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An analysis on the controllability and stability to some fractional delay dynamical systems on time scales with impulsive effects

Abstract

In this article, we establish a new class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. We investigate the qualitative properties of the considered systems. In fact, the article contains three segments, and the first segment is devoted to investigating the existence and uniqueness results. In the second segment, we study the stability analysis, while the third segment is devoted to investigating the controllability criterion. We use the Leray–Schauder and Banach fixed point theorems to prove our results. Moreover, the obtained results are examined with the help of an example.

Introduction

The notion of fractional differential equations (FDEs) has been a field of intense research for the last few decades. In 1695, the notion of FDEs was initiated with a coincidence between Leibniz and L’Hospital. Nowadays, FDEs play an important role in establishing mathematical modeling of many problems occurring in control theory, bioengineering, mathematical networks, aerodynamics, blood flows, engineering, physics, signal processing, etc. [1, 2].

We can analyze from different experiments that FDEs have innumerable prominent status than integer-order derivatives. Consequently, fractional calculus got incredible interest and received more attention from many specialists and researchers. It also set up a better sketch over hereditary properties of processes and various materials, consequently many monographs and research papers have been reported in this field [322].

Recently, the theory of stability analysis, like Lyapunov, exponential, Mittag-Leffler function, and finite time stability for various kinds of functional equations, has been investigated. Ulam and Hyers introduced most important and interesting kind of stability called Hyers–Ulam stability [23] in 1940. Ulam during his talk at Wisconsin University asked a question about the stability of homomorphisms between groups. In 1941, Hyers [24] replied to Ulam’s problem positively under the hypothesis that groups are considered as Banach spaces \((\mathbb{BS})\), and such a stability was called Ulam–Hyers stability. For more information, see [2530]. Impulsive differential equations are best tools to model a physical situation that contains abrupt changes at certain instants. These equations describe medicine, biotechnology process, population dynamics, biological systems, chemical energy, mathematical economy, pharmacokinetics, etc. [3143].

In the past few decades, because of these applications in various fields of interest, impulsive differential equations got considerable attention. In order to unify the difference and differential calculus, Hilger [44] provided the idea of time scales at the end of the twentieth century, which is now a well-known subject. For more details, see [4551]. Lupulescu and Zada [49] provided the basics and fundamental notions of linear impulsive systems on time scales in 2010.

In 1960, Kalman presented the notion of controllability, which is the principal notion in mathematical control theory. In general, controllability provides steering the state of a control dynamical equation to the desired terminal state from an arbitrary initial state by utilizing a suitable control function. Numerous researchers examined the controllability results of dynamical systems [52, 53]. Moreover, controllability results on time scales is a new area, and few results have been achieved [54, 55]. Especially, there are a few articles that examined the existence, controllability, and Ulam type stability regarding a mixed structure of the impulsive fractional dynamical system on time scales.

Inspired by the research conducted in [56], we study the following mixed integral fractional dynamic systems on the time scale \(\mathbb{T}\):

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )=A(\varsigma )\omega (\varsigma )+\mathcal{F}(\varsigma , \omega (\varsigma )) \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )={}}{} +G (\varsigma ,\omega (\varsigma ), \int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{1}(\varsigma ,s, \omega (s))\Delta s,\int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{2}(\varsigma ,s,\omega (s))\Delta s ), \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )=}\varsigma \in {\mathbb{T}}^{\prime }={\mathbb{T}}\backslash \{\varsigma _{1}, \varsigma _{2},\ldots ,\varsigma _{m}\}, \sigma =(0,1), \\ \omega (\varsigma ^{+}_{k})-\omega (\varsigma ^{-}_{k})=\Xi _{k}( \omega (\varsigma ^{-}_{k}))+\Phi _{k} (\varsigma ^{-}_{k},\omega ( \varsigma ^{-}_{k})), \quad k = 1,\dots , m, \\ \omega (\varsigma _{0})=\omega _{0}, \end{cases} $$
(1)

and

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )=A( \varsigma )\omega (\varsigma )+\mathcal{F}(\varsigma ,\omega ( \varsigma )) \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )={}}{} +\mathcal{G} (\varsigma ,\omega ( \varsigma ),\int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{1}( \varsigma ,s,\omega (s))\Delta s,\int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{2}(\varsigma ,s,\omega (s))\Delta s ), \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )=} \varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{T}, i= 1,\dots , m, \sigma =(0,1), \\ \omega (\varsigma )=\frac{1}{\Gamma (\sigma )}\int _{ \varsigma _{i}}^{\varsigma }(\varsigma -s)^{\sigma -1}\hbar _{i}(s, \omega (s))\Delta s,\quad \varsigma \in (\varsigma _{i}, s_{i}]\cap \mathbb{T}, i = 1,\dots , m, \\ \omega (\varsigma _{0})=\omega _{0}. \end{cases} $$
(2)

Also, we discuss the controllability of the following systems:

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )=A(\varsigma )\omega (\varsigma )+\mathcal{F}(\varsigma , \omega (\varsigma )) \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )={}}{} +G (\varsigma ,\omega (\varsigma ), \int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{1}(\varsigma ,s, \omega (s))\Delta s,\int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{2}(\varsigma ,s,\omega (s))\Delta s )+\mathcal{H} \zeta (\varsigma ), \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )=}\varsigma \in {\mathbb{T}}^{\prime }={\mathbb{T}}\backslash \{\varsigma _{1}, \varsigma _{2},\ldots ,\varsigma _{m}\}, \sigma =(0,1), \\ \omega (\varsigma ^{+}_{k})-\omega (\varsigma ^{-}_{k})=\Xi _{k}( \omega (\varsigma ^{-}_{k}))+\Phi _{k} (\varsigma ^{-}_{k},\omega ( \varsigma ^{-}_{k})), \quad k = 1,\dots , m, \\ \omega (\varsigma _{0})=\omega _{0}, \end{cases} $$
(3)

and

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )=A(\varsigma )\omega (\varsigma )+\mathcal{F}(\varsigma , \omega (\varsigma )) \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )={}}{} +\mathcal{G} (\varsigma ,\omega ( \varsigma ),\int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{1}( \varsigma ,s,\omega (s))\Delta s,\int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{2}(\varsigma ,s,\omega (s))\Delta s )+\mathcal{H} \zeta (\varsigma ), \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega ( \varsigma )=}\varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{T}, i = 1,\dots , m,\sigma =(0,1), \\ \omega (\varsigma )=\frac{1}{\Gamma (\sigma )}\int _{ \varsigma _{i}}^{\varsigma }(\varsigma -s)^{\sigma -1}\hbar _{i}(s, \omega (s))\Delta s,\quad \varsigma \in (\varsigma _{i}, s_{i}]\cap \mathbb{T}, i = 1,\dots , m, \\ \omega (\varsigma _{0})=\omega _{0}, \end{cases} $$
(4)

where \({{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\) represents the classical Caputo derivative [1] of fractional order σ on time scales \(\mathbb{T}\). The regressive square matrix \(A(\varsigma )\) is piecewise continuous, and \(\mathcal{H}:\mathbb{T}\rightarrow \mathbb{T}\) is a bounded linear operator. By assuming \(\mathbb{R}\) as the real number, \(\zeta \in \mathcal{L}^{2}(I,\mathbb{R})\) is a control map, \(\mathbb{T}^{0}:=[\varsigma _{0},\varsigma _{f}]_{\mathbb{T}}\), the pre-fixed numbers are \(\varsigma _{0}=s_{0}<\varsigma _{1}< s_{1}< \varsigma _{2}< \cdots < \varsigma _{m}< s_{m}< \varsigma _{m+1}=\varsigma _{f}\), and \(\mathcal{F}:{\mathbb{T}}^{0}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\), \(\mathcal{F}_{1}, \mathcal{F}_{2}:{\mathbb{T}}^{0}\times {\mathbb{T}}^{0} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\), \({G}:{\mathbb{T}}^{0}\times \mathbb{R}^{n}\times \mathbb{R}^{n} \times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\), \(\hbar _{i}:(\varsigma _{i}, s_{i}]\cap \mathbb{T}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\), \(i = 1,\dots , m\), \(\mathbb{F}:(s_{i}, \varsigma _{i+1}]\cap \mathbb{T}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\), \(i = 0,\dots , m\), \(\mathcal{G}:(s_{i}, \varsigma _{i+1}]\cap \mathbb{T}\times \mathbb{R}^{n}\times \mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\), \(i = 1,\dots , m\), \(\Xi _{k}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\), \(\Phi _{k}:{\mathbb{T}}^{0}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) are continuous mappings. In addition, we define the right limit and the left limit of \(\omega (\varsigma )\) at \(\varsigma _{k}\) as \(\omega (\varsigma ^{+}_{k})=\lim_{\tau \rightarrow 0^{+}}\omega ( \varsigma _{k}+\tau )\) and \(\omega (\varsigma ^{-}_{k})=\lim_{\tau \rightarrow 0^{-}}\omega ( \varsigma _{k}-\tau )\), respectively.

Auxiliary definitions and lemmas

Here, we provide definitions, basic notions, and preliminaries for this manuscript.

We define \(\mathrm{C}(\mathrm{I}, \mathbb{R})\) as a \(\mathbb{BS}\) of all continuous mappings endowed with the norm \(\|\omega \|_{\mathrm{C}}=\sup_{\varsigma \in \mathbb{T}}\|\omega ( \varsigma )\|\). \(\mathrm{PS}=\mathrm{C}(\mathrm{I}, \mathbb{R})\times \mathrm{C}( \mathrm{I}, \mathbb{R})\times \mathrm{C}(\mathrm{I}, \mathbb{R})\) is the product space which is a \(\mathbb{BS}\) furnished with the norm \(\|(\omega , \zeta ,\xi )\|_{\mathrm{C}}=\|\omega \|_{\mathcal{C}}+\| \zeta \|_{\mathcal{C}}+\|\xi \|_{\mathcal{C}}\). Also, we define a \(\mathbb{BS}\) \(\mathrm{C}^{1}(\mathrm{I}, \mathbb{R})=\{\omega \in \mathrm{C}( \mathrm{I}, \mathbb{R}): \omega ^{\Delta }\in \mathrm{C}^{1}( \mathrm{I}, \mathbb{R})\}\) with the norm \(\|\omega \|_{\mathrm{C}^{1}}=\max \{\|\omega \|_{\mathcal{C}},\| \omega ^{\Delta }\|_{\mathcal{C}^{1}} \}\). Moreover, \(\mathrm{PS}^{1}=\mathrm{C}^{1}(\mathrm{I}, \mathbb{R})\times \mathrm{C}^{1}(\mathrm{I}, \mathbb{R})\times \mathrm{C}^{1}( \mathrm{I}, \mathbb{R})\) is the product space via \(\|(\omega , \zeta ,\xi )\|_{\mathrm{C}^{1}}=\|\omega \|_{\mathcal{C}^{1}}+ \|\zeta \|_{\mathcal{C}^{1}}+\|\xi \|_{\mathcal{C}^{1}}\).

A nonempty closed subset of \(\mathbb{R}\) is known as a time scale (\(\mathbb{T}\)). We define a time scale interval as \([c,d]_{\mathbb{T}}=\{\varsigma \in \mathbb{T}: c\leq \varsigma \leq d \}\). Similarly, we can define \((c,d)_{\mathbb{T}}\), \([c,d)_{\mathbb{T}}\).

The forward and backward jump operators \(\sigma :\mathbb{T}\rightarrow \mathbb{T}\), \(\rho :\mathbb{T}\rightarrow \mathbb{T}\) are introduced as

$$ \sigma (\varsigma )=\inf \{s\in \mathbb{T}:s>\varsigma \} \quad \text{and}\quad \rho (\varsigma )=\sup \{s\in \mathbb{T}:s< \varsigma \}, $$

respectively. The operator \(\eta :\mathbb{T}\rightarrow [0,\infty )\) formulated by \(\eta (\varsigma )=\sigma (\varsigma )-\varsigma \) is applied to obtain the existing distance between two consecutive points. Along this, the derived version \({\mathbb{T}}^{k}\) of \(\mathbb{T}\) is

$$ \mathbb{T}^{k}= \textstyle\begin{cases} \mathbb{T}\backslash (\rho (\sup \mathbb{T}), \sup \mathbb{T}], &\text{if } \sup \mathbb{T}< \infty , \\ \mathbb{T}, & \text{if } \sup \mathbb{T}=\infty . \end{cases} $$

The regressive (respectively positively regressive) function \(\mathfrak{H}:\mathbb{T}\rightarrow \mathbb{R}\) is defined as \(1+\eta (\varsigma )\mathfrak{H}(\varsigma )\neq 0\) (respectively \(1+\eta (\varsigma )\mathfrak{H}(\varsigma )>0 \)) for all \(\varsigma \in \mathbb{T}^{k}\).

Definition 2.1

([57])

At a point \(\varsigma \in \mathbb{T}^{k}\), the delta derivative \(g^{\Delta }(\varsigma )\) of a mapping \(g:\mathbb{T}\rightarrow \mathbb{R}\) is a number (provided it exists) if, for \(\epsilon >0\), a neighborhood U of ς exists provided that

$$ \bigl\vert \bigl[g\bigl(\sigma (\varsigma )\bigr)-g(\tau ) \bigr]-g^{\Delta }(\varsigma )\bigl[\sigma ( \varsigma )-\tau \bigr] \bigr\vert \leq \epsilon \bigl\vert \sigma (\varsigma )-\tau \bigr\vert ,\quad \text{for all } \tau \in \mathrm{U}. $$

Theorem 2.2

([57])

Let \(c,d\in \mathbb{T}\) and \(f\in \mathcal{C}_{\mathrm{rd}}(\mathbb{T},\mathbb{R})\), then

  1. 1.

    \(\mathbb{T}=\mathbb{R}\) implies

    $$ \int _{c}^{d}f(\varsigma )\Delta \varsigma = \int _{c}^{d}f(\varsigma )\,d \varsigma . $$
  2. 2.

    If \([c,d)\) consists of only isolated points, then

    $$ \int _{c}^{d}f(\varsigma )\Delta \varsigma = \textstyle\begin{cases} \sum_{\varsigma \in [c,d)}\mu (\varsigma )f( \varsigma ), &\textit{if } c< d, \\ 0, &\textit{if } c=d, \\ -\sum_{\varsigma \in [c,d)}\mu (\varsigma )f(\varsigma ), &\textit{if } c>d. \end{cases} $$
  3. 3.

    \(\mathbb{T}=h\mathbb{Z}=\{hk: k\in \mathbb{Z}\}\), \(h>0\), implies

    $$ \int _{c}^{d}f(\varsigma )\Delta \varsigma = \textstyle\begin{cases} \sum_{k=\frac{a}{h}}^{\frac{b}{h}-1}hf(kh), &\textit{if } c< d, \\ 0, &\textit{if } c=d, \\ -\sum_{k=\frac{a}{h}}^{\frac{b}{h}-1}hf(kh), &\textit{if } c>d. \end{cases} $$

Theorem 2.3

([58])

If \(f:\mathbb{R}\rightarrow \mathbb{R}\) is nondecreasing continuous and \(c,d\in \mathbb{T}\), then

$$ \int _{c}^{d}f(\varsigma )\Delta \varsigma \leq \int _{c}^{d}f( \varsigma )\,d\varsigma . $$
(5)

Definition 2.4

([58])

Let \(\phi :[c,d]_{\mathbb{T}}\rightarrow \mathbb{R}\) be an integrable mapping, then delta fractional integral is

$$ {{}^{\Delta }} {\mathrm{I}^{\sigma }_{a^{+}}}\phi ( \varsigma )= \int _{a}^{ \varsigma }\frac{(\varsigma -s)^{\sigma -1}}{\Gamma (\sigma )}\phi (s) \Delta s. $$
(6)

Definition 2.5

([58])

The fractional Caputo derivative of a mapping \(f:\mathbb{T}\rightarrow \mathbb{R}\) on the time scale is

$$ {{}^{c,\mathbb{T}}} {\mathrm{D}^{\sigma }_{a^{+}}}f(\varsigma )= \int _{a}^{ \varsigma }\frac{(\varsigma -s)^{n-\sigma -1}}{\Gamma (n-\sigma )}f^{ \Delta ^{n}}( \varsigma )\Delta \varsigma , $$
(7)

where \(n=[\sigma ]+1\) and the delta nth derivative of f is denoted by \(f^{\Delta ^{n}}\).

  • When \(\mathbb{T}=\bigcup_{i=0}^{\infty }[2i, 2i+1]\). Then we get

    $$\begin{aligned} {{}^{c,\mathbb{T}}} {\mathrm{D}^{\sigma }_{a^{+}}}f(\varsigma )&= \int _{a}^{ \varsigma }\frac{(\varsigma -s)^{n-\sigma -1}}{\Gamma (n-\sigma )}f^{ \Delta ^{n}}( \varsigma )\Delta \varsigma \\ &=\frac{1}{\Gamma (n-\sigma )} \Biggl[\sum_{k=0}^{i-1} \int _{2k}^{2k+1}( \varsigma -s)^{n-\sigma -1}f^{\Delta ^{n}} \Delta s+ \int _{2i}^{2i+1}( \varsigma -s)^{n-\sigma -1}f^{\Delta ^{n}} \Delta s \Biggr] \end{aligned}$$

    for \(\varsigma \in [2i,2i+1]\), \(i=0,1,\dots \).

  • When \(\mathbb{T}=h\mathbb{Z}\), \(h>0\), we have

    $$\begin{aligned} {{}^{c,\mathbb{T}}} {\mathrm{D}^{\sigma }_{a^{+}}}f(\varsigma )&= \int _{a}^{ \varsigma }\frac{(\varsigma -s)^{n-\sigma -1}}{\Gamma (n-\sigma )}f^{ \Delta ^{n}}( \varsigma )\Delta \varsigma \\ &=\frac{1}{\Gamma (n-\sigma )}\sum_{k=0}^{\frac{\varsigma }{h}-1}h( \varsigma -ih)^{n-\sigma -1}f^{\Delta ^{n}}(h),\quad \varsigma \in \mathbb{T}. \end{aligned}$$
  • When \(\mathbb{T}=\{p^{n}: p>1, n\in \mathbb{Z}\}\cup \mathbb{Z}\), then

    $$\begin{aligned} {{}^{c,\mathbb{T}}} {\mathrm{D}^{\sigma }_{a^{+}}}f(\varsigma )&= \int _{a}^{ \varsigma }\frac{(\varsigma -s)^{n-\sigma -1}}{\Gamma (n-\sigma )}f^{ \Delta ^{n}}( \varsigma )\Delta \varsigma \\ &=\frac{1}{\Gamma (n-\sigma )}\sum_{\varsigma \in \mathbb{T}}\mu (s) ( \varsigma -s)^{n-\sigma -1}f^{\Delta ^{n}}(s). \end{aligned}$$

Regard the Mittag-Leffler function as

$$\begin{aligned} E_{\sigma ,\beta }(\varsigma )=\sum_{k=0}^{\infty } \frac{\varsigma ^{k}}{\Gamma (k\sigma +\beta )}\quad \text{for }\sigma ,\beta >0. \end{aligned}$$

For \(\beta =1\),

$$\begin{aligned} E_{\sigma ,1}\bigl(\lambda \varsigma ^{\sigma }\bigr)=E_{\sigma } \bigl(\lambda \varsigma ^{\sigma }\bigr)=\sum_{k=0}^{\infty } \frac{\lambda ^{k}\varsigma ^{k\sigma }}{\Gamma (\sigma k+1)},\quad \lambda ,\varsigma \in \mathbb{C} \end{aligned}$$

has the interesting property \({{}^{c}}{\mathrm{D}_{0+}^{\sigma }}E_{\sigma }(\lambda \varsigma ^{ \sigma })=\lambda E_{\sigma }(\lambda \varsigma ^{\sigma })\).

Remark 2.1

([59])

The solution of system (1) is of the form

$$ \omega (\varsigma )= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s, \\ \quad \varsigma \in (\varsigma _{0},\varsigma _{1}], \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\sum_{j=1}^{i} (\Xi _{j}(\omega (\varsigma ^{-}_{j}))+\Phi _{j} (\varsigma ^{-}_{j},\omega (\varsigma ^{-}_{j})) ),\quad \varsigma \in (\varsigma _{i},\varsigma _{i+1}], i = 1,\dots , m, \end{cases} $$

where \(E_{\sigma }(A\varsigma ^{\sigma })\) is the matrix representation of the aforesaid Mittag-Leffler function given by

$$ E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr)=\sum _{j=1}^{i} \frac{A^{\sigma }{\varsigma ^{\kappa \sigma }}}{\Gamma (1+\kappa \sigma )}. $$

To achieve our results, we consider the following:

\(\mathbf{(A)}\)::

The mappings \({G}, \mathcal{G}:{\mathbb{T}}^{0}\times \mathbb{R}^{n}\times \mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) are continuous, and there exist \({L}_{\hbar _{i}}\), \(\mathcal{L}_{\hbar _{i}}\), \(i=1,2,3\), as the positive constants such that (\(i=1,2,3\))

$$\begin{aligned}& \bigl\vert G(\varsigma , q_{1}, q_{2},q_{3})-G( \varsigma , p_{1}, p_{2},p_{3}) \bigr\vert \leq \sum_{i=1}^{3}{L}_{\hbar _{i}} \vert q_{i}-p_{i} \vert \quad \text{for all } \varsigma \in \mathrm{I}, q_{i},p_{i}\in \mathbb{R}^{n}, \\& \bigl\vert \mathcal{G}(\varsigma , q_{1}, q_{2},q_{3})- \mathcal{G}(\varsigma , p_{1}, p_{2},p_{3}) \bigr\vert \leq \sum_{i=1}^{3}{ \mathcal{L}}_{\mathcal{G}_{i}} \vert q_{i}-p_{i} \vert \quad \text{for all } \varsigma \in \mathrm{I}, q_{i},p_{i}\in \mathbb{R}^{n}. \end{aligned}$$
\(\mathbf{(B)}\)::

The mappings \({G}, \mathcal{G}:{\mathbb{T}}^{0}\times \mathbb{R}^{n}\times \mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) are continuous, and there exist \({l}_{i}\), \({m}_{i}\), \(i=1,2,3\), positive constants such that

$$\begin{aligned}& \bigl\vert G(\varsigma , u, v, w) \bigr\vert \leq l_{0}+{l}_{1} \vert u \vert +{l}_{2} \vert v \vert +{l}_{3} \vert w \vert \quad\text{for all } \varsigma \in \mathrm{I}, u,v,w \in \mathbb{R}^{n}, \\& \bigl\vert \mathcal{G}(\varsigma , u, v, w) \bigr\vert \leq m_{0}+{m}_{1} \vert u \vert +{m}_{2} \vert v \vert +{n}_{3} \vert w \vert \quad\text{for all } \varsigma \in \mathrm{I}, u,v,w\in \mathbb{R}^{n}. \end{aligned}$$
\(\mathbf{(W)}\)::

The linear operator \(({{}^{\sigma }}{\mathcal{W}^{T}_{\varsigma _{0}}}):L^{2}(I, \mathbb{R}) \rightarrow \mathbb{R}\), defined by

$$ {{}^{\sigma }} {\mathcal{W}^{T}_{\varsigma _{0}}} \zeta = \int _{\varsigma _{0}}^{ \varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{ \sigma } \bigr)\mathcal{H}\zeta (s)\Delta s, $$
(8)

possesses a bounded invertible operator \(({{}^{\sigma }}{\mathcal{W}^{T}_{\varsigma _{0}}})^{-1}\), and these operators admit values in \(L^{2}(I, \mathbb{R})\backslash \ker ({{}^{\sigma }}{\mathcal{W}^{T}_{ \varsigma _{0}}})\). Also, there exists a positive constant provided that \(\|({{}^{\sigma }}{\mathcal{W}^{T}_{\varsigma _{0}}})^{-1}\|\leq M_{ \mathcal{W}}^{\sigma }\). Also, \(\mathcal{H}:\mathbb{T}\rightarrow \mathbb{T}\) is a continuous operator, and there exists a positive constant \(M_{\mathcal{H}}\) provided that \(\|\mathcal{H}\|\leq M_{\mathcal{H}}\).

Using Theorem 2.2, equation (8) can be calculated for different \(\mathbb{T}\).

  • When \(\mathbb{T}=h\mathbb{Z}\), \(h>0\):

    $$ {{}^{\sigma }} {\mathcal{W}^{T}_{\varsigma _{0}}}\zeta = \int _{\varsigma _{0}}^{ \varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{ \sigma } \bigr)\mathcal{H}\zeta (s)\Delta s=\sum _{k=0}^{ \frac{\varsigma }{h}-1}h(T-sh)^{\sigma -1} \mathcal{H}\zeta (sh). $$
  • When \(\mathbb{T}=\bigcup_{i=0}^{\infty }[2i,2i+1]\). Let \(\varsigma \in [4,5]\), then we have

    $$\begin{aligned} {{}^{\sigma }} {\mathcal{W}^{T}_{\varsigma _{0}}}\zeta &= \int _{ \varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr)\mathcal{H}\zeta (s)\Delta s \\ &= \int _{0}^{1}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{ \sigma } \bigr)\mathcal{H}\zeta (s)\Delta s \\ &\quad {}+ \int _{2}^{3}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{ \sigma } \bigr)\mathcal{H}\zeta (s)\Delta s \\ &\quad {}+ \int _{4}^{T}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{ \sigma } \bigr)\mathcal{H}\zeta (s)\Delta s. \end{aligned}$$
  • When \(\mathbb{T}=\{q^{m}:q>1,m\in \mathbb{Z}\}\cup \mathbb{Z}\), then

    $$ {{}^{\sigma }} {\mathcal{W}^{T}_{\varsigma _{0}}}\zeta = \int _{\varsigma _{0}}^{ \varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{ \sigma } \bigr)\mathcal{H}\zeta (s)\Delta s=\sum _{\varsigma \in [0,T]} \mu (T-\varsigma )^{\sigma -1}\mathcal{H} \zeta (\varsigma ). $$

Throughout the manuscript, we set

$$\begin{aligned}& Q_{1}:=a_{3} \bigl(a_{2}{L}_{\mathcal{F}}+ a_{2}{L}_{{G}_{1}}+ a_{2}{L}_{{G}_{2}}{L}_{ \mathcal{\mathcal{F}}_{1}}({s_{f}}-{s_{0}}) + a_{2}{L}_{{G}_{3}}{L}_{ \mathcal{\mathcal{F}}_{2}}({s_{f}}-{s_{0}})+M_{G} \bigr) (\varsigma _{f}- \varsigma _{0}); \\& Q_{2}:=a_{3} \bigl(a_{2}L_{\mathcal{F}}+a_{2}L_{{G}_{1}}+a_{2}L_{{G}_{2}}L_{{ \mathcal{F}_{1}}}({s_{f}}-{s_{0}})+ a_{2}L_{{G}_{3}}L_{{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) (\varsigma _{f}-\varsigma _{0}); \\& Q^{*}_{1}:= a_{3} \bigl(a_{2}{L}_{F}+ a_{2}{L}_{\mathcal{G}_{1}}+ a_{2}{L}_{ \mathcal{G}_{2}}{L}_{F_{1}}({s_{f}}-{s_{0}}) + a_{2}{L}_{\mathcal{G}_{3}}{L}_{F_{2}}({s_{f}}-{s_{0}})+ \widehat{M}_{\mathcal{G}} \bigr) (\varsigma _{f}-\varsigma _{0}); \\& Q^{*}_{2}:= a_{3} \bigl(a_{2}L_{{F}}+a_{2}L_{{G}_{1}}+a_{2}L_{{ \mathcal{G}}_{2}}L_{{{F}_{1}}}({s_{f}}-{s_{0}})+ a_{2}L_{{\mathcal{G}}_{3}}L_{{ \mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) (\varsigma _{f}-\varsigma _{0}); \\& Q_{3}:=N_{2}+Q_{2};\qquad Q^{*}_{3}=N_{4}+Q^{*}_{2}; \\& N_{1}:=\sum_{j=1}^{i}{L}_{\Xi } \delta +\sum_{j=1}^{i}{L}_{\Phi } \delta +a_{1};\qquad N_{2}:=\sum _{j=1}^{i}L_{\Xi }+\sum _{j=1}^{i}L_{\Phi }; \\& N_{3}:=\frac{1}{\Gamma (\sigma )}a_{3}L_{g}({ \varsigma _{f}}-{ \varsigma _{0}})+a_{1};\qquad N_{4}:=\frac{1}{\Gamma (\sigma )}a_{3}L_{g}({ \varsigma _{f}}-{\varsigma _{0}}); \\& a_{1}:=\sup_{\varsigma \in \mathbb{T}} \bigl\Vert E_{\sigma }\bigl(A\varsigma ^{ \sigma }\bigr)\omega _{0} \bigr\Vert ;\qquad a_{2}:=\sup_{\varsigma \in \mathbb{T}} \bigl\Vert E_{ \sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr) \bigr\Vert ;\qquad a_{3}=\sup_{ \varsigma \in \mathbb{T}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert . \end{aligned}$$

Existence of solution

Existence criteria are investigated here.

Theorem 3.1

The mixed impulsive system (1) admits a unique solution if assertion \(\mathbf{(A)}\) holds and

$$ \max_{1\leq i\leq 3}\{Q_{i}\}< 1. $$
(9)

Proof

Let \(\Omega \subseteq \mathrm{PS}\) and \(\Omega =\{(\mathbb{X},\mathbb{Y},\mathbb{Z})\in \mathrm{PS}:\|( \mathbb{X},\mathbb{Y},\mathbb{Z})\|_{C}\leq \delta _{2}\}\) also \(\delta _{2}=\max \{\delta ,\delta _{1}\}\) and \(\delta ,\delta _{1}\in (0,1)\) provided that

$$ \delta >\max \{N_{1},N_{2},N_{3} \}, $$

and the remaining constants are introduced in the sequel. Now, we define \(\Lambda _{\sigma }:\Omega \rightarrow \Omega \) as

$$ \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr)= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s, \\ \quad \varsigma \in (\varsigma _{0},\varsigma _{1}], \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\sum_{j=1}^{i} (\Xi _{j}(\omega (\varsigma ^{-}_{j}))+\Phi _{j} (\varsigma ^{-}_{j},\omega (\varsigma ^{-}_{j})) ),\quad \varsigma \in (\varsigma _{i},\varsigma _{i+1}], i = 1,\dots , m. \end{cases} $$
(10)

Assume that

$$ \bigl\Vert \mathcal{F}(s,\omega ) \bigr\Vert \leq \bigl\Vert \mathcal{F}(s,\omega )-\mathcal{F}(s,0) \bigr\Vert + \bigl\Vert \mathcal{F}(s,0) \bigr\Vert \leq L_{\mathcal{F}} \Vert \omega \Vert +M_{\mathcal{F}} $$

and

$$\begin{aligned} \bigl\Vert G(\varsigma ,\mathbb{X},\mathbb{Y},\mathbb{Z}) \bigr\Vert &\leq \bigl\Vert G( \varsigma ,\mathbb{X},\mathbb{Y},\mathbb{Z})-G(\varsigma ,0,0,0) \bigr\Vert + \bigl\Vert G( \varsigma ,0,0,0) \bigr\Vert \\ &\leq L_{\hbar _{1}} \Vert \mathbb{X} \Vert +L_{\hbar _{2}} \Vert \mathbb{Y} \Vert +L_{ \hbar _{3}} \Vert \mathbb{Z} \Vert +M_{G}, \end{aligned}$$

where \(M_{\mathcal{F}}=\sup_{\varsigma \in \mathbb{T}}\|\mathcal{F}(s,0)\|\), \(M_{G}=\sup_{\varsigma \in \mathbb{T}}\|G(\varsigma ,0,0,0)\|\), and \(\widehat{M}_{G}=M_{\mathcal{F}}+M_{G}\). In addition, \(\mathbb{X}=\omega (s)\), \(\mathbb{Y}=\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u\), and \(\mathbb{Z}=\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u, \omega (u))\Delta u\).

Now, we prove that \(\Lambda _{\sigma }:\Omega \rightarrow \Omega \) is a self-mapping.

For \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), one has

$$\begin{aligned} \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert \leq& \sum_{j=1}^{i} \bigl\Vert \Xi _{j}\bigl(\omega \bigl(\varsigma ^{-}_{j} \bigr)\bigr) \bigr\Vert +\sum_{j=1}^{i} \bigl\Vert \Phi _{j} \bigl(\varsigma ^{-}_{j}, \omega \bigl(\varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert + \bigl\Vert E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert \\ &{} + \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl(\mathcal{F} \bigl(s,\omega (s)\bigr) \\ &{} +G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ \leq& \sum_{j=1}^{i}{L}_{\Xi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \sum_{j=1}^{i}{L}_{\Phi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \bigl\Vert E_{ \sigma }\bigl(A\varsigma ^{\sigma }\bigr) \omega _{0} \bigr\Vert \\ &{} + \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl( \bigl\Vert \mathcal{F}\bigl(s,\omega (s)\bigr) \bigr\Vert \\ &{} + \biggl\Vert G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr\Vert +\widehat{M}_{G} \biggr)\Delta s \\ \leq& \sum_{j=1}^{i}{L}_{\Xi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \sum_{j=1}^{i}{L}_{\Phi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \bigl\Vert E_{ \sigma }\bigl(A\varsigma ^{\sigma }\bigr) \omega _{0} \bigr\Vert \\ &{} + \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl({L}_{ \mathcal{F}} \bigl\Vert \omega (s) \bigr\Vert +{L}_{\hbar _{1}} \bigl\Vert \omega (s) \bigr\Vert \\ &{} +{L}_{\hbar _{2}} \int _{s_{0}}^{s_{f}} \bigl\Vert \mathcal{F}_{1}\bigl(s,u, \omega (u)\bigr) \bigr\Vert \Delta u+{L}_{\hbar _{3}} \int _{s_{0}}^{s_{f}} \bigl\Vert \mathcal{F}_{2}\bigl(s,u,\omega (u)\bigr) \bigr\Vert \Delta u+ \widehat{M}_{G} \biggr)\Delta s \\ \leq& \sum_{j=1}^{i}{L}_{\Xi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \sum_{j=1}^{i}{L}_{\Phi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \bigl\Vert E_{ \sigma }\bigl(A\varsigma ^{\sigma }\bigr) \omega _{0} \bigr\Vert \\ &{} + \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl({L}_{ \mathcal{F}} \bigl\Vert \omega (s) \bigr\Vert +{L}_{{G}_{1}} \bigl\Vert \omega (s) \bigr\Vert \\ &{} +{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}} \int _{s_{0}}^{s_{f}} \bigl\Vert \omega (u) \bigr\Vert \Delta u+{L}_{{G}_{3}}{L}_{ \mathcal{\mathcal{F}}_{2}} \int _{s_{0}}^{s_{f}} \bigl\Vert \omega (u) \bigr\Vert \Delta u+\widehat{M}_{G} \biggr)\Delta s \\ =&\sum_{j=1}^{i}{L}_{\Xi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \sum_{j=1}^{i}{L}_{\Phi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \bigl\Vert E_{ \sigma }\bigl(A\varsigma ^{\sigma }\bigr) \omega _{0} \bigr\Vert \\ &{} + \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \bigl({L}_{ \mathcal{F}} \bigl\Vert \omega (s) \bigr\Vert +{L}_{{G}_{1}} \bigl\Vert \omega (s) \bigr\Vert \\ &{} +{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}} \bigl\Vert \omega (u) \bigr\Vert ({s_{f}}-{s_{0}})+{L}_{{G}_{3}}{L}_{ \mathcal{\mathcal{F}}_{2}} \bigl\Vert \omega (u) \bigr\Vert ({s_{f}}-{s_{0}})+ \widehat{M}_{G} \bigr)\Delta s \\ \leq& \sum_{j=1}^{i}{L}_{\Xi } \sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert +\sum _{j=1}^{i}{L}_{\Phi }\sup _{ \varsigma \in \mathbb{T}} \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert + \sup_{\varsigma \in \mathbb{T}} \bigl\Vert E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr) \omega _{0} \bigr\Vert \\ &{} + \int _{\varsigma _{0}}^{\varsigma _{f}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \sup_{\varsigma \in \mathbb{T}} \bigl\Vert E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr) \bigr\Vert \Bigl({L}_{\mathcal{F}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (s) \bigr\Vert \\ &{} +{L}_{{G}_{1}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (s) \bigr\Vert +{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (u) \bigr\Vert ({s_{f}}-{s_{0}}) \\ &{}+{L}_{{G}_{3}}{L}_{ \mathcal{\mathcal{F}}_{2}} \sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (u) \bigr\Vert ({s_{f}}-{s_{0}})+\widehat{M}_{G} \Bigr) \Delta s \\ \leq& \sum_{j=1}^{i}{L}_{\Xi } \Vert \omega \Vert _{\infty }+\sum_{j=1}^{i}{L}_{ \Phi } \Vert \omega \Vert _{\infty }+a_{1}+ \int _{\varsigma _{0}}^{\varsigma _{f}}a_{3}a_{2} \bigl({L}_{\mathcal{F}} \Vert \omega \Vert _{\infty }+{L}_{{G}_{1}} \Vert \omega \Vert _{ \infty } \\ &{} +{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}})+{L}_{{G}_{3}}{L}_{ \mathcal{\mathcal{F}}_{2}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}})+ \widehat{M}_{G} \bigr)\Delta s \\ \leq& \sum_{j=1}^{i}{L}_{\Xi } \delta +\sum_{j=1}^{i}{L}_{\Phi } \delta +a_{1} +a_{3} \bigl(\delta a_{2}{L}_{\mathcal{F}}+ \delta a_{2}{L}_{{G}}+ \delta a_{2}{L}_{{G}}{L}_{\mathcal{\mathcal{F}}_{1}}({s_{f}}-{s_{0}}) \\ &{} +\delta a_{2}{L}_{{G}}{L}_{\mathcal{\mathcal{F}}_{2}}({s_{f}}-{s_{0}})+ \widehat{M}_{G} \bigr)\times \int _{\varsigma _{0}}^{\varsigma _{f}} \Delta s \\ \leq& \sum_{j=1}^{i}{L}_{\Xi } \delta +\sum_{j=1}^{i}{L}_{\Phi } \delta +a_{1} +\delta a_{3} \bigl(a_{2}{L}_{\mathcal{F}}+ a_{2}{L}_{{G}_{1}}+ a_{2}{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}}({s_{f}}-{s_{0}}) \\ &{} + a_{2}{L}_{{G}_{3}}{L}_{\mathcal{\mathcal{F}}_{2}}({s_{f}}-{s_{0}})+ \widehat{M}_{G} \bigr) (\varsigma -\varsigma _{f}). \end{aligned}$$

So

$$ \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert \leq N_{1}+\delta Q_{1} \leq \delta +\delta Q_{1} = \delta _{1}, $$

where \(\delta _{1}=\delta +\delta Q_{1}\). Hence

$$ \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert \leq \delta _{2}. $$
(11)

Therefore, from (11), \(\Lambda (\Omega ) \subseteq \Omega \). Also, for \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), with \(\omega _{0}=\widehat{\omega }_{0}\), one has

$$\begin{aligned} & \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Lambda _{\sigma } \bigl(\widehat{\omega }(\varsigma ) \bigr) \bigr\Vert \\ &\quad \leq \sum_{j=1}^{i} \bigl\Vert \Xi _{j}\bigl(\omega \bigl(\varsigma ^{-}_{j} \bigr)\bigr)-\Xi _{j}\bigl(\widehat{{\omega }}\bigl( \varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert \\ &\qquad {}+\sum_{j=1}^{i} \bigl\Vert \Phi _{j} \bigl(\varsigma ^{-}_{j},\omega \bigl( \varsigma ^{-}_{j}\bigr)\bigr)-\Phi _{j} \bigl(\varsigma ^{-}_{j},\widehat{{\omega }}\bigl( \varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl(\mathcal{F} \bigl(s,\omega (s)\bigr) \\ &\qquad {}+G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr) \\ &\qquad {}- \biggl(\mathcal{F}\bigl(s,\widehat{\omega }(s)\bigr) \\ &\qquad {}+G \biggl(s,\widehat{\omega }(s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ &\quad \leq \sum_{j=1}^{i}L_{\Xi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr)- \widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert +\sum_{j=1}^{i}L_{\Phi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr)- \widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl( \bigl\Vert \mathcal{F} \bigl(s,\omega (s)\bigr)-\mathcal{F}\bigl(s,\widehat{\omega }(s)\bigr) \bigr\Vert \\ &\qquad {}+ \biggl\Vert G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \\ &\qquad {}-G \biggl(s,\widehat{\omega }(s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u \biggr) \biggr\Vert \biggr)\Delta s \\ &\quad \leq \sum_{j=1}^{i}L_{\Xi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr)- \widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert +\sum_{j=1}^{i}L_{\Phi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr)- \widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl(L_{F} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert \\ &\qquad {}+L_{\mathcal{G}_{1}} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert +L_{ \mathcal{G}_{2}} \int _{s_{0}}^{s_{f}} \bigl\Vert \mathcal{F}_{1}\bigl(s,u, \omega (u)\bigr)-\mathcal{F}_{1} \bigl(s,u,\widehat{\omega }(u)\bigr) \bigr\Vert \Delta u \\ &\qquad {}+ L_{{G}_{3}} \int _{s_{0}}^{s_{f}} \bigl\Vert \mathcal{F}_{2}\bigl(s,u,\omega (u)\bigr)- \mathcal{F}_{2} \bigl(s,u,\widehat{\omega }(u)\bigr) \bigr\Vert \Delta u \biggr)\Delta s \\ &\quad \leq \sum_{j=1}^{i}L_{\Xi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr)- \widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert +\sum_{j=1}^{i}L_{\Phi } \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr)- \widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl(L_{ \mathcal{F}} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert \\ &\qquad {}+L_{{G}_{1}} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert +L_{{G}_{2}}L_{{ \mathcal{F}_{1}}} \int _{s_{0}}^{s_{f}} \bigl\Vert \omega (u)- \widehat{\omega }(u) \bigr\Vert \Delta u \\ &\qquad {}+ L_{{G}_{3}}L_{{\mathcal{F}}_{2}} \int _{s_{0}}^{s_{f}} \bigl\Vert \omega (u)- \widehat{\omega }(u) \bigr\Vert \Delta u \biggr)\Delta s \\ &\quad \leq \sum_{j=1}^{i}L_{\Xi } \sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr)-\widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert +\sum _{j=1}^{i}L_{\Phi }\sup _{\varsigma \in \mathbb{T}} \bigl\Vert \omega \bigl(\varsigma ^{-}_{j}\bigr)-\widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr) \bigr\Vert \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \sup_{\varsigma \in \mathbb{T}} \bigl\Vert E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr) \bigr\Vert \Bigl(L_{\mathcal{F}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (s)- \widehat{\omega }(s) \bigr\Vert \\ &\qquad {}+L_{{G}_{1}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (s)- \widehat{\omega }(s) \bigr\Vert +L_{{G}_{2}}L_{{\mathcal{F}_{1}}}\sup _{ \varsigma \in \mathbb{T}} \bigl\Vert \omega (u)-\widehat{\omega }(u) \bigr\Vert ({s_{f}}-{s_{0}}) \\ &\qquad {}+ L_{{G}_{3}}L_{{\mathcal{F}}_{2}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (u)-\widehat{\omega }(u) \bigr\Vert ({s_{f}}-{s_{0}}) \Bigr) \Delta s \\ &\quad =\sum_{j=1}^{i}L_{\Xi } \Vert \omega -\widehat{{\omega }} \Vert _{ \infty }+\sum _{j=1}^{i}L_{\Phi } \Vert \omega - \widehat{{\omega }} \Vert _{ \infty } \\ &\qquad {}+a_{3}a_{2} \bigl(L_{\mathcal{F}} \Vert \omega - \widehat{\omega } \Vert _{ \infty }+L_{{G}_{1}} \Vert \omega - \widehat{\omega } \Vert _{\infty }+L_{{G}_{2}}L_{{ \mathcal{F}_{1}}} \Vert \omega -\widehat{\omega } \Vert _{\infty }({s_{f}}-{s_{0}}) \\ &\qquad {}+ L_{{G}_{3}}L_{{\mathcal{F}}_{2}} \Vert \omega -\widehat{\omega } \Vert _{\infty }({s_{f}}-{s_{0}}) \bigr) \int _{\varsigma _{0}}^{ \varsigma _{f}}\Delta s \\ &\quad \leq \Biggl[\sum_{j=1}^{i}L_{\Xi }+ \sum_{j=1}^{i}L_{\Phi } \\ &\qquad {}+a_{3} \bigl(a_{2}L_{\mathcal{F}}+a_{2}L_{{G}_{1}}+a_{2}L_{{G}_{2}}L_{{ \mathcal{F}_{1}}}({s_{f}}-{s_{0}})+ a_{2}L_{{G}_{3}}L_{{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) (\varsigma -\varsigma _{f}) \Biggr] \\ &\qquad {}\times \Vert \omega -\widehat{\omega } \Vert _{\infty }. \end{aligned}$$

It implies

$$ \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Lambda _{\sigma } \bigl(\widehat{\omega }(\varsigma ) \bigr) \bigr\Vert \leq (N_{2}+ Q_{2} ) \Vert \omega -\widehat{\omega } \Vert _{\infty }= Q_{3} \Vert \omega - \widehat{\omega } \Vert _{\infty }. $$

Hence

$$ \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Lambda _{\sigma } \bigl(\widehat{\omega }(\varsigma ) \bigr) \bigr\Vert \leq Q_{3} \Vert \omega - \widehat{\omega } \Vert _{\infty }\quad (Q_{3}< 1). $$
(12)

Therefore, from inequality (12) and (9), the operator \(\Lambda _{\sigma }\) is strictly contractive. Consequently, the mixed impulsive system (1) admits a unique solution via the Banach principle. □

Now, regarding the mixed impulsive system (2), we have the following result.

Theorem 3.2

The mixed impulsive system (2) involves a unique solution if assertion \(\mathbf{(A)}\) holds and

$$ \max_{1\leq i\leq 3}\bigl\{ Q^{*}_{i} \bigr\} < 1. $$
(13)

Proof

Let \(\Omega \subseteq \mathrm{PS}\) and \(\Omega =\{(\mathbb{X},\mathbb{Y},\mathbb{Z})\in \mathrm{PS}:\|( \mathbb{X},\mathbb{Y},\mathbb{Z})\|_{C}\leq \delta _{2}\}\), also \(\delta _{2}=\max \{\delta ,\delta _{1}\}\) and \(\delta ,\delta _{1}\in (0,1)\) provided that

$$ \delta >\max \{N_{1},N_{2},N_{3} \}. $$

Now, we define \(\Psi _{\sigma }:\Omega \rightarrow \Omega \) as

$$ \Psi _{\sigma } \bigl(\omega (\varsigma ) \bigr)= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0} +\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } ){F} (\varsigma ,\omega (\varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} {F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} {F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s, \\ \quad \varsigma \in (\varsigma _{i}, s_{i}]\cap \mathbb{{T}}, i = 1, \dots , m, \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0} +\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } ){F} (\varsigma ,\omega (\varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} {F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} {F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{s_{i}}(\varsigma -s)^{ \sigma -1}\hbar _{i}(s,\omega (s))\Delta s,\quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}, i = 1,\dots , m. \end{cases} $$
(14)

Also, note that

$$ \bigl\Vert {F}(s,\omega ) \bigr\Vert \leq \bigl\Vert {F}(s,\omega )-{F}(s,0) \bigr\Vert + \bigl\Vert {F}(s,0) \bigr\Vert \leq L_{{F}} \Vert \omega \Vert +M_{{F}} $$

and

$$\begin{aligned} \bigl\Vert \mathcal{G}(\varsigma ,\mathbb{X},\mathbb{Y},\mathbb{Z}) \bigr\Vert \leq & \bigl\Vert \mathcal{G}(\varsigma ,\mathbb{X},\mathbb{Y}, \mathbb{Z})-\mathcal{G}( \varsigma ,0,0,0) \bigr\Vert + \bigl\Vert \mathcal{G}(\varsigma ,0,0,0) \bigr\Vert \\ \leq &L_{\mathcal{G}_{1}} \Vert \mathbb{X} \Vert +L_{\mathcal{G}_{2}} \Vert \mathbb{Y} \Vert +L_{\mathcal{G}_{3}} \Vert \mathbb{Z} \Vert +M_{\mathcal{G}}, \end{aligned}$$

where \(M_{{F}}=\sup_{\varsigma \in \mathbb{T}}\|{F}(s,0)\|\), \(M_{ \mathcal{G}}=\sup_{\varsigma \in \mathbb{T}}\|\mathcal{G}(\varsigma ,0,0,0) \|\), and \(\widehat{M}_{\mathcal{G}}=M_{{F}}+M_{\mathcal{G}}\). In addition, \(\mathbb{X}=\omega (s)\), \(\mathbb{Y}=\int _{s_{0}}^{s_{f}} {F}_{1}(s,u,\omega (u)) \Delta u\), and \(\mathbb{Z}=\int _{s_{0}}^{s_{f}} {F}_{2}(s,u,\omega (u)) \Delta u\).

Now, we prove that \(\Psi _{\sigma }:\Omega \rightarrow \Omega \) is a self-mapping.

For \(\varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}\), \(i = 1, \dots , m\), one has

$$\begin{aligned} \bigl\Vert \Psi _{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert &\leq \biggl\Vert \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}}(\varsigma -s)^{ \sigma -1} \hbar _{i}\bigl(s,\omega (s)\bigr)\Delta s \biggr\Vert + \bigl\Vert E_{\sigma }\bigl(A\varsigma ^{ \sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl({F}\bigl(s, \omega (s)\bigr) \\ &\quad {}+\mathcal{G} \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} {F}_{1}\bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} {F}_{2}\bigl(s,u, \omega (u)\bigr) \Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ &\leq \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}} \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert \hbar _{i}\bigl(s, \omega (s)\bigr) \bigr\Vert \Delta s+ \bigl\Vert E_{\sigma }\bigl(A \varsigma ^{\sigma }\bigr) \bigr\Vert \Vert \omega _{0} \Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl( \bigl\Vert {F} \bigl(s,\omega (s)\bigr) \bigr\Vert \\ &\quad {}+ \biggl\Vert \mathcal{G} \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} {F}_{1}\bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} {F}_{2}\bigl(s,u, \omega (u)\bigr) \Delta u \biggr) \biggr\Vert +\widehat{M}_{\mathcal{G}} \biggr)\Delta s \\ &\leq \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}} \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert L_{g} \bigl\Vert \bigl(\omega (s)\bigr) \bigr\Vert \Delta s+ \bigl\Vert E_{ \sigma }\bigl(A\varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl({L}_{{F}} \bigl\Vert \omega (s) \bigr\Vert +{L}_{\mathcal{G}_{1}} \bigl\Vert \omega (s) \bigr\Vert \\ &\quad {}+{L}_{\mathcal{G}_{2}} \int _{s_{0}}^{s_{f}} \bigl\Vert {F}_{1} \bigl(s,u, \omega (u)\bigr) \bigr\Vert \Delta u+{L}_{\mathcal{G}_{3}} \int _{s_{0}}^{s_{f}} \bigl\Vert {F}_{2} \bigl(s,u,\omega (u)\bigr) \bigr\Vert \Delta u+\widehat{M}_{\mathcal{G}} \biggr)\Delta s \\ &\leq \frac{1}{\Gamma (\sigma )} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert L_{g} \bigl\Vert \bigl(\omega (s)\bigr) \bigr\Vert ({s_{i}}-{\varsigma _{i}})+ \bigl\Vert E_{\sigma }\bigl(A \varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl({L}_{{F}} \bigl\Vert \omega (s) \bigr\Vert +{L}_{{\mathcal{G}}_{1}} \bigl\Vert \omega (s) \bigr\Vert \\ &\quad {}+{L}_{{\mathcal{G}}_{2}}{L}_{{F}_{1}} \int _{s_{0}}^{s_{f}} \bigl\Vert \omega (u) \bigr\Vert \Delta u+{L}_{{\mathcal{G}}_{3}}{L}_{{F}_{2}} \int _{s_{0}}^{s_{f}} \bigl\Vert \omega (u) \bigr\Vert \Delta u+\widehat{M}_{\mathcal{G}} \biggr)\Delta s \\ &=\frac{1}{\Gamma (\sigma )} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert L_{g} \bigl\Vert \bigl( \omega (s)\bigr) \bigr\Vert ({s_{i}}-{\varsigma _{i}})+ \bigl\Vert E_{\sigma }\bigl(A\varsigma ^{ \sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \bigl({L}_{{F}} \bigl\Vert \omega (s) \bigr\Vert +{L}_{{\mathcal{G}}_{1}} \bigl\Vert \omega (s) \bigr\Vert \\ &\quad {}+{L}_{\mathcal{G}_{2}}{L}_{{F}_{1}} \bigl\Vert \omega (u) \bigr\Vert ({s_{f}}-{s_{0}})+{L}_{ \mathcal{G}_{3}}{L}_{F_{2}} \bigl\Vert \omega (u) \bigr\Vert ({s_{f}}-{s_{0}}) + \widehat{M}_{\mathcal{G}} \bigr)\Delta s \\ &\leq \frac{1}{\Gamma (\sigma )}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert L_{g}\sup _{\varsigma \in \mathbb{T}} \bigl\Vert \bigl(\omega (s)\bigr) \bigr\Vert ({s_{i}}-{\varsigma _{i}})+\sup_{\varsigma \in \mathbb{T}} \bigl\Vert E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \sup_{\varsigma \in \mathbb{T}} \bigl\Vert E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr) \bigr\Vert \Bigl({L}_{F}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (s) \bigr\Vert \\ &\quad {}+{L}_{ \mathcal{G}_{1}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (s) \bigr\Vert +{L}_{\mathcal{G}_{2}}{L}_{F_{1}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (u) \bigr\Vert ({s_{f}}-{s_{0}}) \\ &\quad {}+{L}_{\mathcal{G}_{3}}{L}_{F_{2}} \sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (u) \bigr\Vert ({s_{f}}-{s_{0}})+ \widehat{M}_{\mathcal{G}} \Bigr) \Delta s \\ &\leq \frac{1}{\Gamma (\sigma )}a_{3}L_{g} \Vert \omega \Vert _{\infty }({s_{i}}-{ \varsigma _{i}})+a_{1}+ \int _{\varsigma _{0}}^{\varsigma _{f}}a_{3}a_{2} \bigl(L_{F} \Vert \omega \Vert _{\infty } +{L}_{\mathcal{G}_{1}} \Vert \omega \Vert _{ \infty } \\ &\quad {}+{L}_{\mathcal{G}_{2}}{L}_{F_{1}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}})+{L}_{ \mathcal{G}_{3}}{L}_{F_{2}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}})+ \widehat{M}_{\mathcal{G}} \bigr)\Delta s \\ &\leq \frac{1}{\Gamma (\sigma )}a_{3}L_{g}\delta ({s_{i}}-{\varsigma _{i}})+a_{1}+a_{3} \bigl(\delta a_{2}{L}_{F}+\delta a_{2}{L}_{\mathcal{G}}+ \delta a_{2}{L}_{ \mathcal{G}}{L}_{F_{1}}({s_{f}}-{s_{0}}) \\ &\quad {}+\delta a_{2}{L}_{\mathcal{G}}{L}_{F_{2}}({s_{f}}-{s_{0}})+ \widehat{M}_{\mathcal{G}} \bigr)\times \int _{\varsigma _{0}}^{ \varsigma _{f}}\Delta s \\ &\leq \frac{1}{\Gamma (\sigma )}a_{3}L_{g}\delta ({s_{i}}-{\varsigma _{i}})+a_{1}+ \delta a_{3} \bigl(a_{2}{L}_{F}+ a_{2}{L}_{\mathcal{G}_{1}}+ a_{2}{L}_{ \mathcal{G}_{2}}{L}_{F_{1}}({s_{f}}-{s_{0}}) \\ &\quad {}+ a_{2}{L}_{\mathcal{G}_{3}}{L}_{F_{2}}({s_{f}}-{s_{0}})+ \widehat{M}_{ \mathcal{G}} \bigr) (\varsigma -\varsigma _{f}). \end{aligned}$$

Thus

$$ \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert \leq N_{3}+\delta Q^{*}_{1} \leq \delta +\delta Q^{*}_{1}=\delta _{1}, $$

where \(\delta _{1}=\delta +\delta Q^{*}_{1}\). Hence

$$ \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert \leq \delta _{2}. $$
(15)

Therefore, from (15), \(\Psi _{\sigma }(\Omega )\subseteq \Omega \). Also, for \(\varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}\), \(i = 1, \dots , m\), with \(\omega _{0}=\widehat{\omega }_{0}\), we have

$$\begin{aligned} &\bigl\Vert \Psi _{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Psi _{\sigma } \bigl( \widehat{\omega }(\varsigma ) \bigr) \bigr\Vert \\ &\quad \leq \biggl\Vert \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}}(\varsigma -s)^{ \sigma -1} \bigl(\hbar _{i}\bigl(s,\omega (s)\bigr)-\hbar _{i} \bigl(s,\widehat{\omega }(s)\bigr) \bigr)\Delta s \biggr\Vert \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl({F}\bigl(s, \omega (s)\bigr) \\ &\qquad {}+\mathcal{G} \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} {F}_{1}\bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} {F}_{2}\bigl(s,u, \omega (u)\bigr) \Delta u \biggr) \biggr) \\ &\qquad {}- \biggl({F}\bigl(s,\widehat{\omega }(s)\bigr) \\ &\qquad {}+\mathcal{G} \biggl(s,\widehat{\omega }(s), \int _{s_{0}}^{s_{f}} {F}_{1}\bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} {F}_{2}\bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ &\quad \leq \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}} \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert \hbar _{i}\bigl(s, \omega (s)\bigr)-\hbar _{i}\bigl(s, \widehat{\omega }(s)\bigr) \bigr\Vert \Delta s \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl( \bigl\Vert F\bigl(s, \omega (s)\bigr)-F\bigl(s,\widehat{\omega }(s)\bigr) \bigr\Vert \\ &\qquad {}+ \biggl\Vert \mathcal{G} \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} {F}_{1}\bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} {F}_{2}\bigl(s,u, \omega (u)\bigr) \Delta u \biggr) \\ &\qquad {}-\mathcal{G} \biggl(s,\widehat{\omega }(s), \int _{s_{0}}^{s_{f}} {F}_{1}\bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} F_{2}\bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u \biggr) \biggr\Vert \biggr)\Delta s \\ &\quad \leq \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}} \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert L_{g} \bigl\Vert \omega (s)- \widehat{\omega }(s) \bigr\Vert \Delta s \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl(L_{F} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert \\ &\qquad {}+L_{\mathcal{G}_{1}} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert +L_{ \mathcal{G}_{2}} \int _{s_{0}}^{s_{f}} \bigl\Vert F_{1} \bigl(s,u,\omega (u)\bigr)-F_{1}\bigl(s,u, \widehat{\omega }(u)\bigr) \bigr\Vert \Delta u \\ &\qquad {}+ L_{\mathcal{G}_{3}} \int _{s_{0}}^{s_{f}} \bigl\Vert F_{2} \bigl(s,u,\omega (u)\bigr)-F_{2}\bigl(s,u, \widehat{\omega }(u)\bigr) \bigr\Vert \Delta u \biggr)\Delta s \\ &\quad \leq \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}} \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert L_{g} \bigl\Vert \omega (s)- \widehat{\omega }(s) \bigr\Vert \Delta s \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl(L_{F} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert \\ &\qquad {}+L_{\mathcal{G}_{1}} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert +L_{ \mathcal{G}_{2}}L_{F_{1}} \int _{s_{0}}^{s_{f}} \bigl\Vert \omega (u)- \widehat{\omega }(u) \bigr\Vert \Delta u \\ &\qquad {}+ L_{\mathcal{G}_{3}}L_{F_{2}} \int _{s_{0}}^{s_{f}} \bigl\Vert \omega (u)- \widehat{\omega }(u) \bigr\Vert \Delta u \biggr)\Delta s \\ &\quad \leq \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}}\sup_{ \varsigma \in \mathbb{T}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert L_{g} \bigl\Vert \omega (s)-\widehat{\omega }(s) \bigr\Vert \Delta s \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \sup_{\varsigma \in \mathbb{T}} \bigl\Vert E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr) \bigr\Vert \Bigl(L_{F}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (s)- \widehat{\omega }(s) \bigr\Vert \\ &\qquad {}+L_{\mathcal{G}_{1}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (s)- \widehat{\omega }(s) \bigr\Vert +L_{\mathcal{G}_{2}}L_{F_{1}} \sup _{ \varsigma \in \mathbb{T}} \bigl\Vert \omega (u)-\widehat{\omega }(u) \bigr\Vert ({s_{f}}-{s_{0}}) \\ &\qquad {}+ L_{\mathcal{G}_{3}}L_{F_{2}}\sup_{\varsigma \in \mathbb{T}} \bigl\Vert \omega (u)-\widehat{\omega }(u) \bigr\Vert ({s_{f}}-{s_{0}}) \Bigr)\Delta s \\ &\quad =\frac{1}{\Gamma (\sigma )}a_{3}L_{g} \Vert \omega - \widehat{\omega } \Vert _{\infty } \int _{\varsigma _{0}}^{\varsigma _{f}}\Delta s \\ &\qquad {}+a_{3}a_{2} \bigl(L_{F} \Vert \omega - \widehat{\omega } \Vert _{\infty }+L_{{ \mathcal{G}}_{1}} \Vert \omega - \widehat{\omega } \Vert _{\infty } +L_{{ \mathcal{G}}_{2}}L_{{{F}_{1}}} \Vert \omega -\widehat{\omega } \Vert _{ \infty }({s_{f}}-{s_{0}}) \\ &\qquad {}+ L_{{\mathcal{G}}_{3}}L_{{{F}}_{2}} \Vert \omega -\widehat{\omega } \Vert _{\infty }({s_{f}}-{s_{0}}) \bigr) \int _{\varsigma _{0}}^{ \varsigma _{f}}\Delta s \\ &\quad \leq \biggl[\frac{1}{\Gamma (\sigma )}a_{3}L_{g}({ \varsigma _{f}}-{ \varsigma _{0}})+a_{3} \bigl(a_{2}L_{{F}}+a_{2}L_{{G}_{1}}+a_{2}L_{{ \mathcal{G}}_{2}}L_{{{F}_{1}}}({s_{f}}-{s_{0}}) \\ &\qquad {}+ a_{2}L_{{\mathcal{G}}_{3}}L_{{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) (\varsigma -\varsigma _{f}) \biggr] \Vert \omega - \widehat{\omega } \Vert _{\infty }. \end{aligned}$$

It implies

$$ \bigl\Vert \Psi _{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Psi _{\sigma } \bigl( \widehat{\omega }(\varsigma ) \bigr) \bigr\Vert \leq (N_{4}+ Q_{2} ) \Vert \omega -\widehat{\omega } \Vert _{\infty }\leq Q^{*}_{3} \Vert \omega - \widehat{\omega } \Vert _{\infty }. $$

Hence

$$ \bigl\Vert \Psi _{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Psi _{\sigma } \bigl( \widehat{\omega }(\varsigma ) \bigr) \bigr\Vert \leq Q^{*}_{3} \Vert \omega - \widehat{\omega } \Vert _{\infty }. $$
(16)

Therefore, from (16) and (13), the operator \(\Psi _{\sigma }\) is strictly contractive. Consequently, the second impulsive system (2) admits a unique solution via the Banach principle. □

Next, for both mixed impulsive systems (1) and (2), we investigate the existence of at least one solution via the weaker condition \((\mathbf{B})\) and the Leray–Schauder alternative fixed point method.

Theorem 3.3

The mixed impulsive system (1) has at least one solution provided assumption \((\mathbf{B})\) holds and \(\mathcal{K}>0\) exists so that

$$ a_{1}+Q_{3}\mathcal{K}< \mathcal{K}. $$
(17)

Proof

Firstly, we prove that \(\Lambda _{\sigma }\) defined by (10) is a completely continuous operator. We see that the continuity of the mappings Ξ, Φ, \(\mathcal{F}\), and G provides that \(\Lambda _{\sigma }\) is a continuous operator. Also, assume that \(\Omega _{1}{\subseteq }PS\) along with the fact that the operators Ξ, Φ, \(\mathcal{F}\), and G are bounded. Then there exist \(L_{1}\), \(L_{2}\), \(M_{1}\), and \(M_{2}\) (positive constants) such that \(\sum_{j=1}^{i}\Xi _{j}(\omega )\leq L_{1}\), \(\sum_{j=1}^{i}\Phi _{j}(\omega )\leq L_{2}\), \(\mathcal{F} (\varsigma , \omega (\varsigma ) )\leq M_{1}\), and \(G(\varsigma ,\mathbb{X},\mathbb{Y},\mathbb{Z})\leq M_{2}\), where \(p=\omega (s)\),

$$ q= \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u,\omega (u)\bigr) \Delta u, $$

and

$$ r= \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u. $$

Note that we take \(\mathcal{L}=L_{4}+L_{5}+a_{1}\), \(\mathcal{M}=M_{1}+M_{2}\), \(\|(\varsigma -s)^{\sigma -1}\|\leq \mathcal{L}_{1}\), and \(\mathcal{L}+\mathcal{L}_{1}a_{2}\mathcal{M}({\varsigma _{f}}-{ \varsigma _{0}})=\mathfrak{G}\).

Then, for any \(\omega \in \Omega _{1}\) and \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), we have

$$\begin{aligned} \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert &\leq \sum_{j=1}^{i} \bigl\Vert \Xi _{j}\bigl(\omega \bigl(\varsigma ^{-}_{j} \bigr)\bigr) \bigr\Vert +\sum_{j=1}^{i} \bigl\Vert \Phi _{j} \bigl(\varsigma ^{-}_{j}, \omega \bigl(\varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert + \bigl\Vert E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl(\mathcal{F} \bigl(s,\omega (s)\bigr) \\ &\quad {}+G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ &\leq L_{4}+L_{5}+a_{1}+ \mathcal{L}_{1}a_{2}(M_{1}+M_{2}) \int _{ \varsigma _{0}}^{\varsigma _{f}}\Delta s \\ &= \mathcal{L}+\mathcal{L}_{1}a_{2}\mathcal{M}({ \varsigma _{f}}-{ \varsigma _{0}}). \end{aligned}$$

It implies

$$ \bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert \leq \mathfrak{G}. $$
(18)

Thus, from (18), we conclude that Λ is uniformly bounded.

Now, we prove that \(\Lambda _{\sigma }\) is completely continuous. For this, we discuss the following possibilities.

Case 1: Assume that all points on \(\mathbb{T}\) are isolated, i.e., time scales consist of discrete points. Using Theorem 2.2, \(\Lambda _{\sigma }\) becomes

$$ \Lambda _{\sigma } \bigl(\omega (\varsigma ) \bigr)= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\sum_{\varsigma \in \mathbb{T}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\sum_{\varsigma \in \mathbb{T}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ), \\ \quad \varsigma \in (\varsigma _{0},\varsigma _{1}], \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\sum_{\varsigma \in \mathbb{T}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\sum_{\varsigma \in \mathbb{T}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\sum_{j=1}^{i} (\Xi _{j}(\omega (\varsigma ^{-}_{j}))+\Phi _{j} (\varsigma ^{-}_{j},\omega (\varsigma ^{-}_{j})) ),\quad \varsigma \in (\varsigma _{i},\varsigma _{i+1}], i = 1,\dots , m. \end{cases} $$
(19)

Clearly, on a discrete finite set, (19) is a collection of summation operators. Further, the continuity of \(\Xi _{j}\), \(\Phi _{j}\), \(\mathcal{F}\), and G implies that \(\Lambda _{\sigma }\) is completely continuous.

Case 2: Assume that all the points of \(\mathbb{T}\) are dense, i.e., \(\mathbb{T}\) is continuous. Now, let \(\varsigma _{{f}_{1}}, \varsigma _{{f}_{2}}\in (\varsigma _{i}, \varsigma _{i+1}]\), \(i = 1,\dots , m\), such that \(\varsigma _{{f}_{1}}<\varsigma _{{f}_{2}}\), then

$$\begin{aligned} &\bigl\Vert \Lambda _{\sigma } \bigl(\omega (\varsigma _{f_{2}}) \bigr)-\Lambda _{ \sigma } \bigl({\omega }(\varsigma _{f_{1}}) \bigr) \bigr\Vert \\ &\quad \leq \Biggl\| \sum_{j=1}^{i} \bigl[\Xi _{j}\bigl(\omega (\varsigma _{{f_{2}}^{-}_{j}})\bigr)-\Xi _{j}\bigl({{ \omega }}(\varsigma _{{f_{1}}^{-}_{j}})\bigr) \bigr] \\ &\qquad {}+\sum_{j=1}^{i} \bigl[\Phi _{j} \bigl(\varsigma _{{f_{2}}^{-}_{j}}, \omega (\varsigma _{{f_{2}}^{-}_{j}})\bigr)-\Phi _{j} \bigl(\varsigma _{{f_{1}}^{-}_{j}},{{ \omega }}(\varsigma _{{f_{1}}^{-}_{j}})\bigr) \bigr] \\ &\qquad {}+ \biggl[ \biggl( \int _{\varsigma _{0}}^{\varsigma _{f_{2}}}(\varsigma _{f_{2}}-s)^{ \sigma -1} E_{\sigma ,\sigma } \bigl(A(\varsigma _{f_{2}}-s)^{\sigma } \bigr) \biggl(\mathcal{F}\bigl(s,\omega (s)\bigr) \\ &\qquad {}+G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr)\Delta s \biggr) \\ &\qquad {}- \biggl( \int _{\varsigma _{0}}^{\varsigma _{f_{1}}}(\varsigma _{f_{1}}-s)^{ \sigma -1} E_{\sigma ,\sigma } \bigl(A(\varsigma _{f_{1}}-s)^{\sigma } \bigr) \biggl(\mathcal{F}\bigl(s,\omega (s)\bigr) \\ &\qquad {}+G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr) \biggr) \biggr]\Delta s\Biggr\| \\ &\quad \leq \sum_{j=1}^{i}\bigl\| \Xi _{j}\bigl(\omega (\varsigma _{{f_{2}}^{-}_{j}})\bigr)- \Xi _{j}\bigl({{\omega }}(\varsigma _{{f_{1}}^{-}_{j}})\bigr) \bigr\Vert \\ &\qquad {}+\sum_{j=1}^{i}\bigl\| \Phi _{j} \bigl(\varsigma _{{f_{2}}^{-}_{j}}, \omega (\varsigma _{{f_{2}}^{-}_{j}})\bigr)-\Phi _{j} \bigl(\varsigma _{{f_{1}}^{-}_{j}},{{ \omega }}(\varsigma _{{f_{1}}^{-}_{j}})\bigr) \bigr\Vert \\ &\qquad {}+ \biggl\Vert \int _{\varsigma _{0}}^{\varsigma _{f_{2}}} \bigl[(\varsigma _{f_{1}}-s)^{ \sigma -1} E_{\sigma ,\sigma } \bigl(A(\varsigma _{f_{2}}-s)^{\sigma } \bigr)-(\varsigma _{f_{1}}-s)^{\sigma -1} E_{\sigma ,\sigma } \bigl(A( \varsigma _{f_{1}}-s)^{\sigma } \bigr) \bigr] \\ &\qquad {}\times \mathcal{F}\bigl(s,\omega (s)\bigr)\Delta s \biggr\Vert \\ &\qquad {}+ \biggl\Vert \int _{\varsigma _{0}}^{\varsigma _{f_{2}}} \bigl[(\varsigma _{f_{1}}-s)^{ \sigma -1} E_{\sigma ,\sigma } \bigl(A(\varsigma _{f_{2}}-s)^{\sigma } \bigr)-(\varsigma _{f_{1}}-s)^{\sigma -1} E_{\sigma ,\sigma } \bigl(A( \varsigma _{f_{1}}-s)^{\sigma } \bigr) \bigr] \\ &\qquad {}\times G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr)\Delta s \biggr\Vert \\ &\qquad {}+ \biggl\Vert \int _{\varsigma _{f_{1}}}^{\varsigma _{f_{2}}} \bigl[( \varsigma _{f_{2}}-s)^{\sigma -1} E_{\sigma ,\sigma } \bigl(A(\varsigma _{f_{2}}-s)^{ \sigma } \bigr) \bigr]\mathcal{F}\bigl(s,\omega (s) \bigr)\Delta s \biggr\Vert \\ &\qquad {}+ \biggl\Vert \int _{\varsigma _{f_{1}}}^{\varsigma _{f_{2}}} \bigl[( \varsigma _{f_{2}}-s)^{\sigma -1} E_{\sigma ,\sigma } \bigl(A(\varsigma _{f_{2}}-s)^{ \sigma } \bigr) \bigr] \\ &\qquad {}\times G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr)\Delta s \biggr\Vert . \end{aligned}$$

Clearly, we observe from the above that it approaches 0 as \(\varsigma _{f_{2}}\rightarrow \varsigma _{f_{1}}\). Hence, the operator \(\Lambda _{\sigma }\) is equicontinuous. Finally, using the Arzela–Ascoli theorem, we conclude that \(\Lambda _{\sigma }\) is completely continuous.

Case 3: Assume that \(\mathbb{T}\) involves isolated points along with dense ones, i.e., continuous and discrete. Now, utilizing Theorem 2.2 for the isolated points, we can write \(\Lambda _{\sigma }\) as the summation operator which is completely continuous (discussed in case 1). For the dense points, one can prove that \(\Lambda _{\sigma }\) is a completely continuous operator (discussed in case 2). Consequently, \(\Lambda _{\sigma }\) can be written as a sum of two operators for isolated and dense points. As a result, we know that the sum of two operators which are completely continuous is also completely continuous. Thus, the operator \(\Lambda _{\sigma }\) is a completely continuous operator. Hence, by summarizing the above three possibilities, we arrive at the conclusion that \(\Lambda _{\sigma }\) is a completely continuous operator.

Finally, let \(\beta \in [0,1]\), and there exists ω provided that \(\omega (\varsigma )=\beta (\Lambda _{\sigma }(\omega )(\varsigma ) )\). Then, for \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), one obtains

$$\begin{aligned} \bigl\Vert \omega (\varsigma ) \bigr\Vert &= \bigl\Vert \beta \bigl( \Lambda _{\sigma } \bigl( \omega (\varsigma _{f}) \bigr) \bigr) \bigr\Vert \\ &\leq \Biggl\Vert \beta \Biggl[\sum_{j=1}^{i} \Xi _{j}\bigl(\omega (\varsigma _{{f}^{-}_{j}})\bigr)+ \sum _{j=1}^{i}\Phi _{j} \bigl( \varsigma _{{f_{2}}^{-}_{j}},\omega ( \varsigma _{{f_{2}}^{-}_{j}})\bigr) \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f_{2}}}(\varsigma _{f_{2}}-s)^{ \sigma -1} E_{\sigma ,\sigma } \bigl(A(\varsigma _{f_{2}}-s)^{\sigma } \bigr) \biggl(\mathcal{F}\bigl(s,\omega (s)\bigr) \\ &\quad {}+G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr)\Delta s \Biggr] \Biggr\Vert \\ &\leq \sum_{j=1}^{i}{L}_{\Xi } \Vert \omega \Vert _{\infty }+\sum_{j=1}^{i}{L}_{ \Phi } \Vert \omega \Vert _{\infty }+a_{1}+ \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert a_{2} \\ &\quad {}\times \bigl({L}_{\mathcal{F}} \Vert \omega \Vert _{\infty }+{L}_{{G}_{1}} \Vert \omega \Vert _{\infty }+{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}}) \\ &\quad {}+{L}_{{G}_{3}}{L}_{\mathcal{\mathcal{F}}_{2}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}})+ \widehat{M}_{G} \bigr) ({\varsigma _{f}}-{\varsigma _{0}}) \\ &\leq a_{1}+ \Biggl[\sum_{j=1}^{i}{L}_{\Xi }+ \sum_{j=1}^{i}{L}_{\Phi }+ \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert a_{2} \\ &\quad {}\times \bigl({L}_{\mathcal{F}}+{L}_{{G}_{1}}+{L}_{{G}_{2}}{L}_{ \mathcal{\mathcal{F}}_{1}}({s_{f}}-{s_{0}})+{L}_{{G}_{3}}{L}_{ \mathcal{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) ({\varsigma _{f}}-{ \varsigma _{0}}) \Biggr] \Vert \omega \Vert _{\infty } \\ &\leq a_{1}+ [N_{2}+Q_{2} ] \Vert \omega \Vert _{\infty } \\ &\leq a_{1}+Q_{3} \Vert \omega \Vert _{\infty }. \end{aligned}$$

Hence

$$ \frac{ \Vert \omega \Vert _{\infty }}{ a_{1}+Q_{3} \Vert \omega \Vert _{\infty }} \leq 1. $$

Now, from (17), we get \(\mathcal{K}>0\) such that \(\|\omega \|_{\infty }\neq \mathcal{K}\). Let us assume that

$$ \Re =\bigl\{ \omega \in \mathbb{T}, \Vert \omega \Vert _{\infty }< \mathcal{K}\bigr\} . $$

Then the operator \(\Lambda _{\sigma }:\Re \rightarrow \mathbb{T}\) is continuous as well as completely continuous. Thus, from the choice of , there is no \(\omega \in \chi (\Re )\) provided that \(\omega =\beta (\Lambda _{\sigma } ((\omega )(\varsigma ) ) )\), \(\beta \in [0,1]\).

Therefore in the light of fixed point criterion due to nonlinear alternative of Leray–Schauder, \(\Lambda _{\sigma }\) admits a fixed point which is the solution of the mixed impulsive system (1). □

We have a similar conclusion for the mixed impulsive system (2).

Theorem 3.4

The mixed impulsive system (2) admits at least one solution if assumption \((\mathbf{B})\) is satisfied and \(\mathcal{K}^{*}> 0\) exists such that

$$ a_{1}+Q^{*}_{3} \mathcal{K}^{*}< \mathcal{K}^{*}. $$
(20)

Proof

It is similar to the previous argument for \(\Psi _{\sigma }\) in Theorem 3.3. □

Stability analysis

Now, to start this section, we first consider the following inequalities:

$$ \textstyle\begin{cases} \Vert {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )-A( \varsigma )\omega (\varsigma )-{\mathcal{F}}(\varsigma ,\omega ( \varsigma )) \\ \quad {} -{G} (\varsigma ,\omega (\varsigma ),\int _{\varsigma _{0}}^{ \varsigma _{f}} \mathcal{F}_{1}(\varsigma ,s,\omega (s))\Delta s, \int _{\varsigma _{0}}^{\varsigma _{f}} \mathcal{F}_{2}(\varsigma ,s, \omega (s))\Delta s ) \Vert \leq \epsilon ; \quad \varsigma \in { \mathbb{{T}}}', \\ \Vert \omega (\varsigma ^{+}_{k})-\omega (\varsigma ^{-}_{k})- \Xi _{k}(\omega (\varsigma ^{-}_{k}))-\Phi _{k} (\varsigma ^{-}_{k}, \omega (\varsigma ^{-}_{k})) \Vert \leq \epsilon , \quad k = 1, \dots , m, \end{cases} $$
(21)

and

$$ \textstyle\begin{cases} \Vert {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )-A( \varsigma )\omega (\varsigma )-{F}(\varsigma ,\omega (\varsigma )) \\ \quad {} -{\mathcal{G}} (\varsigma ,\omega (\varsigma ),\int _{\varsigma _{0}}^{ \varsigma _{f}} F_{1}(\varsigma ,s,\omega (s))\Delta s,\int _{ \varsigma _{0}}^{\varsigma _{f}} F_{2}(\varsigma ,s,\omega (s)) \Delta s ) \Vert \leq \epsilon , \\ \quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap {\mathbb{T}}, i = 1, \dots , m, \\ \Vert \omega (\varsigma )-\frac{1}{\Gamma (\sigma )}\int _{ \varsigma _{i}}^{\varsigma }(\varsigma -s)^{\sigma -1}\hbar _{i}(s, \omega (s))\Delta s \Vert \leq \epsilon , \quad \varsigma \in ( \varsigma _{i}, s_{i}]\cap {\mathbb{T}}, i = 1,\dots , m, \end{cases} $$
(22)

for each \(\epsilon >0\).

Definition 4.1

The mixed impulsive system (1) is said to be UH stable on \({{\mathbb{T}}}\) if, for any \(\omega \in {P}{C}^{1}({{\mathbb{T}}},\mathbb{R}^{n})\) fulfilling (21), there exists \(\widehat{\omega }\in P{C}^{1}({{\mathbb{T}}},\mathbb{R}^{n})\) as a solution of (1) such that \(\|\widehat{\omega }(s)-\omega (s)\|\leq {C}\epsilon \) for \({C}>0\), \(s\in {{\mathbb{T}}}\).

Definition 4.2

The mixed impulsive system (2) is termed as UH stable if, for any \(\epsilon >0\) and \(\omega \in {P}{C}^{1}({D}, \mathbb{R}^{n})\) that fulfills (22), there exists, \(\widehat{\omega }\in {P}{C}^{1}({D}, \mathbb{R}^{n})\) as a solution of (2) provided \(\|\widehat{\omega }(\varsigma )-\omega (\varsigma )\|\leq {\varrho } \epsilon \) for all \(\varsigma \in D\). Here, \({\varrho }>0\), and its value depends upon ϵ.

Remark 4.1

The solution \(\omega \in P{C}^{1}({{\mathbb{T}}},\mathbb{R}^{n})\) satisfies (21) iff \({f}\in P{C}({{\mathbb{T}}},\mathbb{R}^{n})\) together with the sequence \({f_{k}}\) provided \(\|{f_{k}}\|\leq \epsilon \) so that

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )=A( \varsigma )\omega (\varsigma )+{F}(\varsigma ,\omega (\varsigma )) \\ \hphantom{ {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )={}}{} +{G} (\varsigma ,\omega (\varsigma ),\int _{\varsigma _{0}}^{ \varsigma _{f}} F_{1}(\varsigma ,s,\omega (s))\Delta s,\int _{ \varsigma _{0}}^{\varsigma _{f}} F_{2}(\varsigma ,s,\omega (s)) \Delta s ) +{f}(\varsigma ), \\ \omega (\varsigma _{0})=\omega _{0}, \quad \varsigma \in {{\mathbb{T}}}', \\ \omega (\varsigma _{k}^{+})-\omega (\varsigma _{k}^{-})=\Xi _{k}( \omega (\varsigma ^{-}_{k}))+\Phi _{k} (\varsigma _{k}^{-},\omega ( \varsigma ^{-}_{k}))+{f_{k}}. \end{cases} $$

Lemma 4.3

Each function \(\omega \in P{C}^{1}({{\mathbb{T}}},\mathbb{R}^{n})\) that fulfills (21) also satisfies the following inequality:

$$ \textstyle\begin{cases} \Vert \omega (\varsigma )-E_{\sigma }(A\varsigma ^{\sigma }) \omega _{0}-\sum_{k=1}^{m} (\Xi _{k}(\omega (\varsigma ^{-}_{k}))+ \Phi _{k} (\varsigma ^{-}_{k},\omega (\varsigma ^{-}_{k})) ) \\ \quad {} -\int _{\varsigma _{0}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ){F}(s,\omega (s)) \Delta s \\ \quad {} -\int _{\varsigma _{0}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times {G} (s,\omega (s),\int _{s_{0}}^{s_{f}} F_{1}(s,u,\omega (u)) \Delta u,\int _{s_{0}}^{s_{f}} F_{2}(s,u,\omega (u))\Delta u ) \Delta s \Vert \leq \delta \epsilon \end{cases} $$

for \(\varsigma \in (\varsigma _{k},\varsigma _{k+1}]\subset {{\mathbb{T}}}\), where \(\|E_{\sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\|\leq a_{2}\) and \(\delta = (m+a_{3}a_{2}(\varsigma _{f}-\varsigma _{0}) )\).

Proof

If \(\omega \in P{C}^{1}({{\mathbb{T}}},\mathbb{R}^{n})\) satisfies (21), then via Remark 4.1

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )=A( \varsigma )\omega (\varsigma )+{F}(\varsigma ,\omega (\varsigma )) \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )={}}{} +{G} (\varsigma ,\omega (\varsigma ),\int _{\varsigma _{0}}^{ \varsigma _{f}} F_{1}(\varsigma ,s,\omega (s))\Delta s,\int _{ \varsigma _{0}}^{\varsigma _{f}} F_{2}(\varsigma ,s,\omega (s)) \Delta s )+{f}(\varsigma ), \\ \omega (\varsigma _{0})=\omega _{0},\quad \varsigma \in {{\mathbb{T}}}', \\ \omega (\varsigma _{k}^{+})-\omega (\varsigma _{k}^{-})=\Xi _{k}( \omega (\varsigma ^{-}_{k}))+\Phi _{k} (\varsigma _{k}^{-},\omega ( \varsigma ^{-}_{k}))+{f_{k}}, \quad k = 1,\dots , m, \end{cases} $$

implies

$$\begin{aligned} \omega (\varsigma )&=E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr) \omega _{0}+ \sum_{j=1}^{m} \bigl(\Xi \bigl(\omega \bigl(\varsigma ^{-}_{j}\bigr) \bigr)+\Phi \bigl( \varsigma ^{-}_{j}, \omega \bigl( \varsigma ^{-}_{j}\bigr)\bigr) \bigr)+\sum _{i=1}^{m}{f_{i}} \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr){F}\bigl(s,\omega (s)\bigr) \Delta s \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr) \\ &\quad {}\times {G} \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} F_{1}\bigl(s,u, \omega (u)\bigr) \Delta u, \int _{s_{0}}^{s_{f}} F_{2}\bigl(s,u, \omega (u)\bigr)\Delta u \biggr) \Delta s \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr){f(s)}\Delta s. \end{aligned}$$

So,

$$\begin{aligned} &\Biggl\Vert \omega (\varsigma )-E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr) \omega _{0}-\sum _{j=1}^{m} \bigl(\Xi \bigl(\omega \bigl(\varsigma ^{-}_{j}\bigr)\bigr)+ \Phi \bigl(\varsigma ^{-}_{j}, \omega \bigl(\varsigma ^{-}_{j} \bigr)\bigr)\bigr) \\ &\qquad {}- \int _{\varsigma _{0}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr){F}\bigl(s,\omega (s)\bigr) \Delta s \\ &\qquad {}- \int _{\varsigma _{0}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr) \\ &\qquad {}\times {G} \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} F_{1}\bigl(s,u, \omega (u)\bigr) \Delta u, \int _{\varsigma _{0}}^{\varsigma _{f}} F_{2}\bigl(s,u,\omega (u)\bigr) \Delta u \biggr)\Delta \omega \Biggr\Vert \\ &\quad \leq \int _{\varsigma _{0}}^{\varsigma } \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \bigl\Vert {f}(s) \bigr\Vert \Delta s+\sum_{i=1}^{m} \Vert {f_{i}} \Vert \\ &\quad \leq \delta \epsilon , \end{aligned}$$

and the argument is finished. □

Remark 4.2

The map \(\omega \in {PC}^{1}({D}, \mathbb{R}^{n})\) fulfills inequality (22) iff there are \({f} \in {P}{C}^{1}({D}, \mathbb{R}^{n})\) as a map and bounded sequences \(\{{f_{i}}:i = 1,\dots , m\}\subset \mathbb{R}^{n}\) (depending upon ω) provided that \(\|{f}(\varsigma )\|\leq \epsilon \) for each \(\varsigma \in D\) and \(\|{f_{i}}\|\leq \epsilon \) \(\forall i = 1,\dots , m\) such that

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )=A( \varsigma )\omega (\varsigma )+{F}(\varsigma ,\omega (\varsigma )) \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )={}}{} +\mathcal{G} (\varsigma ,\omega (\varsigma ),\int _{ \varsigma _{0}}^{\varsigma _{f}} F_{1}(\varsigma ,s,\omega (s)) \Delta s,\int _{\varsigma _{0}}^{\varsigma _{f}} F_{2}(\varsigma ,s, \omega (s))\Delta s )+{f}(\varsigma ), \\ \omega (\varsigma _{0})=\omega _{0},\quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap {\mathbb{T}}, i = 1,\dots , m, \\ \omega (\varsigma )=\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{\varsigma } (\varsigma -s)^{\sigma -1}\hbar _{i}(s,\omega (s))\Delta s+{f_{i}},\quad i = 1,\dots , m. \end{cases} $$

Lemma 4.4

Each map \(\omega \in {PC}^{1}({D} ,\mathbb{R}^{n})\) that fulfills (22) also satisfies the inequalities given below:

$$ \textstyle\begin{cases} \Vert \omega (\varsigma )-E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}- \int _{s_{i}}^{\varsigma }E_{\sigma ,\sigma } (A(\varsigma -s)^{ \sigma } ){F}(s,\omega (s))\Delta s \\ \quad {} -\int _{s_{i}}^{\varsigma }E_{\sigma ,\sigma } (A(\varsigma -s)^{ \sigma } )\mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} F_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} F_{2}(s,u,\omega (u)) \Delta u )\Delta s \\ \quad {} -\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{\varsigma }( \varsigma -s)^{\sigma -1}\hbar _{i}(s,\omega (s))\Delta s \Vert \leq (a_{2} \varsigma _{f}-a_{2}s_{i}+m)\epsilon , \\ \quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap {\mathbb{T}}, i = 1, \dots , m, \end{cases} $$

and

$$ \textstyle\begin{cases} \Vert \omega (\varsigma )-\frac{1}{\Gamma (\sigma )} \int _{ \varsigma _{i}}^{\varsigma }(\varsigma -s)^{\sigma -1}\hbar _{i}(s, \omega (s))\Delta s \Vert \leq m\epsilon \quad (\textit{respectively } m \kappa ), \\ \quad \varsigma \in (\varsigma _{i}, s_{i}]\cap {\mathbb{T}}, i = 1,\dots , m, \end{cases} $$

in which \(\|E_{\sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\|\leq a_{2}\).

Proof

If \(\omega \in {P}{C}^{1}({D}, \mathbb{R}^{n})\) satisfies (22), in this case, by virtue of Remark 4.2,

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )=A( \varsigma )\omega (\varsigma )+ {F}(\varsigma ,\omega (\varsigma )) \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )={}}{} +\mathcal{G} (\varsigma ,\omega (\varsigma ),\int _{ \varsigma _{0}}^{\varsigma _{f}} F_{1}(\varsigma ,s,\omega (s)) \Delta s,\int _{\varsigma _{0}}^{\varsigma _{f}} F_{2}(\varsigma ,s, \omega (s))\Delta s ) \\ \hphantom{{{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )={}}{} +{f}(\varsigma ), \omega (\varsigma _{0})=\omega _{0},\quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap {\mathbb{T}}, i = 1,\dots , m, \\ \omega (\varsigma )=\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{\varsigma } (\varsigma -s)^{\sigma -1}\hbar _{i}(s,\omega (s))\Delta s+{f_{i}},\quad s, \varsigma \in (\varsigma _{i}, s_{i}]\cap {\mathbb{T}}, i = 1, \dots , m. \end{cases} $$
(23)

Clearly, equation (23) implies that

$$ \omega (\varsigma )= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{i}}^{ \varsigma }E_{\sigma ,\sigma } (A(\varsigma -s)^{\sigma } ){F}(s, \omega (s))\Delta s \\ \quad {} +\int _{s_{i}}^{\varsigma }E_{\sigma ,\sigma } (A(\varsigma -s)^{ \sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} F_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} F_{2}(s,u,\omega (u)) \Delta u )\Delta s \\ \quad {} +\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{\varsigma }( \varsigma -s)^{\sigma -1}\hbar _{i}(s,\omega (s))\Delta s,\quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap {\mathbb{T}}, i = 1, \dots , m, \\ \frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{\varsigma }( \varsigma -s)^{\sigma -1}\hbar _{i}(s,\omega (s))\Delta s+{f_{i}},\quad \varsigma \in (\varsigma _{i}, s_{i}]\cap {\mathbb{T}}, i = 1, \dots , m. \end{cases} $$

For \(\varsigma \in (s_{i}, \varsigma _{i+1}]\cap {\mathbb{T}}\), \(i = 1, \dots , m\), one has

$$\begin{aligned} &\biggl\Vert \omega (\varsigma )-E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr)\omega _{0}- \int _{s_{i}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr)F\bigl(s,\omega (s)\bigr) \\ &\qquad{} - \int _{s_{i}}^{\varsigma }(\varsigma -s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr) \\ &\qquad {}\times \mathcal{G} \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} F_{1}\bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} F_{2}(s,u,\omega ) \Delta u \biggr)\Delta s \\ &\qquad {}-\frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{\varsigma }( \varsigma -s)^{\sigma -1} \hbar _{i}\bigl(s,\omega (s)\bigr)\Delta s \biggr\Vert \\ &\quad \leq \int _{s_{i}}^{\varsigma } \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{ \sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \bigl\Vert {f(s)} \bigr\Vert \Delta s+ \sum_{i=1}^{m} \Vert {f_{i}} \Vert \\ &\quad \leq \bigl(a_{3}a_{2}(\varsigma -s_{i})+m \bigr)\epsilon . \end{aligned}$$

Using a similar method, we get

$$ \biggl\Vert \omega (\varsigma )-\frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{\varsigma }(\varsigma -s)^{\sigma -1} \hbar _{i}\bigl(s,\omega (s)\bigr)\Delta s \biggr\Vert \leq m \epsilon , \quad \varsigma \in (\varsigma _{i}, s_{i}]\cap { \mathbb{T}}, i = 1,\dots , m, $$

and this ends the argument. □

Now, we provide a sufficient condition for the UH stability of mixed impulsive systems (1) and (2).

Theorem 4.5

The mixed impulsive system (1) is UH stable provided assumption (A) and inequality (9) are satisfied.

Proof

Let ω be the solution of the mixed impulsive system (1) and ω̃ be the solution of inequality (21). Therefore, from Theorem 3.1, we have

$$ \omega (\varsigma )= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s, \\ \quad \varsigma \in (\varsigma _{0},\varsigma _{1}], \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\sum_{j=1}^{i} (\Xi _{j}(\omega (\varsigma ^{-}_{j}))+\Phi _{j} (\varsigma ^{-}_{j},\omega (\varsigma ^{-}_{j})) ), \quad \varsigma \in (\varsigma _{i},\varsigma _{i+1}], i = 1,\dots , m, \end{cases} $$

where \(E_{\sigma }(A\varsigma ^{\sigma })\) stands for the matrix representation of the Mittag-Leffler function. Using a similar approach as that in Theorem 3.1, we get

$$\begin{aligned} \bigl\Vert \widetilde{{\omega }}(\varsigma )-\omega (\varsigma ) \bigr\Vert &\leq \sum_{j=1}^{i} \bigl\| \Xi _{j}\bigl(\widetilde{\omega }\bigl(\varsigma ^{-}_{j} \bigr)\bigr)-\Xi _{j}\bigl({{ \omega }}\bigl(\varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert \\ &\quad {}+\sum_{j=1}^{i}\bigl\| \Phi _{j} \bigl(\varsigma ^{-}_{j}, \widetilde{ \omega }\bigl(\varsigma ^{-}_{j}\bigr)\bigr)-\Phi _{j} \bigl(\varsigma ^{-}_{j},{{ \omega }} \bigl(\varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}\bigl\| (\varsigma -s)^{\sigma -1} \bigr\| \bigl\| E_{\sigma ,\sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr)\bigr\| \biggl\| \biggl( \mathcal{F}\bigl(s,\widetilde{\omega }(s)\bigr) \\ &\quad {}+G \biggl(s,\widetilde{\omega }(s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \widetilde{\omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u, \widetilde{\omega }(u)\bigr)\Delta u \biggr) \biggr) \\ &\quad {}- \biggl(\mathcal{F}\bigl(s,{\omega }(s)\bigr) \\ &\quad {}+G \biggl(s,{\omega }(s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u,{ \omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,{\omega }(u)\bigr) \Delta u \biggr) \biggr)\biggr\| \Delta s \\ &\leq a_{3}a_{2}(\varsigma _{f}-\varsigma _{i})\epsilon +Q_{3} \Vert \widetilde{\omega }-{ \omega } \Vert _{\infty }, \end{aligned}$$

which implies that

$$ \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }\leq a_{3}a_{2}( \varsigma _{f}-\varsigma _{i})\epsilon +Q_{3} \Vert \widetilde{\omega }-{ \omega } \Vert _{\infty }. $$

Hence

$$\begin{aligned} \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }&\leq \bigl(a_{3}a_{2}( \varsigma _{f}-\varsigma _{i}) \bigr)\frac{\varepsilon }{1-Q_{3}} \\ &\leq \mathcal{H}_{(a_{3},a_{2},L_{\mathcal{F}},L_{{G}_{1}},L_{{G}_{2}}L_{{ \mathcal{F}_{1}}},L_{{G}_{3}},L_{{\mathcal{F}}_{2}},L_{\Xi },L_{\Phi })} \epsilon , \end{aligned}$$

where \(\mathcal{H}_{(a_{3},a_{2},L_{\mathcal{F}},L_{{G}_{1}},L_{{G}_{2}}L_{{ \mathcal{F}_{1}}},L_{{G}_{3}},L_{{\mathcal{F}}_{2}},L_{\Xi },L_{\Phi })}= \frac{a_{3}a_{2}(\varsigma _{f}-\varsigma _{i})}{1-Q_{3}}\). Hence, the mixed impulsive system (1) is UH stable. Furthermore, if

$$ \mathcal{\overline{H}}_{(a_{3},a_{2},L_{\mathcal{F}},L_{{G}_{1}},L_{{G}_{2}}L_{{ \mathcal{F}_{1}}},L_{{G}_{3}},L_{{\mathcal{F}}_{2}},L_{\Xi },L_{\Phi })}( \epsilon ) =\mathcal{ \overline{H}}_{(a_{3},a_{2},L_{\mathcal{F}},L_{{G}_{1}},L_{{G}_{2}}L_{{ \mathcal{F}_{1}}},L_{{G}_{3}},L_{{\mathcal{F}}_{2}},L_{\Xi },L_{\Phi })}(0)=0, $$

then our impulsive system (1) becomes generalized UH stable. □

Theorem 4.6

The mixed impulsive system (2) is UH stable provided that assumption (A) and inequality (13) are satisfied.

Proof

Let ω be the solution of the mixed impulsive system (2) and ω̃ be the solution of inequality (22). Therefore, from Theorem 3.2, we have

$$\begin{aligned} \omega (\varsigma )= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1} E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } ){F} (\varsigma ,\omega (\varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} {F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} {F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad \varsigma \in (\varsigma _{i}, s_{i}]\cap {\mathbb{T}}, i = 1, \dots , m, \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } ) {F} (\varsigma ,\omega (\varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} {F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} {F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{s_{i}}(\varsigma -s)^{ \sigma -1}\hbar _{i}(s,\omega (s))\Delta s, \quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap {\mathbb{T}}, i = 1,\dots , m, \end{cases}\displaystyle \end{aligned}$$

where \(E_{\sigma }(A\varsigma ^{\sigma })\) stands for the matrix representation of the Mittag-Leffler function. Using the similar approach as in Theorem 3.2, we get

$$\begin{aligned} \bigl\Vert \widetilde{{\omega }}(\varsigma )-\omega (\varsigma ) \bigr\Vert &\leq \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}} \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert \bigl(\hbar _{i} \bigl(s, \widetilde{\omega }(s)\bigr)-\hbar _{i}\bigl(s,\widetilde{ \omega }(s)\bigr) \bigr) \bigr\Vert \Delta s \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl({F}\bigl(s, \widetilde{\omega }(s)\bigr) \\ &\quad {}+\mathcal{G} \biggl(s,\widetilde{\omega }(s), \int _{s_{0}}^{s_{f}} {F}_{1}\bigl(s,u, \widetilde{\omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} {F}_{2}\bigl(s,u, \widetilde{\omega }(u)\bigr)\Delta u \biggr) \biggr) \\ &\quad {}- \biggl({F}\bigl(s,{\omega }(s)\bigr) \\ &\quad {}+\mathcal{G} \biggl(s,{\omega }(s), \int _{s_{0}}^{s_{f}} {F}_{1}\bigl(s,u,{ \omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} {F}_{2}\bigl(s,u,{ \omega }(u)\bigr) \Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ &\leq a_{3}a_{2}(\varsigma _{f}-\varsigma _{i})\epsilon +Q_{3} \Vert \widetilde{\omega }-{ \omega } \Vert _{\infty }, \end{aligned}$$

which implies

$$ \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }\leq a_{3}a_{2}( \varsigma _{f}-\varsigma _{i})\epsilon +Q^{*}_{3} \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }. $$

Hence

$$\begin{aligned} \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }&\leq \bigl(a_{3}a_{2}( \varsigma _{f}-\varsigma _{i}) \bigr)\frac{\varepsilon }{1-Q^{*}_{3}} \\ &\leq \mathcal{H}_{(a_{3},a_{2},L_{{F}},L_{{\mathcal{G}}_{1}},L_{{ \mathcal{G}}_{2}}L_{{{F}_{1}}},L_{{\mathcal{G}}_{3}},L_{{{F}}_{2}},L_{g})} \epsilon , \end{aligned}$$

where \(\mathcal{H}_{(a_{3},a_{2},L_{{F}},L_{{\mathcal{G}}_{1}},L_{{ \mathcal{G}}_{2}}L_{{{F}_{1}}}, L_{{\mathcal{G}}_{3}},L_{{{F}}_{2}},L_{g})}= \frac{a_{3}a_{2}(\varsigma _{f}-\varsigma _{i})}{1-Q^{*}_{3}}\). Hence, the mixed impulsive system (2) is UH stable. Furthermore, if we take

$$ \mathcal{\overline{H}}_{(a_{3},a_{2},L_{{F}},L_{{\mathcal{G}}_{1}},L_{{ \mathcal{G}}_{2}}L_{{{F}_{1}}},L_{{\mathcal{G}}_{3}},L_{{{F}}_{2}},L_{g})}( \epsilon ) =\mathcal{ \overline{H}}_{(a_{3},a_{2},L_{{F}},L_{{ \mathcal{G}}_{1}},L_{{\mathcal{G}}_{2}}L_{{{F}_{1}}},L_{{\mathcal{G}}_{3}},L_{{{F}}_{2}},L_{g})}(0)=0, $$

then the mentioned system (2) is generalized UH stable. □

Controllability analysis

In the sequel, controllability analysis of given impulsive systems is conducted. At first, we have some definitions in this direction.

Definition 5.1

The function \(\omega \in \mathbb{T}\) is said to be the solution of (3) if ω satisfies \(\omega (0)=\omega _{0}\) and ω is the solution of the following integral equations:

$$ \omega (\varsigma )= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{H}\zeta ( \varsigma )\Delta s,\quad \varsigma \in (\varsigma _{0},\varsigma _{1}], \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{H}\zeta ( \varsigma )\Delta s \\ \quad {} +\sum_{j=1}^{i} (\Xi _{j}(\omega (\varsigma ^{-}_{j}))+\Phi _{j} (\varsigma ^{-}_{j},\omega (\varsigma ^{-}_{j})) ), \quad \varsigma \in (\varsigma _{i},\varsigma _{i+1}], i = 1,\dots , m, \end{cases} $$
(24)

where \(E_{\sigma }(A\varsigma ^{\sigma })\) stands for the matrix representation of the Mittag-Leffler function.

Definition 5.2

The mixed impulsive system (3) is controllable on \(\mathbb{T}\) if, for every \(\omega _{0},\omega _{T}\in \mathbb{T}\) where \(\varsigma _{i+1}=T\), there exists an rd-continuous function \(\zeta \in \mathcal{L}^{2}(I, \mathbb{R})\) such that the corresponding solution of (3) satisfies \(\omega (0)=\omega _{0}\) and \(\zeta (T)=\zeta _{T}\).

We set the following for simplicity:

$$\begin{aligned}& \begin{aligned} Q_{5}&:=a_{3} \bigl(a_{2}{L}_{\mathcal{F}}+ a_{2}{L}_{{G}_{1}}+ a_{2}{L}_{{G}_{2}}{L}_{ \mathcal{\mathcal{F}}_{1}}({s_{f}}-{s_{0}}) + a_{2}{L}_{{G}_{3}}{L}_{ \mathcal{\mathcal{F}}_{2}}({s_{f}}-{s_{0}})+ \widehat{M}_{G} \bigr) \\ &\quad {}\times (\varsigma _{f}-\varsigma _{0}) \bigl(1+M_{\mathcal{H}}M_{ \mathcal{W}}^{\sigma }(\varsigma _{f}-\varsigma _{0}) \bigr); \end{aligned} \\& Q_{6}:=a_{3} \bigl( a_{2}{L}_{F}+ a_{2}{L}_{\mathcal{G}}+ a_{2}{L}_{ \mathcal{G}}{L}_{F_{1}}({s_{f}}-{s_{0}}) + a_{2}{L}_{\mathcal{G}}{L}_{F_{2}}({s_{f}}-{s_{0}}) \bigr)\times ({\varsigma _{f}}-{\varsigma _{0}}); \\& \begin{aligned} Q_{7} &:= \bigl(1+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma }({ \varsigma _{f}}-{ \varsigma _{0}}) \bigr) \Biggl[\sum _{j=1}^{i}L_{\Xi }+\sum _{j=1}^{i}L_{ \Phi } \\ &\quad {}+ \bigl(a_{2}L_{\mathcal{F}}+a_{2}L_{{G}_{1}}+a_{2}L_{{G}_{2}}L_{{ \mathcal{F}_{1}}}({s_{f}}-{s_{0}})+ a_{2}L_{{G}_{3}}L_{{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) ({\varsigma _{f}}-{\varsigma _{0}}) \Biggr]; \end{aligned} \\& \begin{aligned} Q_{8}&:= \bigl(1+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma }( \varsigma - \varsigma _{f}) \bigr) \biggl[\frac{1}{\Gamma (\sigma )}a_{3}L_{g}({s_{i}}-{ \varsigma _{i}}) \\ &\quad {}+a_{3} \bigl( a_{2}{L}_{F}+ a_{2}{L}_{\mathcal{G}}+ a_{2}{L}_{ \mathcal{G}}{L}_{F_{1}}({s_{f}}-{s_{0}}) + a_{2}{L}_{\mathcal{G}}{L}_{F_{2}}({s_{f}}-{s_{0}}) \bigr)\times ({\varsigma _{f}}-{\varsigma _{0}}) \biggr]; \end{aligned} \\& N_{5}:= \bigl(1+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma } \bigr) \Biggl( \sum_{j=1}^{i}{L}_{\Xi } \delta +\sum_{j=1}^{i}{L}_{\Phi } \delta +a_{1} \Biggr)+M_{\mathcal{H}}M_{\mathcal{H}}^{\sigma } \Vert \omega _{T} \Vert ; \\& N_{6}:= \bigl(1+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma } \bigr) \biggl( \frac{1}{\Gamma (\sigma )}a_{3}L_{g}\delta ^{\prime \prime }({s_{i}}-{\varsigma _{i}})+a_{1}+ \widehat{M}_{G} \biggr)+M_{\mathcal{H}}M_{\mathcal{H}}^{\sigma } \Vert \omega _{T} \Vert . \end{aligned}$$

Lemma 5.3

If assertions \((\mathbf{A})\) and \((\mathbf{W})\) hold and \(\omega _{T}\in \mathbb{T}\), where \(\varsigma _{i+1}=T\) is any arbitrary point, then ω is a solution of (3) on \(\mathbb{T}\) defined by (24) along with the control function

$$ \zeta (\varsigma )= \textstyle\begin{cases} ({{}^{\sigma }}{\mathcal{W}^{T}_{\varsigma _{0}}})^{-1} [\omega _{T}-E_{ \sigma }(A\varsigma ^{\sigma })\omega _{0}-\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} -\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s ] \\ \quad \varsigma \in (\varsigma _{0},\varsigma _{1}], \\ ({{}^{\sigma }}{\mathcal{W}^{T}_{\varsigma _{0}}})^{-1} [\omega _{T}-E_{ \sigma }(A\varsigma ^{\sigma })\omega _{0} \\ \quad {} -\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{F} ( \varsigma ,\omega (\varsigma ) )\Delta s \\ \quad {} -\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} -\sum_{j=1}^{i} (\Xi _{j}(\omega (\varsigma ^{-}_{j}))-\Phi _{j} (\varsigma ^{-}_{j},\omega (\varsigma ^{-}_{j})) ) ],\quad \varsigma \in (\varsigma _{i},\varsigma _{i+1}], i = 1,\dots , m, \end{cases} $$
(25)

where \(E_{\sigma }(A\varsigma ^{\sigma })\) stands for the matrix representation of the Mittag-Leffler function, and \(\omega (T)=\omega _{T}\) holds in which \(\varsigma _{i+1}=T\). Also, the control function \(\zeta (\varsigma )\) has the estimate \(\|\zeta (\varsigma )\|\leq \aleph _{\zeta }\), where for \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), we define

$$\begin{aligned} \aleph _{\zeta }&=\aleph _{\mathcal{W}}^{\sigma } \Biggl[ \Vert \omega _{T} \Vert + \bigl\Vert E_{\sigma }\bigl(A \varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert +\sum _{j=1}^{i}{L}_{ \Xi } \Vert \omega \Vert _{\infty } +\sum_{j=1}^{i}{L}_{\Phi } \Vert \omega \Vert _{ \infty } \\ &\quad {}+ \bigl(a_{3}a_{2} \bigl({L}_{\mathcal{F}} \Vert \omega \Vert _{\infty }+{L}_{{G}_{1}} \Vert \omega \Vert _{\infty } +{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}}) \\ &\quad {}+{L}_{{G}_{3}}{L}_{\mathcal{\mathcal{F}}_{2}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}})+ \widehat{M}_{G} \bigr) ({\varsigma _{f}}-{\varsigma _{0}}) \bigr) \Biggr]. \end{aligned}$$

Proof

Let ω be the solution of (3) on \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), defined by (24). Then, for \(\varsigma =T\), we have

$$\begin{aligned} \omega (T)&=E_{\sigma }\bigl(AT^{\sigma }\bigr)\omega _{0}+ \int _{\varsigma _{0}}^{ \varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(T-s)^{\sigma } \bigr)\mathcal{F} \bigl(s,\omega (s) \bigr)\Delta s \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(T-s)^{\sigma } \bigr) \\ &\quad {}\times G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \Delta s \\ &\quad {}+\sum_{j=1}^{i} \bigl(\Xi _{j}\bigl(\omega \bigl(T^{-}_{j}\bigr) \bigr)+\Phi _{j} \bigl(T^{-}_{j}, \omega \bigl(T^{-}_{j}\bigr)\bigr) \bigr) \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr)\mathcal{H}\zeta (\varsigma ) \Delta s \\ &=E_{\sigma }\bigl(AT^{\sigma }\bigr)\omega _{0}+ \int _{\varsigma _{0}}^{ \varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(T-s)^{\sigma } \bigr)\mathcal{F} \bigl(s,\omega (s) \bigr)\Delta s \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(T-s)^{\sigma } \bigr) \\ &\quad {}\times G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \Delta s \\ &\quad {}+\sum_{j=1}^{i} \bigl(\Xi _{j}\bigl(\omega \bigl(T^{-}_{j}\bigr) \bigr)+\Phi _{j} \bigl(T^{-}_{j}, \omega \bigl(T^{-}_{j}\bigr)\bigr) \bigr) \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(\varsigma -s)^{\sigma } \bigr)\mathcal{H} \\ &\quad {}\times \bigl({{}^{\sigma }} {\mathcal{W}^{T}_{\varsigma _{0}}} \bigr)^{-1} \Biggl[ \omega _{T}-E_{\sigma } \bigl(AT^{\sigma }\bigr)\omega _{0} \\ &\quad {}- \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(T-s)^{\sigma } \bigr)\mathcal{F} \bigl(s,\omega (s) \bigr) \Delta s \\ &\quad {}- \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(T-s)^{\sigma } \bigr) \\ &\quad {}\times G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \Delta s \\ &\quad {}-\sum_{j=1}^{i} \bigl(\Xi _{j}\bigl(\omega \bigl(T^{-}_{j}\bigr) \bigr)-\Phi _{j} \bigl(T^{-}_{j}, \omega \bigl(T^{-}_{j}\bigr)\bigr) \bigr) \Biggr]\Delta \tau \\ &=E_{\sigma }\bigl(AT^{\sigma }\bigr)\omega _{0}+ \int _{\varsigma _{0}}^{ \varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma ,\sigma } \bigl(A(T-s)^{\sigma } \bigr)\mathcal{F} \bigl(s,\omega (s) \bigr)\Delta s \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(T-s)^{\sigma } \bigr) \\ &\quad {}\times G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \Delta s \\ &\quad {}+\sum_{j=1}^{i} \bigl(\Xi _{j}\bigl(\omega \bigl(T^{-}_{j}\bigr) \bigr)+\Phi _{j} \bigl(T^{-}_{j}, \omega \bigl(T^{-}_{j}\bigr)\bigr) \bigr) \\ &\quad {}\times \bigl({{}^{\sigma }} {\mathcal{W}^{T}_{\varsigma _{0}}} \bigr) \bigl({{}^{\sigma }} { \mathcal{W}^{T}_{\varsigma _{0}}} \bigr)^{-1} \Biggl[\omega _{T}-E_{\sigma } \bigl(AT^{ \sigma }\bigr)\omega _{0} \\ &\quad {}- \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(T-s)^{\sigma } \bigr)\mathcal{F} \bigl(s,\omega (s) \bigr) \Delta s \\ &\quad {}- \int _{\varsigma _{0}}^{\varsigma _{f}}(T-s)^{\sigma -1}E_{\sigma , \sigma } \bigl(A(T-s)^{\sigma } \bigr) \\ &\quad {}\times G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \Delta s \\ &\quad {}-\sum_{j=1}^{i} \bigl(\Xi _{j}\bigl(\omega \bigl(T^{-}_{j}\bigr) \bigr)-\Phi _{j} \bigl(T^{-}_{j}, \omega \bigl(T^{-}_{j}\bigr)\bigr) \bigr) \Biggr] \\ &=\omega _{T}. \end{aligned}$$

Also, for \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), the estimation

$$\begin{aligned} \bigl\Vert \zeta (\varsigma ) \bigr\Vert &=\aleph _{\mathcal{W}}^{\sigma } \Biggl[ \Vert \omega _{T} \Vert - \bigl\Vert E_{\sigma } \bigl(A\varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}- \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \bigl\Vert \mathcal{F} \bigl( \varsigma ,\omega (\varsigma ) \bigr) \bigr\Vert \Delta s \\ &\quad {}- \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \\ &\quad {}\times \biggl\Vert G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr\Vert \Delta s \\ &\quad {}-\sum_{j=1}^{i} \bigl( \bigl\Vert \Xi _{j}\bigl(\omega \bigl(\varsigma ^{-}_{j} \bigr)\bigr) \bigr\Vert + \bigl\Vert \Phi _{j} \bigl(\varsigma ^{-}_{j},\omega \bigl(\varsigma ^{-}_{j} \bigr)\bigr) \bigr\Vert \bigr) \Biggr] \end{aligned}$$

implies that

$$\begin{aligned} \bigl\Vert \zeta (\varsigma ) \bigr\Vert &=\aleph _{\mathcal{W}}^{\sigma } \Biggl[ \Vert \omega _{T} \Vert + \bigl\Vert E_{\sigma } \bigl(A\varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert +\sum _{j=1}^{i}{L}_{ \Xi } \Vert \omega \Vert _{\infty } +\sum_{j=1}^{i}{L}_{\Phi } \Vert \omega \Vert _{ \infty } \\ &\quad {}+ \bigl(a_{3}a_{2} \bigl({L}_{\mathcal{F}} \Vert \omega \Vert _{\infty }+{L}_{{G}_{1}} \Vert \omega \Vert _{\infty } +{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}}) \\ &\quad {}+{L}_{{G}_{3}}{L}_{\mathcal{\mathcal{F}}_{2}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}})+ \widehat{M}_{G} \bigr) ({\varsigma _{f}}-{\varsigma _{0}}) \bigr) \Biggr] \\ &=\aleph _{\zeta }, \end{aligned}$$

and the argument is completed. □

Definition 5.4

The function \(\omega \in \mathbb{T}\) is said to be the solution of the mixed impulsive system (4) if ω satisfies \(\omega (0)=\omega _{0}\) and ω is the solution of the following integral equations:

$$ \omega (\varsigma )= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } ){F} (\varsigma ,\omega (\varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} {F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} {F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{H}\zeta ( \varsigma )\Delta s, \\ \quad \varsigma \in (\varsigma _{i}, s_{i}]\cap \mathbb{{T}}, i = 1,\dots , m, \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u,\omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u))\Delta u ) \Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{H}\zeta ( \varsigma )\Delta s \\ \quad {} +\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{s_{i}}(\varsigma -s)^{ \sigma -1}\hbar _{i}(s,\omega (s))\Delta s,\quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}, i = 1,\dots , m, \end{cases} $$
(26)

where \(E_{\sigma }(A\varsigma ^{\sigma })\) stands for the matrix representation of the Mittag-Leffler function.

Definition 5.5

The mixed impulsive system (4) is named as a controllable system on \(\mathbb{T}\) if, for every \(\omega _{0},\omega _{T}\in \mathbb{T}\), where \(\varsigma _{i+1}=T\), there exists an rd-continuous function \(\zeta \in \mathcal{L}^{2}(I,\mathbb{R})\) such that the corresponding solution of (4) satisfies \(\omega (0)=\omega _{0}\) and \(\omega (T)=\omega _{T}\).

Lemma 5.6

If assertions \((\mathbf{A})\) and \((\mathbf{W})\) hold and \(\omega _{T}\in \mathbb{T}\), where \(\varsigma _{i+1}=T\) is any arbitrary point, then ω is the solution of (4) on \(\varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}\), \(i = 1, \dots , m\), defined by (26) along with the control function

$$ \zeta (\varsigma )= \textstyle\begin{cases} ({{}^{\sigma }}{\mathcal{W}^{T}_{\varsigma _{0}}})^{-1} [\omega _{T}-E_{ \sigma }(A\varsigma ^{\sigma })\omega _{0} -\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )F (\varsigma ,\omega (\varsigma ) ) \Delta s \\ \quad {}-\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {}\times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} F_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} F_{2}(s,u,\omega (u)) \Delta u ) \Delta s ] \\ \quad \varsigma \in (\varsigma _{i}, s_{i}]\cap \mathbb{{T}}, i = 1, \dots , m, \\ ({{}^{\sigma }}{\mathcal{W}^{T}_{\varsigma _{0}}})^{-1} [\omega _{T}-E_{ \sigma }(A\varsigma ^{\sigma })\omega _{0}-\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )F (\varsigma ,\omega (\varsigma ) ) \Delta s \\ \quad {}-\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {}\times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} F_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} F_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {}-\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{s_{i}}(\varsigma -s)^{ \sigma -1}\hbar _{i}(s,\omega (s))\Delta s, \quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}, i = 1,\dots , m, \end{cases} $$
(27)

where \(E_{\sigma }(A\varsigma ^{\sigma })\) stands for the matrix representation of the Mittag-Leffler function and \(\omega (T)=\omega _{T} \) holds, where \(\varsigma _{i+1}=T\). Also, the control function \(\zeta (\varsigma )\) has the estimate \(\|\zeta (\varsigma )\|\leq \aleph ^{*}_{\zeta }\), where for \(\varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}\), \(i = 1, \dots , m\), we define

$$\begin{aligned} \aleph ^{*}_{\zeta }&=\aleph _{\mathcal{W}}^{\sigma } \biggl[ \Vert \omega _{T} \Vert + \bigl\Vert E_{\sigma } \bigl(A\varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert + \frac{1}{\Gamma (\sigma )}({s_{i}}-{\varsigma _{i}})a_{1}L_{g} \Vert \omega \Vert _{\infty } \\ &\quad {}+ \bigl(a_{3}a_{2} \bigl({L}_{F} \Vert \omega \Vert _{\infty }+{L}_{{ \mathcal{G}}_{1}} \Vert \omega \Vert _{\infty } +{L}_{{\mathcal{G}}_{2}}{L}_{F_{1}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}}) \\ &\quad {}+{L}_{{\mathcal{G}}_{3}}{L}_{F_{2}} \Vert \omega \Vert _{\infty }({s_{f}}-{s_{0}}) + \widehat{M}_{\mathcal{G}} \bigr) ({\varsigma _{f}}-{\varsigma _{0}}) \bigr) \biggr]. \end{aligned}$$

Proof

The proof is similar to that of Lemma 5.3. □

Theorem 5.7

The mixed impulsive system (3) is controllable on \(\mathbb{T}\) such that hypotheses \(\mathbf{(A)}\) and \(\mathbf{(W)}\) are satisfied and the following inequality holds:

$$ \max_{5\leq i\leq 7}\{Q_{i}\}< {1}. $$
(28)

Proof

Let \(\Omega ^{\prime \prime }\subseteq \mathrm{PS}\), provided that \(\Omega ^{\prime \prime }=\{(\mathbb{X},\mathbb{Y},\mathbb{Z})\in \mathrm{PS}:\|( \mathbb{X},\mathbb{Y},\mathbb{Z})\|_{C}\leq \delta ^{\prime \prime }_{2}\}\), where \(\delta ^{\prime \prime }_{2}=\max \{\delta ^{\prime \prime },\delta ^{\prime \prime }_{1}\}\) such that \(\delta {''},\delta ^{\prime \prime }_{1}\in (0,1)\), and also \(\delta ^{\prime \prime }>N_{5}\). Now, we define \(\Lambda _{\sigma }^{\prime \prime }:\Omega ^{\prime \prime }\rightarrow \Omega ^{\prime \prime }\) as

$$ \Lambda _{\sigma }''\bigl(\omega (\varsigma )\bigr)= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{H}\zeta ( \varsigma )\Delta s,\quad \varsigma \in (\varsigma _{0},\varsigma _{1}], \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0}+\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } )\mathcal{F} (\varsigma ,\omega ( \varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times G (s,\omega (s),\int _{s_{0}}^{s_{f}} \mathcal{F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} \mathcal{F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{H}\zeta (s) \Delta s \\ \quad {} +\sum_{j=1}^{i} (\Xi _{j}(\omega (\varsigma ^{-}_{j}))+\Phi _{j} (\varsigma ^{-}_{j},\omega (\varsigma ^{-}_{j})) ),\quad \varsigma \in (\varsigma _{i},\varsigma _{i+1}], i = 1,\dots , m. \end{cases} $$
(29)

Now, we prove \(\Lambda _{\sigma }^{\prime \prime }:\Omega ^{\prime \prime }\rightarrow \Omega ^{\prime \prime }\) is a self-mapping.

For \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), we get

$$\begin{aligned} \bigl\Vert \Lambda ^{\prime \prime }_{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert &\leq \sum_{j=1}^{i} \bigl\Vert \Xi _{j}\bigl(\omega \bigl(\varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert +\sum _{j=1}^{i} \bigl\Vert \Phi _{j} \bigl(\varsigma ^{-}_{j},\omega \bigl(\varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert + \bigl\Vert E_{\sigma }\bigl(A\varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl(\mathcal{F} \bigl(s,\omega (s)\bigr) \\ &\quad {}+G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \Vert \mathcal{H} \Vert \bigl\Vert \zeta (s) \bigr\Vert \Delta s \\ &\leq \sum_{j=1}^{i}{L}_{\Xi } \delta ^{\prime \prime }+\sum_{j=1}^{i}{L}_{\Phi } \delta ^{\prime \prime }+a_{1}+\delta ^{\prime \prime } a_{3} \bigl(a_{2}{L}_{\mathcal{F}}+ a_{2}{L}_{{G}_{1}} \\ &\quad {}+ a_{2}{L}_{{G}_{2}}{L}_{\mathcal{\mathcal{F}}_{1}}({s_{f}}-{s_{0}}) + a_{2}{L}_{{G}_{3}}{L}_{\mathcal{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) (\varsigma _{f}-\varsigma _{0}) \\ &\quad {}+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma } \Biggl[\sum _{j=1}^{i}{L}_{ \Xi }\delta ^{\prime \prime }+\sum_{j=1}^{i}{L}_{\Phi } \delta ^{\prime \prime }+ \Vert \omega \Vert _{T}+a_{1} \\ &\quad {}+\delta ^{\prime \prime } a_{3} \bigl(a_{2}{L}_{\mathcal{F}}+ a_{2}{L}_{{G}_{1}}+ a_{2}{L}_{{G}_{2}}{L}_{ \mathcal{\mathcal{F}}_{1}}({s_{f}}-{s_{0}}) \\ &\quad {}+ a_{2}{L}_{{G}_{3}}{L}_{\mathcal{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) (\varsigma _{f}-\varsigma _{0}) \Biggr] \\ &\leq N_{5}+\delta ^{\prime \prime }Q_{5} \\ &\leq \delta '' + \delta '' Q_{5} = \delta ''_{1}. \end{aligned}$$

Hence,

$$ \bigl\Vert \Lambda ^{\prime \prime }_{\sigma } \bigl( \omega (\varsigma ) \bigr) \bigr\Vert \leq \delta ''_{2}. $$
(30)

Therefore, from (30), \(\Lambda _{\sigma }^{\prime \prime } (\Omega ^{\prime \prime }) \subseteq \Omega ^{\prime \prime }\), also when \(\varsigma \in (\varsigma _{i},\varsigma _{i+1}]\), \(i = 1,\dots , m\), with \(\omega _{0}=\widehat{\omega }_{0}\), we have

$$\begin{aligned} &\bigl\Vert \Lambda ^{\prime \prime }_{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Lambda ^{\prime \prime }_{ \sigma } \bigl(\widehat{\omega }( \varsigma ) \bigr) \bigr\Vert \\ &\quad \leq \sum_{j=1}^{i} \bigl\Vert \Xi _{j}\bigl(\omega \bigl(\varsigma ^{-}_{j}\bigr)\bigr)-\Xi _{j}\bigl( \widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert \\ &\qquad {}+\sum_{j=1}^{i} \bigl\Vert \Phi _{j} \bigl(\varsigma ^{-}_{j},\omega \bigl( \varsigma ^{-}_{j}\bigr)\bigr)-\Phi _{j} \bigl(\varsigma ^{-}_{j},\widehat{{\omega }}\bigl( \varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl(\mathcal{F} \bigl(s,\omega (s)\bigr) \\ &\qquad {}+G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr) \\ &\qquad {}- \biggl(\mathcal{F}\bigl(s,\widehat{\omega }(s)\bigr) \\ &\qquad {}+G \biggl(s,\widehat{\omega }(s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigl\Vert \| \mathcal{H} \Vert \bigr\Vert \bigl({{}^{\sigma }} {\mathcal{W}^{T}_{\varsigma _{0}}} \bigr)^{-1}\bigr\| ( \varsigma -\varsigma _{f}) \\ &\qquad {}\times \Biggl[\sum_{j=1}^{i} \bigl\Vert \Xi _{j}\bigl(\omega \bigl(\varsigma ^{-}_{j} \bigr)\bigr)- \Xi _{j}\bigl(\widehat{{\omega }}\bigl(\varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert \\ &\qquad {}+\sum_{j=1}^{i} \bigl\Vert \Phi _{j} \bigl(\varsigma ^{-}_{j},\omega \bigl( \varsigma ^{-}_{j}\bigr)\bigr)-\Phi _{j} \bigl(\varsigma ^{-}_{j},\widehat{{\omega }}\bigl( \varsigma ^{-}_{j}\bigr)\bigr) \bigr\Vert \\ &\qquad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl(\mathcal{F} \bigl(s,\omega (s)\bigr) \\ &\qquad {}+G \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u,\omega (u)\bigr) \Delta u \biggr) \biggr) \\ &\qquad {}- \biggl(\mathcal{F}\bigl(s,\widehat{\omega }(s)\bigr) \\ &\qquad {}+G \biggl(s,\widehat{\omega }(s), \int _{s_{0}}^{s_{f}} \mathcal{F}_{1} \bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} \mathcal{F}_{2} \bigl(s,u, \widehat{\omega }(u)\bigr)\Delta u \biggr) \biggr) \biggr\Vert \Delta s \Biggr]\Delta \tau \\ &\quad \leq \Biggl[\sum_{j=1}^{i}L_{\Xi }+ \sum_{j=1}^{i}L_{\Phi }+ \bigl(a_{2}L_{ \mathcal{F}}+a_{2}L_{{G}_{1}}+a_{2}L_{{G}_{2}}L_{{\mathcal{F}_{1}}}({s_{f}}-{s_{0}}) \\ &\qquad {}+a_{2}L_{{G}_{3}}L_{{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) ({ \varsigma _{f}}-{\varsigma _{0}}) \Biggr] \times \Vert \widetilde{\omega }-{\omega } \Vert _{\infty } \\ &\qquad {}+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma }({\varsigma _{f}}-{ \varsigma _{0}}) \Biggl[\sum _{j=1}^{i}L_{\Xi }+\sum _{j=1}^{i}L_{\Phi } \\ &\qquad {}+ \bigl(a_{2}L_{\mathcal{F}}+a_{2}L_{{G}_{1}}+a_{2}L_{{G}_{2}}L_{{ \mathcal{F}_{1}}}({s_{f}}-{s_{0}}) \\ &\qquad {} +a_{2}L_{{G}_{3}}L_{{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) ({ \varsigma _{f}}-{\varsigma _{0}}) \Biggr] \times \Vert \widetilde{\omega }-{\omega } \Vert _{\infty } \\ &\quad \leq \bigl(1+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma }({\varsigma _{f}}-{ \varsigma _{0}}) \bigr) \Biggl[\sum _{j=1}^{i}L_{\Xi }+\sum _{j=1}^{i}L_{ \Phi } \\ &\qquad {}+ \bigl(a_{2}L_{\mathcal{F}}+a_{2}L_{{G}_{1}}+a_{2}L_{{G}_{2}}L_{{ \mathcal{F}_{1}}}({s_{f}}-{s_{0}}) \\ &\qquad {}+ a_{2}L_{{G}_{3}}L_{{\mathcal{F}}_{2}}({s_{f}}-{s_{0}}) \bigr) ({ \varsigma _{f}}-{\varsigma _{0}}) \Biggr] \times \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }. \end{aligned}$$

Hence

$$ \bigl\Vert \Lambda ^{\prime \prime }_{\sigma } \bigl( \omega (\varsigma ) \bigr)-\Lambda ^{\prime \prime }_{ \sigma } \bigl( \widehat{\omega }(\varsigma ) \bigr) \bigr\Vert \leq Q_{7} \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }. $$
(31)

Therefore, the operator \(\Lambda _{\sigma }^{\prime \prime }\) is strictly contractive. Therefore, by using the Banach fixed point theorem method, \(\Lambda _{\sigma }^{\prime \prime }\) has only one fixed point, i.e., the mixed impulsive system (3) has a unique solution. Also, using Lemma 5.3, we conclude that \(\omega (\varsigma )\) fulfills \(\omega (T)=\omega _{T}\). Consequently, we conclude that the mixed impulsive system (3) is controllable. □

One can indicate a similar theorem for the mixed impulsive system (4).

Theorem 5.8

The mixed impulsive system (4) is controllable on \(\mathbb{T}\) such that hypotheses \(\mathbf{(A)}\) and \(\mathbf{(W)}\) are satisfied and the following inequality holds:

$$ \max \{Q_{i}\}< 1\quad \textit{where }i=6,8. $$
(32)

Proof

Let \(\Omega ^{\prime \prime }\subseteq \mathrm{PS}\), provided that \(\Omega ^{\prime \prime }=\{(\mathbb{X},\mathbb{Y},\mathbb{Z})\in \mathrm{PS}:\|( \mathbb{X},\mathbb{Y},\mathbb{Z})\|_{C}\leq \delta ^{\prime \prime }_{2}\}\), where \(\delta ^{\prime \prime }_{2}=\max \{\delta ^{\prime \prime },\delta ^{\prime \prime }_{1}\}\) such that \(\delta {''},\delta ^{\prime \prime }_{1}\in (0,1)\), and also \(\delta ^{\prime \prime }>\{N_{6}\}\). Now, we define \(\Lambda _{\sigma }^{**}:\Omega ^{**}\rightarrow \Omega ^{**}\) as

$$ \Lambda ^{**}_{\sigma } \bigl(\omega ( \varsigma ) \bigr)= \textstyle\begin{cases} E_{\sigma }(A\varsigma ^{\sigma })\omega _{0} +\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } ){F} (\varsigma ,\omega (\varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} {F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} {F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s, \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{H}\zeta (s) \Delta s \\ \quad \varsigma \in (\varsigma _{i}, s_{i}]\cap \mathbb{{T}}, i = 1, \dots , m, \\ E_{\sigma }(A\varsigma ^{\sigma })\omega _{0} +\int _{\varsigma _{0}}^{ \varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{\sigma ,\sigma } (A( \varsigma -s)^{\sigma } ){F} (\varsigma ,\omega (\varsigma ) )\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } ) \\ \quad {} \times \mathcal{G} (s,\omega (s),\int _{s_{0}}^{s_{f}} {F}_{1}(s,u, \omega (u))\Delta u,\int _{s_{0}}^{s_{f}} {F}_{2}(s,u,\omega (u)) \Delta u ) \Delta s \\ \quad {} +\frac{1}{\Gamma (\sigma )}\int _{\varsigma _{i}}^{s_{i}}(\varsigma -s)^{ \sigma -1}\hbar _{i}(s,\omega (s))\Delta s \\ \quad {} +\int _{\varsigma _{0}}^{\varsigma _{f}}(\varsigma -s)^{\sigma -1}E_{ \sigma ,\sigma } (A(\varsigma -s)^{\sigma } )\mathcal{H}\zeta (s) \Delta s, \\ \quad \varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}, i = 1, \dots , m, \end{cases} $$
(33)

where \(E_{\sigma }(A\varsigma ^{\sigma })\) stands for the matrix representation of the Mittag-Leffler function. Now, we prove \(\Lambda _{\sigma }^{**}:\Omega ^{**}\rightarrow \Omega ^{**}\) is a self-mapping.

For \(\varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}\), \(i = 1, \dots , m\), we have

$$\begin{aligned} \bigl\Vert \Lambda ^{**}_{\sigma } \bigl(\omega (\varsigma ) \bigr) \bigr\Vert &\leq \frac{1}{\Gamma (\sigma )} \int _{\varsigma _{i}}^{s_{i}} \bigl\Vert ( \varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert \hbar _{i}\bigl(s, \omega (s)\bigr) \bigr\Vert \Delta s+ \bigl\Vert E_{\sigma }\bigl(A \varsigma ^{\sigma }\bigr)\omega _{0} \bigr\Vert \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \biggl\Vert \biggl(F\bigl(s, \omega (s)\bigr) \\ &\quad {}+\mathcal{G} \biggl(s,\omega (s), \int _{s_{0}}^{s_{f}} F_{1}\bigl(s,u, \omega (u)\bigr)\Delta u, \int _{s_{0}}^{s_{f}} F_{2}\bigl(s,u, \omega (u)\bigr) \Delta u \biggr) \biggr) \biggr\Vert \Delta s \\ &\quad {}+ \int _{\varsigma _{0}}^{\varsigma _{f}} \bigl\Vert (\varsigma -s)^{\sigma -1} \bigr\Vert \bigl\Vert E_{\sigma ,\sigma } \bigl(A( \varsigma -s)^{\sigma } \bigr) \bigr\Vert \Vert \mathcal{H} \Vert \bigl\Vert \zeta (s) \bigr\Vert \Delta s \\ &\leq \frac{1}{\Gamma (\sigma )}a_{3}L_{g}\delta ({s_{i}}-{\varsigma _{i}})+a_{1} \\ &\quad {}+a_{3} \bigl(\delta ^{\prime \prime } a_{2}{L}_{F}+ \delta ^{\prime \prime } a_{2}{L}_{ \mathcal{G}}+\delta ^{\prime \prime } a_{2}{L}_{\mathcal{G}}{L}_{F_{1}}({s_{f}}-{s_{0}}) \\ &\quad {}+\delta ^{\prime \prime } a_{2}{L}_{\mathcal{G}}{L}_{F_{2}}({s_{f}}-{s_{0}}) \bigr)\times ({\varsigma _{f}}-{\varsigma _{0}}) \\ &\quad {}+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma } \biggl[ \frac{1}{\Gamma (\sigma )}a_{3}L_{g}\delta ^{\prime \prime }({s_{i}}-{\varsigma _{i}})+a_{1}+ \Vert \omega \Vert _{T} \\ &\quad {}+\delta ^{\prime \prime } a_{3} \bigl( a_{2}{L}_{F}+ a_{2}{L}_{\mathcal{G}}+ a_{2}{L}_{ \mathcal{G}}{L}_{F_{1}}({s_{f}}-{s_{0}}) \\ &\quad {}+ a_{2}{L}_{\mathcal{G}}{L}_{F_{2}}({s_{f}}-{s_{0}}) \bigr)\times ({ \varsigma _{f}}-{\varsigma _{0}}) \biggr] \\ &\leq N_{6}+\delta ^{\prime \prime }Q_{6} \leq \delta '' + \delta '' Q_{6} = \delta ''_{1}. \end{aligned}$$

Hence,

$$ \bigl\Vert \Lambda ^{**}_{\sigma } \bigl( \omega (\varsigma ) \bigr) \bigr\Vert \leq \delta ''_{2}. $$
(34)

Therefore, from (34), \(\Lambda _{\sigma }^{**}(\Omega ^{**})\subseteq \Omega ^{**}\). Also, for \(\varsigma \in (s_{i}, \varsigma _{i+1}]\cap \mathbb{{T}}\), \(i = 1, \dots , m\), with \(\omega _{0}=\widehat{\omega }_{0}\), we have

$$\begin{aligned} &\bigl\Vert \Lambda ^{**}_{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Lambda ^{**}_{ \sigma } \bigl(\widehat{\omega }( \varsigma ) \bigr) \bigr\Vert \\ &\quad \leq \biggl[ \frac{1}{\Gamma (\sigma )}a_{3}L_{g}({s_{i}}-{ \varsigma _{i}}) +a_{3} \bigl( a_{2}{L}_{F}+ a_{2}{L}_{\mathcal{G}} \\ &\qquad {}+ a_{2}{L}_{\mathcal{G}}{L}_{F_{1}}({s_{f}}-{s_{0}})+ a_{2}{L}_{ \mathcal{G}}{L}_{F_{2}}({s_{f}}-{s_{0}}) \bigr) ({\varsigma _{f}}-{ \varsigma _{0}}) \biggr] \Vert \widetilde{\omega }-{\omega } \Vert _{ \infty } \\ &\qquad {}+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma }(\varsigma -\varsigma _{f}) \biggl[\frac{1}{\Gamma (\sigma )}a_{3}L_{g}({s_{i}}-{ \varsigma _{i}}) +a_{3} \bigl( a_{2}{L}_{F}+ a_{2}{L}_{\mathcal{G}} \\ &\qquad {}+ a_{2}{L}_{\mathcal{G}}{L}_{F_{1}}({s_{f}}-{s_{0}}) + a_{2}{L}_{ \mathcal{G}}{L}_{F_{2}}({s_{f}}-{s_{0}}) \bigr) ({\varsigma _{f}}-{ \varsigma _{0}}) \biggr] \Vert \widetilde{\omega }-{\omega } \Vert _{ \infty } \\ &\quad \leq \bigl(1+M_{\mathcal{H}}M_{\mathcal{W}}^{\sigma }(\varsigma - \varsigma _{f}) \bigr) \biggl[\frac{1}{\Gamma (\sigma )}a_{3}L_{g}({s_{i}}-{ \varsigma _{i}})+a_{3} \bigl( a_{2}{L}_{F}+ a_{2}{L}_{\mathcal{G}} \\ &\qquad {}+ a_{2}{L}_{\mathcal{G}}{L}_{F_{1}}({s_{f}}-{s_{0}}) + a_{2}{L}_{ \mathcal{G}}{L}_{F_{2}}({s_{f}}-{s_{0}}) \bigr)\times ({\varsigma _{f}}-{ \varsigma _{0}}) \biggr] \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }. \end{aligned}$$

Hence

$$ \bigl\Vert \Lambda ^{**}_{\sigma } \bigl(\omega (\varsigma ) \bigr)-\Lambda ^{**}_{ \sigma } \bigl(\widehat{\omega }( \varsigma ) \bigr) \bigr\Vert \leq Q_{8} \Vert \widetilde{\omega }-{\omega } \Vert _{\infty }. $$
(35)

Therefore, the operator \(\Lambda _{\sigma }^{**}\) is strictly contractive. Thus, using the Banach fixed point theorem method, \(\Lambda _{\sigma }^{**}\) has a unique fixed point, which is the unique solution of the mixed impulsive system (4). Also, using Lemma 5.6, we conclude that \(\omega (\varsigma )\) fulfills \(\omega (T)=\omega _{T}\). Consequently, the mixed impulsive system (4) is controllable. □

Illustrative example

The last part of the manuscript is devoted to examining our results established in the previous steps.

Example 6.1

Consider the following mixed impulsive system:

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\omega (\varsigma )= \frac{2}{\varsigma -1}\omega (\varsigma )+\frac{2}{\varsigma -1.2} \mathbb{U}(\varsigma ) +e_{p}(\varsigma ,\chi (\omega (\varsigma ))) + \int _{0}^{\varsigma }E_{\sigma ,\beta }(\omega )\Delta \mathrm{s}+ \mathbb{U}(\varsigma ), \\ \omega (0)=1, \quad \varsigma \in {[0, 3]}_{\mathbb{T}}\backslash \{1,1.2 \}, \\ \omega (\varsigma _{k})=\Xi (\omega (\varsigma ^{-}_{k}))+\Gamma _{k} (\varsigma ^{-}_{k},\omega (\varsigma ^{-}_{k}),\mathbb{U}(\varsigma ^{-}_{k})),\quad k=1,2, \end{cases} $$
(36)

and its relevant inequality

$$ \textstyle\begin{cases} \vert {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\widehat{\omega }( \varsigma )-\frac{2}{\varsigma -1}\widetilde{\omega }(\varsigma ) - \frac{2}{\varsigma -1.2}\mathbb{U}(\varsigma ) \\ \quad {} -e_{p}(\varsigma ,\chi (\widetilde{\omega }(\varsigma )))- \int _{0}^{\varsigma }E_{\sigma ,\beta }(\widehat{\omega })\Delta \mathrm{s}-\mathbb{U}(\varsigma ) \vert \leq 1, \\ \quad \varsigma \in {[0, 3]}_{\mathbb{T}}\backslash \{1,1.2\}, \\ \vert \Delta \widetilde{\omega }(\varsigma _{k})-\Xi ( \widetilde{\omega }(\varsigma ^{-}_{k}))-\Gamma _{k} (\varsigma ^{-}_{k}, \widetilde{\omega }(\varsigma ^{-}_{k})) \vert \leq 1,\quad k=1,2. \end{cases} $$
(37)

We set \({{\mathbb{T}}}^{k}={[0, 3]}_{\mathbb{T}}\backslash \{1,1.2\}\), \(\varsigma _{1}=1\), \(\varsigma _{2}=1.2\), \(\mathbb{X}(\varsigma )=\frac{2}{\varsigma -1}\), and \(\mathbb{Y}(\varsigma )=\frac{2}{\varsigma -1.2}\), \(E_{\sigma ,\beta }( \omega )=\sum_{k=0}^{\infty } \frac{\omega ^{k}}{\Gamma (k\sigma +\beta )}\text{for}\sigma \), \(\beta >0 \). In addition, we set

$$ F\bigl(\varsigma ,\omega (\varsigma ),J_{\omega }(\varsigma ), \mathbb{U}( \varsigma )\bigr)=e_{p}\bigl(\varsigma ,\chi \bigl( \omega (\varsigma )\bigr)\bigr)+ \int _{0}^{t}E_{\sigma ,\beta }(\omega )\Delta \mathrm{s}+\mathbb{U}(\varsigma ), $$

where \(J_{\omega }(\varsigma )=\int _{0}^{t}E_{\sigma ,\beta }( \omega ) \Delta \mathrm{s}\) and \(\mathbb{U}(\varsigma )\) is a control map for \(\varsigma \in {{\mathbb{T}}}^{k}\) and substitute \(\epsilon =1\). Let \(\widetilde{\omega }\in PC^{1}([0,2]_{{{\mathbb{T}}}}, \mathbb{R})\) fulfill (37), then there exists \(\hbar \in PC^{1}([0,2]_{{{\mathbb{T}}}^{k}}, \mathbb{R})\) with \(\hbar _{0}\in \mathbb{R}\) such that \(|\hbar (\varsigma )|\leq 1\) \(\varsigma \in {{\mathbb{T}}}^{k}\) and \(|\hbar _{0}|\leq 1\), and so (37) implies that

$$ \textstyle\begin{cases} {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\widetilde{\omega }(\varsigma )= \frac{2}{\varsigma -1}\widetilde{\omega }(\varsigma )+ \frac{2}{\varsigma -1.5}\mathbb{U}(\varsigma ) +e_{p}(\varsigma , \chi (\widetilde{\omega }(\varsigma ))) \\ \hphantom{ {{}^{c,\mathbb{T}}}{\mathrm{D}^{\sigma }}\widetilde{\omega }(\varsigma )={}}{} + \int _{0}^{\varsigma }E_{\sigma , \beta }(\omega )\Delta \mathrm{s}+\mathbb{U}(\varsigma )+\hbar ( \varsigma ), \quad \varsigma \in {{\mathbb{T}}}^{k}, \\ \widetilde{\omega }(\varsigma _{k})=\Xi _{k}(\widetilde{\omega }( \varsigma ^{-}_{k}))-\Gamma _{k} (\varsigma ^{-}_{k}, \widetilde{\omega }(\varsigma ^{-}_{k}),\mathbb{U}(\varsigma ^{-}_{k}))+ \hbar _{0},\quad k=1,2. \end{cases} $$

So the solution of (36) is

$$\begin{aligned} \omega (\varsigma )&=\Xi _{1}\bigl(\omega \bigl(\varsigma ^{-}_{1}\bigr)\bigr)+\Xi _{2}\bigl( \omega \bigl(\varsigma ^{-}_{2}\bigr)\bigr) +\Gamma _{1}\bigl(\varsigma ^{-}_{1},\omega \bigl( \varsigma ^{-}_{1}\bigr),\mathbb{U}\bigl(\varsigma ^{-}_{1}\bigr)\bigr)+\Gamma _{2}\bigl( \varsigma ^{-}_{2},\omega \bigl(\varsigma ^{-}_{2}\bigr),\mathbb{U}\bigl(\varsigma ^{-}_{2}\bigr)\bigr) \\ &\quad {}+ \int _{0}^{t} e_{p}\bigl( \mathrm{s},\chi (\mathrm{s})\bigr) \biggl(e_{p}\bigl( \mathrm{s}, \chi \bigl(\omega (\mathrm{s})\bigr)\bigr)+ \int _{0}^{s}E_{\sigma ,\beta }( \omega )\Delta u+\mathbb{U}(\mathrm{s}) \biggr)\Delta \mathrm{s}. \end{aligned}$$

By our obtained results, the mixed impulsive system (36) has only one solution in \(PC^{1}([0,2]_{{{\mathbb{T}}}}, \mathbb{R})\) and is UH stable on \({{\mathbb{T}}}^{k}\).

Conclusion

In this article, we conducted our research on some mixed integral dynamic systems with impulsive effects on times scales in the fractional settings. We studied the existence and uniqueness successfully using a fixed point method for the considered systems. We established our results by using the Leray–Schauder and Banach fixed point theorems in this regard. In the next step, Ulam–Hyers stability and a generalized version of it were proved for the mentioned mixed impulsive systems. After that, we investigated the controllability property for the aforesaid systems. Lastly, an illustrative example was proposed to examine the results established in the previous sections. For future projects, the main aim of the authors is that these qualitative specifications can be checked and established on some real-world impulsive systems arising in mathematical models of brain.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

The third and fourth authors were supported by Azarbaijan Shahid Madani University. We would like to thank the dear reviewers for giving constructive and useful comments to improve the final version of this paper.

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The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

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Correspondence to Akbar Zada or Shahram Rezapour.

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Pervaiz, B., Zada, A., Etemad, S. et al. An analysis on the controllability and stability to some fractional delay dynamical systems on time scales with impulsive effects. Adv Differ Equ 2021, 491 (2021). https://doi.org/10.1186/s13662-021-03646-9

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MSC

  • 26A33
  • 34A08
  • 34B27

Keywords

  • Time scales
  • Dynamic system
  • Hyers–Ulam stability
  • Controllability
  • Impulsive effects