Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

Abstract

This article aims to explore the existence and stability of solutions to differential equations involving a $\psi$-Hilfer fractional derivative in the Caputo sense, which, compared to classical $\psi$-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the $\psi$-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the $\psi$-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a $\psi$-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a $\psi$-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

1. Introduction

The history of fractional calculus can be traced back to 1965, and after many years of development, it has become an important mathematical tool [1]. As science and technology advance, the realm of fractional calculus is seeing an ever-widening scope of application. Researchers are increasingly turning to this sophisticated mathematical toolkit as a solution to the limitations inherent in traditional calculus methods when it comes to capturing the intricacies of certain complex phenomena and challenges. Currently, fractional calculus is mainly applied in viscoelastic materials [2], electrochemistry [3], control engineering [4], fluid mechanics [5], statistical mechanics [6], numerical schemes [7], etc.

Recently, research on the Hilfer fractional derivative has become a popular direction in fractional calculus. The Hilfer fractional derivative generalizes the Riemann–Liouville (R-L) and Caputo fractional derivatives [8]. Drawing inspiration from the $\psi$-Caputo fractional derivative [9], Sousa et al. introduced a novel fractional operator termed the $\psi$-Hilfer fractional derivative in the R-L sense, which aims to unify the properties of the majority of fractional derivatives [10].

With the extensive usage of the $\psi$-Hilfer fractional derivative in the R-L sense, there has been a problem regarding the selection of the order $\beta$. This arises from the lack of clear physical meaning for the initial conditions associated with this derivative, unless $\beta =1$. To clarify this confusion, Jajarmi et al. started from the definition of the $\psi$-Hilfer fractional derivative in the R-L sense, extended the concept to include the $\psi$-Hilfer fractional derivative in the Caputo sense, and referred to it as the regularized $\psi$-Hilfer derivative [11].

On the other hand, the study of Ulam–Hyers stability and Ulam–Hyers–Rassias stability for fractional differential equations has garnered significant attention from many researchers; see [12,13,14]. After Sousa et al. proposed the $\psi$-Hilfer fractional derivative in the Riemann–Liouville sense, they studied the Ulam–Hyers stability and Ulam–Hyers–Rassias stability of Volterra integro-differential equations involving this derivative, making it one of the earliest articles to focus on the stability of the $\psi$-Hilfer fractional derivative [15].

In the literature [16], Oliveira et al. investigated the classical integro-differential equations containing the $\psi$-Hilfer fractional derivative in the R-L sense below:

$$\left\{\begin{array}{c}{}^{H}D_{a+}^{\alpha ,\beta ;\psi }y\left(x\right)=f\left(x,y\left(x\right),{\int }_a^{x}K(x,\tau ,y\left(\tau \right),y\left(\delta \left(\tau \right)\right))d\tau \right),x\in [a,b]\hfill \\ I_{a+}^{1-\gamma ;\psi }y\left(a\right)=c,\gamma =\alpha +\beta (1-\alpha ),\hfill \end{array}\right.$$

where ${}^{H}D_{a+}^{\alpha ,\beta ;\psi }(·)$ is a $\psi$-Hilfer fractional derivative in the R-L sense, and $I_{a+}^{1-\gamma ;\psi }(·)$ is a $\psi$-R-L fractional integral, with $0<\alpha <1,0<\beta <1,y\in C^1[a,b]$. The authors analyzed the Ulam–Hyers-Rassias, Ulam–Hyers, and semi-Ulam–Hyers–Rassias stability of the integro-differential equations on the interval $[a,b]$ using the Banach fixed-point theorem, as well as the stability on the semi-infinite interval $[a,\infty )$.

Motivated by the above discussions, in this paper, we address the initial-value problem (IVP) for nonlinear differential equations with a $\psi$-Hilfer fractional derivative in the Caputo sense:

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }x\left(t\right)=f(t,x\left(t\right)),t\in (a,b],\hfill \\ {x\left(t\right)|}_{t=a}=x_0,\hfill \end{array}\right.$$

(1)

where $\alpha \in (0,1),\beta \in [0,1],a<b$, and ${}^C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }x(·)$ is the $\psi$-Hilfer fractional derivative in the Caputo sense; see Definition 5. Let $I=[a,b]$, X be a Banach space in which a norm $\Vert ·\Vert$, then $x\in C^1((a,b],X)$. Let $f:I\times X\to X$ be a given continuous function and $x_0$ be the initial value of x.

In this article, we explore the relationship between the $\psi$-Hilfer fractional derivative in the Caputo sense and the $\psi$-R-L fractional integral, identifying them as inverse operators. The flexibility in selecting $\psi$ considerably extends the applications of this derivative. Building on this property, we have demonstrated the existence of solutions for differential equations that incorporate linear $\psi$-Hilfer fractional derivatives in the Caputo sense. Differing from the strict Lipschitz condition employed by Jajarmi et al. to establish solution existence [11], our approach in Theorem 1 employs more relaxed conditions. We assume the nonlinear term $f(t,x(t\left)\right)$ is well defined across the interval I and regulated by the continuous positive function $h\left(t\right)(1+\parallel x(t)\parallel )$, ensuring the existence of solutions. In addition, we use Banach’s fixed-point theorem to demonstrate the stability of the differential equation involving the $\psi$-Hilfer fractional derivative in the Caputo sense. Unlike the equations studied by Oliveira et al. [16], our equation incorporates the $\psi$-Hilfer fractional derivative in the Caputo sense, which gives our initial conditions a clear physical meaning. Additionally, the solution operator is bounded at the initial conditions, resulting in significant differences between our approach to proving stability and that of [16].

This article will discuss the following aspects. In Section 2, the properties of the measure of non-compactness, the definitions of several fractional operators, and the relationships among these fractional operators are provided. Besides that, the knowledge related to Ulam–Hyers–Rassias stability, semi-Ulam–Hyers–Rassias stability and fixed-point theorems will also be elucidated in this section. In Section 3, we provide the proof process of the existence of the solution to Equation (1). In Section 4, we discuss the Ulam–Hyers–Rassias and semi-Ulam–Hyers–Rassias stability results for the solution of Equation (1). In the final section, we illustrate our results through two examples.

2. Preliminaries

Definition 1

([17]). Let B be a metric space, and let J denote a bounded subset of B. The Kuratowski measure of non-compactness is defined in the following manner:

$$\mu \left(J\right)=inf\left\{\xi >0|J=\bigcup _{i=1}^t{J}_i,diam{J}_i\le \xi \right\}.$$

Lemma 1

([18]). In the Banach space, there exist subsets J, $J_1$ and $J_2$. We provide the following properties:

1.

$0\le \mu \left(J\right)<+\infty ;$

2.

$\mu \left(J\right)=\mu \left(\overline{J}\right);$

3.

$\mu \left(J\right)=0\iff J$ is relatively compact;

4.

$J_1\subset J_2\Rightarrow \mu \left(J_1\right)\le \mu \left(J_2\right);$

5.

$\mu (J_1+J_2)\le \mu \left(J_1\right)+\mu \left(J_2\right),$ where $J_1+J_2=\left\{t\right|t=m+n,m\in J_{1}n\in J_2\};$

6.

$\mu \left(cJ\right)=\parallel c\parallel \mu \left(J\right),c\in \mathbb{R},$ where $cJ=\left\{t\right|t=cq,q\in J\}.$

In the sections that follow, we will explore various properties and outcomes associated with fractional calculus that incorporates the kernel function $\psi$.

Definition 2

([1]). Let $\alpha \in (0,+\infty )$, $t\in I$, $x\left(t\right)$ be an integrable function and $\psi \left(t\right)\in C^1(I,X)$ be a strictly increasing function; the αth order ψ-R-L integral of $x\left(t\right)$ is defined as follows:

$$I_{a+}^{\alpha ;\psi }x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}x\left(\tau \right)d\tau .$$

The definition of the ψ-R-L fractional derivate of order $\alpha >0$ with respect to t for an integrable function x is given as follows:

$$\begin{array}{c}\hfill D_{a+}^{\alpha ;\psi }x\left(t\right)={\delta }_{\psi }^n{I}_{a+}^{n-\alpha ,\psi }x\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\delta }_{\psi }^n{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{n-\alpha -1}x\left(\tau \right)d\tau ,\end{array}$$

where $n=\left[\alpha \right]+1,{\delta }_{\psi }^n={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n$ and this expression is defined in the same manner throughout the subsequent text.

Lemma 2

([1]). Let ${\alpha }_1,{\alpha }_2\in (0,+\infty )$. Then, the ψ-R-L integral has the following semigroup property:

$$I_{a+}^{{\alpha }_1;\psi }{I}_{a+}^{{\alpha }_2;\psi }x\left(t\right)=I_{a+}^{{\alpha }_1+{\alpha }_2;\psi }x\left(t\right).$$

Lemma 3

([1]). For any $\alpha ,\beta >0$ and $x\left(t\right)={(\psi \left(t\right)-\psi \left(a\right))}^{\beta -1}$, the following equality holds:

$$I_{a+}^{\alpha ;\psi }x\left(t\right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}}{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha +\beta -1}.$$

Definition 3

([9]). Let $\alpha >0,n\in \mathbb{N},t\in I,x,\psi \in C^n(I,X)$, and $\psi \left(t\right)\in C^1(I,X)$ be a strictly increasing function. Then, the αth order ψ-Caputo fractional derivative with respect to x can be expressed as

$${}^{C}D_{a+}^{\alpha ;\psi }x\left(t\right)=I_{a+}^{n-\alpha ,\psi }{\delta }_{\psi }^n\left[x\left(t\right)\right],$$

where $n=\left[\alpha \right]+1,{\delta }_{\psi }^n\left[x\left(t\right)\right]={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n}x\left(t\right).$ In particular, if $\alpha \in (0,1]$, then

$$\begin{array}{c}\hfill {}^{C}D_{a+}^{\alpha ;\psi }x\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\delta }_{\psi }\left[x\left(\tau \right)\right]d\tau .\end{array}$$

Lemma 4

([9]). If $r\in (0,1)$ and $x\in C^n(I,X)$, then

$$\begin{array}{c}\hfill I_{a^{+}}^{r;\psi }{}^{C}D_{a^{+}}^{r;\psi }x\left(t\right)=x\left(t\right)-x\left(a\right).\end{array}$$

(2)

Lemma 5

([11]). Let $\alpha \in (0,1)$ and $x\in C^1(I,X)$; then,

$${}^{C}D_{a+}^{\alpha ;\psi }x\left(t\right)=D_{a+}^{\alpha ;\psi }x\left(t\right)-{\displaystyle \frac{1}{\Gamma (1-q)}}{\delta }_{\psi }\left(x_0\right){(\psi \left(t\right)-\psi \left(a\right))}^{-\alpha }.$$

Next, we will introduce the $\psi$-Hilfer fractional derivative in the R-L sense and the $\psi$-Hilfer fractional derivative in the Caputo sense, and compare these two types of fractional derivatives.

Definition 4

([10]). Let $\alpha \in (n-1,n)$ with $n\in \mathbb{N},\psi \in C^n(I,X)$, and $\psi \left(t\right)\in C^1(I,X)$ be a strictly increasing function, for all $t\in I$. The ψ-Hilfer fractional derivative of order α and type $\beta \in [0,1]$ in the R-L sense can be written as

$$\begin{array}{c}\hfill {}^{H}D_{a+}^{\alpha ,\beta ;\psi }x\left(t\right)=I_{a+}^{\beta (n-\alpha );\psi }{\delta }_{\psi }^n{I}_{a+}^{(1-\beta )(n-\alpha );\psi }x\left(t\right).\end{array}$$

Building upon the definition of the ψ-Hilfer fractional derivative in the R-L sense provided earlier, it can also be articulated in the following manner:

$$\begin{array}{cc}\hfill {}^{H}D_{a+}^{\alpha ,\beta ;\psi }x\left(t\right)& =I_{a+}^{r-\alpha ;\psi }{D}_{a+}^{r;\psi }x\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma (r-\alpha )}}{\int }_a^t{(\psi \left(t\right)-\psi \left(\tau \right))}^{r-\alpha -1}{\psi }^{\prime }\left(\tau \right)D_{a^{+}}^{r;\psi }x\left(\tau \right)d\tau ,\hfill \end{array}$$

(3)

where $r=\alpha +\beta (n-\alpha ).$

Definition 5

([11]). Let $\alpha \in (0,+\infty )$, $t\in I$, $x\left(t\right)$ be an integrable function and $\psi \left(t\right)\in C^1(I,X)$ be a strictly increasing function. Then, the ψ-Hilfer fractional derivative in the Caputo sense is defined by

$$\begin{array}{cc}\hfill {}^C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }x\left(t\right)& =I_{a^{+}}^{r-\alpha ;\psi }{}^{C}D_{a^{+}}^{r;\psi }x\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma (r-\alpha )}}{\int }_a^t{(\psi \left(t\right)-\psi \left(\tau \right))}^{r-\alpha -1}{\psi }^{\prime }\left(\tau \right){}^{C}D_{a^{+}}^{r;\psi }x\left(\tau \right)d\tau ,\hfill \end{array}$$

where $r=\alpha +\beta (n-\alpha ).$

Lemma 6.

If $\alpha \in (0,1)$, $\beta \in [0,1]$ and $x\in C^1(I,X)$ then

$$\begin{array}{cc}\hfill {}^C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }{I}_{a^{+}}^{\alpha ;\psi }x\left(t\right)& =I_{a^{+}}^{r-\alpha ;\psi }{}^{C}D_{a^{+}}^{r;\psi }{I}_{a^{+}}^{\alpha ;\psi }x\left(t\right)\hfill \\ \hfill & =I_{a^{+}}^{r-\alpha ;\psi }{I}_{a^{+}}^{n-r;\psi }{\delta }_{\psi }^n{I}_{a^{+}}^{\alpha ;\psi }x\left(t\right)\hfill \\ \hfill & =I_{a^{+}}^{r-\alpha ;\psi }{I}_{a^{+}}^{n-r;\psi }{I}_{a^{+}}^{\alpha -n;\psi }x\left(t\right)\hfill \\ \hfill & =x\left(t\right).\hfill \end{array}$$

(4)

Remark 1.

This property is the same as that of the ψ-Caputo fractional derivative, as detailed in Theorem 5 of [9]. Utilizing this property, we can derive other important conclusions related to the ψ-Hilfer derivative in the Caputo sense.

By Lemma 5, Lemma 3, and Equation (3), we have established the relationship between the order $\alpha \in (0,1)$$\psi$-Hilfer fractional derivative in the Caputo sense and the $\psi$-Hilfer fractional derivative in the R-L sense:

$${}^C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }x\left(t\right)={}^{H}D_{a^{+}}^{\alpha ,\beta ;\psi }x\left(t\right)-{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\delta }_{\psi }\left[x_0\right]{(\psi \left(t\right)-\psi \left(a\right))}^{-\alpha }.$$

Definition 6

([15]). If for each $x\left(t\right)\in C^1((a,b],X)$ satisfying

$$\parallel x\left(t\right)-I_{a+}^{\alpha ;\psi }f(t,x\left(t\right))\parallel \le I_{a+}^{\alpha ;\psi }\Phi \left(t\right),$$

where $\Phi \left(t\right)>0$ consistently increases as a continuous function across all t, and assuming the presence of a solution $x_1\left(t\right)$ to Equation (1), alongside a chosen positive constant C, this ensures the fulfillment of the subsequent inequality for all t

$$\parallel x\left(t\right)-x_1\left(t\right)\parallel \le C\Phi \left(t\right),$$

then, it implies that Equation (1) has Ulam–Hyers–Rassias stability. In this case, if $\Phi \left(t\right)$ takes on a constant value, then Equation (1) exhibits Ulam–Hyers stability.

Definition 7

([16]). If for each $x\left(t\right)\in C^1((a,b],X)$ satisfying

$$\parallel x\left(t\right)-I_{a+}^{\alpha ;\psi }f(t,x\left(t\right))\parallel \le \sigma ,$$

where $\sigma >0$, it follows that Equation (1) possesses a solution $x_1\left(t\right)$, and a positive constant C can be identified such that

$$\parallel x\left(t\right)-x_1\left(t\right)\parallel \le C\Phi \left(t\right),$$

where $\Phi \left(t\right)>0$ for all t, which implies that Equation (1) has semi-Ulam–Hyers–Rassias stability.

Lemma 7

([19]). Suppose $(G,d)$ constitutes a generalized complete metric space, and consider Q as a self-mapping on G with a Lipschitz constant $N<1$. Given any $g\in G$, should there be an $i\in \mathbb{N}$ for which $d(Q^{i+1}g,Q^{i}g)<\infty$, then

(i)    

The sequence denoted by $\left\{Q^i{g}_0\right\}$ tends toward a fixed point, $g^{*}$, of the mapping Q, starting from an initial point $g_0$ within G.

(ii)   

In the set $G^{*}=\{k\in G\mid d(Q^{i}k,g)<\infty \}$, $k^{*}$ represents the sole fixed point of the function Q.

(iii) 

If $g\in G^{*}$,then $d(g,k^{*})\le {\displaystyle \frac{1}{1-N}}d(Tg,g)$.

To obtain various stability results for Ulam–Hyers on the interval I, we establish the following Banach spaces $d_1(·)$ and $d_2(·)$ in a manner similar to that in reference [20].

$$\begin{array}{c}\hfill d_1(x,y)=inf\left\{C|\parallel x\left(t\right)-y\left(t\right)\parallel \le C{\Phi }_1\left(t\right),t\in I,C>0\right\},\end{array}$$

where ${\Phi }_1\left(t\right)>0$ is a non-decreasing continuous function on I.

$$\begin{array}{c}\hfill d_2(x,y)=sup\left\{C|{\displaystyle \frac{\parallel x\left(t\right)-y\left(t\right)\parallel }{{\Phi }_2\left(t\right)}}\le C,t\in I,C>0\right\},\end{array}$$

where ${\Phi }_2\left(t\right)>0$ is a non-increasing continuous function on I.

For ease of computation in subsequent calculations, we assume

$$D=\underset{t\in I}{sup}{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }}{\Gamma (\alpha +1)}}.$$

Next, let us consider the linear problem corresponding to Equation (1).

Lemma 8.

Let $\alpha \in (0,1)$ and $p\left(t\right)$ be a continuous function on $(a,b]$. Then, the linear initial value problem for Equation (1)

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }x\left(t\right)=p\left(t\right),t\in (a,b],\hfill \\ {x\left(t\right)|}_{t=a}=x_0,\hfill \end{array}\right.$$

has a unique solution as follows:

$$x\left(t\right)=I_{a^{+}}^{\alpha ;\psi }p\left(t\right)+x_0,t\in I.$$

(5)

Proof. 

According to the definition of the $\psi$-Hilfer fractional derivative in the Caputo sense and using semigroup property, the following equation holds:

$$I_{a^{+}}^{\alpha ;\psi }{(}^C{\mathbb{D}}_{a^{+}}^{\alpha ,\beta ;\psi }x\left(t\right))=I_{a^{+}}^{\alpha ;\psi }\left(I_{a^{+}}^{r-\alpha ;\psi }{}^{C}D_{a^{+}}^{r;\psi }x\left(t\right)\right)=I_{a^{+}}^{r;\psi }{}^{C}D_{a^{+}}^{r;\psi }x\left(t\right),$$

where $r=\alpha +\beta (1-\alpha )$, and based on Lemma 4, it can be seen that

$$I_{a^{+}}^{r;\psi }{}^{C}D_{a^{+}}^{r;\psi }x\left(t\right)=x\left(t\right)-x_0.$$

Then, we obtain

$$x\left(t\right)=I_{a^{+}}^{\alpha ;\psi }p\left(t\right)+x_0.$$

Therefore, if $x\left(t\right)$ satisfies Equation (5), it is easy to obtain

$${}^C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }x\left(t\right)=p\left(t\right),t\in I.$$

□

Lemma 9

([21]). Suppose U is a subset that is bounded, closed, and convex within a Banach space, with 0 included in U, and $Q:U\to U$ is continuous. Then, for any subset V of U, Q possesses a fixed point if and only if $V=\overline{conv}Q\left(V\right)$, or $V=Q\left(V\right)\cup \left\{0\right\}\Rightarrow \mu \left(V\right)=0$.

3. Main Result

In this section, we will prove the existence of Equation (1) by applying the knowledge related to Mönch’s fixed-point theorem. Before proving the existence, we need the following assumptions.

Theorem 1.

Consider the following assumptions to be valid:

(H₁) 

The function $f(t,x)$ maintains continuity in relation to X for a.e. $t\in I$, and with respect to $x\in X$, it is measurable on I.

(H₂) 

For a.e. $t\in I$ and $x\in X$, the subsequent inequality holds:

$$\parallel f(t,x)\parallel \le h\left(t\right)(1+\parallel x\parallel ),$$

where h is a continuous function and $h:I\to [0,\infty )$.

(H₃) 

Let E be an arbitrary bounded subset of X; then, f satisfies

$$\mu \left(f\right(t,E\left)\right)\le h\left(t\right)\mu \left(E\right),$$

where $t\in I$.

If

$$M:=h_{m}D<1,$$

(6)

where $h_m={sup}_{t\in I}h\left(t\right)$, then Equation (1) has a solution.

Proof. 

Let operator $Q:C^1(I,X)\to C^1(I,X)$ be defined by

$$\left(Qx\right)\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^{t}f(\tau ,x\left(\tau \right)){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau +x_0.$$

(7)

From the above assumptions, it can be seen that the fixed points of the operator Q correspond to solution of Equation (1).

For each $\omega \in C^1(I,X)$, let $K\ge {\displaystyle \frac{\parallel x_0\parallel +M}{1-M}}>0,$ which satisfies the ball $B_K:={\{\parallel \omega \parallel }_{\infty }\le K\}$.

For any $x\in C^1(I,X)$ and each $t\in I$, we have

$$\begin{array}{cc}\hfill \parallel \left(Qx\right)\left(t\right)\parallel & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\parallel f(\tau ,x\left(\tau \right))\parallel (\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau +\parallel x_0\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{h\left(\tau \right)\parallel 1+x\left(\tau \right)\parallel (\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau +\parallel x_0\parallel \hfill \\ \hfill & \le {\displaystyle \frac{h_m(1+K)}{\Gamma \left(\alpha \right)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau +\parallel x_0\parallel \hfill \\ \hfill & \le h_m(1+K)D+\parallel x_0\parallel \hfill \\ \hfill & =M(1+K)+\parallel x_0\parallel \le K.\hfill \end{array}$$

Thus,

$${\parallel Q\left(x\right)\parallel }_{\infty }\le K.$$

The above process proves that Q is a transformation of $B_K$ onto itself.

Subsequently, we aim to demonstrate that the operator $Q:B_K\to B_K$ fulfills every criterion stipulated by Mönch’s fixed-point theorem, as outlined in Lemma 9.

Consider a sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ converging to x within $B_K$. For every $t\in I$, it follows that

$$\begin{array}{cc}\hfill & \parallel \left(Q{x}_n\right)\left(t\right)-\left(Qx\right)\left(t\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\parallel f(\tau ,x_n\left(\tau \right))-f(\tau ,x\left(\tau \right))\parallel {(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau .\hfill \end{array}$$

(8)

By $\left(H_1\right)$ and the Lebesgue-dominated convergence theorem, the Equation (8) implies

$$\parallel Q{x}_n{-Qx\parallel }_{\infty }\to 0\mathrm{as}n\to \infty .$$

The above proves that $Q:B_K\to B_K$ is continuous.

Next, we aim to demonstrate that $Q\left(B_K\right)$ is both bounded and equicontinuous. Given that $Q\left(B_K\right)$ is a subset of the bounded set $B_K$, it follows that $Q\left(B_K\right)$ is also bounded. Now, let $a_1,a_2\in I,a_1<a_2$ and let $f\in B_K$. Thus, we have

$$\begin{array}{cc}\hfill \parallel \left(Qx\right)\left(a_2\right)& -\left(Qx\right)\left(a_1\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\parallel {\int }_a^{a_2}f(\tau ,x\left(\tau \right)){(\psi \left(a_2\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & -{\int }_a^{a_1}f(\tau ,x\left(\tau \right)){(\psi \left(a_1\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau \parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{a_1}^{a_2}\parallel f(\tau ,x\left(\tau \right))\parallel {(\psi \left(a_2\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^{a_1}\parallel f(\tau ,x\left(\tau \right))\parallel ·|{(\psi \left(a_2\right)-\psi \left(\tau \right))}^{\alpha -1}-{(\psi \left(a_1\right)-\psi \left(\tau \right))}^{\alpha -1}|{\psi }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{a_1}^{a_2}h\left(\tau \right)(1+\parallel x\left(\tau \right)\parallel ){(\psi \left(a_2\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^{a_1}h\left(\tau \right)(1+\parallel x\left(\tau \right)\parallel )·|{(\psi \left(a_2\right)-\psi \left(\tau \right))}^{\alpha -1}-{(\psi \left(a_1\right)-\psi \left(\tau \right))}^{\alpha -1}|{\psi }^{\prime }\left(\tau \right)d\tau .\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill \parallel \left(Qx\right)\left(a_2\right)-\left(Qx\right)\left(a_1\right)\parallel & \le {\displaystyle \frac{h_m(1+K)}{\Gamma (\alpha +1)}}{(\psi \left(a_2\right)-\psi \left(a_1\right))}^{\alpha }\hfill \\ \hfill & +{\displaystyle \frac{h_m(1+K)}{\Gamma (\alpha +1)}}{\int }_a^{a_1}|{(\psi \left(a_2\right)-\psi \left(\tau \right))}^{\alpha -1}-{(\psi \left(a_1\right)-\psi \left(\tau \right))}^{\alpha -1}|{\psi }^{\prime }\left(\tau \right)d\tau .\hfill \end{array}$$

As $a_1\to a_2$, the entire expression tends to zero.

Now, consider W as a subset of $B_K$ such that $W\subset \overline{Q\left(W\right)}\cup \left\{0\right\}$. The function $t\to \omega \left(t\right)=\mu \left(W\right(t\left)\right)$ remains on Q as W is bounded and equicontinuous. By the properties of measures and the content of $H_3$, we can obtain

$$\begin{array}{cc}\hfill \omega \left(t\right)& \le \mu \left(\right(QW\left)\right(t)\cup \{0\left\}\right)\hfill \\ \hfill & \le \mu \left(\right(QW\left)\right(t\left)\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}h\left(\tau \right)\mu \left(W\left(\tau \right)\right)d\tau \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}h\left(\tau \right)\omega \left(\tau \right)d\tau \hfill \\ \hfill & \le h_{m}D{\parallel \omega \parallel }_{\infty },\hfill \end{array}$$

where $t\in I$. Thus,

$${\parallel \omega \parallel }_{\infty }\le M{\parallel \omega \parallel }_{\infty }.$$

According to the value of M in Equation (6), we obtain ${\parallel \omega \parallel }_{\infty }=0$, further leading to $\omega \left(t\right)=\mu \left(W\right(t\left)\right)=0$ for any $t\in I$. Further, by the properties of non-compactness measure, we can directly deduce that $W\left(t\right)$ is relatively compact in X. By virtue of the Ascoli–Arzeà theorem in reference [22], it can be concluded that W is relatively compact in $B_K$. With Lemma 9 applied, we can conclude that Q possesses a fixed point, representing a solution to Equation (1). □

4. Stability Analysis

In this section, we will present the Ulam–Hyers–Rassias stability, Ulam–Hyers stability, and semi-Ulam–Hyers–Rassias stability of the solution to Equation (1). Before proving the stability, we need to impose the following special growth conditions on the function f to ensure a more effective and reliable proof process.

(H₄) 

Let the continuous function $N(·)\ge 0$, the continuous function $f:I\times X\to X$ and for arbitrary $x,g\in X$ satisfying

$$\parallel f(t,x(t\left)\right)-f(t,g(t\left)\right)\parallel \le N\left(t\right)\parallel x\left(t\right)-g\left(t\right)\parallel ,\forall t\in I.$$

(H₅) 

Assume a constant P in the interval $(0,1)$ which satisfies

$$\underset{t\in I}{sup}{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}N\left(\tau \right)d\tau \le P.$$

Theorem 2.

Let $\left(H_4\right)$ and $\left(H_5\right)$ hold, ${\Phi }_1$ be a positive increasing continuous function, and $t\in I,x\in C^1(I,X)$ satisfying

$$\begin{array}{c}\hfill \parallel x\left(t\right)-\left(Tx\right)\left(t\right)\parallel \le I_{a+}^{\alpha ;\psi }{\Phi }_1\left(t\right),\end{array}$$

(9)

where $t\in I$ and $\left(Qx\right)\left(t\right)$ defined by Equation (7), then Equation (1) has a unique solution $x_1\left(t\right)$ satisfying the following inequality

$$\parallel x\left(t\right)-x_1\left(t\right)\parallel \le {\displaystyle \frac{D}{1-N}}{\Phi }_1\left(t\right),$$

where $t\in I,N\in (0,1)$. That means Equation (1) exhibits Ulam–Hyers–Rassias stability.

Proof. 

Considering the operator $Q:C^1(I,X)\to C^1(I,X)$, we have

$$\left(Qx\right)\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^{t}f(\tau ,x\left(\tau \right)){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau +x_0.$$

For $x,y\in I$, from conditions $\left(H_4\right)$, $\left(H_5\right)$ and metric $d_1(·)$, we obtain

$$\begin{array}{cc}\hfill \parallel \left(Qx\right)\left(t\right)-\left(Qy\right)\left(t\right)\parallel & \le {\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}\parallel f(\tau ,x\left(\tau \right))-f(\tau ,y\left(\tau \right))\parallel d\tau \hfill \\ \hfill & \le {\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}L\left(\tau \right)\parallel x\left(\tau \right)-y\left(\tau \right)\parallel d\tau \hfill \\ \hfill & \le PC{\Phi }_1\left(t\right).\hfill \end{array}$$

Therefore,

$$d_1(Qx,Qy)\le PC=P{d}_1(x,y).$$

Otherwise, Equation (9) implies that

$$\begin{array}{cc}\hfill \parallel x\left(t\right)-\left(Qx\right)\left(t\right)\parallel & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\Phi }_1\left(\tau \right)d\tau \hfill \\ \hfill & \le {\displaystyle \frac{{\Phi }_1\left(t\right)}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}d\tau \hfill \\ \hfill & \le sup{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\alpha }}{\Gamma (\alpha +1)}}{\Phi }_1\left(t\right)\hfill \\ \hfill & =D{\Phi }_1\left(t\right).\hfill \end{array}$$

From the above equation, it indicates that

$$d_1(x,Qx)\le D<\infty .$$

By combining $\left(i\right)$ and $\left(ii\right)$ of Lemma 7, we can conclude the existence of a unique fixed point $x_1$, which satisfies $Q{x}_1=x_1$. According to $\left(iii\right)$ of Lemma 7, we can obtain

$$d_1(x,x_1)\le {\displaystyle \frac{1}{1-N}}{d}_1(Qx,x)\le {\displaystyle \frac{D}{1-N}}.$$

In summary, we conclude that Equation (1) possesses Ulam–Hyers–Rassias stability. □

Remark 2.

If ${\Phi }_1\left(t\right)$ represents an arbitrary constant function in Theorem 2, then the Equation (1) has Ulam–Hyers stability.

Theorem 3.

Let $\left(H_4\right)$ and $\left(H_5\right)$ hold, ${\Phi }_2$ be a positive non-increasing continuous function, and the function $x\in C^1(I,X)$ satisfies

$$\begin{array}{c}\hfill \parallel x\left(t\right)-\left(Qx\right)\left(t\right)\parallel \le I_{a+}^{\alpha ;\psi }\xi ,\end{array}$$

(10)

where $t\in I$, $\xi >0$ and $\left(Qx\right)\left(t\right)$ are defined by Equation (7), then Equation (1) has a unique solution $x_1\left(t\right)$ satisfying the following inequality

$$\parallel x\left(t\right)-x_1\left(t\right)\parallel \le {\displaystyle \frac{\xi D{C}_1}{1-N}}{\Phi }_2\left(t\right),$$

where $t\in I,N\in (0,1)$. This indicates that Equation (1) has semi-Ulam–Hyers–Rassias stability.

Proof. 

Considering operator $Q:C^1(I,X)\to C^1(I,X)$, we have

$$\left(Qx\right)\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^{t}f(\tau ,x\left(\tau \right)){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}{\psi }^{\prime }\left(\tau \right)d\tau +x_0.$$

From the contents of $\left(H_4\right),\left(H_5\right)$ and metric $d_2(·)$, it is straightforward to establish that the following inequality holds for any $x,y\in X$.

$$\begin{array}{cc}\hfill {\displaystyle \frac{\parallel \left(Qx\right)\left(t\right)-\left(Qy\right)\left(t\right)\parallel }{{\Phi }_2\left(t\right)}}& \le {\displaystyle \frac{{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}\parallel f(\tau ,x\left(\tau \right))-f(\tau ,y\left(\tau \right))\parallel d\tau }{{\Phi }_2\left(t\right)}}\hfill \\ \hfill & \le {\displaystyle \frac{{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}L\left(\tau \right)\parallel x\left(\tau \right)-y\left(\tau \right)\parallel d\tau }{{\Phi }_2\left(t\right)}}\hfill \\ \hfill & \le PC.\hfill \end{array}$$

It is obvious that

$$d_2(Qx,Qy)\le PC=P{d}_2(x,y).$$

Based on the functional properties of ${\Phi }_2\left(t\right)$, we define $C_1>0$ such that it satisfies the following:

$${\displaystyle \frac{1}{{\Phi }_2\left(t\right)}}\le C_1.$$

From Equation (10), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\parallel x\left(t\right)-\left(Qx\right)\left(t\right)\parallel }{{\Phi }_2\left(t\right)}}& \le {\displaystyle \frac{I^{\alpha ;\psi }\xi }{{\Phi }_2\left(t\right)}}\hfill \\ \hfill & \le {\displaystyle \frac{\xi {\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}d\tau }{\Gamma \left(\alpha \right){\Phi }_2\left(t\right)}}\hfill \\ \hfill & =\xi D{C}_1.\hfill \end{array}$$

From the above equation, it indicates that

$$d_2(x,Qx)\le \xi D{C}_1<\infty .$$

By combining $\left(i\right)$ and $\left(ii\right)$ of Lemma 7, we can conclude the existence of a unique fixed point $x_1$, which satisfies $Q{x}_1=x_1$. According to $\left(iii\right)$ of Lemma 7, we obtain

$$d_2(x,x_1)\le {\displaystyle \frac{1}{1-N}}{d}_2(Qx,x)\le {\displaystyle \frac{\xi D{C}_1}{1-N}}.$$

In summary, we conclude that Equation (1) possesses semi-Ulam–Hyers–Rassias stability. □

5. Examples

We discuss the initial value problem for the $\psi$-Hilfer fractional derivative in the Caputo sense.

$$\left\{\begin{array}{c}{}^C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }x\left(t\right)=f(t,x\left(t\right)),t\in I,\hfill \\ {x\left(t\right)|}_{t=a}=1,\hfill \end{array}\right.$$

(11)

where $\alpha \in (0,1),\beta \in [0,1],I=[a,b]$, ${}_{}{}_C\mathbb{D}_{a^{+}}^{\alpha ,\beta ;\psi }{}_{}x(·)$ is the $\psi$-Hilfer fractional derivative in the Caputo sense and ${x\left(t\right)|}_{t=a}$ is initial value. Below, we discuss the existence and stability of Equation (11) using two specific examples.

Example 1.

Consider the Equation (11) and substitute $X=I,\psi \left(t\right)=t^{{\displaystyle \frac{1}{2}}},a=1,b=2,\alpha =\beta ={\displaystyle \frac{1}{2}},f(t,x\left(t\right))=sinx\left(t\right)$ into $\left(H_2\right)$, which then clearly shows that $h\left(t\right)$ takes 1.

When $h_m=1$, ${max}_{t\in I}\psi \left(t\right)=\sqrt{2}$ and we have

$$M=h_{m}D\le {\displaystyle \frac{{(\sqrt{2}-1)}^{\frac{1}{2}}}{\Gamma \left(\frac{3}{2}\right)}}\approx 0.725.$$

Obviously, $M\le 1$. From Theorem 1, we can deduce that Equation (11) has a solution on I.

Example 2.

We set $X=I$, and $f(t,x\left(t\right))={\displaystyle \frac{t^{\frac{1}{2}}}{(1+\mid x(t)\mid )}},t\in I$.

Now,

$$f(t,s)={\displaystyle \frac{1}{5e^t(1+\mid s\mid )}},$$

where $t\in I$, $s\in [0,\infty )$. In this case, $\psi \left(t\right)=t^2$, a = 1, b = 2, α and β both equal ${\displaystyle \frac{1}{2}}$.

Since the function f is continuous, and for $s,v\in [0,\infty )$ and $t\in I$, then

$$\mid f(t,s)-f(t,v)\mid \le t^{{\displaystyle \frac{1}{2}}}(\mid s-v\mid ).$$

From the above expression, it is evident that $N\left(t\right)=t^{{\displaystyle \frac{1}{2}}}$ in $\left(H_4\right)$.

Thus,

$$\begin{array}{cc}\hfill & \underset{t\in I}{sup}{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{\alpha -1}N\left(\tau \right)d\tau \hfill \\ \hfill & =\underset{t\in I}{sup}{\displaystyle \frac{1}{\Gamma \left(\frac{1}{2}\right)}}{\int }_1^t{\displaystyle \frac{2{\tau }^{\frac{1}{2}}}{{(t^2-{\tau }^2)}^{\frac{1}{2}}}}d\tau \hfill \\ \hfill & \le {\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_1^2{\displaystyle \frac{\tau \frac{1}{2}}{{(4-{\tau }^2)}^{\frac{1}{2}}}}d\tau \hfill \\ \hfill & \approx 0.616.\hfill \end{array}$$

Hence, the condition $\left(H_5\right)$ is satisfied with $Q=0.616<1$.

6. Conclusions

In this paper, we establish the existence of solutions for differential equations involving the $\psi$-Hilfer fractional derivative in the Caputo sense. Our results are derived through growth conditions and Mönch’s fixed-point theorem, providing a rigorous framework for addressing these complex issues. Furthermore, we analyze the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of Equation (1), which enhances the understanding and resolution of practical problems related to complex nonlinear and fractional-order systems. To validate and illustrate the theoretical results, numerical examples are provided, demonstrating the practical applicability of our approach. Furthermore, Equation (1) in this paper does not currently address the stability of nonlocality, which will be a focus of our future research. In the future, we will further investigate various properties of the $\psi$-Hilfer fractional derivative in the Caputo sense, including asymptotic stability, exponential stability, and maximal regularity.