Study of Uniqueness and Ulam-Type Stability of Abstract Hadamard Fractional Differential Equations of Sobolev Type via Resolvent Operators

Abstract

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives.

1. Introduction

It is widely recognized that fractional differential and integral equations have played a crucial role in advancing various branches of mathematics, as well as other fields such as science and engineering. Several references, including Refs. [1,2,3,4] support this fact. The Riemann–Liouville and Caputo derivatives are widely recognized as the two primary definitions of fractional calculus, encompassing both integration and differentiation. The Hadamard derivative, with its integral kernel in the form of $log\left(\frac{x}{t}\right)$, is particularly effective in characterizing ultra-slow diffusion processes. Additionally, the fractional power ${\left(x\frac{d}{dx}\right)}^{\alpha }$ is well-suited for situations involving the half-axis and remains invariant under dilation. Scholars have extensively examined the definition and properties of the Hadamard fractional derivative, as documented in Refs. [1,2,5,6,7,8,9,10,11,12].

In Ref. [13], the authors conducted a study on the existence of and Ulam-type stability concepts related to functional abstract fractional differential inclusions in Banach spaces. Specifically, their investigation focused on cases where there are no instantaneous impulses involved.

Li and Li [3,14], conducted a study on the stability and logarithmic decline of solutions to fractional differential equations (FDEs) in both linear and nonlinear cases. On the other hand, in Ref. [6], the authors Balachandran and Kiruthika focused on investigating the existence of solutions to nonlinear fractional integro-differential equations of the Sobolev type. Their study included a nonlocal condition.

In the past two decades, numerous mathematicians have developed a theory of abstract, impulsive fractional differential equations with nonlocal conditions. This theory has made use of the resolvent operator and other properties specific to fractional differential equations. Refs. [15,16,17,18,19,20,21] provide further insight into this research area. In Ref. [22], a study on the stability analysis of nonlinear Hadamard fractional differential systems. The stability analysis of Hadamard and Caputo–Hadamard fractional nonlinear systems, both with and without delay, was investigated in Ref. [23]. The authors of that paper examined the stability properties of such systems. In Refs. [24,25,26,27,28,29,30,31,32], the authors focused on investigating the Ulam–Hyers and Ulam–Hyers–Rassias stability of linear and nonlinear fractional differential equations involving various fractional derivatives.

In Refs. [33,34], the authors derived fractional inequalities by utilizing a Hadamard fractional operator. This operator was employed to establish the existence and uniqueness of solutions for fractional differential equations.

The aim of this work is an abstract Hadamard fractional differential equation of the Sobolev type. Such equations arise in various physical problems, including the flow of fluid through fissured rocks, thermodynamics, and the propagation of long waves with small amplitudes. Equations of the Sobolev type have been the subject of investigation by numerous researchers in the field.

In Ref. [35], Lightbourne and Rankin used semigroup techniques to study a problem involving a differential equation of the Sobolev type:

$$\left\{\begin{array}{c}{\left(B\phi \left(t\right)\right)}^{\prime }=A\phi \left(t\right)+g(t,\phi \left(t\right)),t>0,\hfill \\ \phi \left(t\right)=\mathsf{\Phi }\left(t\right),-r\le t\le 0,\hfill \end{array}\right.$$

(1)

where A and B are two closed linear operators with domains contained in a Banach space X and g is a continuous function.

In Ref. [6], Balachandran and Kiruthika extended the Sobolev equation of problem (1) by incorporating the Caputo fractional derivative. They introduced a study on the existence of the following problem:

$$\left\{\begin{array}{c}{D}^q\left(B\phi \left(t\right)\right)=A\phi \left(t\right)+f\left(t,\phi \left(t\right),{\int }_0^{t}k(t,s,\phi \left(s\right))ds\right),t\in \left[0,T\right],\hfill \\ \phi \left(0\right)+\mathcal{P}\left(\phi \right)={\phi }_0,\hfill \end{array}\right.$$

where A and B are two closed linear operators; ${\phi }_0$ belongs to a Banach space X; and f, k, and $\mathcal{P}$ are continuous functions. The resolvent operators play a crucial role in the analysis and solution of this problem.

This paper investigates the Ulam–Hyers stability and the Ulam–Hyers–Rassias stability of the following abstract Sobolev equation:

$$\left\{\begin{array}{c}{\mathcal{D}}^{\alpha }\left(\mathcal{B}\phi \left(t\right)\right)=\mathcal{A}\phi \left(t\right)+{ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\phi \left(t\right),\phi \left(t\right)),t\in \left[1,a\right],\hfill \\ \phi \left(1\right)={\phi }_1+G(\omega ,\phi ),\hfill \end{array}\right.$$

(2)

where ${\mathcal{D}}^{\alpha }$ is the Hadamard fractional derivative and ${\mathcal{I}}^{\sigma }$ is the Hadamard fractional integral $\alpha >0,\sigma <1,$ $a>1.$ $\mathcal{A}$ and $\mathcal{B}$ are two closed linear operators with domains $D\left(\mathcal{A}\right),$ $D\left(\mathcal{B}\right)$ contained in a Banach space X. Here, the family of parameter-dependent functions denoted as ${ꜰ}_{\omega }$ depends on $\omega$ and is defined by ${ꜰ}_{\omega }:\left[1,a\right]\times X^2\to X$ and $G(\omega ,.):$ $C(I,X)\to X,$ where ${ꜰ}_{\omega }$, G are continuous functions; $\omega \ge 0$, $I\subset \mathbb{R}$; and ${\phi }_1$ is given element of X.

2. Preliminaries

In this section, we introduce important concepts and characteristics relevant to our research. For more details, please refer to the works cited in Refs. [1,2,3,4].

Definition 1

([2]). The Hadamard fractional derivative of the function $f:[1,\infty )\to \mathbb{R}$ is defined as follows:

$$D^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(n-\alpha \right)}{\left(t\frac{d}{dt}\right)}^n{\int }_1^t{\left(log\frac{t}{s}\right)}^{n-\alpha -1}\frac{f\left(s\right)}{s}ds,n-1<\alpha <n,$$

(3)

where $n=\left[\alpha \right]+1$ with $\left[\alpha \right]$ denotes the integer part of the real number $\alpha$ and $log\left(t\right)=lo{g}_e\left(t\right).$

Definition 2

([2]). The Hadamard fractional integral operator of order $\alpha >0$ for a continuous function f defined on $[1,\infty )$ is given by:

$${\mathcal{I}}^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{f\left(s\right)}{s}ds,\alpha >0,$$

(4)

where $\mathsf{\Gamma }\left(\alpha \right):={\int }_0^{\infty }{e}^{-u}{u}^{\alpha -1}du.$

For comprehensive information regarding fractional integrals, fractional derivatives, and the properties of operators ${\mathcal{D}}^{\delta }$ and ${\mathcal{I}}^{\delta }$, readers are encouraged to refer to articles [1,2,4,12].

In the subsequent discussion, we define the Ulam-type stability for abstract fractional differential equations. For further details on this topic, readers can refer to works [13,27,28,29,36,37,38,39].

Definition 3

([27], page 04). The abstract fractional differential Equation (2) is said to be Ulam–Hyers stable if there exists a positive real number ${\delta }_{{ꜰ}_{\omega }}>0$ such that for every $\mu >0$ and every solution $\psi \in C(\left[1,a\right],X)$ of the inequality

$$|{\mathcal{D}}^{\alpha }\left[\mathcal{B}\psi \left(t\right)\right]-\mathcal{A}\psi \left(t\right)-{ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\psi \left(t\right),\psi \left(t\right))|\le \mu ,t\in \left[1,a\right],$$

(5)

there exists a solution $\phi \in C(\left[1,a\right],X)$ of the abstract fractional differential Equation (2) satisfying

$$\left|\psi \left(t\right)-\phi \left(t\right)\right|\le {\delta }_{{ꜰ}_{\omega }}\mu ,t\in \left[1,a\right].$$

Definition 4

([28], page 31). The abstract fractional differential Equation (2) is said to be generalized Ulam–Hyers stable if there exists a function ${\xi }_{{ꜰ}_{\omega }}\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ with ${\xi }_{{ꜰ}_{\omega }}\left(0\right)=0$, such that for every solution $\psi \in C\left(\right[1,a],X)$ of the Inequation (5), there exists a solution $\phi \in C\left(\right[1,a],X)$ of the abstract fractional differential Equation (2) satisfying

$$\left|\psi \left(t\right)-\phi \left(t\right)\right|\le {\xi }_{{ꜰ}_{\omega }}\left(\mu \right),t\in [1,a].$$

Definition 5

([27], page 04). The abstract fractional differential Equation (2) is said to be Ulam–Hyers–Rassias stable with respect to $\upsilon \in C\left([1,a],{\mathbb{R}}_{+}\right)$ if there exists a positive real number ${\delta }_{{ꜰ}_{\omega }}>0$ such that for every $\mu >0$ and every solution $\psi \in C\left(\right[1,a],X)$ of the inequality

$$|{\mathcal{D}}^{\alpha }\left[\mathcal{B}\psi \left(t\right)\right]-\mathcal{A}\psi \left(t\right)-{ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\psi \left(t\right),\psi \left(t\right))|\le \mu \upsilon \left(t\right),t\in [1,a],$$

(6)

there exists a solution $\phi \in C\left(\right[1,a],X)$ of the abstract fractional differential Equation (2) satisfying

$$\left|\psi \left(t\right)-\phi \left(t\right)\right|\le {\delta }_{{ꜰ}_{\omega }}\mu \upsilon \left(t\right),t\in [1,a].$$

Definition 6

([28], page 31). The abstract fractional differential Equation (2) is said to be generalized Ulam–Hyers–Rassias stable with respect to $\upsilon \in C\left([1,a],{\mathbb{R}}_{+}\right)$ if there exists a positive real number ${\delta }_{{ꜰ}_{\omega },\upsilon }>0$ such that for every solution $\psi \in C\left(\right[1,a],X)$ of the inequality

$$|{\mathcal{D}}^{\alpha }\left[\mathcal{B}\psi \left(t\right)\right]-\mathcal{A}\psi \left(t\right)-{ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\psi \left(t\right),\psi \left(t\right))|\le \upsilon \left(t\right),t\in [1,a],$$

(7)

there exists a solution $\phi \in C\left(\right[1,a],X)$ of the abstract fractional differential Equation (2) satisfying

$$\left|\psi \left(t\right)-\phi \left(t\right)\right|\le {\delta }_{{ꜰ}_{\omega },\upsilon }\upsilon \left(t\right),t\in [1,a].$$

Lemma 1.

Let us consider $f\in C(\left[1,a\right],X)$ and the continuous function $g:$ $C(I,X)\to X$. Then, the problem

$$\left\{\begin{array}{c}{\mathcal{D}}^{\alpha }\left(\mathcal{B}\phi \left(t\right)\right)=\mathcal{A}\phi \left(t\right)+f\left(t\right),0<\alpha <1,t\in \left[1,a\right],\hfill \\ \phi \left(1\right)={\phi }_1+g\left(\phi \right),\hfill \end{array}\right.$$

(8)

is equivalent to

$$\phi \left(t\right)={\phi }_1+g\left(\phi \right)+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}\mathcal{A}\phi \left(s\right)}{s}ds+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}f\left(s\right)}{s}ds.$$

Proof. 

By taking the Hadamard fractional integral of order $\alpha$ for the equation, we obtain

$$\phi \left(t\right)=C+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}\mathcal{A}\phi \left(s\right)}{s}ds+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}f\left(s\right)}{s}ds.$$

Now, using the boundary condition $\phi \left(1\right)={\phi }_1+g\left(\phi \right)$, we can conclude that $C={\phi }_1+g\left(\phi \right)$. This implies that the solution of the problem (8) is given by

$$\phi \left(t\right)={\phi }_1+g\left(\phi \right)+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}\mathcal{A}\phi \left(s\right)}{s}ds+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}f\left(s\right)}{s}ds.$$

 □

Remark 1.

The above problem (8) can also be written as an integral equation of the form:

$$\phi \left(t\right)=h\left(t\right)+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{\mathcal{H}\phi \left(s\right)}{s}ds,t\ge 0,$$

(9)

where $h\left(t\right)={\phi }_1+g\left(\phi \right)+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}{\displaystyle \frac{{\mathcal{B}}^{-1}f\left(s\right)}{s}}ds$ and $\mathcal{H}={\mathcal{B}}^{-1}\mathcal{A}.$

Let us assume that the integral Equation (9) has an associated resolvent operator $T\left(t\right)$, where $t\ge 0$, acting on X.

We define the resolvent operator for the integral Equation (9) similarly to [40] as follows:

Definition 7.

A one-parameter family of bounded linear operators ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ on X is referred to as the resolvent operator for Equation (9) if the following conditions are satisfied:

(i)

$T(.)x\in C\left([0,\infty ),X\right)$ and $T\left(0\right)x=x$ for all $x\in X.$

(ii)

$T\left(t\right)D\left(\mathcal{H}\right)\subset D\left(\mathcal{H}\right)$ and $\mathcal{H}T\left(t\right)x=T\left(t\right)\mathcal{H}x$ for all $x\in D\left(\mathcal{H}\right)$ and $t\ge 0.$

(iii)

For every $x\in D\left(\mathcal{H}\right)$ and $t\ge 0$

$$T\left(t\right)x=x+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{\mathcal{H}T\left(s\right)}{s}xds.$$

In this paragraph, we establish the following definition of a solution.

Definition 8.

A function $\phi \in C(\left[1,a\right],X)$ is called a mild solution of the integral Equation (9) on $\left[1,a\right]$ if

$${\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}{\displaystyle \frac{\phi \left(s\right)}{s}}ds\in D\left(\mathcal{H}\right)$$

for all $t\in \left[1,a\right],$ $h\left(t\right)\in C(\left[1,a\right],X)$ and

$$\phi \left(t\right)=\frac{\mathcal{H}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{\phi \left(s\right)}{s}ds+h\left(t\right),t\ge 0,t\in \left[1,a\right].$$

(10)

The following result can be derived from Ref. [40].

Lemma 2

([40]). Under the above conditions of Definition 7, the following properties are valid:

1. 

If φ is a mild solution of (9) on $\left[1,a\right],$ then the function $t\to {\int }_1^{t}T\left(t-s\right)h\left(s\right)ds$ is continuously differentiable on $\left[1,a\right]$, and we have

$$\phi \left(t\right)=\frac{d}{dt}{\int }_1^{t}T(t-s)h\left(s\right)ds,t\in \left[1,a\right].$$

2. 

If ${\left(T\left(t\right)\right)}_{t\ge 0}$ is differentiable and $h\in C(\left[1,a\right],D\left(\mathcal{H}\right)),$ then the function $\phi :\left[1,a\right]\to X$ defined by

$$\phi \left(t\right)={\int }_1^t{T}^{\prime }(t-s)h\left(s\right)ds+h\left(t\right),t\in \left[1,a\right],$$

is a mild solution of (9) on $\left[1,a\right].$

The paper is organized as follows. In Section 1, we will prove the theorem of uniqueness of the mild solutions and investigate the Ulam-type stability of the problem (2). In Section 2, we focus on the particular case in which $\mathcal{B}$ is equivalent to the identity operator, $\mathcal{B}\equiv \mathcal{I}$. In Section 3, we examine several examples of fractional partial differential equations with Hadamard derivatives.

3. Main Results

In this paper, the essential hypotheses are as follows:

Hypothesis 1.

The operators $\mathcal{A}:D\left(\mathcal{A}\right)\subset X\to X$ and $\mathcal{B}:D\left(\mathcal{B}\right)\subset X\to X$ are closed linear operators with $D\left(\mathcal{A}\right)\subset D\left(\mathcal{B}\right)$.

Hypothesis 2.

$\mathcal{B}$ is bijective, and and its inverse ${\mathcal{B}}^{-1}:X\to D\left(\mathcal{B}\right)$ exists.

Hypothesis 3.

The resolvent operator $T\left(t\right),$ $t\ge 0$ is differentiable, and there exists a function ${\varphi }_{\mathcal{H}}$ in $L_{loc}^1\left([0,\infty );{\mathbb{R}}^{+}\right)$ such that

$$∥T^{\prime }\left(t\right)x∥\le {\varphi }_{\mathcal{H}}\left(t\right){∥x∥}_{D\left(\mathcal{H}\right)},\mathit{for}\mathit{all}t>0.$$

Hypothesis 4.

For $\omega \ge 0,$  the function ${ꜰ}_{\omega }:\left[1,a\right]\times X^2\to X$ is completely continuous, and there exists $L_{\omega }>0$ such that

$$|{ꜰ}_{\omega }(t,x_1,y_1)-{ꜰ}_{\omega }(t,x_2,y_2)|\le L_{\omega }\left(∥x_1-x_2∥+∥y_1-y_2∥\right),\forall \left(t,x_i,y_i\right)\in \left[1,a\right]\times X^2,i=1,2.$$

Hypothesis 5.

There exists $M_{\omega }>0,$ such that

$$\left|G(\omega ,x_1)-G(\omega ,x_2)\right|\le M_{\omega }∥x_1-x_2∥,\forall x_i\in X,i=1,2.$$

Hypothesis 6.

$\exists {\omega }^{*}\ge 0,\forall \omega \ge {\omega }^{*},$

$$\mathsf{{\rm Y}}=\left(M_{\omega }+{\displaystyle \frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}}\left({\displaystyle \frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}}+1\right)\right)(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1})<1,$$

with $N=∥{\mathcal{B}}^{-1}∥.$

In our research, we denote $D\left(\mathcal{H}\right)$ as the domain of the operator $\mathcal{H}$, which is equipped with the graph norm ${∥x∥}_{D\left(\mathcal{H}\right)}=∥x∥+∥\mathcal{H}x∥.$ Additionally, the norm of $C\left(\right[1,a],X)$ is defined as ${∥x∥}_{C\left(\right[1,a],X)}=\underset{t\in \left[1,a\right]}{max}\left|x\left(t\right)\right|.$

3.1. Uniqueness of the Solution

Considering Lemma 1, we can conclude that problem (2) is equivalent to the following integral equation.

$$\begin{array}{ccc}\hfill \phi \left(t\right)& =& {\phi }_1+G(\omega ,\phi )+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}{\displaystyle \frac{\mathcal{H}\phi \left(s\right)}{s}}ds\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}{\displaystyle \frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\phi \left(s\right),\phi \left(s\right))}{s}}ds,\hfill \\ \hfill \mathrm{where}t& \in & \left[1,a\right].\hfill \end{array}$$

(11)

Definition 9. 

We define a function $\phi \in C(\left[1,a\right],X)$ as a mild solution of problem (2) on the interval $\left[1,a\right]$ if the following conditions are met:

First, for every $t\in \left[1,a\right]$, the function $\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}{\displaystyle \frac{\phi \left(s\right)}{s}}ds$ belongs to the domain $D\left(\mathcal{H}\right)$.

Second, the function φ satisfies the integral Equation (11).

Let us now prove the uniqueness result.

Theorem 1. 

Let the conditions of Hypotheses 1–6 be satisfied, and consider ${\phi }_1\in D\left(\mathcal{H}\right)$. Then, there exists a unique mild solution φ to problem (2) on the interval $\left[1,a\right]$.

Proof. 

Suppose there exists a resolvent operator ${\left(T\left(t\right)\right)}_{t\ge 0}$ that is differentiable, and the function ${ꜰ}_{\omega }$ is continuous in X. Based on point two of Lemma 2, we can define the map $\mathsf{\Lambda }:C(\left[1,a\right],X)\to C(\left[1,a\right],X)$ for $t\in \left[1,a\right]$ as follows:

$$\begin{array}{ccc}& & \mathsf{\Lambda }\phi \left(t\right)\hfill \\ & =& {\phi }_1+G(\omega ,\phi )+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\phi \left(s\right),\phi \left(s\right))}{s}ds\hfill \\ & & +{\int }_1^t{T}^{\prime }(t-s)\left({\phi }_1+G(\omega ,\phi )+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))}{s}d\tau \right)ds.\hfill \end{array}$$

(12)

Next, we will prove that $\mathsf{\Lambda }$ is a contraction. Based on the assumption on ${ꜰ}_{\omega }$ and considering $\phi \in C(\left[1,a\right],X)$, we observe the following:

$$\begin{array}{ccc}& & ∥{\int }_1^t{T}^{\prime }(t-s)\left({\phi }_1+G(\omega ,\phi )+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))}{\tau }d\tau \right)ds∥\hfill \\ & \le & {\int }_1^t∥T^{\prime }(t-s)∥∥{\phi }_1+G(\omega ,\phi )∥ds\hfill \\ & & +{\int }_1^t∥T^{\prime }(t-s)∥\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{1}{\tau }∥{\mathcal{B}}^{-1}∥∥{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))∥d\tau ds\hfill \\ & \le & \left(∥{\phi }_1+G(\omega ,\phi )∥+\frac{{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\underset{s\in \left[1,a\right]}{sup}∥{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))∥\right){∥{\varphi }_{\mathcal{H}}∥}_{L^1}.\hfill \end{array}$$

From this, we can deduce that the function

$$s\to T^{\prime }(t-s)\left({\phi }_1+G(\omega ,\phi )+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))}{s}d\tau \right),$$

is integrable on $\left[1,t\right]$ for all $t\in \left[1,a\right]$. This implies that $\mathsf{\Lambda }\phi \in C(\left[1,a\right],X)$, and thus $\mathsf{\Lambda }$, is well-defined. Furthermore, for $\phi ,\psi \in C(\left[1,a\right],X)$ and $t\in \left[1,a\right]$, we obtain:

$$\begin{array}{ccc}& & ∥\mathsf{\Lambda }\phi -\mathsf{\Lambda }\psi ∥\hfill \\ & \le & ∥G(\omega ,\phi )-G(\omega ,\psi )∥\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{∥{\mathcal{B}}^{-1}∥∥{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\phi \left(s\right),\phi \left(s\right))-{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))∥}{s}ds\hfill \\ & & +{\int }_1^t{\varphi }_{\mathcal{H}}(t-s)∥G(\omega ,\phi )-G(\omega ,\psi )∥ds\hfill \\ & & +{\int }_1^t{\varphi }_{\mathcal{H}}(t-s)\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{∥{\mathcal{B}}^{-1}∥∥{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))-{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\psi \left(\tau \right),\psi \left(\tau \right))∥}{\tau }d\tau ds,\hfill \end{array}$$

then

$$\begin{array}{ccc}& & ∥\mathsf{\Lambda }\phi -\mathsf{\Lambda }\psi ∥\hfill \\ & \le & M_{\omega }∥\phi -\psi ∥+\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(∥{\mathcal{I}}^{\sigma }\phi \left(\tau \right)-{\mathcal{I}}^{\sigma }\psi \left(\tau \right)∥+∥\phi -\psi ∥\right)\hfill \\ & & +M_{\omega }∥\phi -\psi ∥{∥{\varphi }_{\mathcal{H}}∥}_{L^1}+\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(∥I^{\sigma }\phi \left(\tau \right)-{\mathcal{I}}^{\sigma }\psi \left(\tau \right)∥+∥\phi -\psi ∥\right){∥{\varphi }_{\mathcal{H}}∥}_{L^1}\hfill \\ & \le & M_{\omega }∥\phi -\psi ∥\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right)+\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(∥{\mathcal{I}}^{\sigma }\phi \left(\tau \right)-{\mathcal{I}}^{\sigma }\psi \left(\tau \right)∥+∥\phi -\psi ∥\right)(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}).\hfill \end{array}$$

Since, we have

$$\begin{array}{ccc}\hfill ∥{\mathcal{I}}^{\sigma }\phi \left(\tau \right)-{\mathcal{I}}^{\sigma }\psi \left(\tau \right)∥& =& ∥\frac{1}{\mathsf{\Gamma }\left(\sigma \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\sigma -1}\frac{\phi \left(s\right)}{s}ds-\frac{1}{\mathsf{\Gamma }\left(\sigma \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\sigma -1}\frac{\psi \left(s\right)}{s}ds∥\hfill \\ & \le & \frac{1}{\mathsf{\Gamma }\left(\sigma \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\sigma -1}\frac{1}{s}∥\phi \left(s\right)-\psi \left(s\right)∥ds\hfill \\ & \le & \frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}∥\phi \left(s\right)-\psi \left(s\right)∥.\hfill \end{array}$$

(13)

then

$$\begin{array}{ccc}& & ∥\mathsf{\Lambda }\phi -\mathsf{\Lambda }\psi ∥\hfill \\ & \le & M_{\omega }∥\phi -\psi ∥\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right)+\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1})∥\phi -\psi ∥\hfill \\ & \le & \left(M_{\omega }+\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)\right)(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1})∥\phi -\psi ∥,\hfill \end{array}$$

Therefore, according to the assumption in Hypothesis 6, there exists ${\omega }^{*}\ge 0$ such that for all $\omega \ge {\omega }^{*}$, $\mathsf{\Lambda }$ is a contraction. By applying the Banach fixed-point theorem, we can conclude that there exists a unique mild solution to problem (2). This completes the proof. □

3.2. Stability

In this section, we will examine the Ulam-type stability of Equation (2).

Theorem 2. 

Assuming the conditions of Hypothesis 1–4, and considering $0<\alpha ,\sigma <1$, if there exists ${\omega }^{*}\ge 0$ such that for all $\omega \ge {\omega }^{*}$

$$\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right){∥{\varphi }_{\mathcal{H}}∥}_{L^1}<1-\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right),$$

(14)

then the abstract fractional problem (2) is Ulam–Hyers stable, and, as a result, it also demonstrates generalized Ulam–Hyers stability.

Proof. 

Using (12), we define the solution of the problem

$$\left\{\begin{array}{c}{\mathcal{D}}^{\delta }\left(\mathcal{B}\phi \left(t\right)\right)=\mathcal{A}\phi \left(t\right)+{ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\phi \left(t\right),\phi \left(t\right)),t\in \left[0,T\right],\hfill \\ \phi \left(1\right)=\psi \left(1\right).\hfill \end{array}\right.$$

(15)

by the function $\phi \in C(\left[1,a\right],X)$ where

$$\begin{array}{ccc}& & \phi \left(t\right)=C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\phi \left(s\right),\phi \left(s\right))}{s}ds\hfill \\ & & +{\int }_1^t{T}^{\prime }(t-s)\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))}{s}d\tau \right)ds.\hfill \end{array}$$

(16)

Referring to Definition 3 of the Ulam–Hyers stability and utilizing the integration of Inequality (17),

$$\left|{\mathcal{D}}^{\delta }\left[\mathcal{B}\psi \left(t\right)\right]-\mathcal{A}\psi \left(t\right)-{ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\psi \left(t\right),\psi \left(t\right))\right|\le \mu ,t\in \left[0,T\right],$$

(17)

we can deduce that

$$\begin{array}{ccc}& & \left|\psi \left(t\right)-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))}{s}ds\right.\hfill \\ & & \left.-{\int }_1^t{T}^{\prime }(t-s)\left(C_1+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\psi \left(\tau \right),\psi \left(\tau \right))}{s}d\tau \right)ds-C_1\right|\hfill \\ & \le & \frac{\mu }{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }.\hfill \end{array}$$

(18)

If $\phi \left(1\right)=\psi \left(1\right),$ then $C_0=C_1$, and we have

$$\begin{array}{ccc}& & \psi \left(t\right)-\phi \left(t\right)\hfill \\ & =& \psi \left(t\right)-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\phi \left(s\right),\phi \left(s\right))}{s}ds\hfill \\ & & -{\int }_1^t{T}^{\prime }(t-s)\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))}{s}d\tau \right)ds-C_0\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}\left[{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))-{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))\right]}{s}ds\hfill \end{array}$$

$$+{\int }_1^t{T}^{\prime }(t-s)\left(\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}\left[{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\psi \left(\tau \right),\psi \left(\tau \right))-{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\psi \left(\tau \right),\psi \left(\tau \right))\right]}{s}d\tau \right)ds.$$

So

$$\begin{array}{ccc}& & \psi \left(t\right)-\phi \left(t\right)\hfill \\ & =& \psi \left(t\right)-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))}{s}ds\hfill \\ & & -{\int }_1^t{T}^{\prime }(t-s)\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\psi \left(\tau \right),\psi \left(\tau \right))}{s}d\tau \right)ds-C_0\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}\left[{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))-{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\phi \left(s\right),\phi \left(s\right))\right]}{s}ds\hfill \end{array}$$

(19)

$$+{\int }_1^t{T}^{\prime }(t-s)\left(\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}\left[{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\psi \left(\tau \right),\psi \left(\tau \right))-{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))\right]}{s}d\tau \right)ds.$$

On the other hand, by using Hypothesis 4, we obtain

$$\begin{array}{ccc}& & \left|\psi \left(t\right)-\phi \left(t\right)\right|\hfill \\ & \le & \left|\psi \left(t\right)-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))}{s}ds\right.\hfill \\ & & \left.-{\int }_1^t{T}^{\prime }(t-s)\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\psi \left(\tau \right),\psi \left(\tau \right))}{s}d\tau \right)ds-C_0\right|\hfill \\ & & +\frac{L_{\omega }}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{∥{\mathcal{B}}^{-1}∥\left(∥{\mathcal{I}}^{\sigma }\psi \left(t\right)-{\mathcal{I}}^{\sigma }\phi \left(t\right)∥+∥\psi \left(t\right)-\phi \left(t\right)∥\right)}{s}ds\hfill \end{array}$$

$$+{\int }_1^t∥T^{\prime }(t-s)∥\left(\frac{L_{\omega }}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{∥{\mathcal{B}}^{-1}∥\left(∥{\mathcal{I}}^{\sigma }\psi \left(t\right)-{\mathcal{I}}^{\sigma }\phi \left(t\right)∥+∥\psi \left(t\right)-\phi \left(t\right)∥\right)}{s}d\tau \right)ds.$$

Then, due to (13) and (18), we obtain

$$\begin{array}{ccc}\hfill \left|\psi \left(t\right)-\phi \left(t\right)\right|& \le & \frac{\mu }{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }+\frac{N{L}_{\omega }}{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)∥\psi \left(t\right)-\phi \left(t\right)∥\hfill \\ & & +{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\frac{N{L}_{\omega }}{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)∥\psi \left(t\right)-\phi \left(t\right)∥\hfill \end{array}$$

$$\le \frac{\mu }{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }+\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right)\frac{N{L}_{\omega }}{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)∥\psi \left(t\right)-\phi \left(t\right)∥.$$

Hence,

$$\left(1-\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right)\frac{N{L}_{\omega }}{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)\right)∥\psi \left(t\right)-\phi \left(t\right)∥\le \frac{\mu }{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }.$$

Consequently, for each $t\in \left[1,a\right],$ we have

$$∥\psi \left(t\right)-\phi \left(t\right)∥\le \frac{{\left(loga\right)}^{\alpha }}{\left(1-\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right){\displaystyle \frac{N{L}_{\omega }}{\mathsf{\Gamma }\left(\alpha +1\right)}}{\left(loga\right)}^{\alpha }\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)\right)\mathsf{\Gamma }\left(\alpha +1\right)}\mu ={\delta }_{{ꜰ}_{\omega }}\mu ,$$

Since condition (14) holds, we can conclude that ${\delta }_{{ꜰ}_{\omega }}>0$. As a result, the abstract fractional problem (2) is Ulam–Hyers stable. By choosing ${\xi }_{{ꜰ}_{\omega }}\left(\mu \right)={\delta }_{{ꜰ}_{\omega }}\mu ,$ ${\xi }_{{ꜰ}_{\omega }}\left(0\right)=0$ we can further affirm that the abstract fractional problem (2) demonstrates generalized Ulam–Hyers stability. □

We study now the Ulam–Hyers–Rassias stability of the abstract fractional problem (2).

Theorem 3. 

Assuming the conditions of Hypotheses 1–H4 and 6 and condition (14), we also have the following hypothesis:

Hypothesis 7: There exists an function $\upsilon \in C(\left[1,a\right],{\mathbb{R}}_{+})$ and there exists ${\theta }_{\upsilon }>0$ such that for any $t\in \left[1,a\right].$

$$\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{\upsilon \left(s\right)}{s}ds\le {\theta }_{\upsilon }\upsilon \left(t\right).$$

Then, the abstract fractional problem (2) is Ulam–Hyers–Rassias stable.

Proof. 

Let us denote by $\psi \in C(\left[1,a\right],X)$ the solution of Inequality (6) from Definition 5, i.e.,

$$\left|{\mathcal{D}}^{\delta }\left[\mathcal{B}\psi \left(t\right)\right]-\mathcal{A}\psi \left(t\right)-{ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\psi \left(t\right),\psi \left(t\right))\right|\le \mu \upsilon \left(t\right),t\in \left[0,T\right].$$

(20)

Let $\phi \in C(\left[1,a\right],X)$ be the unique solution of problem (15). Thanks to the proof of the previous theorem, we have

$$\begin{array}{ccc}& & \phi \left(t\right)=C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\phi \left(s\right),\phi \left(s\right))}{s}ds\hfill \\ & & +{\int }_1^t{T}^{\prime }(t-s)\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))}{s}d\tau \right)ds.\hfill \end{array}$$

(21)

Integrating Inequality (20), we can obtain

$$\begin{array}{ccc}& & \left|\psi \left(t\right)-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))}{s}ds\right.\hfill \\ & & \left.-{\int }_1^t{T}^{\prime }(t-s)\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\phi \left(\tau \right),\phi \left(\tau \right))}{s}d\tau \right)ds-C_0\right|\hfill \\ & \le & \frac{\mu }{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{\upsilon \left(s\right)}{s}ds.\hfill \end{array}$$

(22)

Due to formula (19), we obtain

$$\begin{array}{ccc}& & \left|\psi \left(t\right)-\phi \left(t\right)\right|\hfill \\ & \le & \left|\psi \left(t\right)-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\psi \left(s\right),\psi \left(s\right))}{s}ds\right.\hfill \\ & & \left.-{\int }_1^t{T}^{\prime }(t-s)\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{{\mathcal{B}}^{-1}{ꜰ}_{\omega }(\tau ,{\mathcal{I}}^{\sigma }\psi \left(\tau \right),\psi \left(\tau \right))}{s}d\tau \right)ds-C_0\right|\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{∥{\mathcal{B}}^{-1}∥L_{\omega }\left(∥{\mathcal{I}}^{\sigma }\psi \left(t\right)-{\mathcal{I}}^{\sigma }\phi \left(t\right)∥+∥\psi \left(t\right)-\phi \left(t\right)∥\right)}{s}ds\hfill \\ & & +{\int }_1^t∥T^{\prime }(t-s)∥\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{∥{\mathcal{B}}^{-1}∥L_{\omega }\left(∥{\mathcal{I}}^{\sigma }\psi \left(t\right)-{\mathcal{I}}^{\sigma }\phi \left(t\right)∥+∥\psi \left(t\right)-\phi \left(t\right)∥\right)}{s}d\tau \right)ds\hfill \end{array}$$

So, by (22), we obtain

$$\begin{array}{ccc}& & \left|\psi \left(t\right)-\phi \left(t\right)\right|\hfill \\ & \le & \frac{\mu }{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{\upsilon \left(s\right)}{s}ds\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{∥{\mathcal{B}}^{-1}∥L_{\omega }\left(∥{\mathcal{I}}^{\sigma }\psi \left(t\right)-{\mathcal{I}}^{\sigma }\phi \left(t\right)∥+∥\psi \left(t\right)-\phi \left(t\right)∥\right)}{s}ds\hfill \\ & & +{\int }_1^t∥T^{\prime }(t-s)∥\left(C_0+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^s{\left(log\frac{s}{\tau }\right)}^{\alpha -1}\frac{∥{\mathcal{B}}^{-1}∥L_{\omega }\left(∥{\mathcal{I}}^{\sigma }\psi \left(t\right)-{\mathcal{I}}^{\sigma }\phi \left(t\right)∥+∥\psi \left(t\right)-\phi \left(t\right)∥\right)}{s}d\tau \right)ds.\hfill \end{array}$$

Now, using Hypotheses 6 and 7 and (13), we have

$$\left|\psi \left(t\right)-\phi \left(t\right)\right|\le \mu {\theta }_{\upsilon }\upsilon \left(t\right)+\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right)\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)∥\psi \left(t\right)-\phi \left(t\right)∥,$$

which implies that

$$\left(1-\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right)\frac{N{L}_{\omega }}{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)\right)∥\psi \left(t\right)-\phi \left(t\right)∥\le \mu {\theta }_{\upsilon }\upsilon \left(t\right).$$

Consequently, for any $t\in \left[1,a\right],$ we can write

$$∥\psi \left(t\right)-\phi \left(t\right)∥\le \frac{{\theta }_{\upsilon }}{\left(1-\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right)\frac{N{L}_{\omega }}{\mathsf{\Gamma }\left(\alpha +1\right)}{\left(loga\right)}^{\alpha }\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)\right)}\mu \upsilon \left(t\right).$$

From condition (14), there exists

$${\delta }_{{ꜰ}_{\omega }}={\displaystyle \frac{{\theta }_{\upsilon }}{\left(1-\left(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1}\right){\displaystyle \frac{N{L}_{\omega }}{\mathsf{\Gamma }\left(\alpha +1\right)}}{\left(loga\right)}^{\alpha }\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)\right)}}>0,$$

such that

$$∥\psi \left(t\right)-\phi \left(t\right)∥\le {\delta }_{{ꜰ}_{\omega }}\mu \upsilon \left(t\right).$$

Therefore, based on Definition 5, we can conclude that the abstract fractional boundary value problem (2) is Ulam–Hyers–Rassias stable. □

4. Particular Case $\mathcal{B}\mathbf{\equiv }\mathit{I}$

We consider the following problem:

$$\left\{\begin{array}{c}{\mathcal{D}}^{\alpha }\phi \left(t\right)=\mathcal{A}\phi \left(t\right)+{ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\phi \left(t\right),\phi \left(t\right)),t\in \left[1,a\right],\hfill \\ \phi \left(1\right)={\phi }_1+G(\omega ,\phi ),\hfill \end{array}\right.$$

(23)

with ${\mathcal{D}}^{\alpha }$ is the Hadamard fractional derivative and ${\mathcal{I}}^{\sigma }$ is the Hadamard fractional integral $\alpha >0,\sigma <1.$ $\mathcal{A}$ and $\mathcal{B}$ are two closed linear operators with a domain contained in the Banach space X. The functions ${ꜰ}_{\omega }:\left[1,a\right]\times X^2\to X$, $G:$ $\left[1,a\right]\times X\to X$ are continuous functions where $\omega \ge 0$ is a parameter and $u_1\in X$.

Problem (23) is equivalent to the following integral equation:

$$\begin{array}{ccc}\hfill \phi \left(t\right)& =& {\phi }_1+G(\omega ,\phi )+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{\mathcal{A}\phi \left(s\right)}{s}ds\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{{ꜰ}_{\omega }(s,{\mathcal{I}}^{\sigma }\phi \left(s\right),\phi \left(s\right))}{s}ds.\hfill \end{array}$$

The result of the uniqueness of the solution for problem (23) is as follows:

Corollary 1. 

Assuming Hypotheses 1–5, and if

$$\exists {\omega }^{*}\ge 0,\forall \omega \ge {\omega }^{*},\mathsf{{\rm Y}}=\left(M_{\omega }+{\displaystyle \frac{L_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}}\left({\displaystyle \frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}}+1\right)\right)(1+{∥{\varphi }_{\mathcal{H}}∥}_{L^1})<1.$$

Then, there exists a unique mild solution of problem (23).

The result of the stability of problem (23) is the following:

Corollary 2. 

Assume Hypotheses 1–4 and Hypothesis 7. If $\exists {\omega }^{*}\ge 0,\forall \omega \ge {\omega }^{*},$

$$\frac{L_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right){∥{\varphi }_{\mathcal{H}}∥}_{L^1}<1-\frac{L_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right).$$

Then, we have:

(a) 

The abstract fractional problem (23) is Ulam–Hyers stable and, consequently, generalized Ulam–Hyers stable.

(b) 

The abstract fractional problem (23) is Ulam–Hyers–Rassias stable.

5. Application

In this section, we will study the uniqueness and Ulam-type stability of a mild solution for a partial differential system with Hadamard derivatives, given as follows:

Problem 1. 

Let us consider the following partial differential equation with the Hadamard derivative in the space $X=L^2\left(\left[0,\pi \right]\right)$:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\left(\phi (t,x)\right)={\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}(t,x)+{\displaystyle \frac{1}{\omega }}\left({\displaystyle \frac{1}{\mathsf{\Gamma }\left(\sigma \right)}}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\sigma -1}{\displaystyle \frac{\phi (s,x)}{s}}ds\right)\hfill \\ \phi (t,0)=\phi (t,\pi )=0,t\in \left[1,e\right],\hfill \\ \phi (1,x)=e^x+{\displaystyle \frac{1}{{\omega }^2+1}}sin\phi ,x\in \left[0,\pi \right],\hfill \end{array}\right.$$

(24)

where $\alpha =\frac{1}{2},$ $\sigma =\frac{1}{2},$ and ω is sufficiently large. Let the closed linear operators $\mathcal{B}\phi =\phi$ and $\mathcal{A}\phi ={\phi }^{″}$ with domain

$$D\left(\mathcal{A}\right)=\left\{\phi \in X,{\phi }^{″}\in X,\phi \left(0\right)=\phi \left(\pi \right)=0\right\}.$$

Using [6], then $\mathcal{A}$ can be written as

$$\mathcal{A}\phi =\sum _{n=1}^{\infty }{n}^2⟨\phi ,e_n⟩e_n,\phi \in D\left(\mathcal{A}\right),$$

where

$$e_n\left(x\right)=\sqrt{\frac{2}{\pi }}sin\left(nx\right);x\in \left[0,\pi \right],n=1,2,\dots .$$

The problem is associated with a differential resolvent operator ${\left(T\left(t\right)\right)}_{t\ge 0}$, and there exists a constant $\eta >0$ such that

$$∥T^{\prime }\left(t\right)x∥\le \eta {∥x∥}_{D\left(\mathcal{A}\right)},\mathit{for}x\in D\left(\mathcal{A}\right).$$

On the other hand, we have

$${ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\phi \left(t\right),\phi \left(t\right))={\displaystyle \frac{1}{\omega }}\frac{1}{\mathsf{\Gamma }\left(\sigma \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\sigma -1}{\displaystyle \frac{\phi (s,x)}{s}}ds.$$

So,

$$\left|{ꜰ}_{\omega }(t,{\phi }_1,{\psi }_1)-{ꜰ}_{\omega }(t,{\phi }_2,{\psi }_2)\right|\le \frac{1}{\omega }\left(∥{\phi }_1-{\psi }_1∥+∥{\phi }_2-{\psi }_2∥\right),$$

and we have $G(\omega ,\phi )={\displaystyle \frac{1}{{\omega }^2+1}}sin\phi ,$ so

$$\left|G(\omega ,{\phi }_1)-G(\omega ,{\psi }_1)\right|\le \frac{1}{{\omega }^2+1}\left(∥{\phi }_1-{\psi }_1∥\right).$$

thus $L_{\omega }={\displaystyle \frac{1}{\omega }}$ and $M_{\omega }={\displaystyle \frac{1}{{\omega }^2+1}}$. Also,

$$\begin{array}{ccc}& & \left(M_{\omega }+{\displaystyle \frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}}\left({\displaystyle \frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}}+1\right)\right)(1+∥{\varphi }_{\mathcal{A}}∥)\hfill \\ & =& {\displaystyle \frac{1}{{\omega }^2+1}}+{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}\left({\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}+1\right)\left(\eta +1\right)\hfill \\ & \le & {\displaystyle \frac{1}{\omega }}(1+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}({\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}+1))\left(\eta +1\right)\approx 3.42\times {\displaystyle \frac{\left(\eta +1\right)}{\omega }}.\hfill \end{array}$$

So $\exists {\omega }^{*}=3.42\left(\eta +2\right)$ $>0,$ such that $\forall \omega \ge {\omega }^{*},$

$$\left(M_{\omega }+{\displaystyle \frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}}\left({\displaystyle \frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}}+1\right)\right)(1+∥{\varphi }_{\mathcal{A}}∥)<1.$$

Hence, for all $\omega \ge {\omega }^{*}$, the assumptions of Hypotheses 1–6 are satisfied. Therefore, based on Theorem 1, we can conclude that the problem (24) has a unique solution.

Now, we study the stability of the problem (24). We have

$$\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right){∥{\varphi }_{\mathcal{A}}∥}_{L^1}={\displaystyle \frac{1}{\omega }}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}\left({\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}+1\right)\eta ,$$

and

$$1-\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)=1-{\displaystyle \frac{1}{\omega \mathsf{\Gamma }\left(\frac{3}{2}\right)}}\left({\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}+1\right).$$

Then, using Theorem 2, $\forall \omega \ge {\omega }^{*}=3.42\left(\eta +2\right),$ condition (14)

$$\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right){∥{\varphi }_{\mathcal{A}}∥}_{L^1}<1-\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right),$$

(25)

is satisfied, and the problem (24) is Ulam-Hyers stable.

Let $\upsilon \left(t\right)=\delta ln\left(t\right),$ $\delta \in {\mathbb{R}}_{+}.$ Then, we have

$$\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{\upsilon \left(s\right)}{s}ds=\frac{\delta }{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\alpha -1}\frac{ln(t+1)}{s}ds\le \frac{\delta }{\mathsf{\Gamma }\left(\alpha +2\right)}ln\left(t\right)={\theta }_{\upsilon }\upsilon \left(t\right).$$

Therefore, the condition in Hypothesis 7 is satisfied with $\upsilon =\delta ln\left(t\right)$ and ${\theta }_{\upsilon }=\frac{\delta }{\mathsf{\Gamma }(\alpha +2)}$. Consequently, based on Theorem 3, problem (24) is Ulam–Hyers–Rassias stable.

Problem 2. 

Let us consider the following partial differential equation with the Hadamard derivative in the space $X=L^2\left(\left[0,\pi \right]\right)$:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\left(\phi (t,x)-{\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}(t,x)\right)={\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}(t,x)+{\displaystyle \frac{1}{\omega }}\left(sin\left(t\right)\phi (t,x)+\frac{1}{\mathsf{\Gamma }\left(\sigma \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\sigma -1}{\displaystyle \frac{\phi (s,x)}{s}}ds\right)\hfill \\ \phi (t,0)=\phi (t,\pi )=0,t\in \left[1,e\right],\hfill \\ \phi (1,x)=\chi \left(x\right)+{\displaystyle \frac{1}{\omega +1}}sin\phi ,x\in \left[0,\pi \right].\hfill \end{array}\right.$$

(26)

where $\alpha =\frac{1}{3},$ $\sigma =\frac{1}{2},$ω parameter large enough, and $\chi \in L^2\left(\left[0,\pi \right]\right).$ In this problem, we have the closed linear operators $\mathcal{B}\phi ={\phi }^{″}$ and $\mathcal{A}\phi =\phi -{\phi }^{″}$ with the domain

$$D\left(\mathcal{B}\right)=D\left(\mathcal{A}\right)=\left\{\phi \in X,{\phi }^{″}\in X,\phi \left(0\right)=\phi \left(\pi \right)=0\right\}.$$

Then, $\mathcal{A}$ and $\mathcal{B}$ can be written as

$$\begin{array}{ccc}\hfill \mathcal{A}\phi & =& \sum _{n=1}^{\infty }{n}^2⟨\phi ,e_n⟩e_n,\phi \in D\left(\mathcal{A}\right),\hfill \\ \hfill \mathcal{B}\phi & =& \sum _{n=1}^{\infty }\left(1+n^2\right)⟨\phi ,e_n⟩e_n,\phi \in D\left(\mathcal{B}\right),\hfill \end{array}$$

(see [35]), where

$$e_n\left(x\right)=\sqrt{\frac{2}{\pi }}sin\left(nx\right);x\in \left[0,\pi \right],n=1,2,\dots .$$

Then, we have, for $\phi \in X$

$$\mathcal{H}\phi ={\mathcal{B}}^{-1}\mathcal{A}\phi =\sum _{n=1}^{\infty }\frac{-n^2}{\left(1+n^2\right)}⟨\phi ,e_n⟩e_n,$$

and $∥{\mathcal{B}}^{-1}∥\le {\displaystyle \frac{1}{4}}.$ Since ${\mathcal{B}}^{-1}\mathcal{A}$ is a bounded operator, we have $∥T^{\prime }\left(t\right)∥\le \eta$ for all $t\ge 0$.

On the other hand,

$${ꜰ}_{\omega }(t,{\mathcal{I}}^{\sigma }\phi \left(t\right),\phi \left(t\right))={\displaystyle \frac{1}{\omega }}\left(sin\left(t\right)\phi \left(t\right)+\frac{1}{\mathsf{\Gamma }\left(\sigma \right)}{\int }_1^t{\left(log\frac{t}{s}\right)}^{\sigma -1}{\displaystyle \frac{\phi (s,x)}{s}}ds\right).$$

So,

$$\left|{ꜰ}_{\omega }(t,{\phi }_1,{\psi }_1)-{ꜰ}_{\omega }(t,{\phi }_2,{\psi }_2)\right|\le \frac{1}{\omega }\left(∥{\phi }_1-{\psi }_1∥+∥{\phi }_2-{\psi }_2∥\right),$$

and we have $G(\omega ,\phi )={\displaystyle \frac{1}{\omega }}sin\phi ,$ so

$$\left|G(\omega ,{\phi }_1)-G(\omega ,{\psi }_1)\right|\le \frac{1}{\omega }\left(∥{\phi }_1-{\psi }_1∥\right).$$

Thus $L_{\omega }=M_{\omega }={\displaystyle \frac{1}{\omega }}$. Moreover,

$$\begin{array}{ccc}& & \left(M_{\omega }+{\displaystyle \frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}}\left({\displaystyle \frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}}+1\right)\right)(1+∥{\varphi }_{\mathcal{A}}∥)\hfill \\ & =& {\displaystyle \frac{1}{\omega }}\left(1+{\displaystyle \frac{1}{4}}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{4}{3}\right)}}\left({\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}+1\right)\right)\left(\eta +1\right)\hfill \\ & \approx & 1.6{\displaystyle \frac{\left(\eta +1\right)}{\omega }}.\hfill \end{array}$$

So, $\exists {\omega }^{*}=1.6\left(\eta +2\right)>0,$ such that $\forall \omega \ge {\omega }^{*},$

$$\left(M_{\omega }+{\displaystyle \frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}}\left({\displaystyle \frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}}+1\right)\right)(1+∥{\varphi }_A∥)<1.$$

Therefore, for all $\omega \ge {\omega }^{*}$, the assumptions in Hypotheses 1–6 are satisfied. Consequently, based on—Theorem 1, we can conclude that problem (26) has a unique solution.

Now, we study the stability of problem (26). We have

$$\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right){∥{\varphi }_{\mathcal{A}}∥}_{L^1}={\displaystyle \frac{1}{4\omega }}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{4}{3}\right)}}\left({\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}+1\right)\eta ={\displaystyle \frac{0.6\eta }{\omega }},$$

and

$$\begin{array}{ccc}\hfill 1-\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right)& =& 1-\frac{1}{4\omega }{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{4}{3}\right)}}\left({\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{3}{2}\right)}}+1\right)\hfill \\ & =& 1-{\displaystyle \frac{0.6}{\omega }}.\hfill \end{array}$$

Then, from Theorem 2, $\forall \omega \ge {\omega }^{*}=1.6\left(\eta +2\right)$, condition (14)

$$\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right){∥{\varphi }_{\mathcal{A}}∥}_{L^1}<1-\frac{N{L}_{\omega }{\left(loga\right)}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\left(\frac{{\left(loga\right)}^{\sigma }}{\mathsf{\Gamma }\left(\sigma +1\right)}+1\right),$$

(27)

is satisfied, and the problem (26) is Ulam–Hyers stable.

6. Conclusions

In this study, we have investigated the uniqueness of mild solutions and the Ulam–Hyers stability as well as the Ulam–Hyers–Rassias stability for abstract fractional differential equations of the Sobolev type with nonlocal boundary conditions, utilizing the Hadamard derivative. The resolvent operators have played a crucial role in our analysis. We have established the uniqueness results by applying the Banach contraction principle. To demonstrate the applicability of our results, we provide some specific applications to which our findings can be applied.