Existence of Mild Solutions to Impulsive Fractional Equations with Almost-Sectorial Operators

Abstract

This study investigates the existence of mild solutions to impulsive fractional differential equations involving almost-sectorial operators. Through the application of solution operator techniques, fixed-point theory, and Laplace transform, we demonstrate the existence of a unique mild solution to the considered system.

1. Introduction

In the few past decades, fractional differential equations have attracted the attention of numerous researchers and gained importance due to their extensive applications in scientific disciplines such as engineering, control theory, physics, chemistry, biology, etc. The reader can find, in the following monographs [1,2,3,4,5], plenty of applications of fractional calculus and differential equations in diverse fields.

Impulsive differential equations have exhibited several advantages when used in the mathematical models of physical and biological processes. For more details, see [6,7,8,9] and the references therein. These types of equations can describe different real-world problems more naturally and in greater depth (see, for instance, [5,7,10]). Many studies have investigated the existence of solutions to impulsive fractional differential equations [11,12]. In addition, and more specifically, there are several published papers concerning the study of the existence of mild solutions, such as [7,13,14,15].

In [16], Karthikeyan et al. provide some existence results by applying Schauder’s fixed-point technique to a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost-sectorial operators:

$$\left\{\begin{array}{c}{D}_{0^{+}}^{\alpha ,\gamma }u\left(t\right)+\mathcal{A}u\left(t\right)=g\left(t,u\left(t\right),{\int }_0^{t}k(t,s)f(s,u\left(s\right)ds\right),t\in (0,t]=J,\hfill \\ I_{0^{+}}^{(1-\alpha )(1-\gamma )}{\left[u\left(t\right)\right]|}_{t=0}+h\left(u\left(t\right)\right)=u_0,\hfill \end{array}\right.$$

where $D_{0^{+}}^{\alpha ,\gamma }$ is the Hilfer fractional derivative of the order $\alpha \in (0,1)$ and type $\gamma \in [0,1]$, and $I_{0^{+}}^{(1-\alpha )(1-\gamma )}$ is a Riemann–Liouville fractional integral of the order $(1-\alpha )(1-\gamma ),$ assuming that $\mathcal{A}$ is an almost-sectorial operator in a Banach space.

Fang, in [17], investigated the existence of mild solutions to the following class of fractional differential equations with almost-sectorial operators and infinite delay in a separable complex Banach space X:

$$\left\{\begin{array}{c}{}^{c}D_t^{q}u\left(t\right)=\mathcal{A}u\left(t\right)+f(t,u\left(t\right),u_t),t\in (0,T],\hfill \\ u_0=\mathrm{\Phi }\in \mathcal{P},\hfill \end{array}\right.$$

where $T>0,$ and $0<q<1.$ The fractional derivative is understood here in the Caputo sense. $\mathcal{P}$ is a phase space, and $\mathcal{A}$ is an almost-sectorial operator.

In [18], Zhang et al. studied fractional Cauchy problems with almost-sectorial operators of the form

$$\left\{\begin{array}{c}{(}^L{D}_0^{q}x)\left(t\right)=\mathcal{A}x+f(t,x\left(t\right)),\mathrm{for}\mathrm{almost}\mathrm{all}t\in [0,a]=J,\hfill \\ \left(I_{0^{+}}^{(1-q)}x\right)\left(0\right)=x_0,\hfill \end{array}\right.$$

where ${}^{L}D_0^q$ is the Riemann–Liouville derivative of the order $q,$ $I_{0^{+}}^{(1-q)}$ is the Riemann–Liouville integral of the order $1-q,$$0<q<1,$ $\mathcal{A}$ is an almost-sectorial operator in a complex Banach space, and f is a given function.

Wang et al. [12] studied Cauchy problems for linear and semilinear time fractional evolution equations, and their work was based on the construction of a pair of families of operators in terms of the generalized Mittag–Leffler-type functions and the resolvent operators.

In this work, motivated and inspired by the above papers, we are interested in studying the following equations with an impulsive effect:

$$\left\{\begin{array}{cc}{\mathfrak{D}}_{\zeta }^{\mu }\upsilon \left(\zeta \right)=\mathcal{A}\upsilon \left(\zeta \right)+\mathfrak{g}(\zeta ,\upsilon \left(\zeta \right)),\hfill & 0\le \zeta \le \mathfrak{K},\hfill \\ \upsilon \left(0\right)={\upsilon }_0,\hfill \\ \Delta \upsilon \left({\zeta }_i\right)={\mathfrak{I}}_i\left(\upsilon \left({\zeta }_i^{-}\right)\right),\hfill & i=1,2,3,\dots ,\mathcal{l},\hfill \end{array}\right.$$

(1)

where $\mathfrak{X}$ is a Banach space; ${\mathfrak{D}}_{\zeta }^{\mu }$ denotes the regularized Caputo derivative of the order $\mu \in (0,1)$; and $0\le {\zeta }_0\le {\zeta }_1\le \cdots \le {\zeta }_{\mathcal{l}}=\mathfrak{K}$, $\mathfrak{g}:[0,\mathfrak{K}]\times \mathfrak{X}⟶\mathfrak{X}$ is a continuous function, and we suppose that $\mathcal{A}$ is an almost-sectorial operator on $\mathfrak{X}$, which will be denoted by $\mathcal{A}\in {\mathrm{\Theta }}_{\theta }^r\left(\mathfrak{X}\right)$, for some $-1<r<0$, and $0<\theta <{\displaystyle \frac{\pi }{2}}$.

Additionally, ${\mathfrak{I}}_i\in \mathfrak{C}(\mathfrak{X},\mathfrak{X}),(i=0,1,2,\dots ,\mathcal{l})$ constitute a collection of bounded functions, and we use the following notation:

$$\Delta \upsilon \left({\zeta }_i\right)=\upsilon \left({\zeta }_i^{+}\right)-\upsilon \left({\zeta }_i^{-}\right),\upsilon \left({\zeta }_i^{+}\right)=\underset{\mathfrak{h}\to 0^{+}}{lim}\upsilon ({\zeta }_i+\mathfrak{h}),\mathrm{and}\upsilon \left({\zeta }_i^{-}\right)=\underset{\mathfrak{h}\to 0^{-}}{lim}\upsilon ({\zeta }_i-\mathfrak{h}),$$

where $\upsilon \left({\zeta }_i^{+}\right)$ and $\upsilon \left({\zeta }_i^{-}\right)$ represent the right-hand and left-hand limits of the function $\upsilon \left(\zeta \right)$ at the point $\zeta ={\zeta }_i$, respectively.

The central focus of this research work is to establish conditions for the existence and uniqueness of mild solutions to the considered problem (1).

This work is organized in the following way: Section 2 introduces necessary notations, preliminary concepts, and foundational results, while Section 3 develops and presents the principal theorems and the main results. We also give an illustrative example, and, at the end, conclusions are given to sum up the findings and perspectives.

2. Preliminaries

In this section, we provide some theorems and lemmas that are used in the following parts of this work. We also illustrate some inequalities that will also be helpful for later analysis.

First, we present the concept of a piecewise continuous function, the Laplace transform, and the Mittag–Leffler function.

Definition 1

([19]). A function f is piecewise continuous on the interval $[0,\infty )$ if

1.

${lim}_{\zeta ⟶0^{+}}f\left(\zeta \right)=f\left(0^{+}\right)$ exists,

2.

f is continuous at every finite interval $(0,b)$, except possibly at a finite number of points ${\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n$ in $(0,b)$ at which f has a jump discontinuity.

Definition 2

([7]). The space of piecewise continuous functions is defined as

$$PC(\mathcal{J},\mathbb{R})=\left\{\upsilon :\mathcal{J}\to \mathfrak{X}|\begin{array}{c}\upsilon \in \mathfrak{C}([{\zeta }_i,{\zeta }_{i+1}],\mathfrak{X}),i=0,1,2,\dots ,\mathcal{l},\hfill \\ \mathit{and}\mathit{there}\mathit{exist}\upsilon \left({\zeta }_i^{-}\right),\upsilon \left({\zeta }_i^{+}\right)\mathit{with}\upsilon \left({\zeta }_i^{-}\right)=\upsilon \left({\zeta }_i\right)\hfill \end{array}\right\}$$

with the norm

$${\parallel \upsilon \parallel }_{PC}=\underset{\zeta \in \mathcal{J}}{sup}\left|\upsilon \left(\zeta \right)\right|.$$

The space $(PC(\mathcal{J},\mathbb{R}),{\parallel \upsilon \parallel }_{PC})$ is a Banach space.

Definition 3

([19,20]). Assume that the function f is defined for $t\ge 0.$ Then, the Laplace transform of $f,$ denoted by $\mathcal{L}\left\{f\right\},$ is defined by the improper integral

$$\mathcal{L}\left\{\mathfrak{f}\right\}\left(\lambda \right)=F\left(\lambda \right)={\int }_0^{\infty }{e}^{-\lambda \zeta }\mathfrak{f}\left(\zeta \right)d\zeta ,Re\left(\lambda \right)>0,$$

provided that this integral exists.

Furthermore, the inverse Laplace transform is defined by

$${\mathcal{L}}^{-1}\left\{F\left(s\right)\right\}=\mathfrak{f}\left(t\right)={\int }_{c-\infty }^{c+\infty }{e}^{st}F\left(s\right)ds.$$

Next, we recall the Mittag–Lefler function, which is known to play a fundamental role in the theory and computation of fractional-order differential equations.

Definition 4

([21,22]). The Mittag–Leffler Function $E_{\alpha }$ is expressed as

$$E_{\alpha }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\alpha k+1)}},\alpha >0,z\in \mathbb{C},$$

and the generalized two-parameter Mittag–Leffler Function $E_{\alpha ,\beta }$ is expressed as

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\alpha k+\beta )}},\alpha ,\beta >0,z\in \mathbb{C}.$$

In the following theorem, we state some of the properties of the Mittag–Leffler function, which will be of some use later on in the analysis of ordinary as well as partial differential equations of a fractional order.

Theorem 1

([22,23]). The following properties hold:

(1)

For $\left|z\right|<1$, the generalized Mittag–Leffler function satisfies

$${\displaystyle {\int }_0^{\infty }{e}^{-t}{t}^{\beta -1}{E}_{\alpha ,\beta }\left(t^{\alpha }z\right)dt=\frac{1}{z-1}.}$$

(2)

For $\left|z\right|<1$, the Laplace transform of the Mittag–Leffler function $E_{\alpha }\left(z^{\alpha }\right)$ is given by

$${\displaystyle {\int }_0^{\infty }{e}^{-zt}{E}_{\alpha }\left(z^{\alpha }\right)dt=\frac{1}{z-z^{1-\alpha }}.}$$

(4)

For special values α, the Mittag–Leffler function is given by

(a)

$E_0\left(z\right)={\displaystyle \frac{1}{z-1}},\hspace{1em}$$E_1\left(z\right)=e^z$.

(b)

$E_2\left(z^2\right)=cosh\left(z\right),\hspace{1em}$$E_2(-z^2)=cos\left(z\right)$.

(5)

The generalized Mittag–Leffler function has the following properties:

(i)

${\displaystyle E_{\alpha ,\beta }\left(z\right)=z{E}_{\alpha ,\alpha +\beta }\left(z\right)+\frac{1}{\mathrm{\Gamma }\left(\beta \right)}.}$

(ii)

$E_{\alpha ,\beta }\left(z\right)=\beta E_{\alpha ,\alpha +\beta }\left(z\right)+\alpha z{\displaystyle \frac{d}{dz}}{E}_{\alpha ,\beta +1}\left(z\right).$

(iii)

${\displaystyle E_{\alpha ,\beta }\left(z\right)=\frac{1}{2\pi i}{\int }_{\mathrm{{\rm Y}}}\frac{{\lambda }^{\alpha -\beta }{e}^{\lambda }}{{\lambda }^{\alpha }-z}d\lambda }$, where Υ is a contour that starts and ends at $-\infty$ and encircles the disc $\left|\lambda \right|\le {\left|z\right|}^{{\displaystyle \frac{1}{\alpha }}}$ counterclockwise.

Definition 5

([10,24,25]). An operator $\mathcal{A}$ on a Banach space is called sectorial if it satisfies the following conditions:

(a)

The spectrum $\sigma \left(\mathcal{A}\right)$ is contained in a closed sector

$$S_{\theta }=\left\{z\in \mathbb{C}\backslash \left\{0\right\}\left|\right|argz|\le \theta \right\}\cup \left\{0\right\},$$

for some angle $0\le \theta <\pi$.

(b)

There exists a constant $M>0$ such that the resolvent estimate

$${\parallel (z-\mathcal{A})}^{-1}\parallel \le {\displaystyle \frac{M}{\left|z\right|}}$$

holds for all $z\in \mathbb{C}\backslash S_{\theta }$.

Definition 6

([24,26]). Let $-1<r<0$ and $0<\theta <{\displaystyle \frac{\pi }{2}}$. We define ${\mathrm{\Theta }}_{\theta }^r\left(\mathfrak{X}\right)$ as the collection of all closed linear operators

$$\mathcal{A}:\mathcal{D}\left(\mathcal{A}\right)\subset \mathfrak{X}\to \mathfrak{X}$$

satisfying the following properties:

(1)

The spectrum $\sigma \left(\mathcal{A}\right)$ is contained in the sector

$$S_{\theta }=\left\{z\in \mathbb{C}\backslash \left\{0\right\}\left|\right|\arg z|\le \theta \right\}\cup \left\{0\right\}.$$

(2)

For each $\theta <\nu <\pi$, there exists a constant $C_{\nu }>0$ such that the resolvent estimate

$$\parallel R(z,\mathcal{A})\parallel \le C_{\nu }{\left|z\right|}^r,\mathit{for}\mathit{all}z\in \mathbb{C}\backslash S_{\nu },$$

holds.

An operator $\mathcal{A}$ is calledalmost sectorial on $\mathfrak{X}$ if $\mathcal{A}\in {\mathrm{\Theta }}_{\theta }^r\left(\mathfrak{X}\right)$.

Remark 1.

For an operator $\mathcal{A}\in {\mathrm{\Theta }}_{\theta }^r\left(\mathfrak{X}\right)$, the definition implies the following properties:

(i)

The origin belongs to the resolvent set, i.e., $0\in \rho \left(\mathcal{A}\right)$.

(ii)

The resolvent operator is given by $R(z,\mathcal{A})={(zI-\mathcal{A})}^{-1}$ for $z\in \rho \left(\mathcal{A}\right)$.

(iii)

The resolvent set is $\rho \left(\mathcal{A}\right)=\mathbb{C}\backslash \sigma \left(\mathcal{A}\right)$.

We now summarize some properties of the fractional integrals and derivatives (for more details, see [22,27,28,29,30]).

Definition 7

([28,30]). The Riemann–Liouville fractional integral operator of the order $\mu >0$ for a function $\mathfrak{f}:{\mathbb{R}}_{+}\to \mathbb{R}$ with $\mathfrak{f}\in L^1({\mathbb{R}}_{+},\mathfrak{X})$ is defined as follows:

$$\begin{array}{cc}\hfill I^0\mathfrak{f}\left(\zeta \right)& =\mathfrak{f}\left(\zeta \right),I^{\mu }\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^{\zeta }{(\zeta -s)}^{\mu -1}\mathfrak{f}\left(s\right)ds,\mu >0,\zeta >0,\hfill \end{array}$$

where $\mathrm{\Gamma }\left(\mu \right)$ denotes the Euler gamma function.

Definition 8

([28,30]). The Caputo’s derivative of the order μ for a function $\mathfrak{f}:[0,\infty )⟶\mathbb{R}$ is defined by

$${}^c\mathfrak{D}_{\zeta }^{\mu }\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\mu )}}{\int }_0^{\zeta }{(\zeta -s)}^{n-\mu -1}{\mathfrak{f}}^{\left(n\right)}\left(s\right)ds,if n-1\le \mu <n, for n\in \mathbb{N}.$$

Definition 9

([2,28,30]). The Laplace transform of the Caputo fractional derivative of the order $\mu >0$ for a function $\mathfrak{f}\left(\zeta \right)$, where $n-1\le \mu <n$, for a certain $n\in \mathbb{N}$, is given by

$$\mathcal{L}\left\{{}^C\mathfrak{D}_{\zeta }^{\mu }\mathfrak{f}\left(\zeta \right),\lambda \right\}={\lambda }^{\mu }\mathfrak{f}\left(\lambda \right)-\sum _{i=0}^{n-1}{\lambda }^{\mu -1}{\mathfrak{f}}^{\left(i\right)}\left(0\right).$$

In short, we set

$$E_{\mu }\left(z\right)=E_{\mu ,1}\left(z\right),{\epsilon }_{\mu }\left(z\right)=E_{\mu ,\mu }\left(z\right).$$

Then, we have

$${}^c\mathfrak{D}_{\zeta }^{\mu }E\left(w{\zeta }^{\mu }\right)=wE\left(w{\zeta }^{\mu }\right),{{\mathfrak{I}}_{\zeta }}^{1-\mu }\left({\zeta }^{\mu -1}{\epsilon }_{\mu }\left(w{\zeta }^{\mu }\right)\right)=E_{\zeta }\left(w{\zeta }^{\mu }\right).$$

Also consider the Wright-type function and some of its properties:

Definition 10

([31,32,33]). Wright Function

$${\chi }_{\mu }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-z)}^n}{n!\mathrm{\Gamma }(-\mu n+1-\mu )}}={\displaystyle \frac{1}{\pi }}\sum _{n=1}^{\infty }{\displaystyle \frac{{(-z)}^n}{(n-1)!}}\mathrm{\Gamma }\left(\mu n\right)sin\left(\mu n\pi \right),z\in \mathbb{C},$$

with $0<\mu <1.$

For $-1<\zeta <\infty ,$ $\lambda >0,$ the following results hold:

(1)

Non-negativity property: ${\chi }_{\mu }\left(\zeta \right)\ge 0$, for $\zeta >0.$

(2)

Laplace transform relation: ${\int }_0^{\infty }{\displaystyle \frac{\mu }{{\zeta }^{\mu -1}}}{\chi }_{\mu }\left({\displaystyle \frac{1}{{\zeta }^{\mu }}}\right)e^{-\lambda \zeta }d\zeta =e^{-{\lambda }^{\mu }}.$

(3)

Moment-generating property: ${\int }_0^{\infty }{\chi }_{\mu }\left(\zeta \right){\zeta }^{J}d\zeta ={\displaystyle \frac{\mathrm{\Gamma }(1+J)}{\mathrm{\Gamma }(1+\mu J)}}.$

(4)

Connection to Mittag–Leffler function: ${\int }_0^{\infty }{\chi }_{\mu }\left(\zeta \right)e^{-z\zeta }d\zeta =E_{\mu }(-z),z\in \mathbb{C}.$

(5)

Modified Laplace transform: ${\int }_0^{\infty }\mu \zeta {\chi }_{\mu }\left(\zeta \right)e^{-z\zeta }d\zeta ={\epsilon }_{\mu }(-z),z\in \mathbb{C}.$

Proposition 1

([25,34]). Let $\mathcal{A}$ be an operator in the class defined $\mathcal{A}\in {\mathrm{\Theta }}_{\theta }^r$, where $-1<r<0$, and $0<\theta <{\displaystyle \frac{\pi }{2}}$. We now define two families of operators based on generalized Mittag–Leffler-type functions and the resolvent operators associated with $\mathcal{A}$.

Let us define the operator families ${\left(S_{\mu }\left(\zeta \right)\right)}_{\zeta \in {\mathcal{S}}_{{\displaystyle \frac{\pi }{2}}-\theta }^0}$ and ${\left({\mathcal{P}}_{\mu }\left(\zeta \right)\right)}_{\zeta \in S_{{\displaystyle \frac{\pi }{2}}-\theta }^0}$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{S}}_{\mu }\left(\zeta \right)& :=E_{\mu }(-z{\zeta }^{\mu })\left(\mathcal{A}\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathrm{\Gamma }}_{\vartheta }}{E}_{\mu }(-z{\zeta }^{\mu })R(z,\mathcal{A})dz,\hfill \\ \hfill {\mathcal{P}}_{\mu }\left(\zeta \right)& :={\epsilon }_{\mu }(-z{\zeta }^{\mu })\left(\mathcal{A}\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathrm{\Gamma }}_{\vartheta }}{\epsilon }_{\mu }(-z{\zeta }^{\mu })R(z,\mathcal{A})dz,\hfill \end{array}$$

where ${\mathrm{\Gamma }}_{\vartheta }=\left(R_{+}{e}^{i\vartheta }\right)\cup \left(R_{+}{e}^{-i\vartheta }\right)$ is an appropriate integration contour that is oriented counterclockwise, and

$$\theta <\vartheta <\mu <{\displaystyle \frac{\pi }{2}}-\left|arg\zeta \right|,$$

with

$${\mathcal{S}}_{\mu }^0=\{z\in \mathbb{C}\backslash \left\{0\right\},|arg\left(z\right)|<\mu \};{\mathcal{P}}_{\mu }=\{z\in \mathbb{C}\backslash \left\{0\right\},|arg\left(z\right)|\le \mu \}.$$

Theorem 2

([33]). For each fixed $\zeta \in {\mathcal{S}}_{{\displaystyle \frac{\pi }{2}}-\theta }^0$, ${\mathcal{S}}_{\mu }\left(\zeta \right)$ and ${\mathcal{P}}_{\mu }\left(\zeta \right)$ are linear and bounded operators on X. Moreover, there exist constants ${\mathcal{C}}_s=\mathcal{C}(\mu ,\theta )>0$ and ${\mathcal{C}}_p=\mathcal{C}(\mu ,\theta )>0$ such that, for all $\zeta >0$,

$$∥{\mathcal{S}}_{\mu }\left(\zeta \right)∥\le {\mathcal{C}}_s{\zeta }^{-\mu (1+\theta )},∥{\mathcal{P}}_{\mu }\left(\zeta \right)∥\le {\mathcal{C}}_p{\zeta }^{-\mu (1+\theta )}.$$

Theorem 3

([33]). For all $\zeta >0$, the operators ${\mathcal{S}}_{\mu }\left(\zeta \right)$ and ${\mathcal{P}}_{\mu }\left(\zeta \right)$ are continuous in the uniform operator topology. Furthermore, for every $J>0$, this continuity is uniform in the interval $[0,\infty )$.

Theorem 4

([26]). Let $0<\beta <1-\theta$. Then,

1.

The range $R\left({\mathcal{P}}_{\mu }\left(\zeta \right)\right)$ of ${\mathcal{P}}_{\mu }\left(\zeta \right)$, for $\zeta >0$, is contained in $\mathcal{D}\left({\mathcal{A}}^{\beta }\right)$.

2.

${\mathcal{S}}_{\mu }\left(\zeta \right)\mathfrak{x}=-{\zeta }^{\mu -1}\mathcal{A}{\mathcal{P}}_{\mu }\left(\zeta \right)\mathfrak{x}$ for $\mathfrak{x}\in X$, and ${\mathcal{S}}_{\mu }\left(\zeta \right)\mathfrak{x}$ is locally integrable on $(0,\infty )$ for $\mathfrak{x}\in \mathcal{D}\left(\mathcal{A}\right)$.

3.

For all $\mathfrak{x}\in \mathcal{D}\left(\mathcal{A}\right)$ and $\zeta >0$,

$$\parallel \mathcal{A}{\mathcal{S}}_{\mu }\left(\zeta \right)\mathfrak{x}\parallel \le \mathcal{C}{\zeta }^{-\mu (1+\beta )}\parallel \mathcal{A}\mathfrak{x}\parallel ,$$

where $\mathcal{C}$ is a constant depending on θ and μ.

Theorem 5

([26]). The following properties hold:

1.

Let $\beta >1+\theta$. For all $\mathfrak{x}\in \mathcal{D}\left({\mathcal{A}}^{\beta }\right)$,

$$\underset{\zeta \to 0^{+}}{lim}{\mathcal{S}}_{\mu }\left(\zeta \right)\mathfrak{x}=\mathfrak{x}.$$

2.

For all $\mathfrak{x}\in \mathcal{D}\left(\mathcal{A}\right)$,

$$({\mathcal{S}}_{\mu }\left(\zeta \right)-I)\mathfrak{x}={\int }_0^{\zeta }-s^{\mu -1}\mathcal{A}{\mathcal{P}}_{\mu }\left(s\right)\mathfrak{x}ds.$$

3.

For all $\mathfrak{x}\in \mathcal{D}\left(\mathcal{A}\right)$ and $\zeta >0$,

$$D_{\zeta }^{\mu }{\mathcal{S}}_{\mu }\left(\zeta \right)\mathfrak{x}=-\mathcal{A}{\mathcal{S}}_{\mu }\left(\zeta \right)\mathfrak{x}.$$

4.

For all $\zeta >0$,

$${\mathcal{S}}_{\mu }\left(\zeta \right)\mathfrak{x}={\int }_0^{\infty }\mu s^{\mu -1}{\mathcal{P}}_{\mu }\left(s\right)\mathfrak{x}ds.$$

Lemma 1

([12]). We set

$$R(\lambda ,-\mathcal{A})={\int }_0^{\infty }{e}^{-\lambda t}\mathbb{T}\left(\zeta \right)d\zeta ,$$

and

$$\mathbb{T}\left(\zeta \right)=e^{-z\zeta }\left(\mathcal{A}\right)={\displaystyle \frac{1}{2i\pi }}{\int }_{{\mathrm{\Gamma }}_{\theta }}{e}^{-z\zeta }R(z,\mathcal{A})dz,z\in {\mathcal{S}}_{{\displaystyle \frac{\pi }{2}}-\theta }^0.$$

If $R(\lambda ,-\mathcal{A})$ is compact for every $\lambda >0$, then $\mathbb{T}\left(\zeta \right)$ is compact for every $\zeta >0.$

Theorem 6

([12]). If $R(\lambda ,-\mathcal{A})$ is compact for every $\lambda >0$, then ${\mathcal{S}}_{\mu }\left(\zeta \right) and {\mathcal{P}}_{\mu }\left(\zeta \right)$ are compact for every $\zeta >0$.

3. Main Results

Lemma 2.

Let $\upsilon :[0,\mathfrak{K}]\to \mathfrak{X}$ be a function satisfying $\upsilon \in \mathfrak{C}\left(\right[0,\mathfrak{K}],\mathfrak{X})$, with $g_{1-\mu }★\upsilon \in {\mathfrak{C}}^1([0,\mathfrak{K}],\mathfrak{X})$, $\upsilon \left(t\right)\in \mathcal{D}\left(\mathcal{A}\right)$ for all $t\in [0,\mathfrak{K}]$, and $\mathcal{A}\upsilon \in L^1([0,\mathfrak{K}],\mathfrak{X})$. If υ satisfies

$$\left\{\begin{array}{cc}{\mathfrak{D}}_{\zeta }^{\mu }\upsilon \left(\zeta \right)=\mathcal{A}\upsilon \left(\zeta \right)+\mathfrak{g}(\zeta ,\upsilon \left(\zeta \right)),\hfill & 0\le \zeta \le \mathfrak{K},\hfill \\ \upsilon \left(0\right)={\upsilon }_0,\hfill \\ \Delta \upsilon \left({\zeta }_i\right)={\mathfrak{I}}_i\left(\upsilon \left({\zeta }_i^{-}\right)\right),\hfill & i=1,2,3,\dots ,\mathcal{l},\hfill \end{array}\right.$$

then υ is a mild solution to this problem if υ is a solution to the following fractional integral equation:

$$\upsilon \left(\zeta \right)=\left\{\begin{array}{cc}{\upsilon }_0,\hfill & \zeta \le 0,\hfill \\ {\mathcal{S}}_{\mu }\left(\zeta \right){\upsilon }_0+{\int }_0^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & 0\le \zeta \le {\zeta }_1,\hfill \\ {\mathcal{S}}_{\mu }\left(\zeta \right)\left[\upsilon \left(0\right)\right]+{\mathcal{S}}_{\mu }(\zeta -{\zeta }_1)[\left(\upsilon \left({\zeta }_1^{-}\right)\right)+{\mathfrak{I}}_1\left(\upsilon \left({\zeta }_1^{-}\right)\right)]\hfill & \\ +{\int }_{{\zeta }_1}^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & {\zeta }_1\le \zeta \le {\zeta }_2,\hfill \\ ⋮\hfill & \\ {\mathcal{S}}_{\mu }\left(\zeta \right)\left[\upsilon \left(0\right)\right]+{\displaystyle \sum _{i=1}^{\mathcal{l}}}{\mathcal{S}}_{\mu }(\zeta -{\zeta }_i)[\upsilon \left({\zeta }_i^{-}\right)+{\mathfrak{I}}_i\upsilon \left({\zeta }_i^{-}\right)]\hfill & \\ +{\int }_{{\zeta }_{\mathcal{l}}}^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & {\zeta }_{\mathcal{l}}\le \zeta \le \mathfrak{K}.\hfill \end{array}\right.$$

Proof. 

If $\zeta \in [0,{\zeta }_1]$, then

$$\left\{\begin{array}{cc}{D}_{\zeta }^{\mu }\upsilon \left(\zeta \right)=\mathcal{A}\upsilon \left(\zeta \right)+\mathfrak{f}(\zeta ,\upsilon \left(\zeta \right)),\hfill & 0\le \zeta \le {\zeta }_1,\hfill \\ \upsilon \left(0\right)={\upsilon }_0.\hfill \end{array}\right.$$

By applying the integral operator on both sides, we obtain

$$\upsilon \left(\zeta \right)+c_1={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^{\zeta }{(\zeta -s)}^{\mu -1}[\mathcal{A}\upsilon \left(s\right)+\mathfrak{f}(s,\upsilon \left(s\right))]ds.$$

Using the initial condition, we get $c_1=-\upsilon \left(0\right)$; thus,

$$\upsilon \left(\zeta \right)=\upsilon \left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^{\zeta }{(\zeta -s)}^{\mu -1}[\mathcal{A}\upsilon \left(s\right)+\mathfrak{f}(s,\upsilon \left(s\right))]ds.$$

(2)

If $\zeta \in [{\zeta }_1,{\zeta }_2]$,

$$\left\{\begin{array}{cc}{D}_{\zeta }^{\mu }\upsilon \left(\zeta \right)=\mathcal{A}\upsilon \left(\zeta \right)+\mathfrak{f}(\zeta ,\upsilon \left(\zeta \right)),\hfill & {\zeta }_1\le \zeta \le {\zeta }_2,\hfill \\ \upsilon \left({\zeta }_1^{+}\right)=\upsilon \left({\zeta }_1^{-}\right)+I_1\upsilon \left({\zeta }_1^{-}\right).\hfill \end{array}\right.$$

By applying the fractional integral operator again, we are left with

$$\upsilon \left(\zeta \right)+c_2+\upsilon \left(0\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_{{\zeta }_1}^{\zeta }{(\zeta -s)}^{\mu -1}[\mathcal{A}\upsilon \left(s\right)+\mathfrak{f}(s,\upsilon \left(s\right))]ds.$$

Using the initial condition, we arrive at

$$\begin{array}{cc}\hfill \upsilon \left({\zeta }_1^{+}\right)+c_2& ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_{{\zeta }_1}^{{\zeta }_1}{({\zeta }_1-s)}^{\mu -1}[\mathcal{A}\upsilon \left(s\right)+\mathfrak{f}(s,\upsilon \left(s\right))]ds,\hfill \\ \hfill \upsilon \left({\zeta }_1^{-}\right)+{\mathfrak{I}}_1\upsilon \left({\zeta }_1^{-}\right)+c_2& ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_{{\zeta }_1}^{{\zeta }_1}{({\zeta }_1-s)}^{\mu -1}[\mathcal{A}\upsilon \left(s\right)+\mathfrak{f}(s,\upsilon \left(s\right))]ds.\hfill \end{array}$$

Thus, $c_2=-\upsilon \left(0\right)-\upsilon \left({\zeta }_1^{-}\right)-{\mathfrak{I}}_1\upsilon \left({\zeta }_1^{-}\right)$, yielding

$$\upsilon \left(\zeta \right)=\upsilon \left(0\right)+\upsilon \left({\zeta }_1^{-}\right)+{\mathfrak{I}}_1\upsilon \left({\zeta }_1^{-}\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_{{\zeta }_1}^{\zeta }{(\zeta -s)}^{\mu -1}[\mathcal{A}\upsilon \left(s\right)+\mathfrak{f}(s,\upsilon \left(s\right))]ds.$$

Similarly, for $\zeta \in [{\zeta }_k,{\zeta }_{k+1}]$,

$$\upsilon \left(\zeta \right)=\upsilon \left(0\right)+\sum _{i=1}^k[\upsilon \left({\zeta }_i^{-}\right)+{\mathfrak{I}}_i\upsilon \left({\zeta }_i^{-}\right)]+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_{{\zeta }_m}^{\zeta }{(\zeta -s)}^{\mu -1}[\mathcal{A}\upsilon \left(s\right)+\mathfrak{f}(s,\upsilon \left(s\right))]ds.$$

Equation (2) can be expressed as

$$\upsilon \left(\zeta \right)=\upsilon \left(0\right)+\sum _{i=1}^m{\sigma }_i\left(\zeta \right)[\upsilon \left({\zeta }_i^{-}\right)+{\mathfrak{I}}_i\upsilon \left({\zeta }_i^{-}\right)]+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_{{\zeta }_{\mathcal{l}}}^{\zeta }{(\zeta -s)}^{\mu -1}[\mathcal{A}\upsilon \left(s\right)+\mathfrak{f}(s,\upsilon \left(s\right))]ds,$$

(3)

for $\zeta \in [0,\mathfrak{K}]$, where the indicator function is

$${\sigma }_i\left(\zeta \right)=\left\{\begin{array}{cc}0,\hfill & \zeta \le {\zeta }_i,\hfill \\ 1,\hfill & \zeta >{\zeta }_i.\hfill \end{array}\right.$$

We take the Laplace transform from both sides of Equation (3) and obtain

$$\begin{array}{cc}\hfill \mathcal{L}\left[\upsilon \right(\zeta \left)\right]& ={\displaystyle \frac{1}{\lambda }}\left[\upsilon \left(0\right)\right]+\sum _{i=1}^{\mathcal{l}}{\displaystyle \frac{e^{-{\zeta }_i\lambda }}{\lambda }}[\upsilon \left({\zeta }_i^{-}\right)+{\mathfrak{I}}_i\upsilon \left({\zeta }_i^{-}\right)]+{\displaystyle \frac{1}{{\lambda }^{\mu }}}[\mathcal{A}\mathcal{L}\upsilon \left(s\right)+\mathcal{L}f(\zeta ,\upsilon \left(\zeta \right))]\hfill \\ \hfill & ={\displaystyle \frac{{\lambda }^{\mu -1}}{({\lambda }^{\mu }I-\mathcal{A})}}\left[\upsilon \left(0\right)\right]+\sum _{i=1}^{\mathcal{l}}{\displaystyle \frac{{\lambda }^{\mu -1}}{({\lambda }^{\mu }I-\mathcal{A})}}{e}^{-{\zeta }_i\lambda }[\upsilon \left({\zeta }_i^{-}\right)+{\mathfrak{I}}_i\upsilon \left({\zeta }_i^{-}\right)]\hfill \\ \hfill & +{\displaystyle \frac{1}{({\lambda }^{\mu }I-\mathcal{A})}}\left[\mathcal{L}f(\zeta ,\upsilon \left(\zeta \right))\right].\hfill \end{array}$$

By taking the inverse Laplace transform for both sides, we get

$$\begin{array}{cc}\hfill \upsilon \left(\zeta \right)& =E_{\mu ,1}\left(\mathcal{A}{\zeta }^{\mu }\right)\upsilon \left(0\right)\hfill \\ \hfill & +\sum _{i=1}^{\mathcal{l}}{E}_{\mu ,1}\left(\mathcal{A}{(\zeta -{\zeta }_i)}^{\mu }\right){\sigma }_i\left(\zeta \right)[\upsilon \left({\zeta }_i^{-}\right)+[\upsilon \left({\zeta }_i^{-}\right)+{\mathcal{I}}_i\upsilon \left({\zeta }_i^{-}\right)]]\hfill \\ \hfill & +{\int }_{{\zeta }_{\mathcal{l}}}^{\zeta }{E}_{\mu ,\mu }\left(\mathcal{A}{(\zeta -{\zeta }_i)}^{\mu }\right)\mathfrak{g}(s,\upsilon \left(s\right))ds.\hfill \end{array}$$

By setting

$${\mathcal{S}}_{\mu }\left(\zeta \right)=E_{\mu ,1}\left(\mathcal{A}{\zeta }^{\mu }\right),\mathrm{and}{\mathcal{P}}_{\mu }\left(\zeta \right)=E_{\mu ,\mu }\left(\mathcal{A}{\zeta }^{\mu }\right),$$

we have

$$\upsilon \left(\zeta \right)={\mathcal{S}}_{\mu }\left(\zeta \right)\left[\upsilon \left(0\right)\right]+\sum _{i=1}^{\mathcal{l}}{\mathcal{S}}_{\mu }(\zeta -{\zeta }_i){\sigma }_i\left(\zeta \right)[\upsilon \left({\zeta }_i^{-}\right)+{\mathfrak{I}}_i\upsilon \left({\zeta }_i^{-}\right)]+{\int }_0^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds.$$

□

Let us define the concept of a solution to our problem (1).

Definition 11.

The function $\upsilon (·):[0,\mathfrak{K}]\to \mathfrak{X}$ is a solution to the problem (1) if the restriction ${\upsilon |}_{[0,\mathfrak{K}]\backslash \{{\zeta }_1,\dots ,{\zeta }_m\}}$ is continuous and it satisfies the fractional integral equation:

$$\upsilon \left(\zeta \right)=\left\{\begin{array}{cc}{\upsilon }_0,\hfill & \zeta \le 0,\hfill \\ {\mathcal{S}}_{\mu }\left(\zeta \right){\upsilon }_0+{\int }_0^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & 0\le \zeta \le {\zeta }_1,\hfill \\ {\mathcal{S}}_{\mu }\left(\zeta \right)\left[\upsilon \left(0\right)\right]+{\mathcal{S}}_{\mu }(\zeta -{\zeta }_1)[\left(\upsilon \left({\zeta }_1^{-}\right)\right)+{\mathfrak{I}}_1\left(\upsilon \left({\zeta }_1^{-}\right)\right)]\hfill & \\ +{\int }_{{\zeta }_1}^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & {\zeta }_1\le \zeta \le {\zeta }_2,\hfill \\ ⋮\hfill & \\ {\mathcal{S}}_{\mu }\left(\zeta \right)\left[\upsilon \left(0\right)\right]+{\displaystyle \sum _{i=1}^{\mathcal{l}}}{\mathcal{S}}_{\mu }(\zeta -{\zeta }_i)[\upsilon \left({\zeta }_i^{-}\right)+{\mathfrak{I}}_i\upsilon \left({\zeta }_i^{-}\right)]\hfill & \\ +{\int }_{{\zeta }_{\mathcal{l}}}^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & {\zeta }_{\mathcal{l}}\le \zeta \le \mathfrak{K}.\hfill \end{array}\right.$$

To obtain the main results, we introduce the following assumptions:

(H₁)

The function $\mathfrak{g}:[0,\mathfrak{K}]\times \mathfrak{X}\to \mathfrak{X}$ satisfies the following conditions: $\mathfrak{g}$ is continuous, and there exists ${\mathfrak{L}}_{\mathfrak{g}}>0$ such that, for all $\zeta ,\xi \in [0,\mathfrak{K}]$ and $\upsilon ,\mathfrak{w}\in \mathfrak{X}$:

$$\parallel \mathfrak{g}(\zeta ,\upsilon )-\mathfrak{g}(\xi ,\mathfrak{w})\parallel \le {\mathfrak{L}}_{\mathfrak{g}}\left(|\zeta -\xi |+{\parallel \upsilon -\mathfrak{w}\parallel }_{\mathfrak{X}}\right).$$

(H₂)

The impulse operators ${\mathfrak{I}}_i:\mathfrak{X}\to \mathfrak{X}$ ($i=1,2,\dots ,\mathcal{l}$) satisfy the following conditions: each ${\mathfrak{I}}_i$ is continuous, and there exists ${\mathfrak{L}}_i>0$ such that, for all $\upsilon ,\mathfrak{w}\in \mathfrak{X}$,

$$\parallel {\mathfrak{I}}_i\left(\upsilon \right)-{\mathfrak{I}}_i\left(\mathfrak{w}\right)\parallel \le {\mathfrak{L}}_i{\parallel \upsilon -\mathfrak{w}\parallel }_{\mathfrak{X}}.$$

(H₃)

The following condition holds:

$$\underset{1\le i\le \mathcal{l}}{max}\left\{{\tilde{\mathcal{M}}}_s(1+\mathcal{L})+{\tilde{\mathcal{M}}}_p{\mathfrak{L}}_{\mathfrak{g}}{\displaystyle \frac{{\mathfrak{K}}^{1-\mu (1+\theta )}}{1-\mu (1+\theta )}}\right\}<1,$$

where $\mathcal{L}=\underset{1\le i\le \mathcal{l}}{max}{\mathfrak{L}}_i$, ${\tilde{\mathcal{M}}}_s=\underset{0\le \zeta \le \mathfrak{K}}{sup}{\parallel {\mathcal{S}}_{\mu }\left(\zeta \right)\parallel }_{\mathcal{L}\left(\mathfrak{X}\right)}$, ${\tilde{\mathcal{M}}}_p=\underset{0\le \zeta \le \mathfrak{K}}{sup}{\parallel {\mathcal{P}}_{\mu }\left(\zeta \right)\parallel }_{\mathcal{L}\left(\mathfrak{X}\right)}.$

Theorem 7.

Under assumptions $\left(H_1\right)$–$\left(H_3\right)$, the system admits a unique solution $\upsilon \in PC(J,\mathfrak{X})$.

Proof. 

Consider the operator $\mathfrak{H}:\mathfrak{X}⟶\mathfrak{X}$ defined as

$$\left(\mathfrak{H}\upsilon \right)\left(\zeta \right)=\left\{\begin{array}{cc}{\upsilon }_0,\hfill & \zeta \le 0,\hfill \\ {\mathcal{S}}_{\mu }\left(\zeta \right){\upsilon }_0+{\int }_0^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & 0\le \zeta \le {\zeta }_1,\hfill \\ {\mathcal{S}}_{\mu }\left(\zeta \right)\left[\upsilon \left(0\right)\right]+{\mathcal{S}}_{\mu }(\zeta -{\zeta }_1)[\left(\upsilon \left({\zeta }_1^{-}\right)\right)+{\mathfrak{I}}_1\left(\upsilon \left({\zeta }_1^{-}\right)\right)]\hfill & \\ +{\int }_{{\zeta }_1}^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & {\zeta }_1\le \zeta \le {\zeta }_2,\hfill \\ ⋮\hfill & \\ {\mathcal{S}}_{\mu }\left(\zeta \right)\left[\upsilon \left(0\right)\right]+{\displaystyle \sum _{i=1}^{\mathcal{l}}}{\mathcal{S}}_{\mu }(\zeta -{\zeta }_i)[\upsilon \left({\zeta }_i^{-}\right)+{\mathfrak{I}}_i\upsilon \left({\zeta }_i^{-}\right)]\hfill & \\ +{\int }_{{\zeta }_{\mathcal{l}}}^{\zeta }{\mathcal{P}}_{\mu }(\zeta -s)\mathfrak{g}(s,\upsilon \left(s\right))ds,\hfill & {\zeta }_{\mathcal{l}}\le \zeta \le \mathfrak{K},\hfill \end{array}\right.$$

We show that $\Vert \left(\mathfrak{H}\upsilon \right)\left(\zeta \right)-\left(\mathfrak{Hz}\right)\left(\zeta \right)\Vert \le \Vert \upsilon -\mathfrak{z}\Vert .$ Indeed,

$$\begin{array}{cc}\hfill & \Vert \left(\mathfrak{H}\upsilon \right)\left(\zeta \right)-\left(\mathfrak{Hz}\right)\left(\zeta \right)\Vert \hfill \\ \hfill & \le \left\{\begin{array}{cc}{\parallel {\mathcal{S}}_{\mu }\left(\zeta \right)\parallel }_{\mathcal{L}\left(\mathfrak{X}\right)}\parallel {\upsilon }_0-{\mathfrak{z}}_0\parallel +{\int }_0^{\zeta }\parallel {\mathcal{P}}_{\mu }(\zeta -s)\parallel \times \parallel \mathfrak{g}(s,\upsilon \left(s\right))-\mathfrak{g}(s,\mathfrak{z}\left(s\right))\parallel ds,\hfill & 0\le \zeta \le {\zeta }_1,\hfill \\ \\ {\parallel {\mathcal{S}}_{\mu }\left(\zeta \right)\parallel }_{\mathcal{L}\left(\mathfrak{X}\right)}\parallel {\upsilon }_0-{\mathfrak{z}}_0\parallel \hfill \\ +{\parallel {\mathcal{S}}_{\mu }(\zeta -{\zeta }_1)\parallel }_{\mathcal{L}\left(\mathfrak{X}\right)}\left[\parallel \upsilon \left({\zeta }_1^{-}\right)-\mathfrak{z}\left({\zeta }_1^{-}\right)\parallel +\parallel {\mathfrak{I}}_1\left(\upsilon \left({\zeta }_1^{-}\right)\right)-{\mathfrak{I}}_1\left(\mathfrak{z}\left({\zeta }_1^{-}\right)\right)\parallel \right]\hfill \\ +{\int }_{{\zeta }_1}^{\zeta }\parallel {\mathcal{P}}_{\mu }(\zeta -s)\parallel \times \parallel \mathfrak{g}(s,\upsilon \left(s\right))-\mathfrak{g}(s,\mathfrak{z}\left(s\right))\parallel ds,\hfill & {\zeta }_1\le \zeta \le {\zeta }_2,\hfill \\ ⋮\hfill \\ {\parallel {\mathcal{S}}_{\mu }\left(\zeta \right)\parallel }_{\mathcal{L}\left(\mathfrak{X}\right)}\parallel {\upsilon }_0-{\mathfrak{z}}_0\parallel \hfill \\ +\sum _{i=1}^{\mathcal{l}}{\parallel {\mathcal{S}}_{\mu }(\zeta -{\zeta }_i)\parallel }_{\mathcal{L}\left(\mathfrak{X}\right)}\left[\parallel \upsilon \left({\zeta }_i^{-}\right)-\mathfrak{z}\left({\zeta }_i^{-}\right)\parallel +\parallel {\mathfrak{I}}_1\left(\upsilon \left({\zeta }_i^{-}\right)\right)-{\mathfrak{I}}_1\left(\mathfrak{z}\left({\zeta }_i^{-}\right)\right)\parallel \right]\hfill \\ +{\int }_{{\zeta }_{\mathcal{l}}}^{\zeta }\parallel {\mathcal{P}}_{\mu }(\zeta -s)\parallel \times \parallel \mathfrak{g}(s,\upsilon \left(s\right))-\mathfrak{g}(s,\mathfrak{z}\left(s\right))\parallel ds,\hfill & {\zeta }_{\mathcal{l}}\le \zeta \le \mathfrak{K}.\hfill \end{array}\right.\hfill \end{array}$$

Then,

$$\Vert \left(\mathfrak{H}\upsilon \right)\left(\zeta \right)-\left(\mathfrak{Hz}\right)\left(\zeta \right)\Vert \le \left\{\begin{array}{cc}\left[{\tilde{\mathcal{M}}}_s\left({\tilde{\mathcal{M}}}_p{\mathfrak{L}}_{\mathfrak{g}}{\displaystyle \frac{{\mathfrak{K}}^{1-\mu (1+\theta )}}{1-\mu (1+\theta )}}\right)\right]\parallel \upsilon -\mathfrak{z}\parallel ,\hfill & 0\le \zeta \le {\zeta }_1,\hfill \\ \left({\tilde{\mathcal{M}}}_s(1+L_1)+{\tilde{\mathcal{M}}}_p{\mathfrak{L}}_{\mathfrak{g}}{\displaystyle \frac{{\mathfrak{K}}^{1-\mu (1+\theta )}}{1-\mu (1+\theta )}}\right)\parallel \upsilon -\mathfrak{z}\parallel ,\hfill & {\zeta }_1\le \zeta \le {\zeta }_2,\hfill \\ ⋮\hfill \\ \left({\tilde{\mathcal{M}}}_s(1+L_m)+{\tilde{\mathcal{M}}}_p{\mathfrak{L}}_{\mathfrak{g}}{\displaystyle \frac{{\mathfrak{K}}^{1-\mu (1+\theta )}}{1-\mu (1+\theta )}}\right)\parallel \upsilon -\mathfrak{z}\parallel ,\hfill & {\zeta }_{\mathcal{l}}\le \zeta \le \mathfrak{K}.\hfill \end{array}\right.$$

Thus, for all $\zeta \in [0,\mathfrak{K}]$, we have the estimates

$$\Vert \left(\mathfrak{H}\upsilon \right)\left(\zeta \right)-\left(\mathfrak{Hz}\right)\left(\zeta \right)\Vert \le \left({\tilde{\mathcal{M}}}_s(1+\mathcal{L})+{\tilde{\mathcal{M}}}_p{\mathfrak{L}}_{\mathfrak{g}}{\displaystyle \frac{{\mathfrak{K}}^{1-\mu (1+\theta )}}{1-\mu (1+\theta )}}\right)\Vert \upsilon -\mathfrak{z}\Vert .$$

Then, from $\left(H_3\right)$, it follows that

$$\Vert \left(\mathfrak{H}\upsilon \right)\left(\zeta \right)-\left(\mathfrak{Hz}\right)\left(\zeta \right)\Vert \le \Vert \upsilon -\mathfrak{z}\Vert .$$

This completes the proof of the theorem. □

4. Example

We consider the following example to illustrate the findings of the results proved. Let $X=L^3\left({\mathbb{R}}^2\right),$$\tilde{\mathcal{A}}={(-i\Delta +\sigma )}^{{\displaystyle \frac{1}{2}}},$ and the Sobolev space $D\left(\tilde{\mathcal{A}}\right)=W^{1,3}\left({\mathbb{R}}^2\right)$.

• Then, $i\Delta$ generates a $\beta$-times integrated semigroup $S^{\beta }\left(t\right)$, with $\beta ={\displaystyle \frac{5}{12}}$, on $L^3\left({\mathbb{R}}^2\right)$ such that ${\Vert S^{\beta }\left(t\right)\Vert }_{L(L^3\left({\mathbb{R}}^2\right)}\le \widehat{M}{t}^{\beta }$, for all $t>0$, and some constant $\widehat{M}>0.$ The operator $i\Delta +\sigma$ belongs to ${\mathrm{\Theta }}_{{\displaystyle \frac{\pi }{2}}}^{\beta -1}\left(L^3\left({\mathbb{R}}^2\right)\right)$, which denotes the family of all linear closed operators

$$\mathcal{A}:D\left(\mathcal{A}\right)\subset L^3\left({\mathbb{R}}^2\right)⟶L^3\left({\mathbb{R}}^2\right)$$

satisfying

$$\sigma \left(\mathcal{A}\right)\subset S_{{\displaystyle \frac{\pi }{2}}}=\left\{z\in \mathbb{C}\backslash \left\{0\right\};\left|arg\right|\le {\displaystyle \frac{\pi }{2}}\right\}\cup \left\{0\right\},$$

and, for every ${\displaystyle \frac{\pi }{2}<\mu <\pi }$, there exists a constant $C_{\mu }$ such that $\Vert R(z;\mathcal{A})\Vert \le C_{\mu }|z{|}^{\beta -1}$, for all $z\in \mathbb{C}\backslash S_{\mu }$. Thus, it follows that

$$\tilde{\mathcal{A}}\in {\mathrm{\Theta }}_{{\displaystyle \frac{\pi }{2}}}^{2\beta -1}\left(L^3\left({\mathbb{R}}^2\right)\right),\mathrm{for}\mathrm{some}0<\omega <{\displaystyle \frac{\pi }{2}}.$$

That is, $\tilde{\mathcal{A}}$ is an almost-sectorial operator for some ${\displaystyle 0<\omega <\frac{\pi }{2}}$, and ${\displaystyle \gamma =-\frac{1}{6}.}$

We denote the semigroup associated with $\tilde{\mathcal{A}}$ using $T\left(t\right)$ and $\Vert T\left(t\right)\Vert \le C_0{t}^{-{\displaystyle \frac{1}{6}}},$ where $C_0$ is a constant. Then, we consider the problem

$$\left\{\begin{array}{cc}{D}_t^{{\displaystyle \frac{1}{2}}}u\left(t\right)=\tilde{\mathcal{A}}u\left(t\right)+\mathfrak{f}(t,u\left(t\right)),\hfill & 0\le t\le 1,t\ne {\displaystyle \frac{1}{2}},\hfill \\ u\left(0\right)=0,\hfill \\ {\displaystyle \Delta u\left(\frac{1}{2}\right)=\frac{1}{3}cosx\left(t\right),}\hfill & {\displaystyle t=\frac{1}{2}.}\hfill \end{array}\right.$$

(4)

Here,

$${\displaystyle \mathfrak{f}(t,u\left(t\right))=\frac{e^{-t}}{(9+e^t)(1+x)}arctanx\left(t\right),\mathrm{with}(t,x)\in [0,1]\times X,}$$

and

$${\displaystyle I_1\left(x\right)=\frac{1}{3}cosx\left(t\right),\mathrm{for}x\in X.}$$

Direct computations lead to

$${\displaystyle \Vert \mathfrak{f}(t,x)-\mathfrak{f}(t,y)\Vert \le \frac{1}{10}\Vert x-y\Vert ,}$$

and

$${\displaystyle \Vert I_1\left(x\right)-I_1\left(y\right)\Vert \le \frac{1}{3}\Vert x-y\Vert .}$$

Then, from the condition (H₃), we have

$$\underset{1\le i\le m}{max}\left\{{\tilde{M}}_s(1+L)+{\tilde{M}}_p{L}_f{\displaystyle \frac{T^{1-\alpha (1+\gamma )}}{1-\alpha (1+\gamma )}}\right\}<1.$$

Thus, all the assumptions of Theorem 7 are satisfied; hence, we conclude that the impulsive fractional problem (4) has a unique solution.

5. Conclusions

In this paper, we used the Laplace transform and Mittag–Leffler functions to discuss the existence and uniqueness of mild solutions to a class of impulsive fractional equations in the sense of the Caputo derivative. Our study was based on a fixed-point approach and sectorial operators. Finally, an illustrative example was given to validate the theoretical results. As possible future research work, we will consider a numerical study of a similar problem, or we will generalize the fractional derivative to a variable-order one.