conditions follow directly from evaluating the integral expression and its derivative at the endpoints. □

From Lemma 2, the operator F to consider in this section is the following:

$$\begin{array}{c}F\left[x\right]\left(t\right)=-{\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \hfill \\ +{\displaystyle \frac{\left(\mathcal{K}\left(a\right)+\mathcal{K}\left(b\right)-2\mathcal{K}\left(t\right)\right){\mathcal{K}}^{\prime }\left(b\right)}{2({\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right))}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \hfill \\ +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau .\hfill \end{array}$$

(6)

Observe that

$$\begin{array}{c}{F}^{\prime }\left[x\right]\left(t\right)=-{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(b\right){\mathcal{K}}^{\prime }\left(t\right)}{{\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right)}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \hfill \\ \hfill +{\mathcal{K}}^{\prime }\left(t\right){\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau .\end{array}$$

Theorem 6.

Suppose the function f is Lipschitz-continuous with respect to its second and third variables, with Lipschitz constant $L\in (0,1)$. Assume further that the constant L satisfies the inequality:

$${\displaystyle \frac{L{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]<1,$$

where

$$W=\underset{t\in J}{sup}\left|{\mathcal{K}}^{\prime }\left(t\right)\right|.$$

Then, the boundary value problem (4) admits a unique solution.

Proof. 

Let $R>0$ be such that $R>C_2/(1-C_1)$, where the constants $C_1$ and $C_2$ are defined by

$$C_1:={\displaystyle \frac{L{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]$$

and

$$C_2:={\displaystyle \frac{M{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right],$$

where

$$M:=\underset{t\in J}{max}\left|f(t,0,0)\right|.$$

We consider the functional F as given in (6), defined on the closed ball

$$B_R:=\{x\in A{C}^2(J,\mathbb{R}):\parallel x\parallel \le R\}.$$

We aim to prove that $F:B_R\to B_R$ is well defined. Let $x\in B_R$. Then, for any $\tau \in J$, using the Lipschitz condition on f, we estimate

$$\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|\le {\displaystyle \frac{LR+M}{1-L}}.$$

Then, proceeding as before, we obtain the following estimates:

$$\begin{array}{cc}\hfill \left|F\right[x\left]\right(t\left)\right|& \le {\displaystyle \frac{(LR+M){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L)\Gamma (\alpha +1)}}\left[3+{\displaystyle \frac{\alpha {\mathcal{K}}^{\prime }\left(b\right)}{{\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right)}}\right]\hfill \\ \hfill & \le {\displaystyle \frac{(LR+M){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L)\Gamma (\alpha +1)}}\left[3+\alpha \right]\hfill \end{array}$$

and

$$|F^{\prime }\left[x\right]\left(t\right)|\le {\displaystyle \frac{2(LR+M)W{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L)\Gamma \left(\alpha \right)}}.$$

Combining the two bounds, we get

$$\begin{array}{cc}\hfill \parallel F\left[x\right]\parallel & =\underset{t\in J}{sup}\left|F\left[x\right]\left(t\right)\right|+\underset{t\in J}{sup}\left|F^{\prime }\left[x\right]\left(t\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{(LR+M){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]\hfill \\ \hfill & =C_{1}R+C_2\le R.\hfill \end{array}$$

This proves that $F\left(B_R\right)\subseteq B_R$. We now prove that F is a contraction mapping. Let $x_1,x_2\in B_R$ and $\tau ,t\in J$. Using the Lipschitz condition on f, we estimate the bound:

$$\left|f\left(\tau ,x_1\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_1\left(\tau \right)\right)-f\left(\tau ,x_2\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_2\left(\tau \right)\right)\right|\le {\displaystyle \frac{L}{1-L}}\parallel x_1-x_2\parallel .$$

Using this bound, we obtain the following estimates for the differences of the functional F and its derivative:

$$|F\left[x_1\right]\left(t\right)-F\left[x_2\right]\left(t\right)|\le {\displaystyle \frac{L{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-L)\Gamma (\alpha +1)}}(3+\alpha )\parallel x_1-x_2\parallel ,$$

and

$$|F^{\prime }\left[x_1\right]\left(t\right)-F^{\prime }\left[x_2\right]\left(t\right)|\le {\displaystyle \frac{2LW{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L)\Gamma \left(\alpha \right)}}\parallel x_1-x_2\parallel .$$

Thus, we can bound the following norm:

$$\parallel F\left[x_1\right]-F\left[x_2\right]\parallel \le {\displaystyle \frac{L{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]\parallel x_1-x_2\parallel .$$

This shows that F is a contraction mapping with a contraction constant less than 1. Consequently, the Banach fixed-point theorem is applicable, which guarantees the existence of a unique solution to problem (4). □

Theorem 7.

Suppose there exist constants $L_0\in {\mathbb{R}}^{+}$ and $L_1\in (0,1)$ such that

$$\left|f(t,x,v)\right|\le L_0+L_1\left(\left|x\right|+\left|v\right|\right),\forall t\in J,\forall x,v\in \mathbb{R}.$$

If

$${\displaystyle \frac{L_1{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L_1)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]<1,$$

where

$$W=\underset{t\in J}{sup}\left|{\mathcal{K}}^{\prime }\left(t\right)\right|,$$

then system (4) admits at least one solution.

Proof. 

Let F be the operator defined in (6). We first show that F maps bounded sets into bounded sets. Given $R>0$, define

$$B_R=\left\{x\in A{C}^2(J,\mathbb{R}):\parallel x\parallel \le R\right\}.$$

Using the bound

$$\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|\le {\displaystyle \frac{L_0+L_{1}R}{1-L_1}},\tau \in J,$$

we obtain the following estimates: for $t\in J$,

$$\left|F\left[x\right]\left(t\right)\right|\le {\displaystyle \frac{(L_0+L_{1}R){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2\Gamma (\alpha +1)(1-L_1)}}(\alpha +3),$$

and

$$|F^{\prime }\left[x\right]\left(t\right)|\le {\displaystyle \frac{2W(L_0+L_{1}R){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)(1-L_1)}}.$$

Hence, we deduce the bound

$$∥F\left[x\right]∥\le {\displaystyle \frac{(L_0+L_{1}R){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{\Gamma (\alpha +1)(1-L_1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right].$$

Define

$$\overline{R}={\displaystyle \frac{(L_0+L_{1}R){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{\Gamma (\alpha +1)(1-L_1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right],$$

so that $∥F\left[x\right]∥\le \overline{R}$. This shows that F maps bounded sets into bounded sets.

Next, we prove that F is continuous. Let ${\left(x_n\right)}_n\subseteq A{C}^2(J,\mathbb{R})$ be a sequence such that $x_n\to x$ in $A{C}^2(J,\mathbb{R})$. Then,

$$\begin{array}{cc}\hfill \parallel F\left[x_n\right]-F\left[x\right]\parallel & \le {\displaystyle \frac{3}{2}}{\mathbb{I}}_{a+}^{\alpha ,\mathcal{K}}\left|f\left(b,x_n\left(b\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_n\left(b\right)\right)-f\left(b,x\left(b\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(b\right)\right)\right|\hfill \\ \hfill & +\left[{\displaystyle \frac{\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right)}{2}}+2W\right]{\mathbb{I}}_{a+}^{\alpha -1,\mathcal{K}}\left|f\left(t,x_n\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}{x}_n\left(t\right)\right)-f\left(t,x\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(t\right)\right)\right|.\hfill \end{array}$$

The right-hand side converges to zero as n goes to infinity, proving the continuity of F.

Now, we analyze the equicontinuity of the set $\{F\left[x\right]:x\in B_R\}$. Let $x\in B_R$ and take $t_1,t_2\in J$ with $t_1<t_2$. Then,

$$\begin{array}{cc}\hfill |F\left[x\right]\left(t_2\right)& -F\left[x\right]\left(t_1\right)|\hfill \\ \hfill & \le {\displaystyle \frac{{\mathcal{K}}^{\prime }\left(b\right)}{{\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right)}}(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(t_1\right)){\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +{\int }_a^{t_1}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right)[{(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}-{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}]}{\Gamma \left(\alpha \right)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{L_0+L_{1}R}{(1-L_1)}}\left[{\displaystyle \frac{2{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(t_1\right))}^{\alpha }+{(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(a\right))}^{\alpha }-{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(a\right))}^{\alpha }}{\Gamma (\alpha +1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(t_1\right)){(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right],\hfill \end{array}$$

which clearly tends to zero as $t_2\to t_1$.

Now, we turn to the derivative $F^{\prime }\left[x\right]$. Consider

$$\begin{array}{cc}\hfill & |F^{\prime }\left[x\right]\left(t_2\right)-F^{\prime }\left[x\right]\left(t_1\right)|\hfill \\ \hfill & \le |{\mathcal{K}}^{\prime }\left(t_2\right)-{\mathcal{K}}^{\prime }\left(t_1\right)|{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(b\right)}{{\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right)}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +\left|{\mathcal{K}}^{\prime }\left(t_2\right){\int }_a^{t_2}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \right.\hfill \\ \hfill & \left.-{\mathcal{K}}^{\prime }\left(t_1\right){\int }_a^{t_1}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \right|.\hfill \end{array}$$

We estimate each term separately. For the first,

$$\begin{array}{c}|{\mathcal{K}}^{\prime }\left(t_2\right)-{\mathcal{K}}^{\prime }\left(t_1\right)|{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(b\right)}{{\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right)}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill \le |{\mathcal{K}}^{\prime }\left(t_2\right)-{\mathcal{K}}^{\prime }\left(t_1\right)|{\displaystyle \frac{(L_0+L_{1}R)({(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)(1-L_1)}}\end{array}$$

which tends to zero as $t_2\to t_1$. For the second term, we write

$$\begin{array}{cc}\hfill & \left|{\mathcal{K}}^{\prime }\left(t_2\right){\int }_a^{t_2}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \right.\hfill \\ \hfill & \left.-{\mathcal{K}}^{\prime }\left(t_1\right){\int }_a^{t_1}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)d\tau \right|\hfill \\ \hfill & \le {\mathcal{K}}^{\prime }\left(t_2\right){\int }_a^{t_1}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right)[{(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}-{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}]}{\Gamma (\alpha -1)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +|{\mathcal{K}}^{\prime }\left(t_2\right)-{\mathcal{K}}^{\prime }\left(t_1\right)|{\int }_a^{t_1}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +{\mathcal{K}}^{\prime }\left(t_2\right){\int }_{t_1}^{t_2}{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}\left|f\left(\tau ,x\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}x\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{L_0+L_{1}R}{\Gamma \left(\alpha \right)(1-L_1)}}\left[{\mathcal{K}}^{\prime }\left(t_2\right)[{(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(a\right))}^{\alpha -1}-{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(a\right))}^{\alpha -1}+2{(\mathcal{K}\left(t_2\right)-\mathcal{K}\left(t_1\right))}^{\alpha -1}]\right.\hfill \\ \hfill & \left.+|{\mathcal{K}}^{\prime }\left(t_2\right)-{\mathcal{K}}^{\prime }\left(t_1\right)|{(\mathcal{K}\left(t_1\right)-\mathcal{K}\left(a\right))}^{\alpha -1}\right],\hfill \end{array}$$

which also tends to zero as $t_2\to t_1$. By the Arzelà—Ascoli theorem, we conclude that F maps bounded sets into relatively compact subsets of $A{C}^2(J,\mathbb{R})$.

To complete the proof, consider the set

$$\Lambda =\{x\in A{C}^2(J,\mathbb{R}):x=\lambda F\left[x\right],\mathrm{for}\mathrm{some}\lambda \in (0,1)\}.$$

Let $x\in \Lambda$, so there exists $\lambda \in (0,1)$ such that $x=\lambda F\left[x\right]$. Then,

$$\parallel x\parallel \le \parallel F\left[x\right]\parallel \le {\displaystyle \frac{(L_0+L_1{\parallel x\parallel )(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L_1)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right].$$

Rearranging terms, we obtain the bound

$$\parallel x\parallel \le {\displaystyle \frac{\frac{L_0{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L_1)\Gamma (\alpha +1)}\left[2\alpha W+\frac{\alpha +3}{2}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]}{1-\frac{L_1{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-L_1)\Gamma (\alpha +1)}\left[2\alpha W+\frac{\alpha +3}{2}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]}},$$

which shows that $\Lambda$ is bounded. Therefore, by Schaefer’s fixed-point theorem, we conclude that F admits a fixed point. □

We also refer to several related works, for example, [41], which studied similar problems using the Caputo-Katugampola derivative, and [52], which considered the Caputo derivative with time delays. Anti-periodic boundary conditions were analyzed in [39], while [53] addressed dual anti-periodic boundary conditions. The Hilfer derivative was the focus of the study in [54].

4. Ulam–Hyers Stability Analysis

Definition 3.

Problem (3) is said to be Ulam–Hyers-stable if there exists a constant $C\in {\mathbb{R}}^{+}$ such that, for every $ϵ>0$, and for any function $y\in AC(J,\mathbb{R})$ satisfying

$$\left\{\begin{array}{c}\left|{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)-f\left(t,y\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)\right)\right|\le ϵ,t\in J,\alpha \in (0,1),\hfill \\ y\left(a\right)+y\left(b\right)=0,\hfill \end{array}\right.$$

(7)

there exists a function $x\in AC(J,\mathbb{R})$ that is a solution of (3) and satisfies

$$\parallel x-y\parallel \le Cϵ.$$

Definition 4.

Problem (4) is said to be Ulam–Hyers-stable if there exists a constant $C\in {\mathbb{R}}^{+}$ such that, for every $ϵ>0$, and for any function $y\in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$ satisfying

$$\left\{\begin{array}{c}\left|{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)-f\left(t,y\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)\right)\right|\le ϵ,t\in J,\alpha \in (1,2),\hfill \\ y\left(a\right)+y\left(b\right)=0,\hfill \\ y^{\prime }\left(a\right)+y^{\prime }\left(b\right)=0,\hfill \end{array}\right.$$

(8)

there exists a function $x\in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$ that is a solution of (4) and satisfies

$$\parallel x-y\parallel \le Cϵ.$$

Definition 5.

Problem (3) is said to be generalized Ulam–Hyers-stable if there exists a continuous function $C:[0,+\infty )\to [0,+\infty )$ such that $C\left(0\right)=0$ and, for every $ϵ>0$ and for any function $y\in AC(J,\mathbb{R})$ satisfying

$$\left\{\begin{array}{c}\left|{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)-f\left(t,y\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)\right)\right|\le ϵ,t\in J,\alpha \in (0,1),\hfill \\ y\left(a\right)+y\left(b\right)=0,\hfill \end{array}\right.$$

(9)

there exists a function $x\in AC(J,\mathbb{R})$ that is a solution of (3) and satisfies

$$\parallel x-y\parallel \le C\left(ϵ\right).$$

Definition 6.

Problem (4) is said to be generalized Ulam–Hyers-stable if there exists a continuous function $C:[0,+\infty )\to [0,+\infty )$ such that $C\left(0\right)=0$ and, for every $ϵ>0$ and for any function $y\in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$ satisfying

$$\left\{\begin{array}{c}\left|{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)-f\left(t,y\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)\right)\right|\le ϵ,t\in J,\alpha \in (1,2),\hfill \\ y\left(a\right)+y\left(b\right)=0,\hfill \\ y^{\prime }\left(a\right)+y^{\prime }\left(b\right)=0,\hfill \end{array}\right.$$

(10)

there exists a function $x\in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$ that is a solution of (4) and satisfies

$$\parallel x-y\parallel \le C\left(ϵ\right).$$

Next, we present two results that establish the Ulam–Hyers stability of the problems previously considered.

Theorem 8.

Suppose the the assumptions of Theorem 4 are satisfied. Then, problem (3) is Ulam–Hyers-stable.

Proof. 

Let $x\in AC(J,\mathbb{R})$ be the unique solution of system (3). Let F be as defined in the proof of Theorem 4.

Observe that a function $y\in AC(J,\mathbb{R})$ satisfies system (9) if and only if, for any $ϵ>0$, there exists a perturbation function $\eta \in AC(J,\mathbb{R})$ such that

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)=f\left(t,y\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)\right)+\eta \left(t\right),t\in J,\hfill \\ y\left(a\right)+y\left(b\right)=0.\hfill \end{array}\right.$$

Consequently, the function y satisfies the following integral equation:

$$\begin{array}{c}y\left(t\right)=-{\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left(f\left(\tau ,y\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(\tau \right)\right)+\eta \left(\tau \right)\right)d\tau \hfill \\ +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left(f\left(\tau ,y\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(\tau \right)\right)+\eta \left(\tau \right)\right)d\tau .\hfill \end{array}$$

Since F is a contraction mapping with contraction constant $C_1$, and x is its fixed point, it follows that

$$\parallel x-y\parallel \le {\displaystyle \frac{\parallel F\left[y\right]-y\parallel }{1-C_1}}.$$

Observe that, given $t\in J$,

$$\begin{array}{cc}\hfill \left|F\right[x\left]\right(t)-y(t\left)\right|& \le {\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|\eta \left(\tau \right)\right|d\tau +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|\eta \left(\tau \right)\right|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{3{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2\Gamma (\alpha +1)}}ϵ.\hfill \end{array}$$

Therefore, we obtain the estimate

$$\parallel x-y\parallel \le {\displaystyle \frac{3{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-C_1)\Gamma (\alpha +1)}}ϵ,$$

which completes the proof of the desired result. □

Theorem 9.

Suppose the the assumptions of Theorem 6 are satisfied. Then, problem (4) is Ulam–Hyers-stable.

Proof. 

Let $x\in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$ be the unique solution of system (4), and let F be the operator defined in the proof of Theorem 6.

Note that a function $y\in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$ satisfies system (10) if and only if, for every $ϵ>0$, there exists a function $\eta \in A{C}_{\mathcal{K}}^2(J,\mathbb{R})$ such that

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)=f\left(t,y\left(t\right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(t\right)\right)+\eta \left(t\right),t\in J,\hfill \\ y\left(a\right)+y\left(b\right)=0,\hfill \\ y^{\prime }\left(a\right)+y^{\prime }\left(b\right)=0.\hfill \end{array}\right.$$

As a consequence, the function y satisfies the following integral equation:

$$\begin{array}{c}y\left(t\right)=-{\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left(f\left(\tau ,y\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(\tau \right)\right)+\eta \left(\tau \right)\right)d\tau \hfill \\ +{\displaystyle \frac{\left(\mathcal{K}\left(a\right)+\mathcal{K}\left(b\right)-2\mathcal{K}\left(t\right)\right){\mathcal{K}}^{\prime }\left(b\right)}{2({\mathcal{K}}^{\prime }\left(a\right)+{\mathcal{K}}^{\prime }\left(b\right))}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}\left(f\left(\tau ,y\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(\tau \right)\right)+\eta \left(\tau \right)\right)d\tau \hfill \\ +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left(f\left(\tau ,y\left(\tau \right),{}^C\mathbb{D}_{a+}^{\alpha ,\mathcal{K}}y\left(\tau \right)\right)+\eta \left(\tau \right)\right)d\tau .\hfill \end{array}$$

Moreover, the following inequality holds:

$$\parallel x-y\parallel \le {\displaystyle \frac{\parallel F\left[y\right]-y\parallel }{1-C_1}},$$

where the constant $C_1$ is defined in the proof of Theorem 6.

To estimate

$$\parallel F\left[y\right]-y\parallel =\underset{t\in J}{sup}|F\left[y\right]\left(t\right)-y\left(t\right)|+\underset{t\in J}{sup}|F^{\prime }\left[y\right]\left(t\right)-y^{\prime }\left(t\right)|,$$

we consider, for a fixed $t\in J$,

$$\begin{array}{cc}\hfill & |F\left[x\right]\left(t\right)-y\left(t\right)|\le {\displaystyle \frac{1}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|\eta \left(\tau \right)\right|d\tau \hfill \\ \hfill & +{\displaystyle \frac{\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right)}{2}}{\int }_a^b{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(b\right)-\mathcal{K}\left(\tau \right))}^{\alpha -2}}{\Gamma (\alpha -1)}}\left|\eta \left(\tau \right)\right|d\tau +{\int }_a^t{\displaystyle \frac{{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left|\eta \left(\tau \right)\right|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2\Gamma (\alpha +1)}}(3+\alpha )ϵ.\hfill \end{array}$$

Additionally,

$$|F^{\prime }\left[x\right]\left(t\right)-y^{\prime }\left(t\right)|\le {\displaystyle \frac{2W{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}ϵ,$$

where the constant W is as defined in Theorem 6. Therefore, we conclude that

$$\parallel x-y\parallel \le {\displaystyle \frac{{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-C_1)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]ϵ,$$

which completes the proof. □

As a consequence of the two previous results, we obtain the following:

Suppose that the assumptions of Theorem 4 are satisfied. Then, problem (3) is generalized Ulam–Hyers-stable. It follows by setting

$$C\left(ϵ\right):={\displaystyle \frac{3{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha }}{2(1-C_1)\Gamma (\alpha +1)}}ϵ.$$

Also, suppose that the assumptions of Theorem 6 are satisfied. Then, problem (4) is generalized Ulam–Hyers-stable. It follows from setting

$$C\left(ϵ\right):={\displaystyle \frac{{(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))}^{\alpha -1}}{(1-C_1)\Gamma (\alpha +1)}}\left[2\alpha W+{\displaystyle \frac{\alpha +3}{2}}(\mathcal{K}\left(b\right)-\mathcal{K}\left(a\right))\right]ϵ.$$

5. Examples

In this section, we present two examples to illustrate the previously established results.

Example 1.

Consider the following boundary value problem:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{0+}^{0.7,\mathcal{K}}x\left(t\right)={\displaystyle \frac{x\left(t\right)+{}^C\mathbb{D}_{0+}^{0.7,\mathcal{K}}x\left(t\right)}{(t+1)(t+2)}},t\in [0,1],\hfill \\ x\left(0\right)+x\left(1\right)=0,\hfill \end{array}\right.$$

(11)

with fractional order $\alpha =0.7$ and kernel $\mathcal{K}\left(t\right)=\sqrt{t+1}$. Define

$$f(t,x,v):={\displaystyle \frac{x+v}{(t+1)(t+2)}},t\in [0,1],x,v\in \mathbb{R}.$$

Observe that

$$|f(t,x_1,v_1)-f(t,x_2,v_2)|\le {\displaystyle \frac{|x_1-x_2|+|v_1-v_2|}{(t+1)(t+2)}}\le {\displaystyle \frac{1}{2}}\left(\right|x_1-x_2|+|v_1-v_2\left|\right),$$

so the function f is Lipschitz-continuous with Lipschitz constant $L\in (0,1)$. Furthermore,

$${\displaystyle \frac{3L{(\mathcal{K}\left(1\right)-\mathcal{K}\left(0\right))}^{0.7}}{2(1-L)\Gamma \left(1.7\right)}}\approx 0.89<1,$$

And, hence, by Theorem 4, the problem admits a unique solution. In addition, by Theorem 8, the system is Ulam–Hyers-stable.

Example 2.

Next, we consider a system of fractional order $\alpha =1.5\in (1,2)$:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{0+}^{1.5,\mathcal{K}}x\left(t\right)={\displaystyle \frac{x\left(t\right)}{exp\left(t\right)+5}}+{\displaystyle \frac{{}^C\mathbb{D}_{0+}^{1.5,\mathcal{K}}x\left(t\right)}{t^2+10}}+5t,t\in [0,1],\hfill \\ x\left(0\right)+x\left(1\right)=0,\hfill \\ x^{\prime }\left(0\right)+x^{\prime }\left(1\right)=0,\hfill \end{array}\right.$$

(12)

where the kernel is given by $\mathcal{K}\left(t\right)=arctan\left(t\right)$. Define the function

$$f(t,x,v):={\displaystyle \frac{x}{exp\left(t\right)+5}}+{\displaystyle \frac{v}{t^2+10}}+5t,t\in [0,1],x,v\in \mathbb{R}.$$

It is straightforward to verify that

$$|f(t,x_1,v_1)-f(t,x_2,v_2)|\le {\displaystyle \frac{1}{6}}\left(\right|x_1-x_2|+|v_1-v_2\left|\right),$$

which shows that f is Lipschitz-continuous with Lipschitz constant $L=1/6\in (0,1)$. We also compute

$$W=\underset{t\in J}{sup}\left|{\mathcal{K}}^{\prime }\left(t\right)\right|=1.$$

Now, observe that the constant L satisfies the inequality:

$${\displaystyle \frac{L{(\mathcal{K}\left(1\right)-\mathcal{K}\left(0\right))}^{0.5}}{(1-L)\Gamma \left(2.5\right)}}\left[3+{\displaystyle \frac{4.5}{2}}(\mathcal{K}\left(1\right)-\mathcal{K}\left(0\right))\right]\approx 0.63<1.$$

Hence, by Theorem 6, the system admits a unique solution. Moreover, by Theorem 9, the system is Ulam–Hyers-stable.

Remark 1.

As α approaches 1 or 2, the Caputo derivative converges to the classical first- and second-order derivatives, respectively. Consequently, memory effects are lost, and the dynamics become less influenced by the past, depending primarily on local behavior.

6. Conclusions and Future Directions

In this work, we propose a unified approach to analyze implicit fractional differential equations involving general Caputo-type derivatives, under anti-periodic boundary conditions. Two classes of problems are considered: one with fractional order $\alpha \in (0,1)$, and another with $\alpha \in (1,2)$, each with corresponding boundary conditions. By reformulating the original problems into Volterra integral equations, we employ fixed-point theorems to derive sufficient conditions ensuring the existence and uniqueness of solutions. Furthermore, Ulam–Hyers and generalized Ulam–Hyers stability results are established. The formulation adopted here generalizes several well-known fractional operators as particular cases, making our results widely applicable. The two examples presented demonstrate how the theoretical findings can be effectively applied in concrete scenarios.

Future research may focus on extending this framework to higher-order derivatives, systems with more general boundary conditions, or those incorporating delays.