Total Controllability for a Class of Fractional Hybrid Neutral Evolution Equations with Non-Instantaneous Impulses

Abstract

This study demonstrates the total control of a class of hybrid neutral fractional evolution equations with non-instantaneous impulses and non-local conditions. The boundary value problem with non-local conditions is created using the Caputo fractional derivative of order $1<\alpha \le 2$. In order to create novel, strongly continuous associated operators, the infinitesimal generator of the sine and cosine families is examined. Additionally, two approaches are used to discuss the solution’s total controllability. A compact strategy based on the non-linear Leray–Schauder alternative theorem is one of them. In contrast, a measure of a non-compactness technique is implemented using the Sadovskii fixed point theorem with the Kuratowski measure of non-compactness. These conclusions are applied using simulation findings for the non-homogeneous fractional wave equation.

1. Introduction

Interest in fractional differential equations has risen significantly in recent years [1,2,3]. Numerous writers have investigated fractional equations with varying conditions and fractional evolution equations; for instance, see [4,5,6,7,8]. Fractional differential equations with non-local conditions are commonly seen in real-life scenarios. These equations are used to model complex systems that involve memory and long-range interactions. For example, fractional differential equations can be used to model the movement of a particle in a fluid, the diffusion of heat in a material, or the spread of a disease in a population [9,10,11].

Non-local conditions are also used to describe the behavior of a system when its behavior is affected by factors outside of its immediate environment. For instance, non-local conditions could be used to describe the effect of a pollutant on a distant ecosystem. Fractional differential equations with non-local conditions are powerful tools for understanding and predicting the behavior of complex systems [12,13].

The modeling of intermediate processes is one of the principal applications of fractional calculus (integration and differentiation of arbitrary (fractional) order), see [14,15,16,17,18].

Due to its applications in engineering, biology, physics, and several other domains, the theory of impulsive differential equations has become a popular area of study. The dynamics of phenomena that experience abrupt changes at certain moments were explored by the first category of impulse systems. The second category of impulse systems, namely, dynamical systems, arose to depict the behavior within the finite time period if abrupt changes persisted. Differential equations are often explained by a handful of impulsive effects. Instantaneous impulses are the first type; alterations of this kind last just briefly. The second kind is non-instantaneous impulses, in which the impulsive activity begins at an arbitrarily fixed moment and continues for a certain amount of time [19,20,21].

Numerous dynamical systems occasionally rely not only on the current and prior states but also on derivatives with delays. Neutral functional differential equations are widely used to explain such techniques; see [22,23].

The capacity to influence a system’s behavior through time is referred to as controllability of evolution equations, which is accomplished by introducing outside pressures into the system, such as control inputs. These inputs can be utilized to modify the behavior of the system and produce the desired results. Since it enables modification of the system’s behavior, controllability of evolution equations is a core concept in the study of these equations. Controllability is crucial to the development of future control theory and engineering, which is closely related to quadratic optimization, structural decomposition, and other concepts [24,25,26,27].

Reformulating the problem as a fixed point problem and seeing if it can be treated using a fixed point justification is one of the most common techniques for proving the existence of an operator equation. Non-compactness measures are significant in non-linear functional analysis. They are useful in metric fixed point theory, operator equation theory in Banach spaces, and the characterization of classes of compact operators. Measures of non-compactness yield useful information that is widely employed in the study of integral and integro-differential equations. Furthermore, it is particularly useful in the study of optimization, differential equations, functional equations, fixed point theory, and so on [28,29].

The Kuratowski measure, the Hausdorff measure, and the Istra$\breve{a}$escu measure are three well-known measures of non-compactness that were proposed by Kuratowski [30], Golden$\check{s}$tein et al. [31] (further investigated by Golden$\check{s}$tein and Markus [32]), and Istra$\breve{a}$escu [33], respectively.

Kumar et al. [34] tried looking at the fractional neutral differential equation with non-instantaneous impulses of the form of the sort

$$\left\{\begin{array}{c}{\displaystyle {}^c{D}_{e_m}^{\omega }\left[\mathcal{X}\left(t\right)-\mathcal{Q}(t,\mathcal{X}(a\left(t\right))\right]=\mathcal{A}\left[\mathcal{X}\left(t\right)-\mathcal{Q}(t,\mathcal{X}(a\left(t\right))\right]}\hfill \\ +\mathcal{C}\mathcal{U}\left(t\right)+\mathcal{G}(t,\mathcal{X}(b\left(t\right)\left)\right),\hfill & t\in (e_m,t_{m+1}],\hfill \\ \mathcal{X}\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\omega \right)}{\int }_0^t{(t-\mathcal{V})}^{\omega -1}{\mathcal{J}}_m(\mathcal{V},x\left(t_m^{-}\right))d\mathcal{V},\hfill & t\in (t_m,e_m],\hfill \\ \mathcal{X}\left(0\right)=x_0,\hfill \end{array}\right.$$

where ${}^c{D}_{e_m}^{\omega }$ denotes the Caputo-fractional derivative of order $\omega \in (0,1)$; $\mathcal{X}\in {\mathbb{R}}^n$ is the state variable, $a,b:I\to I$ with $a\left(t\right),b\left(t\right)\le t$ for $t\in I=[0,T],T>0$. $e_m$ and $t_m$ satisfy the relation $0=t_0=e_0<t_1<e_1<t_2<···<e_n<t_{n+1}=T$; $\mathcal{X}\left(t_m\right)={lim}_{h\to 0^{+}}\mathcal{X}(t_m-h)$ denotes the left limit of $\mathcal{X}\left(t\right)$ at $t=t_m$; $\mathcal{A}$ and $\mathcal{C}$ are the $n\times n$ and $n\times m$ matrices, respectively; $\mathcal{U}\in R^m$ is the control function, and the functions $\mathcal{Q},\mathcal{G}$, and ${\mathcal{J}}_m,m=1,2,\dots ;n,$ are some given functions. Here, the researchers studied the total controllability of the neutral system of order $0<\omega <1$.

By using the Mönch fixed point theorem, they examined the controllability outcomes of a mild solution for the fractional evolution system with a non-local with order $1<r<2$ circumstances. Additionally, they used the Banach fixed point theorem to derive the non-local controllability conclusions for fractional integro-differential evolution systems [35].

Moreover, Kumar et al. [36] investigated the total controllability results for the following $ABC$-fractional neutral dynamic system with non-instantaneous impulses

$$\left\{\begin{array}{cc}{\displaystyle {}^{ABC}{D}^{\alpha }\left[z\left(t\right)-\mathcal{Z}(t,z_a\left(t\right)\right]=}\hfill & \mathcal{A}\left[z\left(t\right)-\mathcal{Z}(t,z_a\left(t\right)\right]\hfill \\ +\mathcal{G}(t,z_b\left(t\right))+\mathfrak{B}u\left(t\right),\hfill & t\in {\displaystyle \bigcup _{k=0}^m}(s_k,t_{k+1}],\hfill \\ z\left(t\right)={\mathcal{Z}}_k(t,z\left(t_k^{-}\right)),\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(t_k,s_k],\hfill \\ z\left(0\right)=z_0\hfill \end{array}\right.$$

where ${}^{ABC}{D}^{\alpha }$ denotes the Atangana–Baleanu fractional derivative of order $\alpha \in (0,1)$ in Caputo sense, z is the state variable, $z_a\left(t\right)=z\left(a\left(t\right)\right)$, $z_b\left(t\right)=z\left(b\left(t\right)\right)$, $a,b:[0,T]\to [0,T]$ with $a\left(t\right),b\left(t\right)\le t$, and $z\left(t_k^{-}\right)={lim}_{h\to 0}z(t-h)$ denotes the left limit of $z\left(t\right)$ at $t=t_k$. The operator $\mathcal{A}:D\left(\mathcal{A}\right)\subset X\to X$ is the infinitesimal generator of q-resolvent families defined on a complex Banach space X. The function $u\in (L^2[0,T],U)$ is the control function into a Banach space U and $\mathfrak{B}$ is a linear and bounded operator from U into X. Semi-group theory, functional analysis, measure of non-compactness, and Mönch fixed point theorem have been used to establish their results.

In light of the aforementioned, we analyze in this paper the existence discoveries for a class of hybrid neutral fractional-order non-instantaneous impulses evolution equations which have the underlying structure

$$\left\{\begin{array}{cc}{\displaystyle {}^c{D}_t^{\alpha }\left[u\left(t\right)-\mathcal{P}(t,u(a\left(t\right))\right]=}\hfill & \mathcal{A}\left[u\left(t\right)-\mathcal{P}(t,u(a\left(t\right))\right]\hfill \\ +\mathcal{H}(t,u(b\left(t\right)\left)\right)+\mathfrak{B}x\left(t\right),\hfill & t\in {\displaystyle \bigcup _{k=0}^m}(s_k,t_{k+1}],\hfill \\ u\left(t\right)={\eta }_k(t,u\left(t\right)),\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(t_k,s_k],\hfill \\ u^{\prime }\left(t\right)={\xi }_k(t,u\left(t\right)),\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(t_k,s_k],\hfill \\ u\left(0\right)+g_0\left(u\right)=u_0\in \mathbb{X},\hfill \\ u^{\prime }\left(0\right)+g_1\left(u\right)=u_1\in \mathbb{X},\hfill \end{array}\right.$$

(1)

where

(i)

${}^c{D}_t^{\alpha }$ is the Caputo fractional derivative of order $1<\alpha \le 2$;

(ii)

$\mathcal{A}$ is an infinitesimal generator of a strongly continuous cosine family;

(iii)

${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$ is a cosine family operator;

(iv)

$\mathfrak{B}:\mathbb{U}\to \mathbb{X}$ is a bounded linear operator;

(v)

$a\left(t\right),b\left(t\right):J\to J$ with $a\left(t\right),b\left(t\right)\le t$ for $t\in J=[0,T]$;

(vi)

$\mathcal{H},\mathcal{P}:J\times \mathbb{X}\to \mathbb{X}$ is a given function satisfying some assumptions;

(vii)

$x\left(t\right)$ is the control function given in ${\mathbb{L}}^2(J,\mathbb{U})$;

(viii)

$g_i\left((J,\mathbb{X});\mathbb{X}\right)$ for $i=0,1$, are continuous functions that will be specified later;

(ix)

The functions ${\mu }_k,{\xi }_k\in C\left((t_k,s_k]\times \mathbb{X};\mathbb{X}\right)$ for all $k=1,2,\dots ,m;m\in \mathbb{N}$ stands for impulsive conditions;

(xi)

$0=t_0=s_0<t_1\le s_1\le t_2<\cdots <t_m\le s_m\le t_{m+1}=T,$ are pre-fixed numbers.

These equations have several applications, the most significant of which are fractional wave equations, which are used in a wide range of physical and engineering disciplines (see [37,38]). There is another important application of our theoretical results in the synchronization of networks. In [39], Xu et al. investigated the problem of exponential bipartite synchronization of fractional-order multilayer signed networks via hybrid impulsive control. The mathematical model of the networks was established by integrating the fractional dynamics of nodes, the multilayer network edges, and the positive and negative weights (antagonistic relationship), which is more pluralistic and practical. Moreover, a hybrid impulsive controller was designed, which is composed of the feedback control part and the impulsive control part, to realize the exponential bipartite synchronization objective. Both positive and negative impulsive effects have been considered and the ranges of the impulsive control gain related to the order of Caputo fractional derivative have been discussed.

In this investigation, the main novelties, advantages, and contributions are listed as follows:

• We demonstrate how to exert control by resolving the earlier system (1), which has the Caputo fractional derivative of order $1<\alpha \le 2$.
• The infinitesimal generator of the cosine and sine families is investigated to construct new, associated, strongly continuous operators.
• Two alternative methods are used to locate it:
  • First, via the measure of non-compactness, and then, rigorously, by applying the Kuratowski measure of non-compactness and the Sadovskii fixed point theorem.
  • Second, the non-linear alternative Leray–Schauder theorem of the fixed point is an alternative approach in the case of compactness.
• Additionally, using the integral term, we demonstrate the results of total controllability for the problem under consideration.
• Finally, we give an illustration to demonstrate the value of the gathered analytical data.

With this research, we hope to extend the knowledge on fractional evolution equations and fill a research gap. Our article is divided into the sections below: In Section 2, we review some basic concepts and preparation outcomes. We show the structure of the mild solution by using the cosine family in Section 3. In Section 4, we present the controllability results for the system (1). Finally, the major outcomes are provided as numerical applications.

2. Preliminaries

The ideas and terminologies connected to this paper’s components, such as fractional calculus and the cosine and sine family operators, are presented in this section. Additionally, a few lemmas that serve to support the major results of this work are given.

Definition 1 ([40]).  

A one-parameter family ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\in \mathbb{R}}$ of bounded linear operators mapping the Banach space $\mathbb{X}$ into itself is called a strongly continuous cosine family if and only if

(i)

$\mathcal{Q}\left(0\right)=I$;

(ii)

$\mathcal{Q}(s+t)+\mathcal{Q}(s-t)=2\mathcal{Q}\left(s\right)\mathcal{Q}\left(t\right)$ for all $s,t\in \mathbb{R}$;

(iii)

$\mathcal{Q}\left(t\right)x$ is a continuous on $\mathbb{R}$ for each $x\in \mathbb{X}$.

A one-parameter family ${\left\{\mathcal{R}\left(t\right)\right\}}_{t\in \mathbb{R}}$ is the sine family associated with the strongly continuous cosine family ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\in \mathbb{R}}$ which is defined by

$$\mathcal{R}\left(t\right)x={\int }_0^t\mathcal{Q}\left(s\right)xds,x\in \mathbb{X},t\in \mathbb{R}.$$

Lemma 1 

([40]). Let ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\in \mathbb{R}}$ be a strongly continuous cosine family in $\mathbb{X}$ satisfying ${\parallel \mathcal{Q}\left(t\right)\parallel }_{\mathcal{B}\left(\mathbb{X}\right)}\le M{e}^{\xi \left|t\right|},t\in \mathbb{R}$, where $\left({\mathcal{B}\left(\mathbb{X}\right);\parallel .\parallel }_{\mathcal{B}\left(\mathbb{X}\right)}\right)$ is the Banach space of all linear and bounded operators from $\mathbb{X}$ to $\mathbb{X}$. Let $\mathcal{A}$ be an infinitesimal generator of ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\in \mathbb{R}}$. Then, for $Re\lambda >\xi$ and $({\xi }^2,\infty )\subset \rho \left(\mathcal{A}\right)$ (the resolvent set of the operator $\mathcal{A}$), we have

$$\lambda R({\lambda }^2;\mathcal{A})x={\int }_0^{\infty }{e}^{-\lambda t}\mathcal{Q}\left(t\right)xdt,R({\lambda }^2;\mathcal{A})x={\int }_0^{\infty }{e}^{-\lambda t}\mathcal{R}\left(t\right)xdt,x\in \mathbb{X}$$

where the operator $R(\lambda ;\mathcal{A})={(\lambda I-\mathcal{A})}^{-1}$ is the resolvent of the operator $\mathcal{A}$.

The infinitesimal generator operator $\mathcal{A}$ of the cosine family ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\in \mathbb{R}}$ is defined by

$$\mathcal{A}x=\underset{t\to 0}{lim}\frac{d^2}{d{t}^2}\mathcal{Q}\left(t\right)x,\mathrm{for}\mathrm{all}x\in \mathcal{D}\left(\mathcal{A}\right)$$

where $\mathcal{D}\left(\mathcal{A}\right)=\{x\in \mathbb{X}:\mathcal{Q}\left(t\right)x\in {\mathcal{C}}^2(\mathbb{R},\mathbb{X})\}$. It is known that the infinitesimal generator $\mathcal{A}$ is closed densely in $\mathbb{X}$.

In order to present our results, we need the following:

Definition 2 

([40]). Supposing that $\tau >0$, the Mainardi’s Wright-type function is defined as

$$M_{\varrho }\left(\tau \right)=\sum _{n=0}^{\infty }\frac{{(-\tau )}^n}{n!\mathsf{\Gamma }(1-\varrho (n+1\left)\right)},\varrho \in (0,1),\tau \in \mathbb{C}$$

and achieves

$$M_{\varrho }\left(\tau \right)\ge 0,{\int }_0^{\infty }{\theta }^{\xi }{M}_{\varrho }\left(\theta \right)d\theta =\frac{\mathsf{\Gamma }(1+\xi )}{\mathsf{\Gamma }(1+\varrho \xi )},\xi >-1.$$

Definition 3 

([41]). Let $x:[a,b]\to \mathbb{R}$ be the nth continuously differentiable function. Then, the left derivative of fractional order $\mathfrak{q}$ due to Caputo is presented as

$$\begin{array}{c}\hfill {}_c{D}_a^{\mathfrak{q}}x\left(s\right)=\frac{1}{\mathsf{\Gamma }(\ell -\mathfrak{q})}{\int }_a^s{(s-t)}^{\ell -\mathfrak{q}-1}{x}^{(\ell )}\left(t\right)dt,\ell -1<\mathfrak{q}\le \ell ;\ell \in \mathbb{N};s\in [a,b].\end{array}$$

Definition 4 

([41]). The left fractional integral of the function x is

$${\mathcal{I}}_a^{\mathfrak{q}}x\left(s\right)=\frac{1}{\mathsf{\Gamma }\left(\mathfrak{q}\right)}{\int }_a^s{\left(s-t\right)}^{\mathfrak{q}-1}x\left(t\right)dt,s>a,\mathfrak{q}>0.$$

Lemma 2 

([42]). Let $\ell \in \mathbb{N},\ell -1<\mathfrak{q}\le \ell$, and x be the nth continuously differentiable function over the interval $[a,b]$. Then,

$$\begin{array}{c}\hfill {\mathcal{I}}_a^{\mathfrak{q}}{}_c{D}_a^{\mathfrak{q}}x\left(s\right)=x\left(s\right)+a_0+a_1(s-a)+\cdots +a_{\ell -1}{(s-a)}^{\ell -1}.\end{array}$$

Lemma 3. 

Let $I_s^{\alpha }$ be the left R-L integral of order α and $f\left(t\right)$ be integrable function defined for $t\ge s\ge 0$. Then,

$${\int }_s^{\infty }{e}^{-\lambda t}{I}_s^{\alpha }f\left(t\right)dt={\lambda }^{-\alpha }{\int }_s^{\infty }{e}^{-\lambda t}f\left(t\right)dt.$$

Definition 5 

([43]). The Kuratowski measure of non-compactness $\mu (·)$ is defined on the bounded set S of Banach space $\mathbb{X}$ as

$$\begin{array}{c}\hfill \mu \left(S\right):=inf\left\{\delta >0:S\subset {\displaystyle \bigcup _{i=1}^m}{S}_i,S_i\subset \mathbb{X},diam\left(S_i\right)<\delta \mathrm{for}i=1,2,\dots ,m;m\in \mathbb{N}\right\}\end{array}$$

where

$$diam\left(S_i\right)=sup\{\parallel x_1-x_2\parallel :x_1,x_2\in S_i\}.$$

The following properties about the Kuratowski measure of non-compactness are well known.

Lemma 4 ([43]). 

Let $\mathcal{T},\mathcal{R}$ be bounded in Banach space $\mathbb{X}$. The following properties are satisfied:

(i)

$\mu \left(\mathcal{T}\right)=0$, if and only if $\overline{\mathcal{T}}$ is compact, where $\overline{\mathcal{T}}$ means the closure hull of $\mathcal{T}$;

(ii)

$\mu \left(\mathcal{T}\right)=\mu \left(\overline{\mathcal{T}}\right)=\mu \left(conv\mathcal{T}\right)$, where $conv\mathcal{T}$ means the convex hull of $\mathcal{T}$;

(iii)

$\mu \left(k\mathcal{T}\right)=\left|k\right|\mu \left(\mathcal{T}\right)$ for any $k\in \mathbb{R}$;

(iv)

$\mathcal{T}\subset \mathcal{R}$ implies $\mu \left(\mathcal{T}\right)\le \mu \left(\mathcal{R}\right)$;

(v)

$\mu (\mathcal{T}+\mathcal{R})\le \mu \left(\mathcal{T}\right)+\mu \left(\mathcal{R}\right)$, where $\mathcal{T}+\mathcal{R}=\left\{x\right|x=y+z,y\in \mathcal{T},z\in \mathcal{R}\}$;

(vi)

$\mu (\mathcal{T}\cup \mathcal{R})=max\{\mu \mathcal{T},\mu \mathcal{R}\}$;

(vii)

If the map $H:\mathcal{D}\left(H\right)\subset \mathbb{X}\to \mathfrak{Y}$ is Lipschitz continuous with constant c, then $\mu \left(H\right(U\left)\right)\le c\mu \left(U\right)$ for any bounded subset $U\in \mathcal{D}\left(H\right)$, where $\mathfrak{Y}$ is another Banach space.

Some interesting topological fixed point results that have been widely used while dealing with the non-linear equations include the following:

Lemma 5 

(Sadovskii’s fixed point theorem [43]). Let $\mathsf{\Psi }$ be a bounded closed and convex subset in Banach space $\mathbb{X}$. If the operator $\mathcal{Q}:\mathsf{\Psi }\to \mathsf{\Psi }$ is continuously μ-condensing, which means that $\mu \left(\mathcal{Q}\right(\mathsf{\Psi }\left)\right)<\mu \left(\mathsf{\Psi }\right)$, then, $\mathcal{Q}$ has at least one fixed point in $\mathsf{\Psi }$.

Theorem 1 

(Non-linear alternative Leray–Schauder theorem [44]). Let H be a Banach space and $D\subset H$ be a convex set, and U be an open subset in D such that $0\in U$. Then, each continuous compact mapping $f:U\to D$ has at least one solution if

1. 

f has a fixed point, or

2. 

There is $(x,\lambda )\in \partial U\times (0,1)$ such that $x=\lambda f\left(x\right)$.

Before presenting the solution, we shall describe the spaces employed in this research.

($\mathcal{C}\left(J,\mathbb{X}\right)$):

The Banach space of continuous and bounded functions from J into $\mathbb{X}$ provided with the topology of uniform convergence with the norm

$${\parallel u\parallel }_{\mathcal{C}}=\underset{t\in J}{sup}\left|u\left(t\right)\right|.$$

($\mathcal{P}\mathcal{C}(J,\mathbb{X})$):

The Banach space of all continuous piecewise functions defined as

$$\begin{array}{cc}\hfill & \mathcal{P}\mathcal{C}(J,\mathbb{X})=\left\{u:J\to \mathbb{X}:u\in \mathcal{C}((t_k,t_{k+1}];\mathbb{X}),k=0,\dots ,m;m\in \mathbb{N}\mathrm{and}\mathrm{there}\mathrm{exist}\right.\hfill \\ \hfill & \left.u\left(t_k^{+}\right)\mathrm{and}u\left(t_k^{-}\right),k=1,\dots ,m;m\in \mathbb{N}\mathrm{with}u\left(t_k^{+}\right)=u\left(t_k^{-}\right)\right\}\hfill \end{array}$$

with the norm

$${\parallel u\parallel }_{\mathcal{P}\mathcal{C}(J,\mathbb{X})}=\underset{t\in J}{sup}\left|u\left(t\right)\right|.$$

(${\mathbb{L}}^p(J,\mathbb{X})$):

The Banach space of the Lebesgue measurable functions from J into $\mathbb{X}$ such that $(1\le p<\infty )$ with the norm

$${\parallel \psi \parallel }_{{\mathbb{L}}^p}={\left({\int }_J{\left|\psi \left(s\right)\right|}^{p}ds\right)}^{\frac{1}{p}}.$$

3. Structure of Mild Solution

We first show the following lemma before giving a formulation of the mild solution of (1).

Lemma 6. 

Let $1<\alpha \le 2$ and $\mathcal{H},\mathcal{P}:J\to \mathbb{X}$ be integrable functions. Then, the mild solution to our problem (1) possesses the form

$$u\left(t\right)=\left\{\begin{array}{c}{\mathcal{Q}}_q\left(t\right)(u_0-g_0\left(u\right)-p\left(0\right))\hfill \\ +\underset{0}{\overset{t}{\int }}{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy\hfill \\ +\underset{0}{\overset{t}{\int }}{(t-y)}^{q-1}{\mathcal{R}}_q(t,y)\left[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)\right]dy,\hfill & t\in [0,t_1],\hfill \\ {\mu }_k(t,u\left(t\right)),\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(t_k,s_k],\hfill \\ {\mathcal{Q}}_q(t-s_k){\mu }_k(s_k,u\left(s_k\right))+\mathcal{P}\left(t\right)\hfill \\ +\underset{s_k}{\overset{t}{\int }}{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy\hfill \\ +\underset{s_k}{\overset{t}{\int }}{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]dy,\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(s_k,t_{k+1}]\hfill \end{array}\right.$$

where $1/2<q=\frac{\alpha }{2}\le 1$,

$$\begin{array}{c}\hfill {\mathcal{Q}}_q\left(t\right)={\int }_0^{\infty }{M}_q\left(\theta \right)\mathcal{Q}\left(t^q\theta \right)d\theta ,{\mathcal{R}}_q(t,s)=q{\int }_0^{\infty }\theta M_q\left(\theta \right)\mathcal{R}\left({(t-s)}^q\theta \right)d\theta \end{array}$$

and $M_q$ is a probability density function defined by Definition 2.

Proof. 

Using Lemma 2 and the operator $I_{s_r}^{\alpha }$ to solve fractional differential equation in (1), we arrive at

$$\begin{array}{c}\hfill u\left(t\right)-\mathcal{P}\left(t\right)=I_{s_k}^{\alpha }\left[\mathcal{A}\left(u\left(t\right)-\mathcal{P}\left(t\right)\right)+\mathcal{H}\left(t\right)+\mathfrak{B}x\left(t\right)\right]+c_{1,k}(t-s_k)+c_{0,k}\end{array}$$

(2)

where $c_{1,k},c_{0,k}\in \mathbb{R},k=0,1,\dots ,m;m\in \mathbb{N}$ are constants to be determined.

• For $t\in [0,t_1]$: By taking $\rho \to 1$ to the results given in Lemma 5 in [45], we have

  $$\begin{array}{cc}\hfill u\left(t\right)-\mathcal{P}\left(t\right)=& {\mathcal{Q}}_q\left(t\right)(u_0-g_0\left(u\right)-p\left(0\right))+{\int }_0^t{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy\hfill \\ \hfill +& {\int }_0^t{(t-y)}^{q-1}{\mathcal{R}}_q(t,y)\left[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)\right]d.\hfill \end{array}$$
• For $t\in (t_1,s_1]$: We obtain
  $u\left(t\right)={\mu }_1(t,u\left(t\right))$ and $u^{\prime }\left(t\right)={\xi }_1(t,u\left(t\right))$
• For $t\in (s_1,t_2]$: The problem (1) becomes

  $$\begin{array}{cc}\hfill {}^c{D}_{s_1}^{\alpha }\left[u\left(t\right)-\mathcal{P}\left(t\right)\right]=& \mathcal{A}\left[u\left(t\right)-\mathcal{P}\left(t\right)\right]+\mathcal{H}\left(t\right)+\mathfrak{B}x\left(t\right),\hfill \\ \hfill u\left(s_1\right)=& {\mu }_1(s_1,u\left(s_1\right)),\hfill \\ \hfill u^{\prime }\left(s_1\right)=& {\xi }_1(s_1,u\left(s_1\right)).\hfill \end{array}$$
  In this interval, Equation (2) becomes

  $$\begin{array}{c}\hfill u\left(t\right)-\mathcal{P}\left(t\right)=I_{s_1}^{\alpha }[\mathcal{A}\left[u\left(t\right)-\mathcal{P}\left(t\right)\right]+\mathcal{H}\left(t\right)+\mathfrak{B}x\left(t\right)]+c_{1,1}(t-s_1)+c_{0,1}.\end{array}$$
  Considering the the past impulsive conditions, we obtain

  $$\begin{array}{c}\hfill c_{0,1}={\mu }_1(s_1,u\left(s_1\right))\mathrm{and}{c}_{1,1}={\xi }_1(s_1,u\left(s_1\right))\end{array}$$

  which imply that

  $$\begin{array}{c}\hfill u\left(t\right)-\mathcal{P}\left(t\right)=I_{s_1}^{\alpha }[\mathcal{A}\left[u\left(t\right)-\mathcal{P}\left(t\right)\right]+\mathcal{H}\left(t\right)+\mathfrak{B}x\left(t\right)]+{\xi }_1(s_1,u\left(s_1\right))(t-s_1)+{\mu }_1(s_1,u\left(s_1\right)).\end{array}$$
  Multiplying both sides by $e^{-\lambda t}$ followed by integrating from $s_1$ to ∞, we achieve

  $$\begin{array}{cc}\hfill U\left(p\right)=& p^{-\alpha }\left\{\mathcal{A}\left[U\left(p\right)\right]+V\left(p\right)]\right\}+p^{-1}{e}^{-p{s}_1}{\mu }_1(s_1,u\left(s_1\right))+p^{-2}{e}^{-p{s}_1}{\xi }_1(s_1,u\left(s_1\right))\hfill \end{array}$$

  where

  $$\begin{array}{cc}\hfill & U\left(p\right)={\int }_{s_1}^{\infty }{e}^{-pt}[u\left(t\right)-\mathcal{P}\left(t\right)]dt\mathrm{and}V\left(p\right)={\int }_{s_1}^{\infty }{e}^{-pt}[\mathcal{H}\left(t\right)+\mathfrak{B}x\left(t\right)]dt.\hfill \end{array}$$
  Given that ${(p^{\alpha }I-A)}^{-1}$ exists, then $p^{\alpha }\in \rho \left(A\right)$, we obtain

  $$\begin{array}{cc}\hfill U\left(p\right)=& {(p^{\alpha }I-A)}^{-1}\left\{p^{\alpha -1}{e}^{-p{s}_1}{\mu }_1(s_1,u\left(s_1\right))+p^{\alpha -2}{e}^{-p{s}_1}{\xi }_1(s_1,u\left(s_1\right))+V\left(p\right)\right\}\hfill \\ \hfill =& p^{q-1}{e}^{-p{s}_1}{\int }_0^{\infty }{e}^{-p^{q}t}\mathcal{Q}\left(t\right){\mu }_1(s_1,u\left(s_1\right))dt\hfill \\ \hfill +& p^{q-2}{e}^{-p{s}_1}{\int }_0^{\infty }{e}^{-p^{q}t}\mathcal{Q}\left(t\right){\xi }_1(s_1,u\left(s_1\right))dt+{\int }_0^{\infty }{e}^{-p^{q}t}\mathcal{R}\left(t\right)V\left(p\right)dt.\hfill \end{array}$$
  Let ${\mathsf{\Psi }}_q\left(\theta \right)=\frac{q}{{\theta }^{q+1}}{M}_q\left({\theta }^{-q}\right)$ be defined for $\theta \in (0,\infty )$ and $q\in (\frac{1}{2},1)$. Then,

  $${\int }_0^{\infty }{e}^{-p\theta }{\mathsf{\Psi }}_q\left(\theta \right)d\theta =e^{-p^q}$$

  which can be used to calculate the first term with replacing t by $s^q$ as

  $$\begin{array}{cc}\hfill p^{q-1}{e}^{-p{s}_1}{\int }_0^{\infty }& e^{-p^{q}t}\mathcal{Q}\left(t\right){\mu }_1(s_1,u\left(s_1\right))dt\hfill \\ \hfill =& q{\int }_0^{\infty }{\left(ps\right)}^{q-1}{e}^{-{\left(ps\right)}^q}\mathcal{Q}\left(s^q\right)e^{-p{s}_1}({\mu }_1{s}_1,u\left(s_1\right))ds\hfill \\ \hfill =& \frac{-1}{p}{\int }_0^{\infty }\frac{d}{ds}\left(e^{-{\left(ps\right)}^q}\right)\mathcal{Q}\left(s^q\right)e^{-p{s}_1}({\mu }_1{s}_1,u\left(s_1\right))ds\hfill \\ \hfill =& {\int }_0^{\infty }{\int }_0^{\infty }\theta {\mathsf{\Psi }}_q\left(\theta \right)e^{-ps\theta }\mathcal{Q}\left(s^q\right)e^{-p{s}_1}({\mu }_1{s}_1,u\left(s_1\right))d\theta ds\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-p(x+s_1)}\left\{{\int }_0^{\infty }{\mathsf{\Psi }}_q\left(\theta \right)\mathcal{Q}\left({\left(\frac{x}{\theta }\right)}^q\right){\mu }_1(s_1,u\left(s_1\right))d\theta \right\}dx\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-p(x+s_1)}\left\{{\int }_0^{\infty }{M}_q\left(\theta \right)\mathcal{Q}\left(x^q\theta \right){\mu }_1(s_1,u\left(s_1\right))d\theta \right\}dx\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-p(x+s_1)}{\mathcal{Q}}_q\left(x\right){\mu }_1(s_1,u\left(s_1\right))dx\hfill \\ \hfill =& {\int }_{s_1}^{\infty }{e}^{-\lambda t}{\mathcal{Q}}_q(t-s_1){\mu }_1(s_1,u\left(s_1\right))dt.\hfill \end{array}$$
  By using Lemma 3 with $\alpha =1$, we obtain

  $$\begin{array}{cc}\hfill & p^{-1}{p}^{q-1}{\int }_0^{\infty }{e}^{-p^{q}t}{\mathcal{Q}}_q\left(t\right)e^{-p{s}_1}{\xi }_1(s_1,u\left(s_1\right))dt={\int }_{s_1}^{\infty }{e}^{-pt}\left\{{\int }_{s_1}^t{\mathcal{Q}}_q(y-s_1){\xi }_1(s_1,u\left(s_1\right))dy\right\}dt.\hfill \end{array}$$
  Finally, we can write

  $$\begin{array}{cc}\hfill & {\int }_0^{\infty }{e}^{-p^{q}t}\mathcal{R}\left(t\right)V\left(p\right)dt=q{\int }_0^{\infty }{e}^{-{\left(ps\right)}^q}\mathcal{R}\left(s^q\right)s^{q-1}V\left(p\right)ds\hfill \\ \hfill =& q{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-ps\theta }{\mathsf{\Psi }}_q\left(\theta \right)\mathcal{R}\left(s^q\right)s^{q-1}V\left(p\right)d\theta ds\hfill \\ \hfill =& q{\int }_0^{\infty }{\int }_{s_1}^{\infty }{\int }_0^{\infty }{\theta }^{-q}{e}^{-px}{\mathsf{\Psi }}_q\left(\theta \right)\mathcal{R}\left({\left(\frac{x}{\theta }\right)}^q\right)x^{q-1}{e}^{-py}[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]d\theta dydx\hfill \\ \hfill =& q{\int }_0^{\infty }{\int }_{s_1}^{\infty }{\int }_0^{\infty }{e}^{-p(x+y)}\theta M_q\left(\theta \right)\mathcal{R}\left(x^q\theta \right)x^{q-1}[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]d\theta dydx\hfill \\ \hfill =& q{\int }_{s_1}^{\infty }{\int }_y^{\infty }{\int }_0^{\infty }{e}^{-pt}\theta M_q\left(\theta \right)\mathcal{R}\left({(t-y)}^q\theta \right){(t-y)}^{q-1}[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]d\theta dtdy\hfill \\ \hfill =& {\int }_{s_1}^{\infty }{\int }_y^{\infty }{e}^{-pt}{(t-y)}^{q-1}{\mathcal{R}}_q\left(t-y\right)[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]dtdy\hfill \\ \hfill =& {\int }_{s_1}^{\infty }{e}^{-pt}\left\{{\int }_{s_1}^t{(t-y)}^{q-1}{\mathcal{R}}_q\left(t-y\right)[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]dy\right\}dt.\hfill \end{array}$$
  In conclusion, we can write

  $$\begin{array}{cc}\hfill {\int }_{s_1}^{\infty }{e}^{-pt}[u\left(t\right)-\mathcal{P}\left(t\right)]dt& ={\int }_{s_1}^{\infty }{e}^{-pt}\{{\mathcal{Q}}_q(t-s_1){\mu }_1(s_1,u\left(s_1\right))\hfill \\ \hfill & +{\int }_{s_1}^t{\mathcal{Q}}_q(y-s_1){\xi }_1(s_1,u\left(s_1\right))dy\hfill \\ \hfill & +{\int }_{s_1}^t{(t-y)}^{q-1}{\mathcal{R}}_q\left(t-y\right)[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]dy\}dt.\hfill \end{array}$$
  Therefore, by taking the inverse Laplace transform which is unique, we have

  $$\begin{array}{cc}\hfill u\left(t\right)=& \mathcal{P}\left(t\right)+{\mathcal{Q}}_q(t-s_1){\mu }_1(s_1,u\left(s_1\right))+{\int }_{s_1}^t{\mathcal{Q}}_q(y-s_1){\xi }_1(s_1,u\left(s_1\right))dy\hfill \\ \hfill +& {\int }_{s_1}^t{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]dy.\hfill \end{array}$$
• For $t\in (s_k,t_{k+1}],k=2,3,\dots ,m;m\in \mathbb{N}$: In a similar manner, we can write

  $$\begin{array}{cc}\hfill u\left(t\right)=& \mathcal{P}\left(t\right)+{\mathcal{Q}}_q(t-s_k){\mu }_k(s_k,u\left(s_k\right))+{\int }_{s_k}^t{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy\hfill \\ \hfill +& {\int }_{s_k}^t{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)[\mathcal{H}\left(y\right)+\mathfrak{B}x\left(y\right)]dy.\hfill \end{array}$$

Consequently, we obtain the solution from the earlier (1). Direct calculation shows the opposite result. The proof is completed. □

Definition 6. 

A function $u\in \mathcal{P}\mathcal{C}(J;\mathbb{X})$ is said to be the mild solution of (1) if it satisfies

$$u\left(t\right)=\left\{\begin{array}{c}{\mathcal{Q}}_q\left(t\right)(u_0-g_0\left(u\right)-p\left(0\right))+\mathcal{P}(t,u\left(a\left(t\right)\right))\hfill \\ +\underset{0}{\overset{t}{\int }}{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy\hfill \\ +\underset{0}{\overset{t}{\int }}{(t-y)}^{q-1}{\mathcal{R}}_q(t,y)\left[\mathcal{H}(y,u(b\left(t\right)\left)\right)+\mathfrak{B}x\left(y\right)\right]dy,\hfill & t\in [0,t_1],\hfill \\ {\mu }_k(t,u\left(t\right)),\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(t_k,s_k],\hfill \\ {\mathcal{Q}}_q(t-s_k){\mu }_k(s_k,u\left(s_k\right))+\mathcal{P}(t,u\left(a\left(t\right)\right))\hfill \\ +\underset{s_k}{\overset{t}{\int }}{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy\hfill \\ +\underset{s_k}{\overset{t}{\int }}{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)[\mathcal{H}(y,u\left(b\left(t\right)\right))+\mathfrak{B}x\left(y\right)]dy,\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(s_k,t_{k+1}].\hfill \end{array}\right.$$

(3)

Remark 1. 

It is obvious to infer from the linearity of $\mathcal{Q}\left(t\right)$ and $\mathcal{R}\left(t\right)$ for any $t\ge 0$ that ${\mathcal{Q}}_q\left(t\right)$ and ${\mathcal{R}}_q(t,y)$ are also linear operators where $0<y<t$.

As a corollary, when $\rho$ approaches 1, the proofs of all subsequent Lemmas are identical.

Lemma 7 

([45]). The following estimates for ${\mathcal{Q}}_q\left(t\right)$ and ${\mathcal{R}}_q(t,y)$ are verified for any fixed $t\ge 0$ and $0<y<t$, where $\parallel \mathcal{Q}\left(t\right)\parallel \le \varpi$, such that ϖ is positive constant $\varpi \ge 1$

$$∥{\mathcal{Q}}_q\left(t\right)x∥\le \varpi \left|x\right|and∥{\mathcal{R}}_q(t,y)x∥\le \frac{\varpi }{\mathsf{\Gamma }\left(2q\right)}\underset{t\in J}{sup}\left|t\right|\left|x\right|.$$

Lemma 8 

([45]). For any $0<y<t$ and $t>0$, the operators ${\mathcal{Q}}_q\left(t\right)$ and ${\mathcal{R}}_q(t,y)$ are strongly continuous.

Lemma 9 

([45]). Pretend that $\mathcal{Q}\left(t\right)$ and $\mathcal{R}(t,y)$ are compact for every $0<y<t$. In that case, for any $0<y<t$, the operators ${\mathcal{Q}}_q\left(t\right)$ and ${\mathcal{R}}_q(t,y)$ are compact.

4. Total Controllability Results

This section presents our results about the total controllability of problem (1). Using Lemma 6, the non-instantaneous fractional neutral evolution problem with non-local circumstances is turned into a fixed point problem.

Before we discuss our key findings, we suggest the following hypotheses:

$\left(\mathbb{A}{\mathbb{E}}_1\right)$

The linear operator $\mathfrak{B}$ is bounded. Then there exists a positive constant $\mathfrak{K}$ such that

$$\parallel \mathfrak{B}\parallel \le \mathfrak{K}.$$

Moreover, let ${\mathcal{G}}_k:{\mathbb{L}}^2\left([s_k,t_{k+1}],\mathbb{X}\right)\to \mathbb{X},k=0,1,\dots ,m;m\in \mathbb{N}$ defined by

$${\mathcal{G}}_{k}x={\int }_{s_k}^{t_{k+1}}{(t_{k+1}-y)}^{q-1}{\mathcal{R}}_q(t_{k+1}-y)\mathfrak{B}x\left(y\right)dy,k=0,1,\dots ,m;m\in \mathbb{N}$$

and have bounded invertible operators

$${\mathcal{G}}_k^{-1}:\mathbb{X}\to {\mathbb{L}}^2\left([s_k,t_{k+1}],\mathbb{X}\right)/ker{\mathcal{G}}_k,k=0,1,\dots ,m;m\in \mathbb{N}.$$

Thus, there are positive constants ${\mathfrak{D}}^k,k=0,1,\dots ,m;m\in \mathbb{N}$ such that

$$\parallel {\mathcal{G}}_k^{-1}\parallel \le {\mathfrak{D}}^k,k=0,1,\dots ,m;m\in \mathbb{N}.$$

$\left(\mathbb{A}{\mathbb{E}}_2\right)$

The functions $\mathcal{P},\mathcal{H}:J\times \mathbb{X}\to \mathbb{X}$ are continuous and there exist positive constants $\mathcal{N},\mathcal{V}$ such that for all $u_i\in \mathbb{X},i=1,2,t\in J$, we find that

$$\parallel \mathcal{P}(t,u_1)-\mathcal{P}(t,u_2)\parallel \le \mathcal{N}\parallel u_1-u_2\parallel ,$$

$$\parallel \mathcal{H}(t,u_1)-\mathcal{H}(t,u_2)\parallel \le \mathcal{V}\parallel u_1-u_2\parallel$$

which imply that

$$\parallel \mathcal{P}(t,u)\parallel \le \mathcal{N}\parallel u\parallel +{\mathcal{N}}_0,\mathrm{where}{\mathcal{N}}_0=\underset{t\in J}{sup}\left|\mathcal{P}(t,0)\right|,$$

$$\parallel \mathcal{H}(t,u)\parallel \le \mathcal{V}\parallel u\parallel +{\mathcal{V}}_0,\mathrm{where}{\mathcal{V}}_0=\underset{t\in J}{sup}\left|\mathcal{H}(t,0)\right|.$$

$\left(\mathbb{A}{\mathbb{E}}_3\right)$

The non-instantaneous impulse ${\eta }_k,{\xi }_k:J_k\times \mathbb{X}\to \mathbb{X},J_k=[t_k,s_k],k=1,2,\dots ,m;m\in \mathbb{N}$ are continuous and there exist positive constants ${\gamma }_{\eta },{\gamma }_{\xi }$ such that

$$\parallel {\eta }_k(t,u_1)-{\eta }_k(t,u_2)\parallel \le {\gamma }_{\eta }^k\parallel u_1-u_2\parallel ,\forall u_i\in \mathbb{X},i=1,2,t\in J_k,$$

$$\parallel {\xi }_k(t,u_1)-{\xi }_k(t,u_2)\parallel \le {\gamma }_{\xi }^k\parallel u_1-u_2\parallel ,\forall u_i\in \mathbb{X},i=1,2,t\in J_k.$$

Furthermore, we have

$$\parallel {\eta }_k(t,u)\parallel \le {\gamma }_{\eta }^k\parallel u\parallel +{\gamma }_{\eta }^0,\mathrm{where}{\gamma }_{\eta }^0=\underset{t\in J}{sup}\left|{\eta }_k(t,0)\right|,$$

$$\parallel {\xi }_k(t,u)\parallel \le {\gamma }_{\xi }^k\parallel u\parallel +{\gamma }_{\xi }^0,\mathrm{where}{\gamma }_{\xi }^0=\underset{t\in J}{sup}\left|{\xi }_k(t,0)\right|.$$

$\left(\mathbb{A}{\mathbb{E}}_4\right)$

The non-local functions $g_i:\mathbb{X}\to \mathbb{X}$ are continuous and there exist positive constants $\delta ,\zeta$ such that

$$\parallel g_0\left(u_1\right)-g_0\left(u_2\right)\parallel \le \delta \parallel u_1-u_2\parallel ,\forall u_i\in \mathbb{X},i=1,2,$$

$$\parallel g_1\left(u_1\right)-g_1\left(u_2\right)\parallel \le \zeta \parallel u_1-u_2\parallel ,\forall u_i\in \mathbb{X},i=1,2.$$

These signify

$$\parallel g_0(t,u)\parallel \le \delta \parallel u\parallel +{\delta }_0,\mathrm{where}{\delta }_0=\underset{t\in J}{sup}\left|\right|g_0\left(0\right)|,$$

$$\parallel g_1(t,u)\parallel \le \zeta \parallel u\parallel +{\zeta }_0,\mathrm{where}{\zeta }_0=\underset{t\in J}{sup}\left|\right|g_1\left(0\right)|.$$

Furthermore, on the basis of the above theories, we shall provide some crucial definitions and some results.

Definition 7. 

If a continuous function $x\in {\mathbb{L}}^2(J,\mathbb{X})$ exists for any $u_0,u_T\in \mathbb{X}$, such that the mild solution (3) meets the starting condition $u\left(0\right)=u_0$ and the goal point $u\left(T\right)=u_T$, the impulsive system (1) is said to be controlled on J.

Definition 8. 

If for any $u_0,u_{t_{k_{m+1}}}\in \mathbb{X},k=0,1,\dots ,m;m\in \mathbb{N}$, there exists a piece-wise continuous function $x\in {\mathbb{L}}^2(J,\mathbb{X})$ such that mild the solution (3) meets initial condition $u\left(0\right)=u_0$ and target points $u\left(t_{k+1}\right)=u_{t_{k+1}}$ for $k=0,1,\dots ,m;m\in \mathbb{N}$, then the impulsive system (1) is described as totally controlled on J, i.e., the impulsive system (1) is controllable on $(0,t_1]$ and $(s_k,t_{k+1}]$.

Lemma 10. 

If the assumptions $\left(\mathbb{A}{\mathbb{E}}_1\right)$, $\left(\mathbb{A}{\mathbb{E}}_2\right)$, and $\left(\mathbb{A}{\mathbb{E}}_4\right)$ hold, then the control function

$$\begin{array}{cc}\hfill x\left(t\right)=& {\mathcal{G}}_0^{-1}\left[u_{t_1}+{\mathcal{Q}}_q\left(t_1\right)(u_0-g_0\left(u\right)-p\left(0\right))+\mathcal{P}(t_1,u\left(a\left(t_1\right)\right))\right.\hfill \\ \hfill & +{\int }_0^{t_1}{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy\hfill \\ \hfill & \left.+{\int }_0^{t_1}{(t_1-y)}^{q-1}{\mathcal{R}}_q(t_1,y)\mathcal{H}(y,u\left(b\left(t_1\right)\right))dy\right],t\in (0,t_1]\hfill \end{array}$$

(4)

steers the state $u\left(t\right)$ of the system (1) from initial points $u_0,u_1\in \mathbb{X}$ to target point $u_{t_1}$. Moreover, the control function $x\left(t\right)$ has an estimate $\parallel x\left(t\right)\parallel \le \mathfrak{Y}$, where

$$\begin{array}{cc}\hfill \mathfrak{Y}& ={\mathfrak{D}}^0\left(\parallel u_{t_1}\parallel +\overline{\chi }+\varrho \parallel u\parallel +\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}{\mathcal{V}}_0)\right),\hfill \\ \hfill \overline{\chi }& =\varpi \left[\parallel u_0\parallel +{\delta }_0+\parallel p\left(0\right)\parallel +\left(\parallel u_1\parallel +{\zeta }_0+\parallel p\left(0\right)\parallel \right)t_1+{\mathcal{N}}_0\right],\hfill \\ \hfill \varrho & =\varpi (\delta +\zeta t_1)+\mathcal{N}+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\mathcal{V}{t}_1^q.\hfill \end{array}$$

Proof. 

Consider the solution $u\left(t\right)$ of the system (1) on $(0,t_1]$ defined by (3). For $t=t_1$, we obtain

$$\begin{array}{cc}\hfill & u\left(t_1\right)={\mathcal{Q}}_q\left(t_1\right)(u_0-g_0\left(u\right)-p\left(0\right))+\mathcal{P}(t_1,u\left(a\left(t\right)\right))+{\int }_0^{t_1}{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy\hfill \\ \hfill & +{\int }_0^{t_1}{(t_1-y)}^{q-1}{\mathcal{R}}_q(t_1,y)\mathcal{H}(y,u\left(b\left(t_1\right)\right))dy+{\int }_0^{t_1}{(t_1-y)}^{q-1}{\mathcal{R}}_q(t_1,y)\mathfrak{B}{\mathcal{G}}_0^{-1}\hfill \\ \hfill & \left[u_{t_1}+{\mathcal{Q}}_q\left(t_1\right)(u_0-g_0\left(u\right)-p\left(0\right))+\mathcal{P}(t_1,u\left(a\left(t_1\right)\right))+{\int }_0^{t_1}{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy\right.\hfill \\ \hfill & \left.+{\int }_0^{t_1}{(t_1-y)}^{q-1}{\mathcal{R}}_q(t_1,y)\mathcal{H}(y,u\left(b\left(t_1\right)\right))dy\right]\hfill \\ \hfill & ={\mathcal{Q}}_q\left(t_1\right)(u_0-g_0\left(u\right)-p\left(0\right))+\mathcal{P}(t_1,u\left(a\left(t\right)\right))+{\int }_0^{t_1}{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy\hfill \\ \hfill & +{\int }_0^{t_1}{(t_1-y)}^{q-1}{\mathcal{R}}_q(t_1,y)\mathcal{H}(y,u\left(b\left(t_1\right)\right))dy\hfill \\ \hfill & +{\mathcal{G}}_0{\mathcal{G}}_0^{-1}[u_{t_1}+{\mathcal{Q}}_q\left(t_1\right)(u_0-g_0\left(u\right)-p\left(0\right))+\mathcal{P}(t_1,u\left(a\left(t_1\right)\right))\hfill \\ \hfill & +{\int }_0^{t_1}{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy+{\int }_0^{t_1}{(t_1-y)}^{q-1}{\mathcal{R}}_q(t_1,y)\mathcal{H}(y,u\left(b\left(t_1\right)\right))dy]=u_{t_1}.\hfill \end{array}$$

The control estimate is also computed as follows:

$$\begin{array}{cc}\hfill \parallel x\left(t\right)\parallel \le & {\mathfrak{D}}^0[\parallel u_{t_1}\parallel +\varpi \left(\parallel u_0\parallel +\delta \parallel u\parallel +{\delta }_0+\parallel p\left(0\right)\parallel \right)+\mathcal{N}\parallel u\parallel +{\mathcal{N}}_0\hfill \\ \hfill & +\varpi \left(\parallel u_1\parallel +\zeta \parallel u\parallel +{\zeta }_0+\parallel p\left(0\right)\parallel \right)t_1+\frac{\varpi t_1^q}{q\mathsf{\Gamma }\left(2q\right)}\left(\mathcal{V}\parallel u\parallel +{\mathcal{V}}_0\right)]\hfill \\ \hfill & ={\mathfrak{D}}^0\left(\parallel u_{t_1}\parallel +\overline{\chi }+\varrho \parallel u\parallel +\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}{t}_1^q{\mathcal{V}}_0\right)\hfill \\ \hfill & =\mathfrak{Y}.\hfill \end{array}$$

This ends the proof. □

Lemma 11. 

If the assumptions $\left(\mathbb{A}{\mathbb{E}}_1\right)$, $\left(\mathbb{A}{\mathbb{E}}_1\right)$, and $\left(\mathbb{A}{\mathbb{E}}_3\right)$ hold, and $u_{t_{k+1}}\in \mathbb{X},k=1,2,\dots ,m;m\in \mathbb{N}$ are arbitrary target points. Let the solution $u\left(t\right)$ of the system (1) on $(s_k,t_{k+1}],k=1,2,\dots ,m;m\in \mathbb{N}$ defined by (3). Then, for $t=t_{k+1}$, we obtain

$$\begin{array}{cc}\hfill x\left(t\right)=& {\mathcal{G}}_k^{-1}\left[u_{t_{k+1}}+{\mathcal{Q}}_q(t_{k+1}-s_k){\mu }_k(s_k,u\left(s_k\right))\right.\hfill \\ \hfill & +\mathcal{P}(t_{k+1},u\left(a\left(t\right)\right))+{\int }_{s_k}^{t_{k+1}}{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy\hfill \\ \hfill & \left.+{\int }_{s_k}^{t_{k+1}}{(t_{k+1}-y)}^{q-1}{\mathcal{R}}_q(t_{k+1}-y)\mathcal{H}(y,u\left(b\left(t_{k+1}\right)\right))dy\right],t\in (s_k,t_{k+1}],\hfill \end{array}$$

(5)

steer the state $u\left(t\right)$ of the system (1) from the initial point to target points $u_{t_{m+1}}$. Moreover, the control function $x\left(t\right)$ has an estimate $\parallel x\left(t\right)\parallel \le {\mathfrak{Y}}^k$, where

$$\begin{array}{cc}\hfill {\mathfrak{Y}}^k& ={\mathfrak{D}}^k\left(\parallel u_{t_{k+1}}\parallel +{\overline{\chi }}_k+{\varrho }_k\parallel u\parallel +\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}{\mathcal{V}}_0{(T-s_k)}^q\right),\hfill \\ \hfill {\overline{\chi }}_k& =\varpi ({\gamma }_{\eta }^0+{\gamma }_{\xi }^0(T-s_k))+{\mathcal{N}}_0),\hfill \\ \hfill {\varrho }_k& =\varpi ({\gamma }_{\eta }^k+{\gamma }_{\xi }^k(T-s_k))+\mathcal{N}+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\mathcal{V}{(T-s_k)}^q.\hfill \end{array}$$

Proof. 

Consider the solution $u\left(t\right)$ of the system (1) on $(s_k,t_{k+1}]$ defined by (3). For $t=t_{k+1}$, we obtain

$$\begin{array}{cc}\hfill & u\left(t_{k+1}\right)={\mathcal{Q}}_q(t_{k+1}-s_k){\mu }_k(s_k,u\left(s_k\right))+\mathcal{P}(t_{k+1},u\left(a\left(t\right)\right))\hfill \\ \hfill & +{\int }_{s_k}^{t_{k+1}}{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy+{\int }_{s_k}^{t_{k+1}}{(t-y)}^{q-1}{\mathcal{R}}_q(t_{k+1}-y)\mathcal{H}(y,u\left(b\left(t_{k+1}\right)\right))dy\hfill \\ \hfill & +{\int }_{s_k}^{t_{k+1}}{(t-y)}^{q-1}{\mathcal{R}}_q(t_{k+1}-y)\mathfrak{B}{\mathcal{G}}_k^{-1}[u_{t_{k+1}}+{\mathcal{Q}}_q(t_{k+1}-s_k){\mu }_k(s_k,u\left(s_k\right))+\mathcal{P}(t_{k+1},u\left(a\left(t\right)\right))\hfill \\ \hfill & +{\int }_{s_k}^{t_{k+1}}{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy+{\int }_{s_k}^{t_{k+1}}{(t_{k+1}-y)}^{q-1}{\mathcal{R}}_q(t_{k+1}-y)\mathcal{H}(y,u\left(b\left(t_{k+1}\right)\right))dy]\hfill \\ \hfill & ={\mathcal{Q}}_q(t_{k+1}-s_k){\mu }_k(s_k,u\left(s_k\right))+\mathcal{P}(t_{k+1},u\left(a\left(t\right)\right))+{\int }_{s_k}^{t_{k+1}}{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy\hfill \\ \hfill & +{\int }_{s_k}^{t_{k+1}}{(t-y)}^{q-1}{\mathcal{R}}_q(t_{k+1}-y)\mathcal{H}(y,u\left(b\left(t_{k+1}\right)\right))dy\hfill \\ \hfill & +{\mathcal{G}}_k{\mathcal{G}}_k^{-1}[u_{t_{k+1}}+{\mathcal{Q}}_q(t_{k+1}-s_k){\mu }_k(s_k,u\left(s_k\right))+\mathcal{P}(t_{k+1},u\left(a\left(t\right)\right))\hfill \\ \hfill & +{\int }_{s_k}^{t_{k+1}}{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy+{\int }_{s_k}^{t_{k+1}}{(t_{k+1}-y)}^{q-1}{\mathcal{R}}_q(t_{k+1}-y)\mathcal{H}(y,u\left(b\left(t_{k+1}\right)\right))dy]\hfill \\ \hfill & =u_{t_{k+1}}.\hfill \end{array}$$

The control estimate is also computed as follows:

$$\begin{array}{cc}\hfill \parallel x\left(t\right)\parallel \le & {\mathfrak{D}}^k[\parallel u_{t_{k+1}}\parallel +\varpi ({\gamma }_{\eta }^k\parallel u\parallel +{\gamma }_{\eta }^0)+\mathcal{N}\parallel u\parallel +{\mathcal{N}}_0\hfill \\ \hfill & +\varpi ({\gamma }_{\xi }^k\parallel u\parallel +{\gamma }_{\xi }^0)(t_{k+1}-s_k)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\mathcal{V}\parallel u\parallel +{\mathcal{V}}_0){(t_{k+1}-s_k)}^q]\hfill \\ \hfill & \le {\mathfrak{D}}^k[\parallel u_{t_{k+1}}\parallel +{\overline{\chi }}_k+\varpi ({\gamma }_{\eta }^k+{\gamma }_{\xi }^k(t_{k+1}-s_k))\parallel u\parallel +\mathcal{N}\parallel u\parallel \hfill \\ \hfill & +\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\mathcal{V}\parallel u\parallel +{\mathcal{V}}_0){(t_{k+1}-s_k)}^q]\hfill \\ \hfill & \le {\mathfrak{D}}^k\left(\parallel u_{t_{k+1}}\parallel +{\overline{\chi }}_k+{\varrho }_k\parallel u\parallel +\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}{\mathcal{V}}_0{(T-s_k)}^q\right)\hfill \\ \hfill & ={\mathfrak{Y}}^k.\hfill \end{array}$$

This ends the proof. □

Now, let us define an operator $\mathsf{\Phi }:\mathcal{P}\mathcal{C}(J,\mathbb{X})\to \mathcal{P}\mathcal{C}(J,\mathbb{X})$ as follows:

$$\left(\mathsf{\Phi }u\right)\left(t\right)=\left\{\begin{array}{c}{\mathcal{Q}}_q\left(t\right)(u_0-g_0\left(u\right)-p\left(0\right))+\mathcal{P}(t,u\left(a\left(t\right)\right))\hfill \\ +\underset{0}{\overset{t}{\int }}{\mathcal{Q}}_q\left(y\right)(u_1-g_1\left(u\right)-p\left(0\right))dy\hfill \\ +\underset{0}{\overset{t}{\int }}{(t-y)}^{q-1}{\mathcal{R}}_q(t,y)\left[\mathcal{H}(y,u(b\left(t\right)\left)\right)+\mathfrak{B}x\left(y\right)\right]dy,\hfill & t\in [0,t_1],\hfill \\ {\mu }_k(t,u\left(t\right)),\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(t_k,s_k],\hfill \\ {\mathcal{Q}}_q(t-s_k){\mu }_k(s_k,u\left(s_k\right))+\mathcal{P}(t,u\left(a\left(t\right)\right))\hfill \\ +\underset{s_k}{\overset{t}{\int }}{\mathcal{Q}}_q(y-s_k){\xi }_k(s_k,u\left(s_k\right))dy\hfill \\ +\underset{s_k}{\overset{t}{\int }}{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)[\mathcal{H}(y,u\left(b\left(t\right)\right))+\mathfrak{B}x\left(y\right)]dy,\hfill & t\in {\displaystyle \bigcup _{k=1}^m}(s_k,t_{k+1}].\hfill \end{array}\right.$$

(6)

Consequently, $x\left(t\right)$ is given by Equations (10) and (11) in the intervals $(0,t_1]$ and $(s_k,t_{k+1}],k=1,2,\dots ,m;m\in \mathbb{N}$, respectively. The existence of a mild solution to the system (1) must now be demonstrated in order to demonstrate its controllability.

4.1. Non-Compactness Case

It is feasible to further explore the presence of mild solutions in a scenario of non-compactness by employing Kuratowski’s measure of non-compactness and Sadovskii’s fixed point Theorem 5. This issue could be handled by taking into account the subsequent actual result.

Theorem 2. 

Assume that the assumptions $\left(\mathbb{A}{\mathbb{E}}_1\right)$–$\left(\mathbb{A}{\mathbb{E}}_4\right)$ with $\mathsf{\Xi }<1$ hold, where

$$\mathsf{\Xi }=\underset{k}{max}\left\{\varrho ,{\gamma }_{\eta }^k,{\varrho }_k\right\}.$$

Then, the system (1) is totally controllable on J.

Proof. 

Whenever ε is a positive number, we stipulate

$${\mathsf{\Omega }}_{\epsilon }=\left\{u\in {\mathcal{P}\mathcal{C}(J,\mathbb{X}):\parallel u\parallel }_{\mathcal{P}\mathcal{C}}\le \epsilon \right\}\subseteq \mathcal{P}\mathcal{C}(J,\mathbb{X})$$

where

$$\epsilon >\underset{k}{max}\left\{\frac{\overline{\chi }+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}({\mathcal{V}}_0+\mathfrak{Y}\mathfrak{K})t_1^q}{1-\varrho },\frac{{\gamma }_{\eta }^0}{1-{\gamma }_{\eta }^k},\frac{{\overline{\chi }}_k+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}({\mathcal{V}}_0+\mathfrak{Y}\mathfrak{K}){(T-s_k)}^q}{1-{\varrho }_k}\right\}.$$

Plainly, the subset ${\mathsf{\Omega }}_{\epsilon }$ is a closed, bounded, and convex non-empty subset of Banach space $\mathcal{P}\mathcal{C}(J,\mathbb{X})$. Firstly, we show that $\mathsf{\Phi }:{\mathsf{\Omega }}_{\epsilon }\to {\mathsf{\Omega }}_{\epsilon }$ is continuous. Let the sequence ${\left\{u^n\right\}}_{n\in \mathbb{N}}$ of a Banach space $\mathcal{P}\mathcal{C}(J,\mathbb{X})$ such that $u^n\to u$ as $n\to \infty$. Correspondingly, three situations are taken into consideration.

• Case I: Whenever $t\in [0,t_1]$, we have

  $$\begin{array}{cc}\hfill \parallel \left(\mathsf{\Phi }{u}^n\right)\left(t\right)-\left(\mathsf{\Phi }u\right)\left(t\right)\parallel & \le \varpi \parallel g_0\left(u^n\right)-g_0\left(u\right)\parallel +\mathcal{N}\parallel u^n-u\parallel \hfill \\ \hfill & +{\int }_0^t\parallel {\mathcal{Q}}_q\left(y\right)\parallel g_1\left(u^n\right)-g_1\left(u\right)\parallel dy\hfill \\ \hfill & +{\int }_0^t{(t-y)}^{q-1}\parallel {\mathcal{R}}_q(t,y)\parallel \mathcal{H}(y,u^n\left(b\left(t\right)\right))-\mathcal{H}(y,u\left(b\left(t\right)\right))\parallel dy\hfill \\ \hfill & \le \varpi \delta \parallel u^n-u\parallel +\mathcal{N}\parallel u^n-u\parallel +\varpi \zeta {\int }_0^t\parallel u^n-u\parallel dy\hfill \\ \hfill & +\frac{\varpi \mathcal{V}}{\mathsf{\Gamma }\left(2q\right)}{\int }_0^t{(t-y)}^{q-1}\parallel u^n-u\parallel dy\hfill \\ \hfill & \le \varrho \parallel u^n-u\parallel \to 0\mathrm{as}n\to \infty .\hfill \end{array}$$
• Case II: For any $t\in (t_k,s_k],k=1,\dots ,m;m\in \mathbb{N}$, there still are

  $$\begin{array}{c}\hfill \parallel \left(\mathsf{\Phi }{u}^n\right)\left(t\right)-\left(\mathsf{\Phi }u\right)\left(t\right)\parallel \le {\gamma }_{\eta }^k\parallel u^n-u\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$
• Case III: For any $t\in (s_k,t_{k+1}],k=1,\dots ,m;m\in \mathbb{N}$, we have

  $$\begin{array}{cc}\hfill \parallel \left(\mathsf{\Phi }{u}^n\right)\left(t\right)-\left(\mathsf{\Phi }u\right)\left(t\right)\parallel & \le \varpi \parallel {\eta }_k(s_k,u^n\left(s_k\right))-{\eta }_k(s_k,u\left(s_k\right))\parallel +\mathcal{N}\parallel u^n-u\parallel \hfill \\ \hfill & +\varpi {\int }_{s_k}^t\parallel {\xi }_k(s_k,u^n\left(s_k\right))-{\xi }_k(s_k,u\left(s_k\right))\parallel dy\hfill \\ \hfill & +\frac{\varpi }{\mathsf{\Gamma }\left(2q\right)}{\int }_{s_k}^t{(t-y)}^{q-1}\parallel \mathcal{H}(y,u^n\left(b\left(t\right)\right))-\mathcal{H}(y,u\left(b\left(t\right)\right))\parallel dy\hfill \\ \hfill & \le \varpi {\gamma }_{\eta }^k\parallel u^n-u\parallel +\mathcal{N}\parallel u^n-u\parallel +\varpi {\gamma }_{\xi }^k{\int }_{s_k}^t\parallel u^n-u\parallel dy\hfill \\ \hfill & +\frac{\varpi \mathcal{V}}{\mathsf{\Gamma }\left(2q\right)}{\int }_{s_k}^t{(t-y)}^{q-1}\parallel u^n-u\parallel dy\hfill \\ \hfill & \le {\varrho }_k\parallel u^n-u\parallel \to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

For the aforementioned, we may write that $\mathsf{\Phi }:{\mathsf{\Omega }}_{\epsilon }\to {\mathsf{\Omega }}_{\epsilon }$ is continuous. We now demonstrate how Φ maps ${\mathsf{\Omega }}_{\epsilon }$ onto itself. Then, for any $u\in {\mathsf{\Omega }}_{\epsilon }$ and in light of Lemmas 10 and 11, three circumstances are consequently taken into account.

• Case I: Whenever $t\in [0,t_1]$, we have

  $$\begin{array}{cc}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)\parallel & \le \parallel {\mathcal{Q}}_q\left(t\right)\parallel (\parallel u_0\parallel +\parallel g_0\left(u\right)\parallel +\parallel p\left(0\right)\parallel )+\parallel \mathcal{P}(t,u\left(a\left(t\right)\right))\parallel \hfill \\ \hfill & +{\int }_0^t\parallel {\mathcal{Q}}_q\left(y\right)\parallel (\parallel u_1\parallel +\parallel g_1\left(u\right)\parallel +\parallel p\left(0\right)\parallel )dy\hfill \\ \hfill & +{\int }_0^t{(t-y)}^{q-1}\parallel {\mathcal{R}}_q(t,y)\parallel \left[\parallel \mathcal{H}(y,u(b\left(t\right)\left)\right)\parallel +\parallel \mathfrak{B}x\left(y\right)\parallel \right]dy\hfill \\ \hfill & \le \varpi \left(\parallel u_0\parallel +\delta \parallel u\parallel +{\delta }_0+\parallel p\left(0\right)\parallel \right)+\mathcal{N}\parallel u\parallel \hfill \\ \hfill & +\varpi \left(\parallel u_1\parallel +\zeta \parallel u\parallel +{\zeta }_0+\parallel p\left(0\right)\parallel \right)t_1+{\mathcal{N}}_0\hfill \\ \hfill & +\frac{\varpi }{\mathsf{\Gamma }\left(2q\right)}(\mathcal{V}\parallel u\parallel +{\mathcal{V}}_0){\int }_0^t{(t-y)}^{q-1}dy+\frac{\varpi \mathfrak{Y}\mathfrak{K}}{\mathsf{\Gamma }\left(2q\right)}{\int }_0^t{(t-y)}^{q-1}dy\hfill \\ \hfill & \le \varpi \left(\parallel u_0\parallel +\delta \epsilon +{\delta }_0+\parallel p\left(0\right)\parallel \right)+\mathcal{N}\epsilon \hfill \\ \hfill & +\varpi \left(\parallel u_1\parallel +\zeta \epsilon +{\zeta }_0+\parallel p\left(0\right)\parallel \right)t_1+{\mathcal{N}}_0\hfill \\ \hfill & +\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}{t}_1^q\left(\mathcal{V}\epsilon +{\mathcal{V}}_0+\mathfrak{Y}\mathfrak{K}\right)\hfill \\ \hfill & \le \overline{\chi }+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}({\mathcal{V}}_0+\mathfrak{Y}\mathfrak{K})t_1^q+\varrho \epsilon \le \epsilon .\hfill \end{array}$$
• Case II: For any $t\in (t_k,s_k],k=1,\dots ,m;m\in \mathbb{N}$, there still are

  $$\begin{array}{c}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)\parallel \le {\gamma }_{\eta }^k\parallel u\parallel +{\gamma }_{\eta }^0\le \epsilon .\end{array}$$
• Case III: For any $t\in (s_k,t_{k+1}],k=1,\dots ,m;m\in \mathbb{N}$, we have

  $$\begin{array}{cc}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)\parallel & \le \varpi \parallel {\mu }_k(s_k,u\left(s_k\right))\parallel +\parallel \mathcal{P}(t,u\left(a\left(t\right)\right))\parallel +\varpi {\int }_{s_k}^t\parallel {\xi }_k(s_k,u\left(s_k\right))\parallel dy\hfill \\ \hfill & +{\int }_{s_k}^t{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)\parallel \mathcal{H}(y,u\left(b\left(t\right)\right))\parallel dy+{\int }_{s_k}^t{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)\parallel \mathfrak{B}x\left(y\right)\parallel dy\hfill \\ \hfill & \le \varpi ({\gamma }_{\eta }^k\parallel u\parallel +{\gamma }_{\eta }^0)+(\mathcal{N}\parallel u\parallel +{\mathcal{N}}_0)+\varpi ({\gamma }_{\xi }^k\parallel u\parallel +{\gamma }_{\xi }^0)(T-s_k)+\frac{\varpi }{\mathsf{\Gamma }\left(2q\right)}(\mathcal{V}\parallel u\parallel \hfill \\ \hfill & +{\mathcal{V}}_0){\int }_{s_k}^t{(t-y)}^{q-1}dy+\frac{\varpi \mathfrak{Y}\mathfrak{K}}{\mathsf{\Gamma }\left(2q\right)}{\int }_{s_k}^t{(t-y)}^{q-1}dy\hfill \\ \hfill & \le \varpi ({\gamma }_{\eta }^k\epsilon +{\gamma }_{\eta }^0+({\gamma }_{\xi }^k\epsilon +{\gamma }_{\xi }^0)(T-s_k))+(\mathcal{N}\epsilon +{\mathcal{N}}_0)\hfill \\ \hfill & +\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\mathcal{V}\epsilon +{\mathcal{V}}_0+\mathfrak{Y}\mathfrak{K}){(T-s_k)}^q\hfill \\ \hfill & \le {\overline{\chi }}_k+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}({\mathcal{V}}_0+\mathfrak{Y}\mathfrak{K}){(T-s_k)}^q+{\varrho }_k\epsilon \le \epsilon .\hfill \end{array}$$

For the aforementioned, we acquire $\parallel \mathsf{\Phi }\parallel \le \epsilon$. Thus, the operator Φ maps bounded subset into bounded subset in ${\mathsf{\Omega }}_{\epsilon }$. The next step of this argument is to demonstrate that the operator Φ satisfies the inequality of the Kuratowski measure of non-compactness in Lemma 5. Consider the expressions $u,u^{*}\in {\mathsf{\Omega }}_{\epsilon }$. Using the assumptions $\left(\mathbb{A}{\mathbb{E}}_1\right)$–$\left(\mathbb{A}{\mathbb{E}}_4\right)$, we obtain

• Case I: Whenever $t\in [0,t_1]$, we have

  $$\begin{array}{cc}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)-\left(\mathsf{\Phi }{u}^{*}\right)\left(t\right)\parallel & \le \varpi \parallel g_0\left(u\right)-g_0\left(u^{*}\right)\parallel +\mathcal{N}\parallel u-u^{*}\parallel \hfill \\ \hfill & +{\int }_0^t\parallel {\mathcal{Q}}_q\left(y\right)\parallel g_1\left(u\right)-g_1\left(u^{*}\right)\parallel dy\hfill \\ \hfill & +{\int }_0^t{(t-y)}^{q-1}\parallel {\mathcal{R}}_q(t,y)\parallel \mathcal{H}(y,u\left(b\left(t\right)\right))-\mathcal{H}(y,u^{*}\left(b\left(t\right)\right))\parallel dy\hfill \\ \hfill & \le \varpi \delta \parallel u-u^{*}\parallel +\mathcal{N}\parallel u-u^{*}\parallel +\varpi \zeta {\int }_0^t\parallel u-u^{*}\parallel dy\hfill \\ \hfill & +\frac{\varpi \mathcal{V}}{\mathsf{\Gamma }\left(2q\right)}{\int }_0^t{(t-y)}^{q-1}\parallel u-u^{*}\parallel dy\hfill \\ \hfill & \le \left(\varpi (\delta +t_1\zeta )+\mathcal{N}+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\mathcal{V}{t}_1^q\right)\parallel u-u^{*}\parallel \hfill \\ \hfill & =\varrho \parallel u-u^{*}\parallel .\hfill \end{array}$$
• Case II: For any $t\in (t_k,s_k],k=1,\dots ,m;m\in \mathbb{N}$, there still are

  $$\begin{array}{c}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)-\left(\mathsf{\Phi }{u}^{*}\right)\left(t\right)\parallel \le {\gamma }_{\eta }^k\parallel u-u^{*}\parallel .\end{array}$$
• Case III: For any $t\in (s_k,t_{k+1}],k=1,\dots ,m;m\in \mathbb{N}$, we have

  $$\begin{array}{cc}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)-\left(\mathsf{\Phi }{u}^{*}\right)\left(t\right)\parallel & \le \varpi \parallel {\eta }_k(s_k,u\left(s_k\right))-{\eta }_k(s_k,u^{*}\left(s_k\right))\parallel +\mathcal{N}\parallel u-u^{*}\parallel \hfill \\ \hfill & +\varpi {\int }_{s_k}^t\parallel {\xi }_k(s_k,u\left(s_k\right))-{\xi }_k(s_k,u^{*}\left(s_k\right))\parallel dy\hfill \\ \hfill & +\frac{\varpi }{\mathsf{\Gamma }\left(2q\right)}{\int }_{s_k}^t{(t-y)}^{q-1}\parallel \mathcal{H}(y,u\left(b\left(t\right)\right))-\mathcal{H}(y,u^{*}\left(b\left(t\right)\right))\parallel dy\hfill \\ \hfill & \le \varpi {\gamma }_{\eta }^k\parallel u-u^{*}\parallel +\mathcal{N}\parallel u-u^{*}\parallel +\varpi {\gamma }_{\xi }^k{\int }_{s_k}^t\parallel u-u^{*}\parallel dy\hfill \\ \hfill & +\frac{\varpi \mathcal{V}}{\mathsf{\Gamma }\left(2q\right)}{\int }_{s_k}^t{(t-y)}^{q-1}\parallel u-u^{*}\parallel dy\hfill \\ \hfill & \le \left(\varpi \left({\gamma }_{\eta }^k+{\gamma }_{\xi }^k(T-s_k)\right)+\mathcal{N}+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\mathcal{V}{(T-s_k)}^q\right)\parallel u-u^{*}\parallel \hfill \\ \hfill & ={\varrho }_k\parallel u-u^{*}\parallel .\hfill \end{array}$$

For the aforementioned, we may write

$$\parallel \left(\mathsf{\Phi }u\right)\left(t\right)-\left(\mathsf{\Phi }{u}^{*}\right)\left(t\right)\parallel \le \mathsf{\Xi }\parallel u-u^{*}\parallel .$$

Let $S\subset {\mathsf{\Omega }}_{\epsilon }$ be closed such that there are $S_i,i=1,2,\dots ,n;n\in \mathbb{N}$ and $S\subseteq {\bigcup }_{i=1}^n{S}_i$. Then, according to the definitions of diameter and Kuratowski measure of non-compactness, we conclude that

$$\begin{array}{cc}\hfill \mu \left(\mathsf{\Phi }S\right)& =inf\left\{\iota :diam\left(\mathsf{\Phi }{S}_i\right)\le \iota ,S\subseteq {\displaystyle \bigcup _{i=1}^n}{S}_i\right\}\hfill \\ \hfill & =inf\left\{\iota :sup\left\{\parallel \left(\mathsf{\Phi }u\right)\left(t\right)-\left(\mathsf{\Phi }{u}^{*}\right)\left(t\right){\parallel }_{\mathsf{\Omega }}\right\}\le \iota ,u,u^{*}\in S_i,S\subseteq {\displaystyle \bigcup _{i=1}^n}{S}_i\right\}\hfill \\ \hfill & \le \mathsf{\Xi }inf\left\{\iota :sup\left\{\parallel u\left(t\right)-u^{*}{\left(t\right)\parallel }_{\mathsf{\Omega }}\right\}\le \iota ,u,u^{*}\in S_i,S\subseteq {\displaystyle \bigcup _{i=1}^n}{S}_i\right\}\hfill \\ \hfill & =\mathsf{\Xi }inf\left\{\iota :diam\left(S_i\right)\le \iota ,S\subseteq {\displaystyle \bigcup _{i=1}^n}{S}_i\right\}\hfill \\ \hfill & =\mathsf{\Xi }\mu \left(S\right).\hfill \end{array}$$

According to Lemma 4 (vii), we are aware that, for each bounded $S\in {\mathsf{\Omega }}_{\epsilon },$

$$\begin{array}{c}\hfill \mu \left(\mathsf{\Phi }\right(S\left)\right)\le \mathsf{\Xi }\mu \left(S\right).\end{array}$$

This indicates that the operator $\mathsf{\Phi }:{\mathsf{\Omega }}_{\epsilon }\to {\mathsf{\Omega }}_{\epsilon }$ is μ-condensing. The operator Φ has at least one fixed point $u\in {\mathsf{\Omega }}_{\epsilon }$, which is merely a mild solution of problem (1), and it follows from Sadovskii’s fixed point theorem. Therefore, we conclude that the system (1) is totally controllable on J. □

4.2. Compactness Case

In this subsection, we assume the compactness of controllability of a mild solution and investigate the existence of it by employing the Leray–Schauder non-linear alternative theorem to deduce the first result about the existence of solution of the problem (1). We first propose these hypotheses before we explain and demonstrate our main results:

$\left(\mathbb{A}{\mathbb{E}}_5\right)$

There exists a constant $\mathcal{Y}>0$, satisfying $\parallel u\parallel \ne \mathcal{Y}$ for some $u\in \mathcal{P}\mathcal{C}(J,\mathbb{X})$.

$\left(\mathbb{A}{\mathbb{E}}_6\right)$

There exist ${\kappa }_i\in \mathcal{C}(J,{\mathbb{R}}^{+})$ and ${\mathbf{m}}_i:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$, $i=1,2$ are non-decreasing functions, such that the continuous functions $\mathcal{H}(t,u)$ and $\mathcal{P}(t,u)$ satisfy

$$\begin{array}{c}\hfill \parallel \widehat{\mathcal{H}}\left(t\right)\parallel =\parallel \mathcal{H}(t,u)\parallel \le {\kappa }_1\left(t\right){\mathbf{m}}_1\left(\right|u\left|\right),\\ \hfill \parallel \widehat{\mathcal{P}}\left(t\right)\parallel =\parallel \mathcal{P}(t,u)\parallel \le {\kappa }_2\left(t\right){\mathbf{m}}_2\left(\right|u\left|\right).\end{array}$$

$\left(\mathbb{A}{\mathbb{E}}_7\right)$

There exist constants ${\vartheta }_k,{\vartheta }_k^{*}>0,k=1,2\dots ,m;m\in \mathbb{N}$, such that the continuous non-instantaneous impulses satisfy

$$\begin{array}{c}\hfill \parallel {\eta }_k(t,u)\parallel \le {\vartheta }_k,\\ \hfill \parallel {\xi }_k(t,u)\parallel \le {\vartheta }_k^{*}.\end{array}$$

Moreover, there exist positive constants ${\aleph }_i,i=0,1$, such that the continuous non-local functions satisfy

$$\begin{array}{c}\hfill \parallel g_0\left(u\right)\parallel \le {\aleph }_0,\\ \hfill \parallel g_1\left(u\right)\parallel \le {\aleph }_1.\end{array}$$

Lemma 12. 

Let the control functions be defined as in Lemmas 10 and 11. Then, by assumptions $\left(\mathbb{A}{\mathbb{E}}_1\right)$, $\left(\mathbb{A}{\mathbb{E}}_6\right)$ and $\left(\mathbb{A}{\mathbb{E}}_7\right)$, we obtain

$$\begin{array}{cc}\hfill \parallel x\left(t\right)\parallel & \le \left\{\begin{array}{cc}\mathfrak{J}={\mathfrak{D}}^0\left[\parallel u_{t_1}\parallel +\mathcal{M}+\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\right|u\left|\right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right))t_1\right],\hfill & t\in [0,t_1],\hfill \\ {\mathfrak{J}}_k={\mathfrak{D}}^k\left[\parallel u_{t_{k+1}}\parallel +{\mathcal{M}}_k+\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\right|u\left|\right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right){(T-s_k)}^q\right],\hfill & t\in (s_k,t_{k+1}].\hfill \end{array}\right.\hfill \\ \hfill \mathrm{where},\\ \hfill \mathcal{M}& =\varpi \left[\parallel u_0\parallel +{\aleph }_0+\parallel p\left(0\right)\parallel +\left(\parallel u_1\parallel +{\aleph }_1+\parallel p\left(0\right)\parallel \right)t_1\right],{\mathcal{M}}_k=\varpi {\vartheta }_k+\varpi {\vartheta }_k^{*}(T-s_k).\hfill \end{array}$$

Proof. 

The control estimate is also computed as two cases:

• Case I: Whenever $t\in [0,t_1]$, we have

  $$\begin{array}{cc}\hfill \parallel x\left(t\right)\parallel \le & {\mathfrak{D}}^0[\parallel u_{t_1}\parallel +\varpi (\parallel u_0\parallel +{\aleph }_0+\parallel p\left(0\right)\parallel )+\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\right|u\left|\right)\hfill \\ \hfill +& \varpi (\parallel u_1\parallel +{\aleph }_1+\parallel p\left(0\right)\parallel )t_1+\frac{\varpi t_1^q}{q\mathsf{\Gamma }\left(2q\right)}(\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right)\left)\right]\hfill \\ \hfill \le & {\mathfrak{D}}^0\left[\parallel u_{t_1}\parallel +\mathcal{M}+\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\right|u\left|\right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right))t_1\right]\hfill \\ \hfill =& \mathfrak{J}.\hfill \end{array}$$
• Case II: For any $t\in [s_k,t_{k+1}]k=1,2\dots ,m;m\in \mathbb{N}$, we have

  $$\begin{array}{cc}\hfill \parallel x\left(t\right)\parallel \le & {\mathfrak{D}}^k[\parallel u_{t_{k+1}}\parallel +\varpi {\vartheta }_k+\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\right|u\left|\right)+\varpi {\vartheta }_k^{*}(t_{k+1}-s_k)\hfill \\ \hfill +& \frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right){(t_{k+1}-s_k)}^q]\hfill \\ \hfill & \le {\mathfrak{D}}^k\left[\parallel u_{t_{k+1}}\parallel +{\mathcal{M}}_k+\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\right|u\left|\right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right){(T-s_k)}^q\right]\hfill \\ \hfill & ={\mathfrak{J}}^k.\hfill \end{array}$$

These end the proof. □

Theorem 3. 

Assume that the assumptions $\left(\mathbb{A}{\mathbb{E}}_1\right)$ and $\left(\mathbb{A}{\mathbb{E}}_5\right)–\left(\mathbb{A}{\mathbb{E}}_7\right)$ with $\beth >1$ hold, where

$$\beth =\underset{k}{max}\left\{\frac{\mathcal{Y}}{\mathcal{M}+ϝ},\frac{\mathcal{Y}}{{\vartheta }_k},\frac{\mathcal{Y}}{{\mathcal{M}}_k+{ϝ}_k}\right\}$$

and

$$\begin{array}{cc}\hfill ϝ& =\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\mathcal{Y}\right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\mathcal{Y}\right))t_1,\hfill \\ \hfill {ϝ}_k& =\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\mathcal{Y}\right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\mathcal{Y}\right){(T-s_k)}^q.\hfill \end{array}$$

Then, the system (1) is totally controllable on J.

Proof. 

There are various steps involved in verifying the Leray–Schauder non-linear alternative hypothesis. The first step is to argue that the operation $\mathsf{\Phi }:\mathcal{P}\mathcal{C}(J,\mathbb{X})\to \mathcal{P}\mathcal{C}(J,\mathbb{X})$ described by Equation (6) maps bounded sets into bounded sets in $\mathcal{P}\mathcal{C}(J,\mathbb{X})$. Alternatively, we demonstrate that, given a positive number $\beta$, there exists a positive constant $\mathcal{I}$ such that $\parallel \mathsf{\Phi }u\parallel \le \mathcal{I}$ for any $u\in {ϝ}_{\beta }$, where ${ϝ}_{\beta }$ is a closed bounded set defined as

$${ϝ}_{\beta }={\{u\in \mathcal{P}\mathcal{C}(J,\mathbb{X}):\parallel u\parallel }_{\mathcal{P}\mathcal{C}}\le \beta \}\subseteq \mathcal{P}\mathcal{C}(J,\mathbb{X})$$

with radius $\beta \ge \underset{k}{max}\{\overline{\mathbb{Y}},{\vartheta }_k,{\overline{\mathbb{Y}}}_k\}$ where

$$\begin{array}{cc}\hfill \overline{\mathbb{Y}}& =\mathcal{M}+\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\beta \right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\left(\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\beta \right)+\mathfrak{Y}\mathfrak{J}\right)t_1^q,\hfill \\ \hfill {\overline{\mathbb{Y}}}_k& ={\mathcal{M}}_k+\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\beta \right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\beta \right)+\mathfrak{Y}{\mathfrak{J}}_k){(T-s_k)}^q.\hfill \end{array}$$

Then, in light of $\left(\mathbb{A}{\mathbb{E}}_6\right)$ and $\left(\mathbb{A}{\mathbb{E}}_7\right)$ and by Lemma 12, we have three cases.

• Case I: Whenever $t\in [0,t_1]$, we have

  $$\begin{array}{cc}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)\parallel \le & \parallel {\mathcal{Q}}_q\left(t\right)\parallel (\parallel u_0\parallel +\parallel g_0\left(u\right)\parallel +\parallel p\left(0\right)\parallel )+\parallel \mathcal{P}(t,u\left(a\left(t\right)\right))\parallel \hfill \\ \hfill & +{\int }_0^t\parallel {\mathcal{Q}}_q\left(y\right)\parallel (\parallel u_1\parallel +\parallel g_1\left(u\right)\parallel +\parallel p\left(0\right)\parallel )dy\hfill \\ \hfill & +{\int }_0^t{(t-y)}^{q-1}\parallel {\mathcal{R}}_q(t,y)\parallel \left[\parallel \mathcal{H}(y,u(b\left(t\right)\left)\right)\parallel +\parallel \mathfrak{B}x\left(y\right)\parallel \right]dy\hfill \\ \hfill & \le \varpi \left(\parallel u_0\parallel +{\aleph }_0+\parallel p\left(0\right)\parallel \right)+\parallel \widehat{\mathcal{P}}(t,u)\parallel \hfill \\ \hfill & +\varpi \left(\parallel u_1\parallel +{\aleph }_1+\parallel p\left(0\right)\parallel \right)t_1\hfill \\ \hfill & +\frac{\varpi }{\mathsf{\Gamma }\left(2q\right)}{\int }_0^t{(t-y)}^{q-1}\parallel \widehat{\mathcal{H}}\left(y\right)\parallel dy+\frac{\varpi \mathfrak{Y}\mathfrak{J}}{\mathsf{\Gamma }\left(2q\right)}{\int }_0^t{(t-y)}^{q-1}dy\hfill \\ \hfill & \le \mathcal{M}+\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\beta \right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\left(\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\beta \right)+\mathfrak{Y}\mathfrak{J}\right)t_1^q=\overline{\mathbb{Y}}.\hfill \end{array}$$
• Case II: For any $t\in (t_k,s_k],k=1,\dots ,m;m\in \mathbb{N}$, there still are

  $$\begin{array}{c}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)\parallel \le {\vartheta }_k.\end{array}$$
• Case III: For any $t\in (s_k,t_{k+1}],k=1,\dots ,m;m\in \mathbb{N}$, we have

  $$\begin{array}{cc}\hfill \parallel \left(\mathsf{\Phi }u\right)\left(t\right)\parallel \le & \varpi \parallel {\mu }_k(s_k,u\left(s_k\right))\parallel +\parallel \mathcal{P}(t,u\left(a\left(t\right)\right))\parallel +\varpi {\int }_{s_k}^t\parallel {\xi }_k(s_k,u\left(s_k\right))\parallel dy\hfill \\ \hfill & +{\int }_{s_k}^t{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)\parallel \widehat{\mathcal{H}}(y,u)\parallel dy\hfill \\ \hfill & +{\int }_{s_k}^t{(t-y)}^{q-1}{\mathcal{R}}_q(t-y)\parallel \mathfrak{B}x\left(y\right)\parallel dy\hfill \\ \hfill & \le \varpi {\vartheta }_k+\parallel \widehat{\mathcal{P}}\left(y\right)\parallel \hfill \\ \hfill & +\varpi {\vartheta }_k^{*}(T-s_k)+\frac{\varpi }{\mathsf{\Gamma }\left(2q\right)}(\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\right|\beta \left|\right)){\int }_{s_k}^t{(t-y)}^{q-1}dy\hfill \\ \hfill & +\frac{\varpi \mathfrak{Y}{\mathfrak{J}}_k}{\mathsf{\Gamma }\left(2q\right)}{\int }_{s_k}^t{(t-y)}^{q-1}dy\hfill \\ \hfill & \le {\mathcal{M}}_k+\parallel {\kappa }_1\parallel {\mathbf{m}}_1\left(\beta \right)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\parallel {\kappa }_2\parallel {\mathbf{m}}_2\left(\beta \right)+\mathfrak{Y}{\mathfrak{J}}_k\left){(T-s_k)}^q\right)={\overline{\mathbb{Y}}}_k.\hfill \end{array}$$

From the above three inequalities, we can conclude that $\parallel \mathsf{\Phi }u\parallel \le \mathbb{Y}$ where $\mathbb{Y}={\displaystyle \underset{k}{max}}\{\overline{\mathbb{Y}},{\vartheta }_k,{\overline{\mathbb{Y}}}_k\}.$ Thus, the operator $\mathsf{\Phi }$ maps bounded sets into bounded sets of the space $\mathcal{P}\mathcal{C}(J,\mathbb{X})$.

The next step is to verify that the operator $\mathsf{\Phi }$ in $\mathcal{P}\mathcal{C}(J,\mathbb{X})$ transforms bounded sets into equicontinuous sets. In light of the criteria $\left(\mathbb{A}{\mathbb{E}}_6\right)$ and $\left(\mathbb{A}{\mathbb{E}}_7\right)$, $\mathsf{\Phi }$ is continuous.

• Case I: Whenever $0\le {\tau }_1<{\tau }_2\le t_1$ and $u\in {ϝ}_{\beta }$, we obtain that

  $$\begin{array}{cc}\hfill & \parallel \left(\mathsf{\Phi }u\right)\left({\tau }_2\right)-\left(\mathsf{\Phi }u\right)\left({\tau }_1\right)\parallel \le \left[\parallel u_0\parallel +{\aleph }_0+\parallel P\left(0\right)\parallel \right]∥{\mathcal{Q}}_q\left({\tau }_2\right)-{\mathcal{Q}}_q\left({\tau }_1\right)∥\hfill \\ \hfill & +\parallel \mathcal{P}({\tau }_2,u\left(a\left({\tau }_2\right)\right))-\mathcal{P}({\tau }_1,u\left(a\left({\tau }_1\right)\right))\parallel +\varpi \left[\parallel u_1\parallel +{\aleph }_1+\parallel P\left(0\right)\parallel \right]({\tau }_2-{\tau }_1)\hfill \\ \hfill & +{\int }_{{\tau }_1}^{{\tau }_2}{({\tau }_2-y)}^{q-1}{\mathcal{R}}_q({\tau }_2,y)\left[\parallel \widehat{\mathcal{H}}\left(y\right)\parallel +\parallel \mathfrak{B}x\left(y\right)\parallel \right]dy\hfill \\ \hfill & +{\int }_0^{{\tau }_1}∥{({\tau }_2-y)}^{q-1}\mathcal{R}({\tau }_2,y)-{({\tau }_1-y)}^{q-1}\mathcal{R}({\tau }_1,y)∥\parallel \widehat{\mathcal{H}}\left(y\right)\parallel dy\hfill \\ \hfill & +{\int }_0^{{\tau }_1}∥{({\tau }_2-y)}^{q-1}\mathcal{R}({\tau }_2,y)-{({\tau }_1-y)}^{q-1}\mathcal{R}({\tau }_1,y)∥\parallel \mathfrak{B}x\left(y\right)\parallel dy\hfill \\ \hfill & \le \left[\parallel u_0\parallel +{\aleph }_0+\parallel P\left(0\right)\parallel \right]∥{\mathcal{Q}}_q\left({\tau }_2\right)-{\mathcal{Q}}_q\left({\tau }_1\right)∥\hfill \\ \hfill & +\parallel \mathcal{P}({\tau }_2,u\left(a\left({\tau }_2\right)\right))-\mathcal{P}({\tau }_1,u\left(a\left({\tau }_1\right)\right))\parallel +\varpi \left[\parallel u_1\parallel +{\aleph }_1+\parallel P\left(0\right)\parallel \right]({\tau }_2-{\tau }_1)\hfill \\ \hfill & +\frac{\varpi }{q\mathsf{\Gamma }\left(q\right)}\left[\parallel k_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right)+\mathfrak{Y}\mathfrak{J}\right]\left[({\tau }_2-{\tau }_1)+({\tau }_1^q-{\tau }_2^q)+{({\tau }_2-{\tau }_1)}^q\right]\hfill \\ \hfill & +\left[\parallel k_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right)+\mathfrak{Y}\mathfrak{J}\right]{\int }_0^{{\tau }_1}{({\tau }_2-y)}^{q-1}\parallel \mathcal{R}({\tau }_2,y)-\mathcal{R}({\tau }_1,y)\parallel dy.\hfill \end{array}$$
  Based on the compactness of operator ${\mathcal{Q}}_q\left(y\right)$ and ${\mathcal{R}}_q(t,y)$ and the continuity of $\mathcal{P}(t,u(a\left(t\right))$. When permitting ${\tau }_2\to {\tau }_1,$ it is evident that the aforementioned inequality becomes closer to zero.
• Case II: For each $t_k<{\tau }_1<{\tau }_2\le s_k,k=1,2,\dots ,m;m\in \mathbb{N}$ and $u\in {ϝ}_{\beta }$, we obtain that

  $$\begin{array}{c}\hfill \parallel \left(\mathsf{\Phi }u\right)\left({\tau }_2\right)-\left(\mathsf{\Phi }u\right)\left({\tau }_1\right)\parallel =\parallel \eta ({\tau }_2,u)-\eta ({\tau }_1,u)\parallel .\end{array}$$
  Owing to the continuity of the non-instantaneous impulse, the above inequality approaches zero when letting ${\tau }_2\to {\tau }_1.$
• Case III: For each $s_k<{\tau }_1<{\tau }_2\le t_{k+1},k=1,2,\dots ,m;m\in \mathbb{N}$ and $u\in {ϝ}_{\beta }$, we obtain that

  $$\begin{array}{cc}\hfill & \parallel \left(\mathsf{\Phi }u\right)\left({\tau }_2\right)-\left(\mathsf{\Phi }u\right)\left({\tau }_1\right)\parallel \le {\vartheta }_k∥{\mathcal{Q}}_q\left({\tau }_2\right)-{\mathcal{Q}}_q\left({\tau }_1\right)∥+\parallel \mathcal{P}({\tau }_2,u\left(a\left({\tau }_2\right)\right))-\mathcal{P}({\tau }_1,u\left(a\left({\tau }_1\right)\right))\parallel \hfill \\ \hfill & +{\int }_{{\tau }_1}^{{\tau }_2}{\mathcal{Q}}_q(y-s_k)\xi (s_k,u\left(s_k\right))dy+{\int }_{{\tau }_1}^{{\tau }_2}{({\tau }_2-y)}^{q-1}{\mathcal{R}}_q({\tau }_2,y)\left[\parallel \widehat{\mathcal{H}}\left(y\right)\parallel +\parallel \mathfrak{B}x\left(y\right)\parallel \right]dy\hfill \\ \hfill & +{\int }_{s_k}^{{\tau }_1}∥{({\tau }_2-y)}^{q-1}\mathcal{R}({\tau }_2,y)-{({\tau }_1-y)}^{q-1}\mathcal{R}({\tau }_1,y)∥\parallel \widehat{\mathcal{H}}\left(y\right)\parallel dy\hfill \\ \hfill & +{\int }_{s_k}^{{\tau }_1}∥{({\tau }_2-y)}^{q-1}\mathcal{R}({\tau }_2,y)-{({\tau }_1-y)}^{q-1}\mathcal{R}({\tau }_1,y)∥\parallel \mathfrak{B}x\left(y\right)\parallel dy\hfill \\ \hfill & \le {\vartheta }_k∥{\mathcal{Q}}_q\left({\tau }_2\right)-{\mathcal{Q}}_q\left({\tau }_1\right)∥+\parallel \mathcal{P}({\tau }_2,u\left(a\left({\tau }_2\right)\right))-\mathcal{P}({\tau }_1,u\left(a\left({\tau }_1\right)\right))\parallel \hfill \\ \hfill & +\varpi {\vartheta }_k^{*}({\tau }_2-{\tau }_1)+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\left[\parallel k_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right)+\mathfrak{Y}{\mathfrak{J}}^k\right]({\tau }_2-{\tau }_1)\hfill \\ \hfill & +\frac{\varpi }{q\mathsf{\Gamma }\left(q\right)}\left[\parallel k_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right)+\mathfrak{Y}\mathfrak{J}\right]\left[({\tau }_2-{\tau }_1)+({\tau }_1^q-{\tau }_2^q)+{({\tau }_2-{\tau }_1)}^q\right]\hfill \\ \hfill & +\left[\parallel k_1\parallel {\mathbf{m}}_1\left(\right|u\left|\right)+\mathfrak{Y}\mathfrak{J}\right]{\int }_{s_k}^{{\tau }_1}{({\tau }_2-y)}^{q-1}\parallel \mathcal{R}({\tau }_2,y)-\mathcal{R}({\tau }_1,y)\parallel dy.\hfill \end{array}$$

Due to the compactness of operator ${\mathcal{Q}}_q\left(y\right)$ and ${\mathcal{R}}_q(t,y)$ and the continuity of $\mathcal{P}(t,u(a\left(t\right))$, when permitting ${\tau }_2\to {\tau }_1$, it is evident that the aforementioned inequality becomes closer to zero. The three aforementioned inequalities lead us to the conclusion that $\parallel \left(\mathsf{\Phi }u\right)\left({\tau }_2\right)-\left(\mathsf{\Phi }u\right)\left({\tau }_1\right)\parallel \to 0$ independently of $u\in \mathcal{P}\mathcal{C}(J,\mathbb{X})$ as ${\tau }_2\to {\tau }_1$. The operator $\mathsf{\Phi }:\mathcal{P}\mathcal{C}(J,\mathbb{X})\to \mathcal{P}\mathcal{C}(J,\mathbb{X})$ is completely continuous according to the prior proof and the Arzela–Ascoli theorem.

Finally, we demonstrate the existence of an open set $\mathcal{Z}\subset \mathcal{P}\mathcal{C}(J,\mathbb{X})$ with $u\ne \mathsf{\Lambda }\mathsf{\Phi }u$ for $\mathsf{\Lambda }\in (0,1)$ and $u\in \partial Z$. Consider the formula $u=\mathsf{\Lambda }\mathsf{\Phi }u$ for $\mathsf{\Lambda }\in (0,1)$. Afterward, with Step 1, we have the following cases:

• Case I: Whenever $0\le {\tau }_1<{\tau }_2\le t_1$ and $u\in {ϝ}_{\beta }$, we obtain that

  $$\begin{array}{cc}\hfill \parallel u\left(t\right)\parallel =\parallel \mathsf{\Lambda }\left(\mathsf{\Phi }u\right)\left(t\right)\parallel \le & \varpi \left(\parallel u_0\parallel +{\aleph }_0+\parallel p\left(0\right)\parallel \right)+\parallel {\kappa }_1\parallel {\mathbf{m}}_1(\parallel u\parallel )\hfill \\ \hfill +& \varpi \left(\parallel u_1\parallel +{\aleph }_1+\parallel p\left(0\right)\parallel \right)t_1\hfill \\ \hfill +& \frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\parallel {\kappa }_2\parallel {\mathbf{m}}_2(\parallel u\parallel +\mathfrak{Y}\mathfrak{J})t_1^q\hfill \\ \hfill =& \mathcal{M}+\parallel {\kappa }_1\parallel {\mathbf{m}}_1(\parallel u\parallel )+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\parallel {\kappa }_2\parallel {\mathbf{m}}_2(\parallel u\parallel +\mathfrak{Y}\mathfrak{J})t_1^q,\hfill \end{array}$$

  which indicates that

  $$\begin{array}{c}\hfill \frac{\parallel u\parallel }{\mathcal{M}+\parallel {\kappa }_1\parallel {\mathbf{m}}_1(\parallel u\parallel )+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\parallel {\kappa }_2\parallel {\mathbf{m}}_2(\parallel u\parallel +\mathfrak{Y}\mathfrak{J})t_1^q}\le 1.\end{array}$$

  (7)
• Case II: For each $t_k<{\tau }_1<{\tau }_2\le s_k,k=1,2,\dots ,m,m\in \mathbb{N}$ and $u\in {ϝ}_{\beta }$, we obtain that

  $$\begin{array}{c}\hfill \parallel u\left(t\right)\parallel =\parallel \mathsf{\Lambda }\left(\mathsf{\Phi }u\right)\left(t\right)\parallel \le {\vartheta }_k,\end{array}$$

  which suggests

  $$\begin{array}{c}\hfill \frac{\parallel u\parallel }{{\vartheta }_k}\le 1.\end{array}$$

  (8)
• Case III: For each $s_k<{\tau }_1<{\tau }_2\le t_{k+1},k=1,2,\dots ,m,m\in \mathbb{N}$ and $u\in {ϝ}_{\beta }$, we obtain that

  $$\begin{array}{cc}\hfill \parallel u\left(t\right)\parallel =\parallel \mathsf{\Lambda }\left(\mathsf{\Phi }u\right)\left(t\right)\parallel \le & \varpi {\vartheta }_k+\parallel {\kappa }_1\parallel {\mathbf{m}}_1(\parallel u\parallel )+\varpi {\vartheta }_k^{*}(T-s_k)\hfill \\ \hfill +& \frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}(\parallel {\kappa }_2\parallel {\mathbf{m}}_2(\parallel u\parallel )+\mathfrak{Y}{\mathfrak{J}}_k\left){(T-s_k)}^q\right)\hfill \\ \hfill =& {\mathcal{M}}_k+\parallel {\kappa }_1\parallel {\mathbf{m}}_1(\parallel u\parallel )+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\left(\parallel {\kappa }_2\parallel {\mathbf{m}}_2(\parallel u\parallel )+\mathfrak{Y}{\mathfrak{J}}_k\right){(T-s_k)}^q),\hfill \end{array}$$

  which demonstrates that

  $$\begin{array}{c}\hfill \frac{\parallel u\parallel }{{\mathcal{M}}_k+\parallel {\kappa }_1\parallel {\mathbf{m}}_1(\parallel u\parallel )+\frac{\varpi }{q\mathsf{\Gamma }\left(2q\right)}\left(\parallel {\kappa }_2\parallel {\mathbf{m}}_2(\parallel u\parallel )+\mathfrak{Y}{\mathfrak{J}}_k\right){(T-s_k)}^q)}\le 1.\end{array}$$

  (9)

If (7)–(9) are integrated with $\left(\mathbb{A}{\mathbb{E}}_5\right)$ and the specified circumstance $\beth >1$ is met, it is possible to find a positive number $\mathcal{Y}$ such that $\parallel u\parallel \ne \mathcal{Y}$ may be established. Create the operator $\mathsf{\Phi }:\overline{\mathcal{Z}}\to \mathcal{P}\mathcal{C}(J,\mathbb{X})$ being continuous and completely continuous with a set $\mathcal{Z}=\{u\in \mathcal{P}\mathcal{C}(J,\mathbb{X}):\parallel u\parallel <\mathcal{Y}\}$. Based on the $\mathcal{Y}$ that was chosen, there is no $u\in \partial \mathcal{Z}$ satisfying $u=\mathsf{\Lambda }\mathsf{\Phi }u$ for $\mathsf{\Lambda }\in (0,1)$. According to the Leray–Schauder non-linear alternative, the operator $\mathsf{\Phi }$ has a fixed point $u\in \overline{\mathcal{Z}}$ that identifies a solution to the system (1). □

5. Application

To show how our main results can be applied, we offer an application in this section. Consider the following fractional inhomogeneous wave equation:

$$\left\{\begin{array}{c}{\displaystyle {}^c{D}_t^{\frac{5}{3}}[u(t,y)-\mathcal{P}(t,u\left(a\left(t\right)\right))]=\frac{{\partial }^2}{\partial x^2}[u(t,y)-\mathcal{P}(t,u\left(a\left(t\right)\right))]}\hfill \\ +\mathcal{H}(t,u(b\left(t\right)\left)\right)+\mathfrak{B}x(t,y),\hfill & t\in (0,\frac{3}{7}]\cup (\frac{5}{7},1],y\in [0,\pi ],\hfill \\ u(t,y)=\frac{(1+t^{\frac{3}{2}})}{7}+\frac{sin\frac{3u}{\sqrt{7}}}{10\sqrt{2-t}},\hfill & t\in (\frac{3}{7},\frac{5}{7}]y\in [0,\pi ],\hfill \\ u^{\prime }(t,y)=\frac{3t^{\frac{1}{2}}}{14}+\frac{cos\left(\frac{2u}{3}\right)}{(3+7t)},\hfill & t\in (\frac{3}{7},\frac{5}{7}]y\in [0,\pi ],\hfill \\ u(t,0)=u(t,\pi )=0,\hfill & t\in [0,1]\hfill \\ u(0,y)+\frac{sin3u}{9}=\frac{1}{2},\hfill & t\in [0,1],y\in [0,\pi ],\hfill \\ u^{\prime }(0,y)+\frac{arctan2u}{16\left|u\right|}=\frac{1}{19},\hfill & t\in [0,1],y\in [0,\pi ].\hfill \end{array}\right.$$

Let the space $\mathbb{X}=\mathbb{U}=L^2[0,\pi ]$ be the space of a square-integrable function equipped with the norm

$${∥\mathcal{U}∥}_{\mathbb{X}}={\left({\int }_0^{\pi }{\left|\mathcal{U}\left(\xi \right)\right|}^{2}d\xi \right)}^{\frac{1}{2}}.$$

Furthermore, the operator $\mathbb{A}:D\left(\mathbb{A}\right)\subset \mathbb{X}\to \mathbb{X}$ is defined as $\mathbb{A}=\frac{{\partial }^2}{\partial x^2}$ with a domain

$$D\left(\mathbb{A}\right)=\left\{\mathcal{U}\in \mathbb{X}|\frac{\partial }{\partial x}\mathcal{U},\frac{{\partial }^2}{\partial x^2}\mathcal{U}\in \mathbb{X}\right\}$$

The operator $\mathbb{A}$ is densely defined in $\mathbb{X}$ and is the infinitesimal generator of a resolvent cosine family $\mathcal{K}\left(\xi \right),\xi >0$ on $\mathbb{X}$.

Here, we take $\alpha =\frac{5}{3}$, which implies $q=\frac{5}{6}$ and $\mathbb{A}=\frac{{\partial }^2}{\partial x^2},x\in [0,\pi ]$, and $J=[0,1],0=t_0=s_0<t_1=\frac{3}{7}<s_1=\frac{5}{7}<t_2=1,$ $u_0=\frac{1}{2},u_1=\frac{1}{19},a\left(t\right)=\frac{3t}{2},b\left(t\right)=\frac{t}{9}$. Moreover, we take $∥{\mathcal{Q}}_q\left(t\right)∥\le 1,∥{\mathcal{R}}_q(t,y)∥\le \frac{1}{\mathsf{\Gamma }\left(\frac{5}{3}\right)}\left\{\underset{t\in (0,1]}{sup}\left|t\right|\right\}=\frac{1}{\mathsf{\Gamma }\left(\frac{5}{3}\right)}$.

In our example, the non-instantaneous functions that are continuous on the interval $(\frac{3}{7},\frac{5}{7}]$,

$$\begin{array}{cc}\hfill \eta (t,u)& =\frac{(1+t^{\frac{3}{2}})}{7}+\frac{sin\left(\frac{3u}{\sqrt{7}}\right)}{10\sqrt{2-t}},\hfill \\ \hfill \xi (t,u)& =\frac{3t^{\frac{1}{2}}}{14}+\frac{cos\left(\frac{2u}{3}\right)}{(3+7t)}.\hfill \end{array}$$

Furthermore, the non-local functions given by

$$\begin{array}{cc}\hfill g_0(u\left(t\right)=& \frac{sin3u}{9},\hfill \\ \hfill g_1\left(u\left(t\right)\right)=& \frac{arctan2u}{16\left|u\right|}.\hfill \end{array}$$

Define the control operator $\mathfrak{B}:\mathbb{U}\to \mathbb{X}$ by

$$\mathfrak{B}x=\sqrt{\frac{\pi }{2}}x.$$

with the norm

$$\begin{array}{c}\hfill ∥\mathfrak{B}x∥=\sqrt{\frac{\pi }{2}}\parallel x\parallel \mathrm{which}\mathrm{implies}\sqrt{\frac{\pi }{2}}=\mathfrak{K}.\end{array}$$

Assume the operator $\mathbb{W}:{\mathbb{L}}^2([0,1],\mathbb{U})\to \mathfrak{X}$ is defined as

• Case I: Whenever $t\in [0,\frac{3}{7}]$, if we choose $u_{\frac{3}{7}}=2$, we obtain

  $$\begin{array}{cc}\hfill {\mathcal{G}}_{0}x& ={\int }_0^{\frac{3}{7}}{\left(\frac{3}{7}-y\right)}^{q-1}{\mathcal{R}}_q\left(\frac{3}{7}-y\right)\mathfrak{B}x\left(y\right)dy,\hfill \\ \hfill ∥{\mathcal{G}}_{0}x∥& =\frac{5{\left(\frac{3}{7}\right)}^{\frac{5}{6}}}{6\mathsf{\Gamma }\left(\frac{5}{3}\right)}\sqrt{\frac{\pi }{2}}\parallel x\parallel ,\hfill \\ \hfill ∥{\mathcal{G}}_0^{-1}∥& =\frac{1}{{\mathcal{G}}_0}=0.5710:={\mathfrak{D}}^0.\hfill \end{array}$$
• Case II: For any $t\in (\frac{5}{7},1]$, if we chose $u_1=4$, we have

  $$\begin{array}{cc}\hfill {\mathcal{G}}_{1}x& ={\int }_{\frac{5}{7}}^1{\left(1-y\right)}^{q-1}{\mathcal{R}}_q(1-y)\mathfrak{B}x\left(y\right)dy,\hfill \\ \hfill ∥{\mathcal{G}}_{1}x∥& =\frac{5}{6\mathsf{\Gamma }\left(\frac{5}{3}\right)}\sqrt{\frac{\pi }{2}}{\left(1-\frac{5}{7}\right)}^{\frac{5}{6}}\parallel x\parallel ,\hfill \\ \hfill ∥{\mathcal{G}}_1^{-1}∥& =\frac{1}{{\mathcal{G}}_1}=0.4073:={\mathfrak{D}}^1.\hfill \end{array}$$

• Application to Theorem 2 With the conditions $\left(\mathbb{A}{\mathbb{E}}_1\right)$–$\left(\mathbb{A}\mathbb{E}4\right)$, the Sadovskii fixed point theorem has been applied in Theorem 2. Let us pick a specific example from our study. Let the continue functions $\mathcal{P}(t,u(a\left(t\right)\left)\right),\mathcal{H}(t,u(b\left(t\right)\left)\right):[0,1]\times \mathbb{X}\to \mathbb{X}$ be defined as

  $$\begin{array}{cc}\hfill & \mathcal{P}\left(t,u\left(a\right(t\left)\right)\right)=\frac{|u\left(\frac{3t}{2}\right)|}{15e^{3t+1}\left(1+u\left(\frac{3t}{2}\right)\right)},\hfill \\ \hfill & \mathcal{H}\left(t,u\left(b\right(t\left)\right)\right)=\frac{tsin\left(u\left(\frac{t}{9}\right)\right)}{\left(4+3e^{t^2}\right)u\left(\frac{t}{9}\right)}.\hfill \end{array}$$
  It is clear that the functions $\mathcal{P}(t,u(a\left(t\right)\left)\right),\mathcal{H}(t,u(b\left(t\right)\left)\right)$ are continuous and that these fulfill the hypothesis $\left(\mathbb{A}{\mathbb{E}}_2\right)$

  $$\begin{array}{cc}\hfill ∥\mathcal{P}\left(t,u\left(a\right(t\left)\right)\right)-\mathcal{P}\left(t,v\left(a\right(t\left)\right)\right)∥& \le \frac{∥\left(∥u\left(\frac{3t}{2}\right)∥-∥u\left(\frac{3t}{2}\right)∥\right)∥}{∥15e^{3t+1}∥∥\left(1+u\left(\frac{3t}{2}\right)\right)\left(1+v\left(\frac{3t}{2}\right)\right)∥}\hfill \\ \hfill & \le \frac{1}{15e}∥∥u\left(\frac{3t}{2}\right)∥-∥u\left(\frac{3t}{2}\right)∥∥\hfill \\ \hfill & \le \frac{1}{15e}∥u-v∥\le \frac{1}{15}∥u-v∥.\hfill \end{array}$$
  Furthermore,

  $$\begin{array}{cc}\hfill ∥\mathcal{H}\left(t,u\left(b\right(t\left)\right)\right)-\mathcal{H}\left(t,v\left(b\right(t\left)\right)\right)∥& \le \frac{∥t\left(sin\left(u\left(\frac{t}{9}\right)\right)-sin\left(v\left(\frac{t}{9}\right)\right)\right)∥}{∥\left(4+3e^{t^2}\right)∥∥\left(u\left(\frac{t}{9}\right)-v\left(\frac{t}{9}\right)\right)∥}\hfill \\ \hfill & \le ∥\frac{t}{\left(4+3e^{t^2}\right)}∥∥\left(sin\left(u\left(\frac{t}{9}\right)\right)-sin\left(v\left(\frac{t}{9}\right)\right)\right)∥\hfill \\ \hfill & \le \frac{1}{7}∥sin\left(u\left(\frac{t}{9}\right)\right)-sin\left(v\left(\frac{t}{9}\right)\right)∥\le \frac{1}{7}\parallel u-v\parallel .\hfill \end{array}$$
  Thus, the condition $\left(\mathbb{A}{\mathbb{E}}_2\right)$ of Theorem 2 is satisfied with

  $$\mathcal{N}=\frac{1}{15},\mathcal{V}=\frac{1}{7}.$$
  Moreover, the non-instantaneous function at $t\in (\frac{3}{7},\frac{5}{7}]$ verifies

  $$\begin{array}{cc}\hfill & ∥\eta (t,u)-\eta (t,v)∥=∥\frac{sin\left(\frac{3u}{\sqrt{7}}\right)}{10\sqrt{2-t}}-\frac{sin\left(\frac{3v}{\sqrt{7}}\right)}{10\sqrt{2-t}}∥\le \frac{1}{10}\parallel u-v\parallel ,\hfill \\ \hfill & ∥\xi (t,u)-\xi (t,v)∥=∥\frac{cos\left(\frac{2u}{3}\right)-cos\left(\frac{2v}{3}\right)}{(3+7t)}∥\le \frac{1}{8}∥cos\left(\frac{2u}{3}\right)-cos\left(\frac{2v}{3}\right)∥\le \frac{1}{12}\parallel u-v\parallel .\hfill \end{array}$$
  The assumption $\left(\mathbb{A}{\mathbb{E}}_3\right)$ satisfies

  $${\gamma }_{\eta }^1=\frac{1}{10},{\gamma }_{\xi }^1=\frac{1}{12}.$$
  To verify the last assumption for non-local conditions, we can write

  $$\begin{array}{cc}\hfill & ∥\frac{sin3u}{9}-\frac{sin3v}{9}∥\le \frac{1}{9}∥sin3u-sin3v∥\le \frac{1}{3}\parallel u-v\parallel ,\hfill \\ \hfill & ∥\frac{arctan2u}{16\left|u\right|}-\frac{arctan2v}{16\left|v\right|}∥\le \frac{1}{16}∥arctan2u-arctan2v∥\le \frac{\pi }{16}\parallel u-v\parallel .\hfill \end{array}$$
  Therefore, the assumption $\left(\mathbb{A}{\mathbb{E}}_4\right)$ is achieved by these values

  $$\delta =\frac{1}{3}\zeta =\frac{\pi }{16}.$$
  In essence, we possess

  $$\mathsf{\Xi }=\underset{k}{max}\left\{\varrho ,{\gamma }_{\eta }^k,{\varrho }_k\right\}=\underset{k}{max}\{0.5605,0.1,0.2450\}=0.5473.$$
  As a result, Theorem 2 states that the system (5) is totally controllable on $[0,1].$
• Application to Theorem 3 For the purpose of proving Theorem 3, we take

  $$\begin{array}{cc}\hfill & \mathcal{H}\left(t,u\left(b\right(t\left)\right)\right)=\frac{e^{-2t}cos\left(3u\left(\frac{t}{9}\right)\right)}{18\sqrt{5-t}},\hfill \\ \hfill & \mathcal{P}\left(t,u\left(a\right(t\left)\right)\right)=\frac{2tanh\left(\frac{u}{5}\right)}{\left(1+e^{-t}\right)}.\hfill \end{array}$$
  Clearly, $\mathcal{H},\mathcal{P}:[0,1]\times \mathbb{X}\to \mathbb{X}$ are continuous and satisfy

  $$\begin{array}{cc}\hfill & ∥\widehat{\mathcal{H}}∥=∥\mathcal{H}\left(t,u\left(\frac{t}{9}\right)\right)∥\le \frac{cos\left(3u\right)}{18\sqrt{5-t}}\le \frac{1}{\sqrt{5-t}}\left(\frac{1}{6}∥u∥\right):={\mathbf{m}}_1\left(t\right){\kappa }_1(\parallel u\parallel )\hfill \\ \hfill & ∥\widehat{\mathcal{P}}∥=∥\mathcal{P}\left(t,u\left(\frac{3t}{2}\right)\right)∥\le \frac{tanh\left(\frac{u}{5}\right)}{(1+e^{-t})}\le \frac{1}{(1+e^{-t})}\left(\frac{1}{5}∥u∥\right):={\mathbf{m}}_2\left(t\right){\kappa }_2(\parallel u\parallel )\hfill \end{array}$$

  where

  $$\begin{array}{ccc}\hfill {\mathbf{m}}_1\left(t\right)& =\frac{1}{\sqrt{5-t}},t\in (0,\frac{3}{7}]\cup (\frac{5}{7},1],\hfill & \hfill {\kappa }_1(\parallel u\parallel )=\frac{1}{6}\parallel u\parallel ,\\ \hfill {\mathbf{m}}_2\left(t\right)& =\frac{1}{(1+e^{-t})},t\in (0,\frac{3}{7}]\cup (\frac{5}{7},1],\hfill & \hfill {\kappa }_2(\parallel u\parallel )=\frac{2}{5}\parallel u\parallel .\end{array}$$
  The functions ${\mathbf{m}}_i,i=1,2$ are non-decreasing on $J=[0,1]$, which admits the hypothesis $\left(\mathbb{A}{\mathbb{E}}_5\right)$ where $\parallel {\mathbf{m}}_1\parallel \le \mathbf{m}\left(1\right)=\frac{1}{2}$ and $\parallel {\mathbf{m}}_2\parallel \le \mathbf{m}\left(1\right)=0.731058.$
  Furthermore, to verify the assumption $\left(\mathbb{A}{\mathbb{E}}_6\right)$, we obtain

  $$\begin{array}{cc}\hfill ∥\eta (t,u)∥& \le ∥\frac{(1+t^{\frac{3}{2}})}{7}∥+∥\frac{sin\left(\frac{3u}{\sqrt{7}}\right)}{10\sqrt{2-t}}∥\le \frac{(1+{\left(\frac{5}{7}\right)}^{\frac{3}{2}})}{7}+\frac{\sqrt{7}}{30}∥sin\left(\frac{3u}{\sqrt{7}}\right)∥=0.3291,\hfill \\ \hfill ∥\xi (t,u)∥& \le ∥\frac{3t^{\frac{1}{2}}}{14}∥+∥\frac{cos\left(\frac{2u}{3}\right)}{(3+7t)}∥\le \frac{3{\left(\frac{5}{7}\right)}^{\frac{1}{2}}}{14}+\frac{1}{8}∥cos\left(\frac{2u}{3}\right)∥\le 0.1811+\frac{1}{12}=0.2644.\hfill \end{array}$$
  It is clear that the functions above are continuous on $(\frac{3}{7},\frac{5}{7}]$ and satisfy

  $${\vartheta }_1=0.3291,{\vartheta }_1^{*}=0.2644.$$
  The non-local function is also realized as

  $$\begin{array}{cc}\hfill ∥g_0\left(u\right)∥=& ∥\frac{sin3u}{9}∥\le \frac{1}{3},\hfill \\ \hfill ∥g_1\left(u\right)∥=& ∥\frac{arctan2u}{16\left|u\right|}∥\le \frac{1}{16}∥arctan2u∥\le \frac{\pi }{16},\hfill \end{array}$$

  with

  $${\aleph }_0=\frac{1}{3},{\aleph }_1=\frac{\pi }{16}.$$
  The control estimate is also computed as two cases.
  • Case I: Whenever $t\in [0,\frac{3}{7}]$, if we choose $u_{\frac{3}{7}}=2$, we obtain

    $$\begin{array}{cc}\hfill \parallel x\left(t\right)\parallel \le & \left(0.5710\right)[2+(\frac{1}{2}+\frac{1}{3})+\left(\frac{2}{5}\right)\left(0.731058\right)\hfill \\ \hfill & +(\frac{1}{19}+\frac{1}{8})\frac{3}{7}+\frac{6}{5\mathsf{\Gamma }\left(\frac{5}{3}\right)}\left(\frac{1}{12}\right)]\hfill \\ \hfill & =1.8653:=\mathfrak{J}.\hfill \end{array}$$
  • Case II: For any $t\in (\frac{5}{7},1]$, if we choose $u_1=4$, we have

    $$\begin{array}{cc}\hfill \parallel x\left(t\right)\parallel \le & \left(0.4073\right)[4+0.4291+\left(\frac{2}{5}\right)\left(0.731058\right)+\left(1.6811\right)\left(\frac{2}{7}\right)\hfill \\ \hfill +& \frac{6}{5\mathsf{\Gamma }\left(\frac{5}{3}\right)}\left(\frac{1}{12}\right){\left(\frac{2}{7}\right)}^{\frac{5}{6}}]=2.1346:={\mathfrak{J}}^1.\hfill \end{array}$$
  Equations (7)–(9) and condition $\left(\mathbb{A}{\mathbb{E}}_5\right)$ suggest that

  $$\begin{array}{cc}\hfill \mathcal{Y}>& \underset{k}{max}\{\frac{(\parallel u_0\parallel +{\aleph }_0+\parallel \mathcal{P}\left(0\right)\parallel )+(\parallel u_1\parallel +{\aleph }_1+∥\mathcal{P}\left(0\right)∥)\left(\frac{3}{7}\right)+\frac{6\mathfrak{J}\mathfrak{K}}{5\mathsf{\Gamma }\left(\frac{5}{3}\right)}{\left(\frac{3}{7}\right)}^{5/6}}{1-(\frac{\parallel {\mathbf{m}}_1\parallel }{6}+\frac{12{\left(\frac{3}{7}\right)}^{\frac{5}{6}}}{25\mathsf{\Gamma }\left(\frac{5}{3}\right)}\parallel {\mathbf{m}}_2\parallel )},\hfill \\ \hfill & {\vartheta }_1,\frac{{\vartheta }_1+\left(\frac{2}{7}\right){\vartheta }_1^{*}+\frac{6\mathfrak{Y}{\mathfrak{J}}_1}{5\mathsf{\Gamma }\left(\frac{5}{3}\right)}{\left(\frac{2}{7}\right)}^{\frac{5}{6}}}{1-(\parallel {\kappa }_1\parallel {\mathbf{m}}_1(\parallel u\parallel )+\frac{6}{5\mathsf{\Gamma }\left(\frac{5}{3}\right)}\parallel {\kappa }_2\parallel {\mathbf{m}}_2(\parallel u\parallel ){\left(\frac{2}{7}\right)}^{\frac{5}{6}})}\}.\hfill \end{array}$$
  After the calculations, we obtain

  $$\mathcal{Y}>\left\{3.4131,0.3291,2.1244\right\}>3.4131.$$
  As a result, the requirements of Theorem 2 are met. Then, problem (1) is totally controllable on $[0,1]$.

6. Conclusions

In this article, we have openly discussed how neutral functional evolution equations with non-instantaneous impulses of order $\alpha \in (1,2)$ may be controlled totally. We were able to determine the prerequisites for the existence of the controlled solution by employing two distinct methodologies. The first module makes use of the measure of non-compactness technique, Sadovskii’s theory, and we used the Kuratowski measure for the presence of operator contraction. On the other hand, using the non-linear Alternative Leray–Schauder theorem, we obtained the value of $\mathsf{\Xi }$ that produces the controlled solution when utilizing the compact approach and we established the necessary conditions. Simulation results of the inhomogeneous fractional wave equation were then used to apply these findings. The infinitesimal generator of the cosine and sine families was investigated to construct new, associated, strongly continuous operators which are bounded and used to represent the mild solution of our problem.

In future work, the development of the controllability and Ulam–Hyers stability of non-instantaneous impulsive Caputo–Katugampola fractional differential inclusions will be discussed by applying the non-linear alternative of the Leray–Schauder type and the technique of measures of non-compactness.