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Study of fractional order pantograph type impulsive antiperiodic boundary value problem

Abstract

In this paper, we study existence and stability results of an anti-periodic boundary value problem of nonlinear delay (pantograph) type implicit fractional differential equations with impulsive conditions. Using Schaefer’s fixed point theorem and Banach’s fixed point theorem, we have established results of at least one solution and uniqueness. Also, using the Hyers–Ulam concept, we have derived various kinds of Ulam stability results for the considered problem. Finally, we have applied our obtained results to a numerical problem.

Introduction

The study of non-integer order differential equations has emerged as one of the most active research fields in modern mathematics. The non-integer order derivative is also known as fractional order derivative. The main advantage of non-integer order derivative is that it is a global operator and produces accurate and stable results, while the integer order derivative is a local operator. The non-integer order differential equations have multi-dimensional applications in various fields of modern sciences. For example, the non-integer order oscillators are used to control the phase difference and to achieve independently the high frequency oscillation. In electrical engineering the non-integer order DC-DC converter models are used to get best estimation of the power conversion efficiency. Similarly, the non-integer order bio-impedance models give the best fitting to the measured data obtained from vegetables and fruits. For historical background and some applications of non-integer order derivatives, we refer the readers to study [19].

The impulsive differential equations have impulsive conditions at points of discontinuity. Various physical and evolutionary phenomena which have discontinuous jumps and sudden changes can be modeled via these equations, and therefore, these equations constitute an important class of differential equations. For some recent work, we refer the readers to study work in [1014].

Some processes and phenomena cannot be described at the current time and depend on their previous states. For this purpose various types of delay differential equations are used. There are mainly three types of delay differential equations: discrete delay differential equations, continuous delay differential equations, and proportional delay differential equations. The proportional delay differential equations are known as pantograph differential equations. The pantograph differential equations are used to model numerous processes and phenomena. More specifically these equations have applications in electro-dynamic, quantum mechanics, number theory, biology, etc. We refer the readers to study [1517].

Existence theory and stability analysis are two important aspects of qualitative theory. The Hyers–Ulam stability concept is one of the well-known methods used to study the stability of functional and differential problems. This concept was given by Ulam and Hyers in 1940–41 [18, 19]. Rassias was the first who established Hyers–Ulam stability of linear mapping [20]. Jung studied Hyers–Ulam–Rassias stability results for functional equations in nonlinear analysis [21]. Due to its simple procedure, Hyers–Ulam concept has got a great deal of attention from researchers. Using this concept, they investigated stability of various systems of functional and differential equations. We refer the readers to study the recent work in [2227]. Also, for more related results about the existence, uniqueness, and stability, the readers may consider the work in [2835].

The antiperiodic boundary value problems are of great interest as these problems appear in various fields of science. In [36], Ahmad et al. investigated a class of fractional integro-differential equations with dual anti-periodic boundary conditions. In [37], Agarwal et al. studied a problem of fractional-order differential equations with anti-periodic boundary conditions.

In [38], Wang et al. studied the existence of solution to the following antiperiodic problem with impulsive conditions:

$$ \textstyle\begin{cases} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} w(t)=\mathrm{g}(t, w(t)),\quad t \in [0,1], t\neq t_{k}, 0< \delta < 1, \varsigma \in (2,3], \\ \Delta w(t_{k})=\mathcal{I}_{k}(w(t_{k})),\quad k=1,2,\ldots,n, \\ \Delta w'(t_{k})=\hat{\mathcal{I}}_{k}(w(t_{k})), \qquad \Delta w''(t_{k})= \bar{\mathcal{I}}_{k}(w(t_{k})),\quad k=1,2,\ldots,n, \\ w(0)=-w(1), \qquad w'(0)=-w'(1),\qquad w''(0)=-w''(1), \end{cases} $$
(1)

where \(\mathrm{g}:[0, 1]\times \mathbb {R}\rightarrow [0,\infty )\) and \(\mathcal{I}_{k},\hat{\mathcal{I}}_{k},\bar{\mathcal{I}}_{k}: \mathbb {R}\rightarrow \mathbb {R}\) are continuous functions. In this paper, we generalize problem 1 to an anti-periodic boundary value problem of nonlinear pantograph implicit fractional differential equations with given impulsive conditions as follows:

$$ \textstyle\begin{cases} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} w(t)=\mathrm{g}(t, w(t), w( \delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} w(t)),\quad t\in [0,1], t \neq t_{k}, 0< \delta < 1, \varsigma \in (2,3], \\ \Delta w(t_{k})=\mathcal{I}_{k}(w(t_{k})),\quad k=1,2,\ldots,n, \\ \Delta w'(t_{k})=\hat{\mathcal{I}}_{k}(w(t_{k})), \qquad \Delta w''(t_{k})= \bar{\mathcal{I}}_{k}(w(t_{k})),\quad k=1,2,\ldots,n, \\ w(0)=-w(1), \qquad w'(0)=-w'(1),\qquad w''(0)=-w''(1), \end{cases} $$
(2)

where \(\mathrm{g}:[0, 1]\times \mathbb {R}\times \mathbb {R}\times \mathbb {R} \rightarrow [0,\infty )\) and \(\mathcal{I}_{k},\hat{\mathcal{I}}_{k},\bar{\mathcal{I}}_{k}: \mathbb {R}\rightarrow \mathbb {R}\) are continuous functions. \({_{0}^{C}\mathbb {D}_{t}^{\varsigma }}\) is a standard notion for Caputo type fractional differential operator of order ς. \(\Delta w(t_{k})=w(t_{k}^{+})-w(t_{k}^{-})\), \(\Delta w'(t_{k})=w'(t_{k}^{+})-w'(t_{k}^{-})\), \(\Delta w''(t_{k})=w''(t_{k}^{+})-w''(t_{k}^{-})\); \(w(t_{k}^{+}), w'(t_{k}^{+}), w''(t_{k}^{+})\) are right-hand limits and \(w(t_{k}^{-}), w'(t_{k}^{-}), w''(t_{k}^{-})\) are left-hand limits of the function \(w(t)\) at \(t=t_{k}\). And the sequence \({t_{k}}\) satisfies that \(0=t_{0}< t_{1}< t_{2}<\cdots <t_{n}<t_{n+1}, n\in \mathbb {N}\). We study the existence, uniqueness, and Hyers–Ulam stability of the generalized problem (2).

Let \(I=[0, 1]\), \(I_{0}=[0, t_{1}]\) and \(I_{k}=(t_{k}, t_{k+1}]\). We define the space of piecewise continuous functions by

$$\begin{aligned} \mathscr{Z}=PC(I,\mathbb {R})={}&\bigl\{ w:I\rightarrow \mathbb {R} | w \in C(I),\quad k=1,2,\dots,n, \\ & w\bigl(t_{k}^{+}\bigr), w\bigl(t_{k}^{-} \bigr) \text{ exist for } k=1,2,\dots,n\bigr\} . \end{aligned}$$

\(\mathscr{Z}\) is a Banach space with the norm defined by \(\|w\|_{\mathscr{Z}}=\max_{t\in I}|w(t)|\).

Preliminaries

Definition 1

([24])

The fractional order integral of function \(h\in L^{1}([0,1],\mathbb {R}^{+})\) of order \(\varsigma \in \mathbb {R}^{+}\) is defined by

$$\begin{aligned} {_{0}\mathrm{I}_{t}^{\varsigma }} h(t)= \int _{0}^{t} \frac{(t-\mathit{s})^{\varsigma -1}}{\Gamma (\varsigma )}h(s)\,d s, \end{aligned}$$
(3)

provided that integral on the right-hand side exists.

Definition 2

([25])

For a function \(h\in C^{n}[0, +\infty )\), the Caputo fractional derivative of order ς is defined as

$$\begin{aligned} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} h(t)= \frac{1}{\Gamma (n-\varsigma )} \int _{0}^{t} (t-s)^{n- \varsigma -1} h^{(n)}(s)\,ds,\quad n-1< \varsigma < n, \end{aligned}$$
(4)

where \(n=[\varsigma ]+1\); \([\varsigma ]\) denotes the integer part of ς.

Lemma 1

([24])

Let \(\varsigma >0\), then \({_{0}\mathrm{I}_{t}^{\varsigma }} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} h(t)=h(t)+{c_{0}}+{c_{1}}t+{c_{2}}{t^{2}}+\cdots +{c_{n-1}}t^{n-1}\), \({c_{i}}\in {\mathbb {R}},i=0,1,\ldots,n-1, n=[\varsigma ]+1\).

Let \(\phi \in C(I, \mathbb {R}_{+})\) be a nondecreasing function, \(\varphi \geq 0\), \(\nu \in \mathscr{Z}\) such that, for \(t\in I_{k}\), \(k=1,2,\dots,n\), the following sets of inequalities are satisfied:

$$\begin{aligned} &\textstyle\begin{cases} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t)-\mathrm{g}(t, \nu (t), \nu (\delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t))\leq \epsilon, \quad\varsigma \in (2,3], 0< \delta < 1, \\ \Delta \nu (t_{k})-\mathcal{I}_{k}(\nu (t_{k}))\leq \epsilon, \\ \Delta \nu '(t_{k})-\hat{\mathcal{I}}_{k}(\nu (t_{k}))\leq \epsilon, \\ \Delta \nu ''(t_{k})-\bar{\mathcal{I}}_{k}(\nu (t_{k}))\leq \epsilon; \end{cases}\displaystyle \end{aligned}$$
(5)
$$\begin{aligned} &\textstyle\begin{cases} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t)-\mathrm{g}(t, \nu (t), \nu (\delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t))\leq \phi (t), \quad\varsigma \in (2,3], 0< \delta < 1, \\ \Delta \nu (t_{k})-\mathcal{I}_{k}(\nu (t_{k}))\leq \varphi, \\ \Delta \nu '(t_{k})-\hat{\mathcal{I}}_{k}(\nu (t_{k}))\leq \varphi, \\ \Delta \nu ''(t_{k})-\bar{\mathcal{I}}_{k}(\nu (t_{k}))\leq \varphi; \end{cases}\displaystyle \end{aligned}$$
(6)
$$\begin{aligned} &\textstyle\begin{cases} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t)-\mathrm{g}(t, \nu (t), \nu (\delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t))\leq \epsilon \phi (t), \quad\varsigma \in (2,3], 0< \delta < 1, \\ \Delta \nu (t_{k})-\mathcal{I}_{k}(\nu (t_{k}))\leq \epsilon \varphi, \\ \Delta \nu '(t_{k})-\hat{\mathcal{I}}_{k}(\nu (t_{k}))\leq \epsilon \varphi, \\ \Delta \nu ''(t_{k})-\bar{\mathcal{I}}_{k}(\nu (t_{k}))\leq \epsilon \varphi. \end{cases}\displaystyle \end{aligned}$$
(7)

Definition 3

([39])

If for \(\epsilon >0\) there exists a constant \(C_{\mathrm{g}}>0\) such that, for any solution \(\nu \in \mathscr{Z}\) of inequality (5), there is a unique solution \(w\in \mathscr{Z}\) of system (2) that satisfies

$$ \bigl\vert \nu (t)-w(t) \bigr\vert \leq C_{\mathrm{g}}\epsilon,\quad t\in I, $$

then system (2) is Hyers–Ulam stable.

Definition 4

If for \(\epsilon >0\) and set of positive real numbers \(\mathbb {R}^{+}\) there exists \(\phi \in C(\mathbb {R}^{+},\mathbb {R}^{+})\), with \(\phi (0)=0\) such that, for any solution \(\nu \in \mathscr{Z}\) of inequality (6), there exist \(\epsilon >0\) and a unique solution \(w\in \mathscr{Z}\) of system (2) that satisfy

$$ \bigl\vert \nu (t)-w(t) \bigr\vert \leq \phi (\epsilon ),\quad t\in I, $$

then system (2) is generalized Hyers–Ulam stable.

Definition 5

([39])

If for \(\epsilon >0\) there exists a real number \(C_{\mathrm{g}}>0\) such that, for any solution \(\nu \in \mathscr{Z}\) of inequality (7), there is a unique solution \(w\in \mathscr{Z}\) of system (2) that satisfies

$$ \bigl\vert \nu (t)-w(t) \bigr\vert \leq C_{\mathrm{g}}\epsilon \bigl( \varphi +\phi (t)\bigr),\quad t\in I, $$

then system (2) is Hyers–Ulam–Rassias stable with respect to \((\varphi,\phi )\).

Definition 6

([39])

If there exists a constant \(C_{\mathrm{g}}>0\) such that, for any solution \(\nu \in \mathscr{Z}\) of inequality (6), there is a unique solution \(w\in \mathscr{Z}\) of system (2) that satisfies

$$ \bigl\vert \nu (t)-w(t) \bigr\vert \leq C_{\mathrm{g}}\bigl(\varphi +\phi (t)\bigr),\quad t\in I, $$

then system (2) is generalized Hyers–Ulam–Rassias stable with respect to \((\varphi,\phi )\).

Remark 1

The function \(\nu \in \mathscr{Z}\) is called a solution for inequality (5) if there exists a function \(\psi \in \mathscr{Z}\) together with a sequence \(\psi _{k}\), \(k=1,2\dots,n\) (which depends on ν) such that

  1. (i)

    \(|\psi (t)|\leq \epsilon \), \(|\psi _{k}|\leq \epsilon \), \(t\in I\),

  2. (ii)

    \({_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t)=\mathrm{g}(t, \nu (t), \nu (\delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t))+\psi (t)\), \(\varsigma \in (2,3]\), \(0<\delta <1\), \(t\in I\),

  3. (iii)

    \(\Delta \nu (t_{k})=\mathcal{I}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\),

  4. (iv)

    \(\Delta \nu '(t_{k})=\hat{\mathcal{I}}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\),

  5. (v)

    \(\Delta \nu ''(t_{k})=\bar{\mathcal{I}}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\).

Remark 2

A function \(\nu \in \mathscr{Z}\) is a solution of inequality (6) if there exist a function \(\psi \in \mathscr{Z}\) and a sequence \(\psi _{k}\), \(k=1,2\dots,n\) (which depends on ν) such that

  1. (i)

    \(|\psi (t)|\leq \phi (t)\), \(|\psi _{k}|\leq \varphi \), \(t\in I\),

  2. (ii)

    \({_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t)=\mathrm{g}(t, \nu (t), \nu (\delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t))+\psi (t)\), \(\varsigma \in (2,3]\), \(0<\delta <1\), \(t\in I\),

  3. (iii)

    \(\Delta \nu (t_{k})=\mathcal{I}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\),

  4. (iv)

    \(\Delta \nu '(t_{k})=\hat{\mathcal{I}}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\),

  5. (v)

    \(\Delta \nu ''(t_{k})=\bar{\mathcal{I}}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\).

Remark 3

A function \(\nu \in \mathscr{Z}\) is a solution of inequality (7) if there exist a function \(\psi \in \mathscr{Z}\) and a sequence \(\psi _{k}\), \(k=1,2\dots,n\) (which depends on ν) such that

  1. (i)

    \(|\psi (t)|\leq \epsilon \phi (t)\), \(|\psi _{k}|\leq \epsilon \varphi \), \(t\in I\),

  2. (ii)

    \({_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t)=\mathrm{g}(t, \nu (t), \nu (\delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t))+\psi (t)\), \(\varsigma \in (2,3]\), \(0<\delta <1\), \(t\in I\),

  3. (iii)

    \(\Delta \nu (t_{k})=\mathcal{I}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\),

  4. (iv)

    \(\Delta \nu '(t_{k})=\hat{\mathcal{I}}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\),

  5. (v)

    \(\Delta \nu ''(t_{k})=\bar{\mathcal{I}}_{k}(\nu (t_{k}))+\psi _{k}\), \(t\in I\).

Lemma 2

For a given function \(\vartheta \in C[0, 1]\), function w is the solution of the linear BVP of impulsive differential equations

$$ \textstyle\begin{cases} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} w(t)=\vartheta (t),\quad t\in I', \varsigma \in (2,3], k=1,2,\ldots,n, \\ \Delta w(t_{k})=\mathcal{I}_{k}(w(t_{k})), \quad k=1,2,\ldots,n, \\ \Delta w'(t_{k})=\hat{\mathcal{I}}_{k}(w(t_{k})),\qquad \Delta w''(t_{k})= \bar{\mathcal{I}}_{k}(w(t_{k})),\quad k=1,2,\ldots,n, \\ w(0)=-w(1), \qquad w'(0)=-w'(1),\qquad w''(0)=-w''(1) \end{cases} $$
(8)

if and only if w satisfies the following fractional integral equation:

$$ w(t)=\textstyle\begin{cases} \frac{1}{\Gamma (\varsigma )}\int _{0}^{t}(t-s)^{ \varsigma -1}\vartheta (s)\,ds-\mathcal{A},\quad t\in I_{0}; \\ \frac{1}{\Gamma (\varsigma )}\int _{t_{k}}^{t}(t-s)^{ \varsigma -1}\vartheta (s)\,ds+ \frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k}\int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1}\vartheta (s)\,ds \\ \quad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)}\int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2}\vartheta ( s)\,ds\\ \quad{}+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)}\int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3}\vartheta (s)\,ds \\\quad{} +\sum_{m=1}^{k}\frac{(t-t_{k})}{\Gamma (\varsigma -1)}\int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2}\vartheta ( s)\,ds\\ \quad{}+\sum_{m=1}^{k-1} \frac{(t-t_{k})(t_{k}-t_{m})}{\Gamma (\varsigma -2)}\int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3}\vartheta (s)\,ds \\ \quad{} +\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2\Gamma (\varsigma -2)}\int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3}\vartheta ( s)\,ds+\sum_{m=1}^{k}\mathcal{I}_{m}(w(t_{m}))\\ \quad{}+\sum_{m=1}^{k-1}(t_{k}-t_{m}) \hat{\mathcal{I}}_{m}(w(t_{m})) \\\quad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})^{2}}{2}\bar{\mathcal{I}}_{m}(w(t_{m}))+ \sum_{m=1}^{k}(t-t_{k})\mathcal{I}_{m}(w(t_{m}))\\ \quad{}+\sum_{m=1}^{k-1}(t-t_{k})(t_{k}-t_{m}) \bar{\mathcal{I}}_{m}(w(t_{m})) \\\quad{} +\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2}\bar{\mathcal{I}}_{m}(w(t_{m}))- \mathcal{A},\quad t\in I_{k}, k=1,2,\dots,n, \end{cases} $$
(9)

where

$$\begin{aligned} \mathcal{A}={}&\frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \vartheta ( s)\,ds+\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \vartheta (s)\,ds \\ &{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \vartheta (s)\,ds+\sum_{m=1}^{n} \frac{(1-2t_{n})}{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \vartheta (s)\,ds \\ &{} +\sum_{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \vartheta (s)\,ds\\ &{}-\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \vartheta (s)\,ds \\ &{} -\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2}\vartheta (s)\,ds+ \frac{1}{2} \sum_{m=1}^{n} \mathcal{I}_{m}\bigl(w(t_{m})\bigr)+\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2}\hat{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{}+\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)+ \sum _{m=1}^{n}\frac{(1-2t_{n})}{4}\mathcal{I}_{m} \bigl(w(t_{m})\bigr)\\ &{}+\sum_{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4}\bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{} -\sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)+ \frac{t}{2\Gamma (\varsigma -1)}\sum _{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \vartheta (s)\,ds \\ &{} +\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \vartheta (s)\,ds+\sum_{m=1}^{n} \frac{t(1-2t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \vartheta (s)\,ds \\ &{} -\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3}\vartheta (s)\,ds+ \frac{t}{2} \sum_{m=1}^{n} \mathcal{I}_{m}\bigl(w(t_{m})\bigr)+\sum _{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{} +\sum_{m=1}^{n}\frac{t(1-2t_{n})}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)+ \frac{t^{2}}{4\Gamma (\varsigma -2)}\sum _{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \vartheta (s)\,ds\\ &{}- \frac{t^{2}}{4}\sum_{m=1}^{n} \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr). \end{aligned}$$

Proof

For proof, see [38]. □

Main results

Corollary 1

As a result of Lemma 2, system (2) has the following solution:

$$ w(t)=\textstyle\begin{cases} \frac{1}{\Gamma (\varsigma )}\int _{0}^{t}(t-s)^{ \varsigma -1}\beta _{w}(s)\,ds-\mathcal{M},\quad t\in I_{0}; \\ \frac{1}{\Gamma (\varsigma )}\int _{t_{k}}^{t}(t-s)^{ \varsigma -1}\beta _{w}(s)\,ds+ \frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k}\int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1}\beta _{w}(s)\,ds \\\quad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)}\int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2}\beta _{w}( s)\,ds\\ \quad{}+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)}\int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3}\beta _{w}(s)\,ds \\\quad{} +\sum_{m=1}^{k}\frac{(t-t_{k})}{\Gamma (\varsigma -1)}\int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2}\beta _{w}( s)\,ds\\ \quad{}+\sum_{m=1}^{k-1} \frac{(t-t_{k})(t_{k}-t_{m})}{\Gamma (\varsigma -2)}\int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3}\beta _{w}(s)\,ds \\\quad{} +\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2\Gamma (\varsigma -2)}\int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3}\beta _{w}( s)\,ds\\ \quad{}+\sum_{m=1}^{k}\mathcal{I}_{m}(w(t_{m}))+\sum_{m=1}^{k-1}(t_{k}-t_{m}) \hat{\mathcal{I}}_{m}(w(t_{m})) \\\quad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})^{2}}{2}\bar{\mathcal{I}}_{m}(w(t_{m}))\\ \quad{}+ \sum_{m=1}^{k}(t-t_{k})\mathcal{I}_{m}(w(t_{m}))+\sum_{m=1}^{k-1}(t-t_{k})(t_{k}-t_{m}) \bar{\mathcal{I}}_{m}(w(t_{m})) \\\quad{} +\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2}\bar{\mathcal{I}}_{m}(w(t_{m}))- \mathcal{M},\quad t\in I_{k}, k=1,2,\dots,n, \end{cases} $$
(10)

where

$$\begin{aligned} \beta _{w}=\mathrm{g}\bigl(t, w(t), w(\delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} w(t) \bigr), \end{aligned}$$

and

$$\begin{aligned} \mathcal{M}={}&\frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \beta _{w}( s)\,ds+\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \beta _{w}(s)\,ds \\ & {}+\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{w}( s)\,ds+\sum_{m=1}^{n} \frac{(1-2t_{n})}{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \beta _{w}(s)\,ds \\ & {}+\sum_{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds\\ &{}-\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds \\ &{} -\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2}\beta _{w}(s)\,ds+\frac{1}{2} \sum_{m=1}^{n} \mathcal{I}_{m}\bigl(w(t_{m})\bigr)+\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2}\hat{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)+ \sum _{m=1}^{n}\frac{(1-2t_{n})}{4}\mathcal{I}_{m} \bigl(w(t_{m})\bigr)\\ &{}+\sum_{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4}\bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{} -\sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)+ \frac{t}{2\Gamma (\varsigma -1)}\sum _{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \beta _{w}(s)\,ds \\ &{} +\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{w}( s)\,ds+\sum_{m=1}^{n} \frac{t(1-2t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds \\ &{} -\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3}\beta _{w}(s)\,ds+\frac{t}{2} \sum_{m=1}^{n} \mathcal{I}_{m}\bigl(w(t_{m})\bigr)+\sum _{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{} +\sum_{m=1}^{n}\frac{t(1-2t_{n})}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)+ \frac{t^{2}}{4\Gamma (\varsigma -2)}\sum _{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds\\ &{}- \frac{t^{2}}{4}\sum_{m=1}^{n} \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr). \end{aligned}$$

For \(w \in \mathscr{Z}\), we define an operator \(\mathcal{N}:\mathscr{Z}\rightarrow \mathscr{Z}\) by

$$\begin{aligned} \mathcal{N}w(t)={}&\frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t- s)^{\varsigma -1}\beta _{w}(s)\,ds+ \frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \beta _{w}(s)\,ds \\ &{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \beta _{w}( s)\,ds \\ &{}+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds \\ &{} +\sum_{m=1}^{k}\frac{(t-t_{k})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \beta _{w}( s)\,ds \\ &{}+\sum_{m=1}^{k-1} \frac{(t-t_{k})(t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds \\ &{} +\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{w}( s)\,ds \\ &{}+\sum_{m=1}^{k} \mathcal{I}_{m}\bigl(w(t_{m})\bigr)+\sum _{m=1}^{k-1}(t_{k}-t_{m}) \hat{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})^{2}}{2}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)+ \sum _{m=1}^{k}(t-t_{k})\mathcal{I}_{m} \bigl(w(t_{m})\bigr) \\ &{}+\sum_{m=1}^{k-1}(t-t_{k}) (t_{k}-t_{m}) \bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr) \\ &{}+\sum_{m=1}^{k} \frac{(t-t_{k})^{2}}{2}\bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \beta _{w}(s)\,ds \\ & {}-\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \beta _{w}( s)\,ds \\ &{}-\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds \\ & {}-\sum_{m=1}^{n}\frac{(1-2t_{n})}{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \beta _{w}( s)\,ds \\ &{}-\sum_{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds \\ &{} +\sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{w}( s)\,ds+\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2}\beta _{w}(s)\,ds \\ &{} -\frac{1}{2}\sum_{m=1}^{n} \mathcal{I}_{m}\bigl(w(t_{m})\bigr)-\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2}\hat{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)-\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{} -\sum_{m=1}^{n}\frac{(1-2t_{n})}{4} \mathcal{I}_{m}\bigl(w(t_{m})\bigr)-\sum _{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{}+ \sum _{m=1}^{n}\frac{t_{n}(1-t_{n})}{4}\bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr) \\ &{} -\frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \beta _{w}(s)\,ds \\ &{}-\sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds \\ &{} -\sum_{m=1}^{n}\frac{t(1-2t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{w}( s)\,ds+\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3}\beta _{w}(s)\,ds \\ &{} -\frac{t}{2}\sum_{m=1}^{n} \mathcal{I}_{m}\bigl(w(t_{m})\bigr)-\sum _{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr)-\sum _{m=1}^{n} \frac{t(1-2t_{n})}{4}\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \\ &{} -\frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds+ \frac{t^{2}}{4}\sum_{m=1}^{n} \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr), \\ &\quad t\in I_{k}, k=1,2,\dots,n. \end{aligned}$$
(11)

We take the following assumptions:

\((H_{1})\):

Let \(\mathrm{g}:[0, 1]\times \mathbb {R}\times \mathbb {R}\times \mathbb {R} \rightarrow [0,\infty )\) be a jointly continuous function.

\((H_{2})\):

For any \(x, y, z,\bar{x}, \bar{y}, \bar{z}\in C(I,\mathbb {R}), let the following inquality\)

$$ \bigl\vert \mathrm{g}(t, x, y, z)-\mathrm{g}(t, \bar{x}, \bar{y}, \bar{z}) \bigr\vert \leq L_{\mathrm{g}}\bigl( \vert x-\bar{x} \vert + \vert y- \bar{y} \vert \bigr)+N_{\mathrm{g}} \vert z- \bar{z} \vert $$

hold, where \(L_{\mathrm{g}}>0\) and \(0< N_{\mathrm{g}}<1\).

\((H_{3})\):

There exist \(C_{1}, C_{2}, C_{3}>0\) such that the following relations hold true:

$$\begin{aligned} &\bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr)- \mathcal{I}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \leq C_{1} \bigl\vert w(t_{m})- \bar{w}(t_{m}) \bigr\vert , \\ &\bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \leq C_{2} \bigl\vert w(t_{m})-\bar{w}(t_{m}) \bigr\vert , \\ &\bigl\vert \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \hat{\mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \leq C_{3} \bigl\vert w(t_{m})-\bar{w}(t_{m}) \bigr\vert . \end{aligned}$$
\((H_{4})\):

There exist functions \(\theta _{1}, \theta _{2}, \theta _{3} \in C(I,\mathbb {R}^{+})\), with

$$\begin{aligned} &\bigl\vert \mathrm{g}\bigl(t, w(t), w(\delta t), _{0}^{C}D_{t_{i}}^{\varsigma }w(t) \bigr) \bigr\vert \leq \theta _{1}(t)+\theta _{2}(t) \bigl( \vert w \vert + \bigl\vert w(\delta t) \bigr\vert \bigr)+ \theta _{3}(t) \bigl\vert _{0}^{C}D_{t_{i}}^{\varsigma }w(t) \bigr\vert ,\\ &\quad \text{for } t \in I, w \in \mathrm{E}, \end{aligned}$$

such that \(\theta _{3}^{*}=\max_{t\in I}|\theta _{3}(t)|<1\).

\((H_{5})\):

If \(\mathrm{g},\mathcal{I}_{m},\hat{\mathcal{I}}_{m},\bar{\mathcal{I}}_{m}\) are continuous functions and there exist constants \(C_{4}\), \(C_{5}\), \(C_{6}>0\) such that, for all \(w\in \mathbb {R}\), the following inequalities are satisfied:

$$ \bigl\vert \mathcal{I}_{m}(w) (t) \bigr\vert \leq C_{4}, \qquad\bigl\vert \hat{\mathcal{I}}_{m}(w) (t) \bigr\vert \leq C_{5},\qquad \bigl\vert \bar{\mathcal{I}}_{m}(w) (t) \bigr\vert \leq C_{6}. $$

Theorem 1

If assumptions \((H_{1})\)\((H_{3})\) and the following inequality

$$ \biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr]< 1 $$

are satisfied, then problem (2) has a unique solution.

Proof

We take \(w, \bar{w}\in \mathscr{Z}\) and consider

$$\begin{aligned} & \bigl\vert \mathcal{N}w(t)-\mathcal{N}\bar{w}(t) \bigr\vert \\ &\quad \leq \frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t-s)^{ \varsigma -1} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,d s+ \frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds \\ & \qquad{}+ \sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w}( s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds+ \sum_{m=1}^{k}\frac{(t-t_{k})}{\Gamma (\varsigma -1)} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds \\ & \qquad{}+ \sum_{m=1}^{k-1}\frac{(t-t_{k})(t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}( s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds+\sum_{m=1}^{k} \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr)-\mathcal{I}_{m} \bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{k-1}(t_{k}-t_{m}) \bigl\vert \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \hat{\mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert +\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{k}(t-t_{k}) \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr)- \mathcal{I}_{m}\bigl( \bar{w}(t_{m})\bigr) \bigr\vert \\ & \qquad{}+ \sum_{m=1}^{k-1}(t-t_{k}) (t_{k}-t_{m}) \bigl\vert \bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr)-\bar{\mathcal{I}}_{m}\bigl( \bar{w}(t_{m})\bigr) \bigr\vert \\ & \qquad{}+\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ & \qquad{} + \frac{1}{2\Gamma (\varsigma )} \sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{ \varsigma -1} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,d s \\ & \qquad{}+\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w}( s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds+\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds \\ & \qquad{}+ \sum_{m=1}^{n}\frac{ \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}( s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1} \frac{ \vert 1-2t_{n} \vert (t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}( s) \bigr\vert \,ds+\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds+ \frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}( s) \bigr\vert \,ds \\ & \qquad{}+\frac{1}{2}\sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr)- \mathcal{I}_{m}\bigl( \bar{w}(t_{m})\bigr) \bigr\vert + \sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2} \bigl\vert \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)-\hat{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert + \sum_{m=1}^{n} \frac{ \vert 1-2t_{n} \vert }{4} \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr)- \mathcal{I}_{m}\bigl( \bar{w}(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{ \vert 1-2t_{n} \vert (t_{n}-t_{m})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)-\bar{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ & \qquad{}+ \sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}( s) \bigr\vert \,ds+\sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds \\ & \qquad{}+ \sum_{m=1}^{n}\frac{t \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}( s)-\bar{\beta }_{\bar{w}}(s) \bigr\vert \,ds \\ &\qquad{} +\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}( s) \bigr\vert \,ds+ \frac{t}{2}\sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr)- \mathcal{I}_{m} \bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ & \qquad{}+ \sum_{m=1}^{n} \frac{t \vert 1-2t_{n} \vert }{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s)-\bar{\beta }_{\bar{w}}( s) \bigr\vert \,ds \\ & \qquad{}+\frac{t^{2}}{4}\sum_{m=1}^{n} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr)-\bar{ \mathcal{I}}_{m}\bigl(\bar{w}(t_{m})\bigr) \bigr\vert , \\ & \quad t\in I_{k}, k=1,2,\dots,n, \end{aligned}$$
(12)

where

$$\begin{aligned} &\beta _{w}(t)=\mathrm{g}\bigl(t, w(t), w(\delta t), \beta _{w}(t)\bigr), \\ &\bar{\beta }_{\bar{w}}(t)=\mathrm{g}\bigl(t, \bar{w}(t), \bar{w}( \delta t), \bar{\beta }_{\bar{w}}(t)\bigr). \end{aligned}$$

By \((H_{2})\) we have

$$\begin{aligned} \bigl\vert \beta _{w}(t)-\bar{\beta }_{\bar{w}}(t) \bigr\vert &= \bigl\vert \mathrm{g}\bigl(t, w(t), w( \delta t), \beta _{w}(t)\bigr)-\mathrm{g}\bigl(t, \bar{w}(t), \bar{w}( \delta t), \bar{ \beta }_{\bar{w}}(t)\bigr) \bigr\vert \\ &\leq L_{\mathrm{g}}\bigl( \bigl\vert w(t)-\bar{w}(t) \bigr\vert + \bigl\vert w(\delta t)-\bar{w}(\delta t) \bigr\vert \bigr)+N_{ \mathrm{g}} \bigl\vert \beta _{w}(t)-\bar{\beta }_{\bar{w}}(t) \bigr\vert \\ &\leq \frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \bigl\vert w(t)-\bar{w}(t) \bigr\vert . \end{aligned}$$

Hence using the last inequality and assumption \((H_{3})\), from (12), we get

$$\begin{aligned} \Vert \mathcal{N}w-\mathcal{N}\bar{w} \Vert _{\mathscr{Z}}\leq{}& \biggl[ \frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr) \\ &{}+\frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr] \Vert w-\bar{w} \Vert _{ \mathscr{Z}}. \end{aligned}$$

Since

$$\begin{aligned} &\biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)\\ &\quad {}+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr]< 1 , \end{aligned}$$

therefore, by the Banach contraction principle, problem (2) has a unique solution. □

Theorem 2

If assumptions \((H_{1})\)\((H_{4})\) and the inequality are satisfied, then system (2) has at least one solution.

Proof

The proof is given in the following four steps.

Step 1: We show that the operator \(\mathcal{N}\) defined in (11) is continuous. We take a sequence \(w_{n}\in \mathscr{Z}\) such that \(w_{n}\rightarrow w \in \mathscr{Z}\). Consider

$$\begin{aligned} & \bigl\vert \mathcal{N}w_{n}(t)-\mathcal{N}w(t) \bigr\vert \\ &\quad \leq \frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t- s)^{\varsigma -1} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{\Gamma (\varsigma )} \sum_{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \bigl\vert \beta _{w,n}(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w,n}( s)-\beta _{w}(s) \bigr\vert \,ds+ \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{k} \frac{(t-t_{k})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k-1} \frac{(t-t_{k})(t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds+ \sum_{m=1}^{k} \frac{(t-t_{k})^{2}}{2\Gamma (\varsigma -2)} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}(s) \bigr\vert \,ds+ \sum_{m=1}^{k} \bigl\vert \mathcal{I}_{m}\bigl(w_{n}(t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{k-1}(t_{k}-t_{m}) \bigl\vert \hat{\mathcal{I}}_{m}\bigl(w_{n}(t_{m}) \bigr)- \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w_{n}(t_{m}) \bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{k}(t-t_{k}) \bigl\vert \mathcal{I}_{m}\bigl(w_{n}(t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} + \sum _{m=1}^{k-1}(t-t_{k}) (t_{k}-t_{m}) \bigl\vert \bar{\mathcal{I}}_{m} \bigl(w_{n}(t_{m})\bigr)- \bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ & \qquad{}+\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w_{n}(t_{m}) \bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+\frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \bigl\vert \beta _{w,n}(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w,n}( s)-\beta _{w}(s) \bigr\vert \,ds+ \sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \\ &\qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{n} \frac{(1-2t_{n})}{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w,n}(s) -\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds+ \sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}(s) \bigr\vert \,ds+ \frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{} +\frac{1}{2}\sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(w_{n}(t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2} \bigl\vert \hat{\mathcal{I}}_{m}\bigl(w_{n}(t_{m})\bigr)- \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w_{n}(t_{m}) \bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n}\frac{(1-2t_{n})}{4} \bigl\vert \mathcal{I}_{m}\bigl(w_{n}(t_{m}) \bigr)-\mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ & \qquad{}+\sum_{m=1}^{n-1}\frac{(1-2t_{n})(t_{n}-t_{m})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w_{n}(t_{m}) \bigr)-\bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+ \sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w_{n}(t_{m}) \bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds+ \sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{n} \frac{t(1-2t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds \\ & \qquad{}+\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds+ \frac{t}{2}\sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(w_{n}(t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w_{n}(t_{m}) \bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+\sum_{m=1}^{n}\frac{t(1-2t_{n})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w_{n}(t_{m}) \bigr)-\bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{}+ \frac{t^{2}}{4}\sum_{m=1}^{n} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w_{n}(t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert , \\ & \quad t\in I_{k}, k=1,2,\dots,n; \end{aligned}$$
(13)

where \(\beta _{w,n}(t),\beta _{w}(t)\in C(I,\mathbb {R})\), which satisfy the following function equations:

$$\begin{aligned} &\beta _{w,n}(t)=\mathrm{g}\bigl(t, w_{n}(t), w_{n}(\delta t), \beta _{w,n}(t)\bigr), \\ &\beta _{w}(t)=\mathrm{g}\bigl(t, w(t), w(\delta t), \beta _{w}(t)\bigr). \end{aligned}$$

By using \((H_{2})\), we obtain

$$\begin{aligned} \Vert \beta _{w,n}-\beta _{w} \Vert _{\mathscr{Z}}\leq \frac{2L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \Vert w_{n}-w \Vert _{\mathscr{Z}}. \end{aligned}$$

\(w_{n}\rightarrow w\) as \(n\rightarrow \infty \), this implies that \(\beta _{w,n}\rightarrow \beta _{w}\) as \(n\rightarrow \infty \). Moreover, every convergent sequence is bounded, hence let for each \(t\in PC(I,\mathbb {R})\) there exist \(\ell >0\) such that \(|\beta _{w,n}(t)|\leq \ell \) and \(|\beta _{w}(t)|\leq \ell \). Then

$$\begin{aligned} (t-s)^{\varsigma -1} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert &\leq (t-s)^{\varsigma -1} \bigl( \bigl\vert \beta _{w,n}( s) \bigr\vert + \bigl\vert \beta _{w}(s) \bigr\vert \bigr) \\ &\leq 2\ell (t-s)^{\varsigma -1},\\ (t_{m}-s)^{\varsigma -1} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert &\leq (t_{m}-s)^{\varsigma -1} \bigl( \bigl\vert \beta _{w,n}( s) \bigr\vert + \bigl\vert \beta _{w}(s) \bigr\vert \bigr) \\ &\leq 2\ell (t_{m}-s)^{\varsigma -1},\\ (t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert &\leq (t_{m}-s)^{\varsigma -2} \bigl( \bigl\vert \beta _{w,n}( s) \bigr\vert + \bigl\vert \beta _{w}(s) \bigr\vert \bigr) \\ &\leq 2\ell (t_{m}-s)^{\varsigma -2},\\ (t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert &\leq (t_{m}-s)^{\varsigma -3} \bigl( \bigl\vert \beta _{w,n}( s) \bigr\vert + \bigl\vert \beta _{w}(s) \bigr\vert \bigr) \\ &\leq 2\ell (t_{m}-s)^{\varsigma -3},\\ (1-s)^{\varsigma -2} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert &\leq (1-s)^{\varsigma -2} \bigl( \bigl\vert \beta _{w,n}( s) \bigr\vert + \bigl\vert \beta _{w}(s) \bigr\vert \bigr) \\ &\leq 2\ell (1-s)^{\varsigma -2},\\ (1-s)^{\varsigma -3} \bigl\vert \beta _{w,n}(s)-\beta _{w}( s) \bigr\vert &\leq (1-s)^{\varsigma -3} \bigl( \bigl\vert \beta _{w,n}( s) \bigr\vert + \bigl\vert \beta _{w}(s) \bigr\vert \bigr) \\ &\leq 2\ell (1-s)^{\varsigma -3}. \end{aligned}$$

For \(t\in PC(I,\mathbb {R})\), the functions \(s\rightarrow 2\ell (t_{m}-s)^{\varsigma -1}\), \(s\rightarrow 2\ell (t_{m}-s)^{\varsigma -1}\), \(s\rightarrow 2\ell (t_{m}-s)^{\varsigma -2}\), \(s\rightarrow 2\ell (t_{m}-s)^{\varsigma -3}\), \(s\rightarrow 2\ell (1-s)^{\varsigma -2}\), \(s\rightarrow 2\ell (1-s)^{\varsigma -3}\) are integrable on \([0,t]\). Thus, using the Lebesgue dominated convergent theorem, from (13) we have \(|\mathcal{N}w_{n}(t)-\mathcal{N}w(t)|\rightarrow 0\) as \(t\rightarrow \infty \), which implies \(\|\mathcal{N}w_{n}-\mathcal{N}w\|_{\mathscr{Z}}\rightarrow 0\) as \(t\rightarrow \infty \). Therefore, the operator \(\mathcal{N}\) is continuous.

Step 2: Next we show that \(\mathcal{N}\) is a bounded operator. For each \(w\in \mathbf{S}_{\varrho }=\{w\in \mathscr{Z}: \|w\|_{\mathscr{Z}} \leq \varrho \}\), . For \(t\in \mathrm{J_{k}}\),

$$\begin{aligned} \bigl\vert \mathcal{N}w(t) \bigr\vert \leq{}& \frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t- s)^{\varsigma -1} \bigl\vert \beta _{w}(s) \bigr\vert \,ds+ \frac{1}{\Gamma (\varsigma )}\sum _{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &{}+\sum _{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{} +\sum_{m=1}^{k}\frac{(t-t_{k})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &{}+\sum _{m=1}^{k-1} \frac{(t-t_{k})(t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{} +\sum_{m=1}^{k} \frac{(t-t_{k})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &{}+\sum _{m=1}^{k} \bigl\vert \mathcal{I}_{m} \bigl(w(t_{m})\bigr) \bigr\vert + \sum_{m=1}^{k-1}(t_{k}-t_{m}) \bigl\vert \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})^{2}}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \sum_{m=1}^{k}(t-t_{k}) \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &{}+\sum_{m=1}^{k-1}(t-t_{k}) (t_{k}-t_{m}) \bigl\vert \bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ &{} +\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &{}+\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{} +\sum_{m=1}^{n}\frac{(1-2t_{n})}{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &{}+\sum _{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ & {}+\sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &{}+\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{} +\frac{1}{2}\sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2} \bigl\vert \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &{} +\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &{} +\sum_{m=1}^{n}\frac{(1-2t_{n})}{4} \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \sum_{m=1}^{n-1}\frac{(1-2t_{n})(t_{n}-t_{m})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &{} + \sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ & {}+\frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{}+\sum _{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{} +\sum_{m=1}^{n}\frac{t(1-2t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &{}+\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ & {}+\frac{t}{2}\sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &{}+\sum_{m=1}^{n} \frac{t(1-2t_{n})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &{} +\frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &{} + \frac{t^{2}}{4}\sum _{m=1}^{n} \bigl\vert \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert ,\quad t \in I_{k}, k=1,2,\dots,n. \end{aligned}$$
(14)

Using \((H_{4})\) with \(\theta _{1}^{*}=\max_{t\in I}|\theta _{1}(t)|, \theta _{2}^{*}= \max_{t\in I}|\theta _{2}(t)|\), we have

$$\begin{aligned} \bigl\vert \beta _{w}(t) \bigr\vert &= \bigl\vert \mathrm{g} \bigl(t, w(t), w(\delta t), \beta _{w}(t)\bigr) \bigr\vert \\ &\leq \theta _{1}(t)+\theta _{2}(t) \bigl( \vert w \vert + \bigl\vert w(\delta t) \bigr\vert \bigr)+ \theta _{3}(t) \bigl\vert \beta _{w}(t) \bigr\vert . \end{aligned}$$

Taking maximum, we have

$$\begin{aligned} \max_{t\in I} \bigl\vert \beta _{w}(t) \bigr\vert \leq \theta _{1}^{*}+2\theta _{2}^{*} \varrho +\theta _{3}^{*} \bigl\vert \beta _{w}(t) \bigr\vert , \end{aligned}$$

which implies

$$ \max_{t\in I} \bigl\vert \beta _{w}(t) \bigr\vert \leq \frac{\theta _{1}^{*}+2\theta _{2}^{*}\varrho }{1-\theta _{3}^{*}}=: \mu. $$
(15)

Using the result (15), we obtain from (14)

Thus \(\mathcal{N}\) is a bounded operator.

Step 3: To show that \(\mathcal{N}\) is equicontinuous, we take \(w\in \mathbf{S}_{\xi }=\{w\in \mathscr{Z}: \|w\|_{\mathscr{Z}}\leq \xi \}\) and \(t_{1}, t_{2} \in I_{k}\) such that \(t_{2}>t_{1}\). We have

$$\begin{aligned} & \bigl\vert \mathcal{N}w(t_{2})-\mathcal{N}w(t_{1}) \bigr\vert \\ &\quad\leq \frac{1}{\Gamma (\varsigma )} \int _{t_{1}}^{t_{2}}(t_{2}- s)^{\varsigma -1} \bigl\vert \beta _{w}(s) \bigr\vert \,ds+ \frac{1}{\Gamma (\varsigma )}\sum _{0< t_{k}< t_{2}-t_{1}} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ & \qquad{}+\sum_{0< t_{k-1}< t_{2}-t_{1}} \frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w}(s) \bigr\vert \,ds+\sum _{0< t_{k-1}< t_{2}-t_{1}} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds+\sum _{0< t_{k}< t_{2}-t_{1}} \frac{(t_{2}-t_{1})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} +\sum_{0< t_{k-1}< t_{2}-t_{1}} \frac{(t_{2}-t_{1})(t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{w}(s)\,ds \\ &\qquad{}+\sum_{0< t_{k}< t_{2}-t_{1}} \frac{ \vert (t_{2}-t_{1})(t_{2}+t_{1}-2t_{k}) \vert }{2\Gamma (\varsigma -2)} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds+\sum _{0< t_{k}< t_{2}-t_{1}} \bigl\vert \mathcal{I}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+\sum_{0< t_{k-1}< t_{2}-t_{1}}(t_{k}-t_{m}) \bigl\vert \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{0< t_{k-1}< t_{2}-t_{1}}\frac{(t_{k}-t_{m})^{2}}{2} \bigl\vert \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{0< t_{k}< t_{2}-t_{1}}(t_{2}-t_{1}) \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ & \qquad{}+\sum_{0< t_{k-1}< t_{2}-t_{1}}(t_{k}-t_{m}) (t_{2}-t_{1}) \bigl\vert \bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+\sum_{0< t_{k}< t_{2}-t_{1}} \frac{ \vert (t_{2}-t_{1})(t_{2}+t_{1}-2t_{k}) \vert }{2} \bigl\vert \bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\frac{1}{2\Gamma (\varsigma )}\sum_{0< t_{n+1}< t_{2}-t_{1}} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &\qquad{}+\sum _{0< t_{n-1}< t_{2}-t_{1}} \frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} +\sum_{0< t_{n-1}< t_{2}-t_{1}} \frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds+\sum _{0< t_{n}< t_{2}-t_{1}} \frac{(1-2t_{n})}{4\Gamma (\varsigma -1)} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+\sum _{0< t_{n-1}< t_{2}-t_{1}} \frac{(1-2t_{n})(t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} +\sum_{0< t_{n}< t_{2}-t_{1}} \frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds+ \frac{1}{2}\sum _{0< t_{n}< t_{2}-t_{1}} \bigl\vert \mathcal{I}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ & \qquad{}+\sum_{0< t_{n-1}< t_{2}-t_{1}}\frac{(t_{n}-t_{m})}{2} \bigl\vert \hat{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{0< t_{n-1}< t_{2}-t_{1}} \frac{(t_{n}-t_{m})^{2}}{4} \bigl\vert \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{0< t_{n}< t_{2}-t_{1}}\frac{(1-2t_{n})}{4} \bigl\vert \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \sum _{0< t_{n-1}< t_{2}-t_{1}}\frac{(1-2t_{n})(t_{n}-t_{m})}{4} \bigl\vert \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{0< t_{n}< t_{2}-t_{1}}\frac{t_{n}(1-t_{n})}{4} \bigl\vert \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+ \frac{t}{2\Gamma (\varsigma -1)} \sum_{0< t_{n+1}< t_{2}-t_{1}} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} +\sum_{0< t_{n-1}< t_{2}-t_{1}} \frac{(t_{2}-t_{1})(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}( s) \bigr\vert \,ds \\ &\qquad{}+\sum _{0< t_{n}< t_{2}-t_{1}} \frac{(t_{2}-t_{1})(1-2t_{n})}{4\Gamma (\varsigma -2)} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \frac{(t_{2}-t_{1})}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds+ \frac{ \vert t_{2}-t_{1} \vert }{2}\sum _{0< t_{n}< t_{2}-t_{1}} \bigl\vert \mathcal{I}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ & \qquad{}+\sum_{0< t_{n-1}< t_{2}-t_{1}}\frac{(t_{2}-t_{1})(t_{n}-t_{m})}{2} \bigl\vert \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+\sum _{0< t_{n}< t_{2}-t_{1}} \frac{(t_{2}-t_{1})(1-2t_{n})}{4} \bigl\vert \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\frac{({t_{2}}^{2}-{t_{1}}^{2})}{4\Gamma (\varsigma -2)}\sum_{0< t_{n+1}< t_{2}-t_{1}} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+\frac{({t_{2}}^{2}-{t_{1}}^{2})}{4} \sum _{0< t_{n}< t_{2}-t_{1}} \bigl\vert \bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr) \bigr\vert . \end{aligned}$$
(16)

Using assumptions \((H_{1})\), \((H_{4})\)\((H_{5})\), inequality (15) in (16) and evaluating, we can easily show that as \(t_{1}\) tends to \(t_{2}\) the right-hand side of (16) tends to 0. Thus by the Arzelà–Ascoli theorem the operator \(\mathcal{N}\) is completely continuous.

Step 4: In the final step we define a set \(\mathbf{S}_{\varrho ^{*}}=\{w\in \mathscr{Z}: {\varrho ^{*}} \mathcal{N}w, for 0<{\varrho ^{*}}<1\}\). We need to show that \(\mathbf{S}_{\varrho ^{*}}\) is bounded. Let \(w\in \mathbf{S}_{\varrho ^{*}}\), then by definition, we have \(w={\varrho ^{*}}\mathcal{N}w\). From Step 2, we obtain

Thus set \(\mathcal{E}\) is bounded. Therefore, by Schaefer’s fixed point theorem, system (2) has at least one solution. □

Stability results

In this section we investigate the results related to Hyers–Ulam stability of system (2).

Theorem 3

If assumptions \((H_{1})\)\((H_{3})\) and the inequality

$$ \biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr]< 1 $$

are satisfied, then system (2) is Hyers–Ulam stable.

Proof

Let ν be any solution of inequality (5). Then, by Remark 1, we write

$$ \textstyle\begin{cases} {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t)=\mathrm{g}(t, \nu (t), \nu (\delta t), {_{0}^{C}\mathbb {D}_{t}^{\varsigma }} \nu (t))+\psi (t) \varsigma \in (2,3],\quad 0< \delta < 1, \\ \Delta \nu (t_{k})=\mathcal{I}_{k}(\nu (t_{k}))+\psi _{k} \\ \Delta \nu '(t_{k})-\hat{\mathcal{I}}_{k}(\nu (t_{k}))+\psi _{k} \\ \Delta \nu ''(t_{k})-\bar{\mathcal{I}}_{k}(\nu (t_{k}))+\psi _{k} \\ \nu (0)=-\nu (1), \qquad \nu '(0)=-\nu '(1), \qquad\nu ''(0)=-\nu ''(1). \end{cases} $$
(17)

By Corollary 1, the solution of (17) for \(t\in I_{0}\) is given by

$$ \nu (t)=\frac{1}{\Gamma (\varsigma )} \int _{0}^{t}(t- s)^{\varsigma -1}\beta _{\nu }(s)\,ds+ \frac{1}{\Gamma (\varsigma )} \int _{0}^{t}(t-s)^{ \varsigma -1}\psi (s)\,ds- \mathcal{M}^{*} $$

and the solution of (17) for \(t\in I_{k}\), \(k=1,2,\dots,n\), is given by

$$\begin{aligned} \nu (t)={}&\frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t- s)^{\varsigma -1}\beta _{\nu }(s)\,ds+ \frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t-s)^{ \varsigma -1}\psi (s)\,ds \\ &{}+ \frac{1}{\Gamma (\varsigma )} \sum_{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{ \varsigma -1} \beta _{\nu }(s)\,ds \\ &{} +\frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \psi (s)\,ds+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \beta _{\nu }(s)\,ds \\ &{} +\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \\ &{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \psi (s)\,ds+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{\nu }(s)\,ds \\ &{}+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \\ &{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \psi (s)\,ds+\sum_{m=1}^{k} \frac{(t-t_{k})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \beta _{\nu }(s)\,ds+\sum_{m=1}^{k} \frac{(t-t_{k})}{\Gamma (\varsigma -1)} \\ &{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \psi (s)\,ds+\sum_{m=1}^{k-1} \frac{(t-t_{k})(t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{\nu }(s)\,ds \\ &{} +\sum_{m=1}^{k-1} \frac{(t-t_{k})(t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \psi (s)\,ds \\ &{}+\sum_{m=1}^{k} \frac{(t-t_{k})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{\nu }(s)\,ds \\ &{} +\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \psi ( s)\,ds+\sum_{m=1}^{k} \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr) \\ &{}+ \sum _{m=1}^{k}\psi _{k}+\sum _{m=1}^{k-1}(t_{k}-t_{m}) \hat{ \mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr) \\ &{} +\sum_{m=1}^{k-1}(t_{k}-t_{m}) \psi _{k}+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2}\bar{\mathcal{I}}_{m}\bigl(\nu (t_{m}) \bigr) \\ &{}+\sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2}\psi _{k}+\sum_{m=1}^{k}(t-t_{k}) \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr) \\ &{} +\sum_{m=1}^{k}(t-t_{k})\psi _{k}+\sum_{m=1}^{k-1}(t-t_{k}) (t_{k}-t_{m}) \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)+\sum_{m=1}^{k-1}(t-t_{k}) (t_{k}-t_{m}) \psi _{k} \\ & {}+\sum_{m=1}^{k}\frac{(t-t_{k})^{2}}{2}\bar{ \mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)+ \sum _{m=1}^{k}\frac{(t-t_{k})^{2}}{2}\psi _{k}- \mathcal{M}^{*}, \end{aligned}$$

where

$$\begin{aligned} \beta _{\nu }=\mathrm{g}\bigl(t, \nu (t), \nu (\delta t), {_{0}^{C} \mathbb {D}_{t}^{\varsigma }} \nu (t) \bigr), \end{aligned}$$

and

$$\begin{aligned} \mathcal{M}^{*}={}&\frac{1}{2\Gamma (\varsigma )}\sum _{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \beta _{ \nu }(s)\,ds+\frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1}\psi ( s)\,ds \\ &{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \beta _{\nu }( s)\,ds+\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \psi (s)\,ds \\ &{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{ \nu }(s)\,ds+\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \psi (s)\,ds \\ &{} +\sum_{m=1}^{n}\frac{(1-2t_{n})}{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \beta _{\nu }( s)\,ds+\sum_{m=1}^{n} \frac{(1-2t_{n})}{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \psi (s)\,ds \\ & {}+\sum_{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{\nu }(s)\,ds\\ &{}+\sum_{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \psi (s)\,ds \\ &{} -\sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{\nu }( s)\,ds-\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \psi (s)\,ds \\ &{} -\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2}\beta _{\nu }(s)\,ds- \frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2}\psi (s)\,ds \\ &{}+ \frac{1}{2}\sum_{m=1}^{n} \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)+\frac{1}{2} \sum_{m=1}^{n}\psi _{k} \\ &{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2}\hat{ \mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)+ \sum _{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2}\psi _{k} \\ &{}+\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4}\bar{ \mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)+\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4}\psi _{k} \\ &{} +\sum_{m=1}^{n}\frac{(1-2t_{n})}{4} \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)+ \sum _{m=1}^{n}\frac{(1-2t_{n})}{4}\psi _{k}+\sum _{m=1}^{n-1} \frac{(1-2t_{n})(t_{n}-t_{m})}{4}\bar{ \mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr) \\ &{} +\sum_{m=1}^{n-1}\frac{(1-2t_{n})(t_{n}-t_{m})}{4}\psi _{k}-\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4}\bar{\mathcal{I}}_{m}\bigl(\nu (t_{m}) \bigr)-\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4}\psi _{k} \\ &{} +\frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \beta _{\nu }(s)\,ds+ \frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \psi (s)\,ds \\ &{} +\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{ \nu }(s)\,ds+\sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \psi (s)\,ds \\ & {}+\sum_{m=1}^{n}\frac{t(1-2t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \beta _{\nu }( s)\,ds+\sum_{m=1}^{n} \frac{t(1-2t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \psi (s)\,ds \\ &{} -\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3}\beta _{\nu }(s)\,ds- \frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3}\psi (s)\,ds \\ &{}+ \frac{t}{2}\sum_{m=1}^{n} \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr) \\ & {}+\frac{t}{2}\sum_{m=1}^{n}\psi _{k}+\sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2}\bar{\mathcal{I}}_{m}\bigl(\nu (t_{m}) \bigr)+\sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2}\psi _{k} \\ &{}+\sum_{m=1}^{n} \frac{t(1-2t_{n})}{4}\bar{\mathcal{I}}_{m}\bigl(\nu (t_{m}) \bigr) \\ &{} +\sum_{m=1}^{n}\frac{t(1-2t_{n})}{4}\psi _{k}+ \frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \beta _{\nu }(s)\,ds+ \frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \\ &{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \psi (s)\,ds-\frac{t^{2}}{4}\sum_{m=1}^{n} \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \frac{t^{2}}{4}\sum_{m=1}^{n} \psi _{k}. \end{aligned}$$

We consider, for \(t\in I_{k}\),

$$\begin{aligned} & \bigl\vert \nu (t)-w(t) \bigr\vert \\ &\quad \leq \frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t- s)^{\varsigma -1} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+\frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t- s)^{\varsigma -1} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+ \frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k}\frac{ \vert t-t_{k} \vert }{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{k} \frac{ \vert t-t_{k} \vert }{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k-1} \frac{ \vert t-t_{k} \vert (t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+ \sum_{m=1}^{k-1} \frac{ \vert t-t_{k} \vert (t_{k}-t_{m})}{\Gamma (\varsigma -2)} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds+\sum_{m=1}^{k} \frac{ \vert (t-t_{k})^{2} \vert }{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{k}\frac{ \vert (t-t_{k})^{2} \vert }{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{}+\sum_{m=1}^{k} \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{k} \vert \psi _{k} \vert \\ & \qquad{}+\sum_{m=1}^{k-1}(t_{k}-t_{m}) \bigl\vert \hat{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{k-1}(t_{k}-t_{m}) \vert \psi _{k} \vert + \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2} \\ & \qquad{}\times \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)-\bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert + \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2} \vert \psi _{k} \vert \\ &\qquad{} +\sum _{m=1}^{k} \vert t-t_{k} \vert \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ & \qquad{}+\sum_{m=1}^{k} \vert t-t_{k} \vert \vert \psi _{k} \vert +\sum _{m=1}^{k-1} \vert t-t_{k} \vert (t_{k}-t_{m}) \bigl\vert \bar{\mathcal{I}}_{m} \bigl(\nu (t_{m})\bigr)-\bar{\mathcal{I}}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+ \sum_{m=1}^{k-1} \vert t-t_{k} \vert (t_{k}-t_{m}) \vert \psi _{k} \vert \\ &\qquad{} +\sum_{m=1}^{k}\frac{ \vert (t-t_{k})^{2} \vert }{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl( \nu (t_{m})\bigr)-\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{k} \frac{ \vert (t-t_{k})^{2} \vert }{2} \vert \psi _{k} \vert +\frac{1}{2\Gamma (\varsigma )} \sum_{m=1}^{n+1} \\ &\qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \bigl\vert \beta _{\nu }(s)-\beta _{w}(s) \bigr\vert \,ds+ \frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n}\frac{ \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{n} \frac{ \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1} \frac{ \vert 1-2t_{n} \vert (t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+ \sum_{m=1}^{n-1} \frac{ \vert 1-2t_{n} \vert (t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \\ & \qquad{}\times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds+\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ & \qquad{}+\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds+\frac{1}{2} \sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \frac{1}{2}\sum_{m=1}^{n} \vert \psi _{k} \vert \\ & \qquad{}+\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2} \bigl\vert \hat{\mathcal{I}}_{m}\bigl( \nu (t_{m})\bigr)-\hat{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2} \vert \psi _{k} \vert +\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4} \\ &\qquad{} \times \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)-\bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert + \sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4} \vert \psi _{k} \vert \\ &\qquad{} +\sum _{m=1}^{n} \frac{ \vert 1-2t_{n} \vert }{4} \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)-\mathcal{I}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n}\frac{ \vert 1-2t_{n} \vert }{4} \vert \psi _{k} \vert +\sum_{m=1}^{n-1} \frac{ \vert 1-2t_{n}|(t_{n}-t_{m}) \vert }{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert \\ & \qquad{}+\sum_{m=1}^{n-1}\frac{ \vert 1-2t_{n} \vert (t_{n}-t_{m})}{4} \vert \psi _{k} \vert + \sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert \\ &\qquad{}+\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4} \vert \psi _{k} \vert \\ &\qquad{} +\frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{}+ \frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \psi ( s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{n}\frac{t \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{}+ \sum_{m=1}^{n} \frac{t \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+ \frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{t}{2}\sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \frac{t}{2}\sum_{m=1}^{n} \vert \psi _{k} \vert \\ &\qquad{}+\sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert \\ & \qquad{}+\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2} \vert \psi _{k} \vert +\sum_{m=1}^{n} \frac{t \vert 1-2t_{n} \vert }{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert +\sum_{m=1}^{n} \frac{t \vert 1-2t_{n} \vert }{4} \vert \psi _{k} \vert \\ &\qquad{} +\frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{}+ \frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{} +\frac{t^{2}}{4}\sum_{m=1}^{n} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \frac{t^{2}}{4}\sum_{m=1}^{n} \vert \psi _{k} \vert . \end{aligned}$$
(18)

Taking into account assumptions \((H_{1})\)\((H_{3})\) and taking maximum value, we obtain

$$\begin{aligned} & \Vert \nu -w \Vert _{\mathscr{Z}} \\ &\quad\leq \biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}\\ &\qquad{}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr] \Vert \nu -w \Vert _{ \mathscr{Z}} \\ &\qquad{} + \biggl(\frac{3(n+1)}{2\Gamma (\varsigma +1)}+ \frac{13n-3}{4\Gamma (\varsigma )}+ \frac{15n-8}{4\Gamma (\varsigma -1)}+ \frac{34n-16}{4} \biggr)\epsilon. \end{aligned}$$

This inequality implies

$$ \Vert \nu -w \Vert _{\mathscr{Z}}\leq \frac{ (\frac{3(n+1)}{2\Gamma (\varsigma +1)}+\frac{13n-3}{4\Gamma (\varsigma )}+\frac{15n-8}{4\Gamma (\varsigma -1)}+\frac{34n-16}{4} )\epsilon }{1- [\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} (\frac{3(n+1)}{\Gamma (\varsigma +1)}+\frac{13n-3}{2\Gamma (\varsigma )}+\frac{15n-8}{2\Gamma (\varsigma -1)} )+\frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} ]} .$$

Or

$$ \Vert \nu -w \Vert _{\mathscr{Z}}\leq C_{\mathrm{g}}\epsilon, $$

where

$$ C_{\mathrm{g}}= \frac{ (\frac{3(n+1)}{2\Gamma (\varsigma +1)}+\frac{13n-3}{4\Gamma (\varsigma )}+\frac{15n-8}{4\Gamma (\varsigma -1)}+\frac{34n-16}{4} )}{1- [\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} (\frac{3(n+1)}{\Gamma (\varsigma +1)}+\frac{13n-3}{2\Gamma (\varsigma )}+\frac{15n-8}{2\Gamma (\varsigma -1)} )+\frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} ]} $$

with

$$\begin{aligned} &\biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)} + \frac{13n-3}{2\Gamma (\varsigma )}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)\\ &\quad{}+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr]< 1 . \end{aligned}$$

Therefore, problem (2) is Hyers–Ulam stable. □

Corollary 2

In Theorem 3, if we set \(\phi (\epsilon )=C_{\mathrm{g}}(\epsilon )\) such that \(\phi (0)=0\), then problem (2) becomes generalized Hyers–Ulam stable.

For the next coming result, we assume that

\((H_{6})\):

There exist a non-decreasing function \(\phi \in {C(I, \mathbb {R})}\) and constants \(\lambda _{\phi }>0, \epsilon >0\) such that the following inequality holds:

$$ {_{0}I_{t}}^{\varsigma }\phi (t)\leq \lambda _{\phi }\phi (t). $$

Theorem 4

If assumptions \((H_{1})\)\((H_{3})\), \((H_{6})\) and the inequality

$$ \biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr]< 1 $$

are satisfied, then system (2) is Hyers–Ulam–Rassias stable with respect to \((\varphi,\phi )\), where ϕ is a nondecreasing function and \(\varphi \geq 0\).

Proof

Let ν be any solution of inequality (7) and w be the unique solution of problem (2). Then, from the proof of 3, we have the following inequality:

$$\begin{aligned} & \bigl\vert \nu (t)-w(t) \bigr\vert \\ &\quad \leq \frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t- s)^{\varsigma -1} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+\frac{1}{\Gamma (\varsigma )} \int _{t_{k}}^{t}(t- s)^{\varsigma -1} \bigl\vert \psi (s) \bigr\vert \,ds \\ & \qquad{}+\frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+ \frac{1}{\Gamma (\varsigma )}\sum_{m=1}^{k} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})}{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k-1}\frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k}\frac{ \vert t-t_{k} \vert }{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{k} \frac{ \vert t-t_{k} \vert }{\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k-1} \frac{ \vert t-t_{k} \vert (t_{k}-t_{m})}{\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+ \sum_{m=1}^{k-1} \frac{ \vert t-t_{k} \vert (t_{k}-t_{m})}{\Gamma (\varsigma -2)} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds+\sum_{m=1}^{k} \frac{ \vert (t-t_{k})^{2} \vert }{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{k}\frac{ \vert (t-t_{k})^{2} \vert }{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{k} \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{k} \vert \psi _{k} \vert \\ &\qquad{} +\sum_{m=1}^{k-1}(t_{k}-t_{m}) \bigl\vert \hat{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \hat{\mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{k-1}(t_{k}-t_{m}) \vert \psi _{k} \vert + \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2} \\ & \qquad{}\times \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)-\bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert + \sum_{m=1}^{k-1} \frac{(t_{k}-t_{m})^{2}}{2} \vert \psi _{k} \vert \\ &\qquad{} +\sum _{m=1}^{k} \vert t-t_{k} \vert \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{}+\sum_{m=1}^{k} \vert t-t_{k} \vert \vert \psi _{k} \vert +\sum_{m=1}^{k-1} \vert t-t_{k} \vert (t_{k}-t_{m}) \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)-\bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{k-1} \vert t-t_{k} \vert (t_{k}-t_{m}) \vert \psi _{k} \vert +\sum_{m=1}^{k} \frac{ \vert (t-t_{k})^{2} \vert }{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{k} \frac{ \vert (t-t_{k})^{2} \vert }{2} \vert \psi _{k} \vert \\ & \qquad{}+\frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -1} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{} + \frac{1}{2\Gamma (\varsigma )}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -1} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n}\frac{ \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{n} \frac{ \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -1)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\sum_{m=1}^{n-1} \frac{ \vert 1-2t_{n} \vert (t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{n-1} \frac{ \vert 1-2t_{n} \vert (t_{n}-t_{m})}{4\Gamma (\varsigma -2)} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds+\sum_{m=1}^{n} \frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{} +\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{} +\frac{1}{4\Gamma (\varsigma -1)} \int _{t_{n}}^{1}(1- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds+\frac{1}{2} \sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \frac{1}{2}\sum_{m=1}^{n} \vert \psi _{k} \vert \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{(t_{n}-t_{m})}{2} \bigl\vert \hat{\mathcal{I}}_{m}\bigl( \nu (t_{m})\bigr)-\hat{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{n-1} \frac{(t_{n}-t_{m})}{2} \vert \psi _{k} \vert +\sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4} \\ &\qquad{} \times \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)-\bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert + \sum_{m=1}^{n-1} \frac{(t_{n}-t_{m})^{2}}{4} \vert \psi _{k} \vert \\ &\qquad{} +\sum _{m=1}^{n} \frac{ \vert 1-2t_{n} \vert }{4} \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)-\mathcal{I}_{m} \bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n}\frac{ \vert 1-2t_{n} \vert }{4} \vert \psi _{k} \vert +\sum_{m=1}^{n-1} \frac{ \vert 1-2t_{n}|(t_{n}-t_{m}) \vert }{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n-1} \frac{ \vert 1-2t_{n} \vert (t_{n}-t_{m})}{4} \vert \psi _{k} \vert \\ &\qquad{} +\sum_{m=1}^{n}\frac{t_{n}(1-t_{n})}{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert +\sum _{m=1}^{n} \frac{t_{n}(1-t_{n})}{4} \vert \psi _{k} \vert +\frac{t}{2\Gamma (\varsigma -1)} \sum_{m=1}^{n+1} \\ &\qquad{} \times \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -2} \bigl\vert \beta _{\nu }(s)-\beta _{w}(s) \bigr\vert \,ds+ \frac{t}{2\Gamma (\varsigma -1)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -2} \bigl\vert \psi (s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ & \qquad{}+\sum_{m=1}^{n}\frac{t \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \beta _{ \nu }(s)-\beta _{w}(s) \bigr\vert \,ds \\ &\qquad{} + \sum_{m=1}^{n} \frac{t \vert 1-2t_{n} \vert }{4\Gamma (\varsigma -2)} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ & \qquad{}+\frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds+ \frac{t}{4\Gamma (\varsigma -2)} \int _{t_{n}}^{1}(1- s)^{\varsigma -3} \bigl\vert \psi (s) \bigr\vert \,ds \\ &\qquad{} +\frac{t}{2}\sum_{m=1}^{n} \bigl\vert \mathcal{I}_{m}\bigl(\nu (t_{m})\bigr)- \mathcal{I}_{m}\bigl(w(t_{m})\bigr) \bigr\vert \\ &\qquad{} + \frac{t}{2}\sum_{m=1}^{n} \vert \psi _{k} \vert + \sum_{m=1}^{n-1} \frac{t(t_{n}-t_{m})}{2} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert \\ &\qquad{} +\sum_{m=1}^{n-1}\frac{t(t_{n}-t_{m})}{2} \vert \psi _{k} \vert +\sum_{m=1}^{n} \frac{t \vert 1-2t_{n} \vert }{4} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{\mathcal{I}}_{m}\bigl(w(t_{m}) \bigr) \bigr\vert +\sum_{m=1}^{n} \frac{t \vert 1-2t_{n} \vert }{4} \vert \psi _{k} \vert \\ & \qquad{}+\frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}- s)^{\varsigma -3} \bigl\vert \beta _{\nu }(s)-\beta _{w}( s) \bigr\vert \,ds \\ &\qquad{} + \frac{t^{2}}{4\Gamma (\varsigma -2)}\sum_{m=1}^{n+1} \int _{t_{m-1}}^{t_{m}}(t_{m}-s)^{\varsigma -3} \bigl\vert \psi ( s) \bigr\vert \,ds \\ &\qquad{} +\frac{t^{2}}{4}\sum_{m=1}^{n} \bigl\vert \bar{\mathcal{I}}_{m}\bigl(\nu (t_{m})\bigr)- \bar{ \mathcal{I}}_{m}\bigl(w(t_{m})\bigr) \bigr\vert + \frac{t^{2}}{4}\sum_{m=1}^{n} \vert \psi _{k} \vert . \end{aligned}$$
(19)

Using assumptions \((H_{1})\)\((H_{3})\) and \((H_{6})\), we get the following result in its simplified form:

$$\begin{aligned} & \Vert \nu -w \Vert _{\mathscr{Z}} \\ &\quad\leq \biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}\\ &\qquad{}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr] \Vert \nu -w \Vert _{ \mathscr{Z}} \\ & \qquad{}+\epsilon \bigl(\phi (t)+\varphi \bigr) \biggl( \frac{\lambda _{\phi }(34n-9)+2(17n-8)}{4} \biggr), \end{aligned}$$

which implies

$$\begin{aligned} \Vert \nu -w \Vert _{\mathscr{Z}}\leq \frac{\epsilon (\phi (t)+\varphi ) (\frac{\lambda _{\phi }(34n-9)+2(17n-8)}{4} )}{1- [\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} (\frac{3(n+1)}{\Gamma (\varsigma +1)}+\frac{13n-3}{2\Gamma (\varsigma )}+\frac{15n-8}{2\Gamma (\varsigma -1)} )+\frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} ]}, \end{aligned}$$

or

$$\begin{aligned} \Vert \nu -w \Vert _{\mathscr{Z}}\leq C_{\mathrm{g}}\epsilon \bigl(\phi (t)+ \varphi \bigr), \end{aligned}$$
(20)

where

$$\begin{aligned} C_{\mathrm{g}}=\frac{ (\frac{\lambda _{\phi }(34n-9)+2(17n-8)}{4} )}{1- [\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} (\frac{3(n+1)}{\Gamma (\varsigma +1)}+\frac{13n-3}{2\Gamma (\varsigma )}+\frac{15n-8}{2\Gamma (\varsigma -1)} )+\frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} ]}. \end{aligned}$$

Therefore, problem (2) is Hyers–Ulam–Rassias stable. □

Application

In this section we provide a numerical problem to verify the applications of our main results.

Example 1

Consider the following problem:

$$ \textstyle\begin{cases} {_{0}^{C}\mathbb {D}_{t}^{\frac{5}{2}}}w(t)=\frac{e^{-\pi t}}{12}+ \frac{e^{-t}}{44+t^{2}} (\sin ( \vert w(t) \vert )+w (\frac{1}{3}t )+ \sin ( \vert {_{0}^{C}\mathbb {D}_{t}^{\frac{5}{2}}}w(t) \vert ) ),\\ \quad t\in [0,1], t\neq \frac{1}{3}, k=1, \\\Delta w(\frac{1}{3})=\mathcal{I}_{1}(w(\frac{1}{3}))= \frac{ \vert w(\frac{1}{3}) \vert }{26+ \vert w(\frac{1}{3}) \vert }, \\\Delta w'(\frac{1}{3})=\hat{\mathcal{I}}_{1}(w(\frac{1}{3}))= \frac{ \vert w(\frac{1}{3}) \vert }{25+ \vert w(\frac{1}{3}) \vert }, \\\Delta w''(\frac{1}{3})=\bar{\mathcal{I}}_{1}(w(\frac{1}{3}))= \frac{ \vert w(\frac{1}{3}) \vert }{20+ \vert w(\frac{1}{3}) \vert }, \\w(0)=-w(1), \\w'(0)=-w'(1), \\w''(0)=-w''(1), \end{cases} $$
(21)

where e is an exponential function.

Here,

$$\begin{aligned} \mathrm{g}\bigl(t,w(t), w(\delta t), {_{0}^{C} \mathbb {D}_{t}^{\frac{5}{2}}}w(t)\bigr)=\frac{e^{-\pi t}}{12}+ \frac{\exp (-t)}{44+t^{2}} \biggl(\sin \bigl( \bigl\vert w(t) \bigr\vert \bigr)+ w \biggl(\frac{1}{3}t \biggr)+\sin \bigl( \bigl\vert {_{0}^{C} \mathbb {D}_{t}^{\frac{5}{2}}}w(t) \bigr\vert \bigr) \biggr), \end{aligned}$$

with \(\varsigma =\frac{5}{2}\), \(\delta =\frac{1}{3}\). The continuity of g is obvious.

By hypothesis \((H_{2})\), for any \(w, \bar{w} \in \mathbb {R}\), we have

$$\begin{aligned} &\bigl\vert \mathrm{g}\bigl(t,w(t), w(\lambda _{\phi } t),{_{0}^{C}\mathbb {D}_{t}^{ \frac{5}{2}}}w(t) \bigr)-\mathrm{g}\bigl(t,\bar{w}(t), \bar{w}(\lambda _{\phi } t),{_{0}^{C} \mathbb {D}_{t}^{\frac{5}{2}}} \bar{w}(t)\bigr) \bigr\vert \\ &\quad \leq \frac{1}{44} \bigl[2 \bigl\vert w(t)- \bar{w}(t) \bigr\vert + \bigl\vert {_{0}^{C}\mathbb {D}_{t}^{\frac{5}{2}}}w(t)-{_{0}^{C} \mathbb {D}_{t}^{ \frac{5}{2}}}\bar{w}(t) \bigr\vert \bigr]. \end{aligned}$$

Hence g satisfies hypothesis \((H_{2})\) with \(L_{\mathrm{g}}=N_{\mathrm{g}}=\frac{1}{44}\). Also hypothesis \((H_{4})\) holds with \(\theta _{0}(t)=\frac{\exp (-\pi t)}{12}\), \(\theta _{1}(t)=\theta _{2}(t)=\frac{\exp (-t)}{44+t}\), where \(\theta _{0}^{*}(t)=\frac{1}{12}\), \(\theta _{1}^{*}(t)=\theta _{2}^{*}(t)=\frac{1}{44}\).

At \(t_{1}=\frac{1}{3}\) the impulsive conditions are given as follows:

$$\begin{aligned} &\mathcal{I}_{1}w\biggl(\frac{1}{3}\biggr)= \frac{ \vert w(\frac{1}{3}) \vert }{26+ \vert w(\frac{1}{3}) \vert }, \\ &\hat{\mathcal{I}}_{1}w'\biggl(\frac{1}{3}\biggr)= \frac{ \vert w(\frac{1}{3}) \vert }{25+ \vert w(\frac{1}{3}) \vert }, \\ &\bar{\mathcal{I}}_{1}w''\biggl( \frac{1}{3}\biggr)=\frac{ \vert w(\frac{1}{3}) \vert }{20+ \vert w(\frac{1}{3}) \vert }. \end{aligned}$$

For any \(w, \bar{w}\in \mathrm{E}\), we have

$$\begin{aligned} &\biggl\vert \mathcal{I}_{1}\biggl(w\biggl(\frac{1}{3}\biggr) \biggr)-\mathcal{I}_{1}\biggl(\bar{w}\biggl(\frac{1}{3}\biggr) \biggr) \biggr\vert = \biggl\vert \frac{ \vert w(\frac{1}{3}) \vert }{26+ \vert w(\frac{1}{3}) \vert }- \frac{ \vert w(\frac{1}{3}) \vert }{26+ \vert \bar{w}(\frac{1}{3}) \vert } \biggr\vert \leq \frac{1}{26} \biggl\vert w\biggl(\frac{1}{3} \biggr)-\bar{w}\biggl(\frac{1}{3}\biggr) \biggr\vert ,\\ &\biggl\vert \hat{\mathcal{I}}_{1}\biggl(w\biggl(\frac{1}{3} \biggr)\biggr)-\hat{\mathcal{I}}_{1}\biggl(\bar{w}\biggl( \frac{1}{3}\biggr)\biggr) \biggr\vert = \biggl\vert \frac{ \vert w(\frac{1}{3}) \vert }{25+ \vert w(\frac{1}{3}) \vert }- \frac{ \vert w(\frac{1}{3}) \vert }{25+ \vert \bar{w}(\frac{1}{3}) \vert } \biggr\vert \leq \frac{1}{25} \biggl\vert w\biggl( \frac{1}{3}\biggr)-\bar{w}\biggl(\frac{1}{3}\biggr) \biggr\vert \end{aligned}$$

and

$$ \biggl\vert \bar{\mathcal{I}}_{1}\biggl(w\biggl(\frac{1}{3} \biggr)\biggr)-\bar{\mathcal{I}}_{1}\biggl(\bar{w}\biggl( \frac{1}{3}\biggr)\biggr) \biggr\vert = \biggl\vert \frac{ \vert w(\frac{1}{3}) \vert }{20+ \vert w(\frac{1}{3}) \vert }- \frac{ \vert w(\frac{1}{3}) \vert }{20+ \vert \bar{w}(\frac{1}{3}) \vert } \biggr\vert \leq \frac{1}{20} \biggl\vert w\biggl( \frac{1}{3}\biggr)-\bar{w}\biggl(\frac{1}{3}\biggr) \biggr\vert ,$$

which satisfy \((H_{3})\) with \(C_{1}=\frac{1}{26}\), \(C_{2}=\frac{1}{25}\), \(C_{3}=\frac{1}{20}\). So we have

$$\begin{aligned} & \biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr] \\ & \quad=\frac{1771}{25\text{,}800\sqrt{\pi }}< 1. \end{aligned}$$

Therefore, by Theorem 1, problem (21) has a unique solution. By Theorem 3, problem (21) is Hyers–Ulam stable. For any \(t\in [0,1]\), we set \(\phi (t)=t\), \(\varphi =1\). Then

$$\begin{aligned} {_{0}\mathrm{I}_{t}}^{\frac{5}{2}}\phi (t)&= \frac{1}{\Gamma (\frac{5}{2})} \int _{0}^{t} (t-s)^{\frac{5}{2}-1}s \,ds \\ &\leq \frac{8}{15\sqrt{\pi }}t. \end{aligned}$$

We see assumption \((H_{6})\) holds with \(\lambda _{\phi }= \frac{8}{15\sqrt{\pi }}\). Also since

$$\begin{aligned} & \biggl[\frac{L_{\mathrm{g}}}{(1-N_{\mathrm{g}})} \biggl( \frac{3(n+1)}{\Gamma (\varsigma +1)}+ \frac{13n-3}{2\Gamma (\varsigma )}+ \frac{15n-8}{2\Gamma (\varsigma -1)} \biggr)+ \frac{13nC_{1}+5(3n-2)C_{2}+6(n-1)C_{3}}{4} \biggr] \\ &\quad =\frac{1771}{25\text{,}800\sqrt{\pi }}< 1. \end{aligned}$$

Therefore, the numerical problem (21) is Hyers–Ulam–Rassias stable with respect to \((\varphi,\phi )\).

Conclusion

In this paper, using Schaefer’s fixed point theorem, we derived a result of at least one solution to system (2). By the application of Banach contraction theorem, we obtained conditions for unique solution of problem (2). Further, by the applications of qualitative theory and nonlinear functional analysis, we investigated Ulam–Hyers stability to the considered system. We applied our obtained results to a numerical problem.

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Acknowledgements

We are really thankful to the reviewers for their useful suggestions.

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All authors equally contributed to this manuscript and approved the final version.

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Correspondence to Thabet Abdeljawad or Hasib Khan.

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Ali, A., Shah, K., Abdeljawad, T. et al. Study of fractional order pantograph type impulsive antiperiodic boundary value problem. Adv Differ Equ 2020, 572 (2020). https://doi.org/10.1186/s13662-020-03032-x

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Keywords

  • Delay differential equation
  • Impulsive conditions
  • Schaefer’s fixed point theorem
  • Hyers–Ulam stability