Existence and Stability Analysis of Anti-Periodic Boundary Value Problems with Generalized Tempered Fractional Derivatives

Abstract

In this study, we investigate implicit fractional differential equations subject to anti-periodic boundary conditions. The fractional operator incorporates two distinct generalizations: the Caputo tempered fractional derivative and the Caputo fractional derivative with respect to a smooth function. We investigate the existence and uniqueness of solutions using fixed-point theorems. Stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias is also considered. Three detailed examples are presented to illustrate the applicability and scope of the theoretical results. Several existing results in the literature can be recovered as particular cases of the framework developed in this work.

1. Introduction

Fractional calculus (FC) generalizes the traditional concepts behind differentiation as well as integration to non-integer orders, providing a versatile framework for describing systems with memory and hereditary characteristics. While FC remained primarily theoretical from its birth in 1695 until the beginning of the 20th century, it has gained considerable attention in recent decades because of its successful application in diverse fields such as multi-agent systems [1], biology [2], control theory [3], viscoelasticity [4], epidemiology [5], physics [6], mechanics [7], and medicine [8]. For more details, we direct the reader to the established literature on fractional calculus and its applications [9,10,11], which provide comprehensive foundations, detailed theoretical developments, with extensive applications in mathematics, physics, and engineering.

There are multiple approaches to defining fractional derivatives, with the most common being those of Riemann–Liouville, Grünwald–Letnikov, Hadamard and Caputo formulations, each suited to different types of problems and initial or boundary conditions. Given the existence of multiple definitions of fractional derivatives, it is essential to consider generalized frameworks that include various classical formulations as special cases. Two notable generalizations include fractional derivatives with respect to a smooth kernel, as introduced in [9,12], and tempered fractional derivatives, introduced in [13].

Fractional derivatives defined with respect to a smooth kernel provide significant advantages, particularly in their ability to generalize and unify classical definitions through appropriate choices of the kernel function. This flexibility enables the study of a broad class of models within a single coherent mathematical framework [1,8,14].

It is well known that classical fractional derivatives effectively capture long-range memory and spatial effects, making them valuable tools for modeling anomalous diffusion and complex dynamical systems. However, in many real-world scenarios, these memory effects tend to decay over time. To address this, an exponential tempering parameter can be introduced to control the decay of the memory tail. This modification allows a more realistic representation of system behavior, improves the applicability of the model to practical problems, and represents one of the main advantages of tempered fractional derivatives [13,15,16].

In the present work, we explore recent advances in fractional derivatives, particularly those introduced in [15]. In this work, the authors introduced new generalized fractional derivatives, formulated in both the Riemann–Liouville and Caputo senses, which integrate the ideas of tempered fractional derivatives with those of fractional derivatives defined via a smooth kernel. This unified framework enables the study of a broad class of fractional operators in a general setting. Furthermore, the authors analyzed the problem of existence and uniqueness of solutions within a certain class of fractional differential equations with prescribed initial data. They also analyzed the stability of the system and examined the attractivity of solutions under appropriate assumptions.

The literature on fractional differential equations is extensive, covering a wide range of topics, including the existence and uniqueness of solutions [15,17,18,19,20,21,22], stability analysis [15,23,24], qualitative properties of solutions [25], and the development of numerical methods to solve fractional differential equations [26,27,28,29]. An important class of fractional differential equations that has attracted considerable attention in recent decades involves anti-periodic boundary conditions, due to their relevance in modeling a variety of physical phenomena. These boundary conditions naturally appear in real-world systems where the solution exhibits alternating or sign-reversing behavior over time or space, such as in the modeling of alternating electric currents and vibrational systems. By capturing the inherent sign reversal after each period, anti-periodic conditions provide a natural framework for analyzing the dynamics and stability of inherently periodic systems. For a more detailed discussion on anti-periodic boundary value problems, see [30,31] and the references therein. We also refer to [32], where implicit fractional differential equations involving a general form of fractional derivative are studied. The results presented there can be recovered as a special case of the framework developed in this work, through an appropriate choice of parameters.

Remark 1.

Periodic and anti-periodic boundary conditions describe different types of repeating behavior in a system. In periodic boundary conditions, the solution repeats after a given period T, so that $u(T)=u(0)$. In contrast, anti-periodic boundary conditions require that the solution at the end of the period is the negative of its initial value, $u(T)=-u(0)$. Anti-periodic conditions naturally model oscillatory systems with sign reversal, such as alternating currents or certain vibrational phenomena, whereas periodic conditions are suited for systems that repeat identically over each cycle.

Motivated by the above observations, this study focuses on fractional differential equations with anti-periodic boundary conditions, involving the tempered Caputo fractional derivative with respect to a smooth kernel, with fractional order $\alpha \in (0,1)$. The fractional differential operator employed in the formulation of these equations involves two significant generalizations: a tempered fractional derivative, which introduces an exponential tempering factor to the classical fractional calculus framework, and the fractional derivative with respect to a smooth function. These extensions provide a flexible framework for modeling complex phenomena in various scientific fields. Since classical fractional derivatives have previously proven to be effective in modeling certain complex phenomena, and in this work we consider a highly generalized form of derivative, we believe it may prove even more useful in describing real-world problems.

The main contributions of this work are (i) the introduction of generalized tempered Caputo operators in the context of anti-periodic boundary value problems; (ii) new existence and uniqueness results obtained via Banach, Schaefer, and Schauder fixed-point theorems; (iii) a comprehensive analysis of Ulam–Hyers and Ulam–Hyers–Rassias type stability; and (iv) illustrative examples that corroborate the theoretical analysis.

The structure of this paper is as follows. Section 2 reviews the basic background and introduces the problem formulation. Section 3 establishes existence and uniqueness results. Section 4 addresses stability properties. Section 5 provides three illustrative examples. Finally, Section 6 presents an overview of this paper and outlines future perspectives. We would like to emphasize that, by specifying the kernel function and adjusting the parameters of the tempered derivative, many particular cases can be derived from the general results presented in this work.

2. Preliminaries and Problem Formulation

We start this section by presenting the fundamental concepts of generalized tempered fractional integrals and derivatives, as well as the key results that will be used. These concepts not only incorporate the exponential function, as inherited from tempered fractional operators, but also depend on a smooth kernel. As a result, it includes other operators—such as those of Riemann–Liouville, Hadamard, and Erdélyi–Kober—as special cases.

We fix a parameter $\lambda \ge 0$. The kernel dependence is characterized by an increasing differentiable function $\psi \in C^1([a,b],\mathbb{R})$ with positive derivative.

Definition 1

([15]). Let $\alpha >0$ be a fractional order, and let $u\in L^1([a,b],\mathbb{R})$. The $(\alpha ,\lambda ,\psi )$-tempered fractional integral of u is defined by the following:

$${\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }u(t)={\displaystyle \frac{e^{-\lambda t}}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){(\psi (t)-\psi (\tau ))}^{\alpha -1}{e}^{\lambda \tau }u(\tau )d\tau ,t\in [a,b],$$

where Γ denotes the Gamma function.

Remark 2.

The $(\alpha ,\lambda ,\psi )$-tempered fractional integral of a function represents a generalized form within the framework of fractional calculus. Several well-known fractional integral operators emerge as special cases of this formulation. Notable examples include the following:

• For $\lambda =0$, we recover the fractional integral with respect to another function (cf. [9]):

$${\mathbb{I}}_{a+}^{\alpha ,\psi }u(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){(\psi (t)-\psi (\tau ))}^{\alpha -1}u(\tau )d\tau ,t\in [a,b].$$

• For $\psi (t)=t$, we obtain the tempered fractional integral (cf. [13]):

$${\mathbb{I}}_{a+}^{\alpha ,\lambda }u(t)={\displaystyle \frac{e^{-\lambda t}}{\Gamma (\alpha )}}{\int }_a^t{(t-\tau )}^{\alpha -1}{e}^{\lambda \tau }u(\tau )d\tau ,t\in [a,b].$$

• When $\lambda =0$ and $\psi (t)=t$, we recover the Riemann–Liouville fractional integral (cf. [9]):

$${\mathbb{I}}_{a+}^{\alpha }u(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_a^t{(t-\tau )}^{\alpha -1}u(\tau )d\tau ,t\in [a,b].$$

• For $\lambda =0$ and $\psi (t)=ln(t)$, we obtain the Hadamard fractional integral (cf. [9]):

$${\mathbb{I}}_{a+}^{\alpha }u(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_a^t{\displaystyle \frac{1}{t}}{(ln(t)-ln(\tau ))}^{\alpha -1}u(\tau )d\tau ,t\in [a,b],a>0.$$

Definition 2

([15]). Let $\alpha \in (0,1)$ be a fractional order, and let u be an absolutely continuous function defined on the interval $[a,b]$. The $(\alpha ,\lambda ,\psi )$-tempered Caputo fractional derivative of u is defined by the following:

$${}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t)={\displaystyle \frac{e^{-\lambda t}}{\Gamma (1-\alpha )}}{\int }_a^t{(\psi (t)-\psi (\tau ))}^{-\alpha }{\displaystyle \frac{d}{d\tau }}(e^{\lambda \tau }u(\tau ))d\tau .$$

We note that, setting $\lambda =0$, the operator ${}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }$ reduces to the one defined in [12]. Furthermore, when $\psi$ is the identity function, we recover the classical Caputo fractional derivative ([9]):

$${}^C\mathbb{D}_{a+}^{\alpha }u(t)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_a^t{(t-\tau )}^{-\alpha }{u}^{\prime }(\tau )d\tau .$$

For example, given $\beta >0$, the following formulas hold (see [15]):

$${\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }(e^{-\lambda t}{(\psi (t)-\psi (a))}^{\beta })=e^{-\lambda t}{\displaystyle \frac{\Gamma (\beta +1)}{\Gamma (\beta +1+\alpha )}}{(\psi (t)-\psi (a))}^{\beta +\alpha },$$

and

$${}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }(e^{-\lambda t}{(\psi (t)-\psi (a))}^{\beta })=e^{-\lambda t}{\displaystyle \frac{\Gamma (\beta +1)}{\Gamma (\beta +1-\alpha )}}{(\psi (t)-\psi (a))}^{\beta -\alpha }.$$

Remark 3.

In this paper, we focus exclusively on what are known as left-sided fractional operators, as evaluating the solution at a given time $t\in [a,b]$ requires knowledge of the process over the interval $[a,t]$. These operators are the most commonly used in applications, as they naturally incorporate memory effects and depend only on past or already known information. For completeness, analogous definitions can be formulated to describe the evolution based on future values. These are referred to as right-sided fractional operators, and are defined as follows:

$${\mathbb{I}}_{b-}^{\alpha ,\lambda ,\psi }u(t)={\displaystyle \frac{e^{\lambda t}}{\Gamma (\alpha )}}{\int }_t^b{\psi }^{\prime }(\tau ){(\psi (\tau )-\psi (t))}^{\alpha -1}{e}^{-\lambda \tau }u(\tau )d\tau ,t\in [a,b].$$

and

$${}^C\mathbb{D}_{b-}^{\alpha ,\lambda ,\psi }u(t)={\displaystyle \frac{-e^{\lambda t}}{\Gamma (1-\alpha )}}{\int }_t^b{(\psi (\tau )-\psi (t))}^{-\alpha }{\displaystyle \frac{d}{d\tau }}(e^{-\lambda \tau }u(\tau ))d\tau ,t\in [a,b].$$

We write $AC([a,b],\mathbb{R})$ for the set of absolutely continuous functions $u:[a,b]\to \mathbb{R}$, with the norm defined as follows:

$$\parallel u\parallel =\underset{t\in [a,b]}{sup}\left|u(t)\right|.$$

The following result establishes the relationship between the fractional operators ${}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }$ and ${\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }$.

Theorem 1

([15]). Let $\alpha \in (0,1)$. For any $u\in AC([a,b],\mathbb{R})$, the following identities hold:

$${}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }u(t)=u(t)$$

and

$${\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t)=u(t)-e^{-\lambda (t-a)}u(a).$$

The aim of this paper is to investigate implicit fractional differential equations involving the $(\alpha ,\lambda ,\psi )$-tempered Caputo fractional derivative, with $\alpha \in (0,1)$, subject to anti-periodic boundary conditions. The problem is formulated as follows:

Consider the following fractional differential equation:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t)=f(t,u(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t)),t\in [a,b],\hfill \\ u(a)=-u(b),\hfill \end{array}\right.$$

(1)

where $f:[a,b]\times {\mathbb{R}}^2\to \mathbb{R}$ is a continuous function. The admissible function space is the space $AC([a,b],\mathbb{R})$.

We say that a function $u\in AC([a,b],\mathbb{R})$ is a solution of the boundary value problem (1) if it satisfies the following equation:

$${}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t)=f(t,u(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t)),t\in [a,b],$$

and fulfills the anti-periodic boundary condition $u(a)=-u(b)$.

To establish the existence and uniqueness of solutions, we apply fixed-point theorems, specifically the Banach, Schaefer, and Schauder fixed-point theorems, which we now recall.

Theorem 2

(Banach’s fixed-point theorem) Consider a complete metric space X with metric d. If $\mathcal{G}:X\to X$ is a mapping satisfying the contraction condition with constant L, then it admits a unique fixed point $u^{★}$. Moreover, this fixed point satisfies the inequality $(1-L)d(u,u^{★})\le d(u,\mathcal{G}u)$, for all $u\in X$.

Theorem 3

(Schaefer’s fixed-point theorem) Let $\mathcal{G}:B\to B$ be a completely continuous operator on the Banach space B, and assume that the following set is bounded:

$$S=\{v\in B:v=\lambda \mathcal{G}v\mathit{for}\mathit{some}0\le \lambda \le 1\}$$

Then $\mathcal{G}$ has a fixed point in B.

Theorem 4

(Schauder’s fixed-point theorem). Let $\Omega \ne \varnothing$ be a closed, convex, and bounded subset of a Banach space B. If the mapping $\mathcal{G}:\Omega \to \Omega$ is completely continuous, then $\mathcal{G}$ possesses at least one fixed point in $\Omega$.

For the reader’s convenience, we recall the following result, which will be used later.

Theorem 5

(Arzelà–Ascoli theorem). A set $\Omega \subseteq AC([a,b],\mathbb{R})$ is relatively compact if and only if it is equicontinuous and uniformly bounded.

To conclude our review section, we present several notions of stability, following the framework established in [33].

Definition 3.

We call Equation (1) Ulam–Hyers stable when there is $K\in {\mathbb{R}}^{+}$ (called the UH constant) such that the following holds:

For any $\delta >0$, if a function $v\in AC([a,b],\mathbb{R})$ satisfies the following:

$$\left\{\begin{array}{c}\left|{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }v(t)-f(t,v(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }v(t))\right|\le \delta ,t\in [a,b],\hfill \\ v(a)=-v(b),\hfill \end{array}\right.$$

(2)

there exists $u\in AC([a,b],\mathbb{R})$, which satisfies (1), such that the following is true:

$$\parallel u-v\parallel \le K\delta .$$

Definition 4.

Equation (1) is said to be generalized Ulam–Hyers stable if a function $g\in C({\mathbb{R}}_0^{+},{\mathbb{R}}_0^{+})$ exists, satisfying $g(0)=0$, such that the following holds:

For any $\delta >0$, if a function $v\in AC([a,b],\mathbb{R})$ satisfies Equation (2), there exists $u\in AC([a,b],\mathbb{R})$, which satisfies (1), such that the following is true:

$$\parallel u-v\parallel \le g(\delta ).$$

Definition 5.

Let $r:[a,b]\to {\mathbb{R}}^{+}$ be a given positive function. Equation (1) is said to be Ulam–Hyers–Rassias stable with respect to r if there exists a positive $K\in {\mathbb{R}}^{+}$, possibly dependent on r, such that the following holds:

For any $\delta >0$, if a function $v\in AC([a,b],\mathbb{R})$ satisfies the following:

$$\left\{\begin{array}{c}\left|{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }v(t)-f(t,v(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }v(t))\right|\le \delta r(t),t\in [a,b],\hfill \\ v(a)=-v(b),\hfill \end{array}\right.$$

(3)

there exists $u\in AC([a,b],\mathbb{R})$, which satisfies (1), such that the following is true:

$$\parallel u-v\parallel \le K\delta \parallel r\parallel .$$

3. Existence and Uniqueness Results

In this section, we address the existence and uniqueness of solutions to Equation (1). We begin by reformulating the fractional differential Equation (1) as an equivalent Volterra integral equation.

Theorem 6.

A function $u\in AC([a,b],\mathbb{R})$ solves the fractional problem (1) if and only if it satisfies the corresponding Volterra integral equation:

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

(4)

Proof. 

Let u be a solution of (1). By applying the operator ${\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }$ to the fractional differential equation, we obtain the following:

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

Given that $u(b)=-u(a)$, it follows that the following is true:

$$u(a)=-{\displaystyle \frac{e^{-\lambda b}}{\Gamma (\alpha )}}{\displaystyle \frac{1}{1+e^{-\lambda (b-a)}}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau ,$$

(5)

proving Formula (4).

To prove the converse, let u be a function satisfying (4). Applying the operator ${}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }$ to both sides yields the following:

$$\begin{array}{c}{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t)\hfill \\ \hspace{1em}=-{\displaystyle \frac{1}{\Gamma (\alpha )}}{\displaystyle \frac{e^{-\lambda (b-a)}}{1+e^{-\lambda (b-a)}}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau {}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{e}^{-\lambda t}\\ \hfill +{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }f(t,u(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t))=f(t,u(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t)),\end{array}$$

since ${}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{e}^{-\lambda t}=0$. The boundary condition is satisfied according to Formula (5). □

Based on Theorem 6, we introduce the operator $\mathcal{G}:AC([a,b],\mathbb{R})\to AC([a,b],\mathbb{R})$ as follows:

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

(6)

Banach’s fixed-point theorem will be used to demonstrate that problem (1) admits a unique solution, provided the function f satisfies certain conditions.

Theorem 7.

Suppose that the function f is Lipschitz continuous with constant $L\in (0,1)$, so that for any $t\in [a,b]$ and $u_i,d_i\in \mathbb{R}$, for $i=1,2$, the following is true:

$$|f(t,u_1,d_1)-f(t,u_2,d_2)|\le L(|u_1-u_2|+|d_1-d_2|).$$

Moreover, suppose that the constant L satisfies the inequality as follows:

$$K_1:={\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·{\displaystyle \frac{L}{1-L}}<1.$$

Then, the Equation (1) has a unique solution.

Proof. 

We will show that $\mathcal{G}:AC([a,b],\mathbb{R})\to AC([a,b],\mathbb{R})$ is a contraction mapping. Let $u_1,u_2\in AC([a,b],\mathbb{R})$ and fix $t\in [a,b]$. We begin with the following estimate:

$$\begin{array}{cc}\hfill |f(t,u_1(t),& {}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_1(t))-f(t,u_2(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_2(t))|\hfill \\ \hfill & \le L|u_1(t)-u_2(t)|+L|{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_1(t)-{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_2(t)|\hfill \\ \hfill & =L|u_1(t)-u_2(t)|+L|f(t,u_1(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_1(t))-f(t,u_2(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_2(t))|.\hfill \end{array}$$

Rearranging terms, we obtain the following:

$$|f(t,u_1(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_1(t))-f(t,u_2(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_2(t))|\le {\displaystyle \frac{L}{1-L}}\parallel u_1-u_2\parallel .$$

Observing the following:

$$\begin{array}{cc}\hfill |(\mathcal{G}{u}_1)(t)-(\mathcal{G}{u}_2)(t)|& \le {\displaystyle \frac{e^{-\lambda a}}{\Gamma (\alpha )}}{\displaystyle \frac{e^{-\lambda (b-a)}}{1+e^{-\lambda (b-a)}}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda b}\hfill \\ \hfill & \times |f(\tau ,u_1(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_1(\tau ))-f(\tau ,u_2(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_2(\tau ))|d\tau \hfill \\ \hfill & +{\displaystyle \frac{e^{-\lambda a}}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){\left(\psi (t)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda b}\hfill \\ \hfill & \times |f(\tau ,u_1(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_1(\tau ))-f(\tau ,u_2(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }{u}_2(\tau ))|d\tau ,\hfill \end{array}$$

and

$${\displaystyle \frac{e^{-\lambda (b-a)}}{1+e^{-\lambda (b-a)}}}\le {\displaystyle \frac{1}{2}},$$

we obtain the following:

$$\begin{array}{cc}\hfill |(\mathcal{G}{u}_1)(t)-(\mathcal{G}{u}_2)(t)|& \le {\displaystyle \frac{1}{2}}{\displaystyle \frac{e^{-\lambda a}}{\Gamma (\alpha )}}{\displaystyle \frac{L}{1-L}}\parallel u_1-u_2\parallel {\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda b}d\tau \hfill \\ \hfill & +{\displaystyle \frac{e^{-\lambda a}}{\Gamma (\alpha )}}{\displaystyle \frac{L}{1-L}}\parallel u_1-u_2\parallel {\int }_a^t{\psi }^{\prime }(\tau ){\left(\psi (t)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda b}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·{\displaystyle \frac{L}{1-L}}\parallel u_1-u_2\parallel .\hfill \end{array}$$

Hence, $\mathcal{G}$ is a contraction on $AC([a,b],\mathbb{R})$. The conclusion now follows from Banach’s fixed-point theorem. □

We next employ Schaefer’s fixed-point theorem to prove the existence of a solution to Equation (1), under a different set of assumptions on the function f.

Theorem 8.

If there exist two constants $L_0\in {\mathbb{R}}^{+}$ and $L_1\in (0,1)$ such that the following is true:

$$\left|f(t,u,d)\right|\le L_0+L_1(|u|+\left|d\right|),$$

for all $t\in [a,b]$ and all $u,d\in \mathbb{R}$, and the following:

$${\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·{\displaystyle \frac{L_1}{1-L_1}}<1,$$

(7)

then, there exists at least one solution to Equation (1).

Proof. 

We employ Schaefer’s fixed-point theorem to demonstrate that the function $\mathcal{G}$ (see (6)) admits at least one fixed point. The demonstration is structured in two stages.

Step 1: To show that $\mathcal{G}$ is a completely continuous operator, we will demonstrate that $\mathcal{G}$ maps bounded sets into relatively compact sets. Let $R>0$ be an arbitrary real number, and define the following:

$$B_R=\{u\in AC([a,b],\mathbb{R}):\parallel u\parallel \le R\}.$$

Let $u\in B_R$ and $t\in [a,b].$ Since the following is true:

$$|f(t,u(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t))|\le L_0+L_1(\parallel u\parallel +\parallel {}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u\parallel )$$

we obtain the following:

$$\parallel f(·,u(·),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(·))\parallel \le {\displaystyle \frac{L_0+L_1\parallel u\parallel }{1-L_1}}.$$

Hence, the following:

$$\begin{array}{cc}\hfill \parallel \mathcal{G}u\parallel & \le {\displaystyle \frac{1}{2}}{\displaystyle \frac{e^{-\lambda a}}{\Gamma (\alpha )}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda b}{\displaystyle \frac{L_0+L_1\parallel u\parallel }{1-L_1}}d\tau \hfill \\ \hfill & +{\displaystyle \frac{e^{-\lambda a}}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){\left(\psi (t)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda b}{\displaystyle \frac{L_0+L_1\parallel u\parallel }{1-L_1}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·{\displaystyle \frac{L_0+L_{1}R}{1-L_1}},\hfill \end{array}$$

proving that $\mathcal{G}(B_R)$ is uniformly bounded. Now, we will prove that $\mathcal{G}(B_R)$ is equicontinuous. Let $u\in B_R$ and $t_1,t_2\in [a,b]$ be such that $t_2>t_1$. Since the following is true:

$$\begin{array}{cc}\hfill |(\mathcal{G}u)(t_2)-(\mathcal{G}u)(t_1)|& =|-{\displaystyle \frac{e^{-\lambda t_2}}{\Gamma (\alpha )}}·{\displaystyle \frac{e^{-\lambda (b-a)}}{1+e^{-\lambda (b-a)}}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau \hfill \\ \hfill & +{\displaystyle \frac{e^{-\lambda t_1}}{\Gamma (\alpha )}}·{\displaystyle \frac{e^{-\lambda (b-a)}}{1+e^{-\lambda (b-a)}}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau \hfill \\ \hfill & +{\displaystyle \frac{e^{-\lambda t_2}}{\Gamma (\alpha )}}{\int }_a^{t_2}{\psi }^{\prime }(\tau ){\left(\psi (t_2)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau \hfill \\ \hfill & -{\displaystyle \frac{e^{-\lambda t_1}}{\Gamma (\alpha )}}{\int }_a^{t_1}{\psi }^{\prime }(\tau ){\left(\psi (t_1)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{e^{\lambda b}}{2\Gamma (\alpha +1)}}·{\displaystyle \frac{L_0+L_{1}R}{1-L_1}}·{(\psi (b)-\psi (a))}^{\alpha }\left(e^{-\lambda t_1}-e^{-\lambda t_2}\right)\hfill \\ \hfill & +\left|{\displaystyle \frac{e^{-\lambda t_2}}{\Gamma (\alpha )}}{\int }_{t_1}^{t_2}{\psi }^{\prime }(\tau ){\left(\psi (t_2)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau \right|\hfill \\ \hfill & +|{\displaystyle \frac{e^{-\lambda t_2}}{\Gamma (\alpha )}}{\int }_a^{t_1}{\psi }^{\prime }(\tau ){\left(\psi (t_2)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau \hfill \\ \hfill & -{\displaystyle \frac{e^{-\lambda t_1}}{\Gamma (\alpha )}}{\int }_a^{t_1}{\psi }^{\prime }(\tau ){\left(\psi (t_2)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau |\hfill \\ \hfill & +|{\displaystyle \frac{e^{-\lambda t_1}}{\Gamma (\alpha )}}{\int }_a^{t_1}{\psi }^{\prime }(\tau ){\left(\psi (t_2)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau \hfill \\ \hfill & -{\displaystyle \frac{e^{-\lambda t_1}}{\Gamma (\alpha )}}{\int }_a^{t_1}{\psi }^{\prime }(\tau ){\left(\psi (t_1)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))d\tau |,\hfill \end{array}$$

then, the following is true:

$$\begin{array}{cc}\hfill |(\mathcal{G}u)(t_2)-(\mathcal{G}u)(t_1)|& \le \left(e^{-\lambda t_1}-e^{-\lambda t_2}\right){\displaystyle \frac{e^{\lambda b}}{\Gamma (\alpha +1)}}·{\displaystyle \frac{L_0+L_{1}R}{1-L_1}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{2}}+{(\psi (t_2)-\psi (a))}^{\alpha }-{(\psi (t_2)-\psi (t_1))}^{\alpha }\right]\hfill \\ \hfill & +{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}·{\displaystyle \frac{L_0+L_{1}R}{1-L_1}}\left[{(\psi (t_2)-\psi (a))}^{\alpha }-{(\psi (t_1)-\psi (a))}^{\alpha }\right].\hfill \end{array}$$

From the last inequality, we conclude that $(\mathcal{G}u)(t_2)-(\mathcal{G}u)(t_1)\to 0$ as $t_2\to t_1$, which proves that $\mathcal{G}(B_R)$ forms an equicontinuous family.

Using the Arzelà–Ascoli Theorem, we infer that $\mathcal{G}(B_R)$ is relatively compact, which shows that $\mathcal{G}$ is completely continuous.

Step 2: Next, we prove that the following set is bounded:

$$S=\{u\in AC([a,b],\mathbb{R}):\mathrm{there}\mathrm{exists}\lambda \in [0,1]\mathrm{such}\mathrm{that}u=\lambda \mathcal{G}u\}$$

For that purpose, consider $u\in S$ and let $\lambda \in [0,1]$ be a real with $u=\lambda \mathcal{G}u$. Since the following is true:

$$\begin{array}{cc}\hfill \parallel u\parallel & =\parallel \lambda \mathcal{G}u\parallel \le \parallel \mathcal{G}u\parallel \hfill \\ \hfill & \le {\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·{\displaystyle \frac{L_0+L_1\parallel u\parallel }{1-L_1}},\hfill \end{array}$$

we obtain the following:

$$\parallel u\parallel \le {\displaystyle \frac{\frac{3}{2}·\frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·\frac{L_0}{1-L_1}}{1-\frac{3}{2}·\frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·\frac{L_1}{1-L_1}}},$$

proving that S is bounded.

It follows from Schaefer’s fixed-point theorem that Equation (1) has at least one solution. □

To conclude this section, we present another existence result, this time using Schauder’s fixed-point theorem.

Theorem 9.

Suppose a positive constant $M\in {\mathbb{R}}^{+}$ exists such that the following is true:

$$|f(t,u,d)|\le M,$$

for every $t\in [a,b]$ and all $u,d\in \mathbb{R}$, then the Equation (1) admits at least one solution.

Proof. 

Let $R>0$ be such that the following is true:

$$R\ge {\displaystyle \frac{3M}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha },$$

and define the following:

$$B_R=\{u\in AC([a,b],\mathbb{R}):\parallel u\parallel \le R\}.$$

We aim to show that, for any $u\in B_R$, $\mathcal{G}u\in B_R$, where $\mathcal{G}$ is the operator defined in (6). Let $u\in B_R$ and $t\in [a,b].$ Since the following is true:

$$|f(t,u(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(t))|\le M$$

we obtain the following:

$$\begin{array}{cc}\hfill |(\mathcal{G}u)(t)|& \le {\displaystyle \frac{M}{2}}{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha )}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}d\tau \hfill \\ \hfill & +M{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){\left(\psi (t)-\psi (\tau )\right)}^{\alpha -1}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{3M}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }\le R.\hfill \end{array}$$

Then, the following is true:

$$\parallel \mathcal{G}u\parallel \le R,$$

which proves that $\mathcal{G}u\in B_R$, proving that $\mathcal{G}:B_R\to B_R$ is well defined. Now, we will prove that the operator $\mathcal{G}:B_R\to B_R$ is completely continuous. Given $t_1,t_2\in [a,b]$, such that $t_1<t_2$, and $u\in B_R$, one obtains the following:

$$\begin{array}{cc}\hfill |(\mathcal{G}u)(t_2)-(\mathcal{G}u)(t_1)|& \le \left(e^{-\lambda t_1}-e^{-\lambda t_2}\right){\displaystyle \frac{e^{\lambda b}}{\Gamma (\alpha +1)}}M\hfill \\ \hfill & \times \left[{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{2}}+{(\psi (t_2)-\psi (a))}^{\alpha }-{(\psi (t_2)-\psi (t_1))}^{\alpha }\right]\hfill \\ \hfill & +{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}M\left[{(\psi (t_2)-\psi (a))}^{\alpha }-{(\psi (t_1)-\psi (a))}^{\alpha }\right],\hfill \end{array}$$

which tends to zero as $t_2\to t_1$, proving that $\mathcal{G}(B_R)$ is equicontinuous. Since $\mathcal{G}(B_R)$ is uniformly bounded by the Arzelà–Ascoli theorem, $\mathcal{G}(B_R)$ is relatively compact and, therefore, $\mathcal{G}$ is completely continuous. Schauder’s fixed-point theorem ensures that it possesses at least one fixed point, which means that Equation (1) has at least one solution. □

4. Ulam–Hyers and Ulam–Hyers–Rassias Stability Analysis

In this section, we establish sufficient conditions for the Ulam–Hyers stability, generalized Ulam–Hyers stability, and Ulam–Hyers–Rassias stability of Equation (1). These stability concepts offer valuable insights into the behavior of solutions under small perturbations, contributing to both the theoretical development and practical reliability of the model.

Theorem 10.

Suppose that the conditions of Theorem 7 are satisfied. Then, Equation (1) exhibits Ulam–Hyers stability.

Proof. 

According to Theorem 7, Equation (1) admits a unique solution, which we denote by $u^{★}$. Observe that $v\in AC([a,b],\mathbb{R})$ is a solution of (2) if and only if, for any $\delta >0$, there exists $h\in AC([a,b],\mathbb{R})$ such that, for all $t\in [a,b]$, $|h(t)|\le \delta$ and

$$\left\{\begin{array}{c}h(t)={}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }v(t)-f(t,v(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }v(t)),\hfill \\ v(a)=-v(b).\hfill \end{array}\right.$$

• By Theorem 6, we conclude the following:

  $$\begin{array}{c}v(t)=-{\displaystyle \frac{e^{-\lambda t}}{\Gamma (\alpha )}}{\displaystyle \frac{e^{-\lambda (b-a)}}{1+e^{-\lambda (b-a)}}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }[f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))+h(\tau )]d\tau \hfill \\ \hfill +{\displaystyle \frac{e^{-\lambda t}}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){\left(\psi (t)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }[f(\tau ,u(\tau ),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }u(\tau ))+h(\tau )]d\tau .\end{array}$$

  (8)
• Since the operator $\mathcal{G}:AC([a,b],\mathbb{R})\to AC([a,b],\mathbb{R})$, defined in (6), is a contraction mapping with the following contraction constant:

  $$K_1={\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·{\displaystyle \frac{L}{1-L}},$$
• it follows from the Banach’s fixed-point theorem that the following is true:

  $$\parallel u^{★}-v\parallel \le {\displaystyle \frac{1}{1-K_1}}\parallel \mathcal{G}v-v\parallel .$$
• Observing that, for all $t\in [a,b]$, the following is true:

  $$\begin{array}{cc}\hfill |(\mathcal{G}v)(t)-v(t)|& =|-{\displaystyle \frac{e^{-\lambda t}}{\Gamma (\alpha )}}{\displaystyle \frac{e^{-\lambda (b-a)}}{1+e^{-\lambda (b-a)}}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }h(\tau )d\tau \hfill \\ \hfill & +{\displaystyle \frac{e^{-\lambda t}}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){\left(\psi (t)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }h(\tau )d\tau |\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha )}}\delta {\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}d\tau \hfill \\ \hfill & +{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha )}}\delta {\int }_a^t{\psi }^{\prime }(\tau ){\left(\psi (t)-\psi (\tau )\right)}^{\alpha -1}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{3\delta }{2}}{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha },\hfill \end{array}$$

  we conclude the following:

  $$\parallel u^{★}-v\parallel \le \left({\displaystyle \frac{1}{1-K_1}}{\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }\right)\delta ,$$

  which proves the desired result. □

Corollary 1.

Suppose that the conditions of Theorem 7 are satisfied. Then, Equation (1) exhibits generalized Ulam–Hyers stability.

Proof. 

This result follows directly from Theorem 10 by setting the following:

$$g(\delta )=\left({\displaystyle \frac{1}{1-K_1}}{\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }\right)\delta .$$

□

Theorem 11.

Suppose the assumptions of Theorem 7 are satisfied and let $r:[a,b]\to {\mathbb{R}}^{+}$ be such that the following is true:

$${\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }r(b)\le r(t),forallt\in [a,b].$$

Then, the Equation (1) is Ulam–Hyers–Rassias stable with respect to the function r.

Proof. 

Let $u^{★}$ be the unique solution of the Equation (1) and let $v\in AC([a,b],\mathbb{R})$ a solution of (3). Since, for any $\delta >0$, there exists $h\in AC([a,b],\mathbb{R})$ such that the following is true:

• $h(t)={}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }v(t)-f(t,v(t),{}^C\mathbb{D}_{a+}^{\alpha ,\lambda ,\psi }v(t)),$ for all $t\in [a,b]$;
• $|h(t)|\le \delta r(t)$, for all $t\in [a,b]$;
• $v(a)=-v(b)$.

• Then, for all $t\in [a,b]$, the following is true:

  $$\begin{array}{cc}\hfill |(\mathcal{G}v)(t)-v(t)|& \le {\displaystyle \frac{1}{2}}\delta {\displaystyle \frac{e^{-\lambda t}}{\Gamma (\alpha )}}{\int }_a^b{\psi }^{\prime }(\tau ){\left(\psi (b)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }r(\tau )d\tau \hfill \\ \hfill & +\delta {\displaystyle \frac{e^{-\lambda t}}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){\left(\psi (t)-\psi (\tau )\right)}^{\alpha -1}{e}^{\lambda \tau }r(\tau )d\tau \hfill \\ \hfill & \le {\displaystyle \frac{3}{2}}\delta {\mathbb{I}}_{a+}^{\alpha ,\lambda ,\psi }r(b)\hfill \\ \hfill & \le {\displaystyle \frac{3}{2}}\delta r(t).\hfill \end{array}$$
• Hence, the following is true:

  $$\parallel \mathcal{G}v-v\parallel \le {\displaystyle \frac{3}{2}}\delta \parallel r\parallel .$$
• From the Banach’s fixed-point theorem, we conclude the following:

  $$\parallel u^{★}-v\parallel \le {\displaystyle \frac{1}{1-K_1}}{\displaystyle \frac{3}{2}}\delta \parallel r\parallel$$

  where

  $$K_1={\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^{\lambda (b-a)}}{\Gamma (\alpha +1)}}{\left(\psi (b)-\psi (a)\right)}^{\alpha }·{\displaystyle \frac{L}{1-L}},$$

  proving that Equation (1) is Ulam–Hyers–Rassias stable with respect to the function r. □

5. Examples

To demonstrate the applicability and effectiveness of the theoretical results, three examples are presented, illustrating how the analytical conditions can be verified in practice.

Example 1.

Consider the following fractional differential equation:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{0+}^{0.5,1,\psi }u(t)={\displaystyle \frac{sin(u(t))}{10}}+{\displaystyle \frac{t}{5}}{}^C\mathbb{D}_{0+}^{0.5,1,\psi }u(t),t\in [0,1],\hfill \\ u(0)=-u(1),\hfill \end{array}\right.$$

(9)

where $\alpha =0.5$, $\lambda =1$, and the following:

$$f(t,u(t),{}^C\mathbb{D}_{0+}^{0.5,1,\psi }u(t))={\displaystyle \frac{sin(u(t))}{10}}+{\displaystyle \frac{t}{5}}{}^C\mathbb{D}_{0+}^{0.5,1,\psi }u(t).$$

A straightforward computation shows the following:

$$|f(t,u_1,d_1)-f(t,u_2,d_2)|\le {\displaystyle \frac{1}{5}}(|u_1-u_2|+|d_1-d_2|),$$

which shows that the function f is Lipschitz continuous with Lipschitz constant $L=1/5$. Now, if we take the kernel function $\psi (t)=\sqrt{t+1}$, then the corresponding constant, as follows:

$$K_1={\displaystyle \frac{3}{2}}·{\displaystyle \frac{e}{\Gamma (1.5)}}{\left(\sqrt{2}-1\right)}^{0.5}·{\displaystyle \frac{1}{4}},$$

as defined in Theorem 7, satisfies $K_1\approx 0.4764<1$. Therefore, all the hypotheses of Theorem 7 are fulfilled, guaranteeing the existence and uniqueness of a solution to problem (9).

It is worth emphasizing the crucial role played by the kernel function in the applicability of the result. For instance, if we instead consider $\psi (t)=exp(t)$, then the constant becomes $K_1\approx 1.9764\ge 1$ and the theorem can no longer be applied in this case.

Regarding the stability of Equation (9), Theorem 10 ensures that the equation is Ulam–Hyers stable, while Corollary 1 further establishes that it is also generalized Ulam–Hyers stable.

To examine the applicability of Theorem 11, consider the function $r(t)=exp(-t)$, defined on the interval $t\in [0,1]$, and the kernel $\psi (t)=\sqrt{t+1}$. A direct computation gives ${\mathbb{I}}_{0+}^{0.5,1,\psi }r(1)\approx 0.2369$ and since $r(t)=exp(-t)$ satisfies $0.3678<r(t)\le 1$ for all $t\in [0,1]$, it follows that the following is true:

$${\mathbb{I}}_{0+}^{0.5,1,\psi }r(1)\le r(t),forallt\in [0,1].$$

Therefore, all the conditions of Theorem 11 are satisfied, and we conclude that the equation is also Ulam–Hyers–Rassias stable with respect to the function r.

Example 2.

We now illustrate the applicability of Theorem 8 with the following example. Consider the following fractional differential equation:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{0+}^{0.7,2,\psi }u(t)=arctan\left(u(t)+{}^C\mathbb{D}_{0+}^{0.7,2,\psi }u(t)\right)+{\displaystyle \frac{u(t)+{}^C\mathbb{D}_{0+}^{0.7,2,\psi }u(t)}{5}},t\in [0,1],\hfill \\ u(0)=-u(1),\hfill \end{array}\right.$$

(10)

where the kernel is given by $\psi (t)=arctan(t+1)$. Define the nonlinear function as follows

$$f(t,u,d)=arctan(u+d)+{\displaystyle \frac{u+d}{5}}.$$

We observe the following:

$$\left|f(t,u,d)\right|\le |arctan(u+d)|+{\displaystyle \frac{1}{5}}(|u+d|)\le {\displaystyle \frac{\pi }{2}}+{\displaystyle \frac{1}{5}}(|u|+\left|d\right|),$$

for all $t\in [0,1]$, $u,d\in \mathbb{R}$. Thus, the function f satisfies the growth condition:

$$\left|f(t,u,d)\right|\le L_0+L_1(|u|+\left|d\right|),$$

with $L_0={\displaystyle \frac{\pi }{2}}>0$ and $L_1={\displaystyle \frac{1}{5}}\in (0,1)$.

Since the left-hand side of (7) is as follows:

$${\displaystyle \frac{3}{2}}·{\displaystyle \frac{e^2}{\Gamma (1.7)}}{\left(arctan(2)-{\displaystyle \frac{\pi }{4}}\right)}^{0.7}·{\displaystyle \frac{1}{4}}\approx 0.9812<1,$$

all the hypotheses of Theorem 8 are fulfilled. Therefore, the problem (10) is guaranteed to have at least one solution.

Example 3.

Consider the fractional differential equation:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{0+}^{0.8,5,\psi }u(t)=t+cos\left(u(t)+{}^C\mathbb{D}_{0+}^{0.8,5,\psi }u(t)\right),t\in [0,1],\hfill \\ u(0)=-u(1),\hfill \end{array}\right.$$

(11)

where ψ is an arbitrary $C^1$ function such that ${\psi }^{\prime }(t)>0$, for all $t\in [0,1].$ Since the following is true:

$$|f(t,u(t),{}^C\mathbb{D}_{0+}^{0.8,5,\psi }u(t))|=\left|t+cos\left(u(t)+{}^C\mathbb{D}_{0+}^{0.8,5,\psi }u(t)\right)\right|\le 2,$$

for all $t\in [0,1]$, Theorem 9 guarantees the existence of at least one solution to the problem (11).

6. Conclusions

This study addresses a class of implicit fractional differential equations involving a generalized fractional derivative under anti-periodic boundary conditions. We proved the existence and uniqueness of solutions by employing suitable fixed-point theorems. Furthermore, we investigated the stability properties of the solutions, deriving sufficient conditions for Ulam–Hyers stability, generalized Ulam–Hyers stability, and Ulam–Hyers–Rassias stability. Our findings generalize earlier results, which are obtained as special cases of the results established in this paper. Overall, the findings contribute to the growing body of research on fractional differential equations, particularly those involving generalized tempered operators and anti-periodic boundary conditions, offering valuable insights for future research in this area.

As directions for future research, it would be interesting to investigate implicit FDEs of order $\alpha$ greater than 1. Another important avenue is the development of numerical methods for obtaining approximate solutions for the boundary value problems studied in this paper. Additionally, it would be valuable to test the generalized operators on previously studied cases involving classical fractional derivatives, in order to evaluate the modeling errors.