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Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

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Abstract

In this paper, we consider a nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses. We study the existence, uniqueness and generalized Ulam–Hyers–Rassias stability of the proposed model with the help of fixed point approach, over generalized complete metric space. We give an example which supports our main result.

Introduction

At Wisconsin university, Ulam raised a question about the stability of functional equations in 1940. The question of Ulam was: Under what conditions does there exist an additive mapping near an approximately additive mapping?; see [30]. In 1941, Hyers was the first mathematician who gave a partial answer to Ulam’s question [12] in a Banach space. Since then, stability of such form is known as Ulam–Hyers stability. In 1978, Rassias [23] provided a remarkable generalization of the Ulam–Hyers stability of mappings by considering variables. For more information about the topic, we refer the reader to [3, 14,15,16, 24, 28, 31, 40, 42].

An equation of the form \(m\,\frac{d^{2}X}{dt^{2}}=\lambda \,\frac{dX}{dt}+ \eta (t)\) is called Langevin equation, introduced by Paul Langevin in 1908. Langevin equations have been widely used to describe stochastic problems in physics, chemistry and electrical engineering. For example, Brownian motion is well described by the Langevin equation when the random fluctuation force is assumed to be white noise. For the removal of noise, mathematicians used fractional order differential equations, which also perform well in reducing the staircase effects compared to integer order differential equations. Thus it is very important to study Langevin equations with fractional derivatives; see, for instance, [2, 10, 20, 21].

Fractional order differential equations are generalizations of the classical integer order differential equations. Fractional calculus has become a fast developing area, and its applications can be found in diverse fields ranging from physical sciences, porous media, electrochemistry, economics, electromagnetics, medicine and engineering to biological sciences. Progressively, fractional differential equations play a very important role in thermodynamics, statistical physics, viscoelasticity, nonlinear oscillation of earthquakes, defence, optics, control, electrical circuits, signal processing, astronomy, etc. There are some outstanding articles which provide the main theoretical tools for the qualitative analysis of this research field, and at the same time, show the interconnection as well as the distinction between integral models, classical and fractional differential equations; see [1, 5, 13, 17, 19, 22, 25,26,27, 29].

Impulsive fractional differential equations are used to describe both physical and social sciences. Also they describe many practical dynamical systems such as evolutionary processes, characterized by abrupt changes of the state at certain instants. In the last few decades, the theory of impulsive fractional differential equations were well utilized in medicine, mechanical engineering, ecology, biology and astronomy, etc. There are some remarkable monographs [8, 11, 18, 32, 33, 35,36,37, 39, 41], which consider fractional differential equations with impulses.

Recently, many mathematicians devoted considerable attention to the existence, uniqueness and different types of Hyers–Ulam stability of the solutions of nonlinear implicit fractional differential equations with Caputo fractional derivative, see [4, 6, 7].

Wang et al. [34] studied generalized Ulam–Hyers–Rassias stability of the following fractional differential equation

$$ \textstyle\begin{cases} {{}^{c}}D_{0,t}^{\alpha }x(t)=f(t,x(t)),\quad t\in (t_{k},s_{k}], k=0,1,\dots ,m, 0< \alpha < 1, \\ x(t)=g_{k}(t,x(t)),\quad t\in (s_{k-1},t_{k}],k=1,2,\dots ,m. \end{cases} $$

Zada et al. [38] studied existence and uniqueness of solutions by using Diaz–Margolis’s fixed point theorem and presented Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability for a class of nonlinear implicit fractional differential equation with noninstantaneous integral impulses and nonlinear integral boundary condition:

$$ \textstyle\begin{cases} {{}^{c}}D_{0,t}^{\alpha }x(t)=f(t,x(t),{^{c}}D_{0,t}^{\alpha }x(t)), \quad t\in (t_{k},s_{k}], k=0,1,\dots ,m, 0< \alpha < 1, t\in (0,1], \\ x(t)=I_{s_{k-1},t_{k}}^{\alpha }(\xi _{k}(t,x(t))),\quad t\in (s_{k-1},t _{k}], k=0,1,\dots ,m, \\ x(0)=\frac{1}{\varGamma {\alpha }}\int _{0}^{T}(T-s)^{\alpha -1}\eta (s,x(s))\,ds. \end{cases} $$

Motivated by [34, 38], we consider the following nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses:

$$ \textstyle\begin{cases} {{}^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(t)\\ \quad =f(t,x(t), {{}^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(t)) \\ \qquad {}+\int _{0}^{t}\frac{(t-s)^{\sigma -1}}{\varGamma (\delta )}f(s,x(s))\,ds, \quad t\in (t_{k},s_{k}], k=0,1,\dots ,m, \\ x(t)=g_{k}(t,x(t)),\quad t\in (s_{k-1},t_{k}],k=1,2,\dots ,m, \\ x(0)=x_{0},\qquad x(T)=\theta \int _{0}^{\eta }\frac{1}{\varGamma {p}}( \eta -s)^{p-1}x(s)\,ds,\quad 0< \eta < T, \end{cases} $$
(1.1)

where \({^{c}}D^{\alpha }_{0,t}\) and \({^{c}}D^{\beta }_{0,t}\) represent classical Caputo derivatives [5] of order α and β with the lower bound zero, \(0=t_{0}< s_{0}< t_{1}< s_{1}<\cdots <t_{m} < s_{m}=\tau \), τ is the free fixed number and \(\lambda \in \mathbb{R}\setminus \{0\}\), \(0<\alpha \), \(\beta <1\), \(0< \alpha +\beta <2\), \(\sigma , p>0\), \(x_{0}\), θ are constants, \(f:[0,\tau ]\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous and \(g_{k}: [s_{k-1},t_{k}]\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous for all \(k=1,2,\dots ,m\).

In Sect. 2, we create a uniform framework to originate appropriate formula of solutions for our proposed model. In Sect. 3, we study the concept of generalized Ulam–Hyers–Rassias stability of Eq. (1.1). Finally, we give an example to illustrate our main result.

Solution framework of linear impulsive fractional Langevin equation

Let \(J=[0,\tau ]\) and \(C(J,\mathbb{R})\) be the space of all continuous functions from J to \(\mathbb{R}\), and the piecewise continuous function space \(PC(J,\mathbb{R})=\{x:f\rightarrow \mathbb{R}: x\in ((t _{k},t_{k-1})],\mathbb{R}), k=0,\dots ,m\mbox{ and there exist } x(t_{k} ^{-}) \textrm{ and } x(t_{k}^{+}), k=1,2,\dots ,m \textrm{ with } x(t _{k}^{-})=x(t_{k}^{+})\}\).

In the current section, we create a uniform framework to originate an appropriate formula for the solution of impulsive fractional differential equation of the form:

$$ \textstyle\begin{cases} {{}^{c}}D^{\alpha }_{0,t}({^{c}}D^{\beta }_{0,t}+\lambda )x(t)=f(t), \quad t\in (t_{k},s_{k}], k=0,1,\dots ,m, 0< \alpha ,\beta < 1, \\ x(t)=g_{k}(t),\quad t\in (s_{k-1},t_{k}],k=1,2,\dots ,m, \\ x(0)=x_{0}, \qquad x(T)=\theta I^{p}x(\eta )\\ \quad \mbox{where } I^{p}x(\eta )=\int _{0}^{\eta }\frac{1}{\varGamma {p}}(\eta -s)^{p-1}x(s)\,ds, \quad 0< \eta < T. \end{cases} $$
(2.1)

We recall some definitions of fractional calculus from [17] as follows.

Definition 2.1

The fractional integral of order α from 0 to t for the function f is defined by

$$ I_{0,t}^{\gamma }f(t)=\frac{1}{\varGamma (\alpha )} \int _{0}^{t} f(s) (t-s)^{ \alpha -1}\,ds,\quad t>0, \alpha >0, $$

where \(\varGamma (\cdot )\) is the Gamma function.

Definition 2.2

The Riemann–Liouville fractional derivative of fractional order α from 0 to t for a function f can be written as

$$ ^{L}D_{0,t}^{\alpha }f(t)=\frac{1}{\varGamma (n-\alpha )}\, \frac{d^{n}}{dt ^{n}} \int _{0}^{t}\frac{f(s)}{(t-s)^{\alpha +1-n}}\,ds,\quad t>0, n-1< \alpha < n, $$

where \(\varGamma (\cdot )\) is the Gamma function.

Definition 2.3

The Caputo derivative of fractional order α from 0 to t for a function f can be defined as

$$ {^{c}}D_{0,t}^{\alpha }f(t)=\frac{1}{\varGamma (n-\alpha )} \int _{0}^{t} (t-s)^{n- \alpha -1}f^{n}(s) \,ds,\quad \mbox{where } n=[\alpha ]+1. $$

Definition 2.4

The general form of classical Caputo derivative of order α of a function f can be given as

$$ {^{c}}D_{0,t}^{\alpha }= {^{L}}D_{0,t}^{\alpha } \Biggl(f(t)-\sum_{k=0} ^{n-1} \frac{t^{k}}{k!}f^{(k)}(0) \Biggr), \quad t>0, n-1< \alpha < n. $$

Remark 2.1

  1. (i)

    If \(f(\cdot )\in C^{m}([0,\infty ),\mathbb{R})\), then

    $$\begin{aligned} ^{L}D_{0,t}^{\alpha }f(t) =&\frac{1}{\varGamma (m-\alpha )} \int _{0}^{t}\frac{f ^{m}(s)}{(t-s)^{\alpha +1-m}} \,ds\\ =&I_{0,t}^{m-\alpha }f^{(m)}(t),\quad t>0, m-1< \alpha < m. \end{aligned}$$
  2. (ii)

    In Definition 2.4, the integrable function f can be discontinuous. This fact can lead us to consider impulsive fractional problems in the sequel.

Lemma 2.1

([22])

Let \(\alpha >0\), \(\beta >0\), and \(f\in L^{1} ([a,b] )\). Then

$$\begin{aligned} I^{\alpha }I^{\beta }f(t)=I^{\alpha +\beta }f(t), {^{c}}D_{0,t}^{ \alpha } \bigl({^{c}}D_{0,t}^{\beta }f(t)\bigr)= {^{c}}D_{0,t}^{\alpha +\beta }f(t) \quad \textit{and}\quad I^{\alpha }D_{0,t}^{\alpha }f(t)=f(t),\quad t\in [a,b]. \end{aligned}$$

Lemma 2.2

Function \(x\in PC(J,\mathbb{R})\) is a solution of (2.1) if and only if

$$ x(t)= \textstyle\begin{cases} \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s)\,ds-\frac{ \lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ \quad {} -\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s)\,ds +\frac{\lambda \Delta t ^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s)\,ds \\ \quad {}-\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ \quad {}- (\frac{\Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{ \varGamma (p+1)\varGamma (\beta +1)}-1 )x_{0}, \quad t\in (0,s_{0}]; \\ \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s)\,ds-\frac{ \lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ \quad {}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s)\,ds -\frac{ \lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {}-\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s)\,ds \\ \quad {}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ \quad {}+ (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-\lambda ) \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1}f(s)\,ds \\ \quad {}- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )\frac{ \lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{\beta -1}x(s)\,ds \\ \quad {}- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g _{k}(t_{k}), \quad t\in (t_{k},s_{k}]; \\ g_{k}(t), \quad t\in (s_{k-1},t_{k}], k=1,2,\dots ,m. \end{cases} $$

Proof

Let x be a solution of problem (2.1).

Case 1. For \(t\in [0,s_{0}]\), we consider

$$ {{}^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)x(t)=f(t) \quad \textrm{with } x(0)=x_{0} \quad \textrm{and} \quad x(T)=\theta I^{p}x(\eta ). $$

After using fractional integrals \(I^{\alpha }\) and \(I^{\beta }\) for the solution of the above fractional Langevin equation, we get

$$\begin{aligned} x(t)=I^{\alpha +\beta } f(t)-\lambda I^{\beta }x(t)- \frac{c_{0}t^{ \beta }}{\varGamma (\beta +1)}-c. \end{aligned}$$
(2.2)

Using boundary conditions, we obtain

$$\begin{aligned} x(t) =&\frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1}f(s)\,ds- \frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{ \beta -1}x(s)\,ds \\ &{}-\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s)\,ds \\ &{}+ \frac{\lambda \Delta t ^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ &{}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s)\,ds \\ &{}-\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ &{}- \biggl(\frac{ \Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{\varGamma (p+1)\varGamma ( \beta +1)}-1 \biggr)x_{0}, \quad t\in [0,s_{0}]. \end{aligned}$$

For \(t\in (s_{0},t_{1}]\), \(x(t)=g_{1}(t)\).

Case 2. For \(t\in (t_{1},s_{1}]\), we consider

$$ {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)x(t)=f(t) \quad \textrm{with } x(t_{1})=g_{1}(t_{1}) \quad \textrm{and} \quad x(T)=\theta I^{p}x(\eta ). $$

Since \(x(t_{1})=g_{1}(t_{1})\), Eq. (2.2) is of the following type:

$$\begin{aligned} g_{1}(t_{1})=I^{\alpha +\beta } f(t_{1})-\lambda I^{\beta }x(t_{1})- \frac{c _{0}t_{1}^{\beta }}{\varGamma (\beta +1)}-c. \end{aligned}$$
(2.3)

Using boundary conditions, we get

$$\begin{aligned} x(t) =&\frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1}f(s)\,ds- \frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{ \beta -1}x(s)\,ds \\ &{}+\frac{\Delta (t_{1}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s)\,ds \\ &{}- \frac{ \lambda \Delta (t_{1}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ &{}-\frac{\theta \Delta (t_{1}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s)\,ds \\ &{}+\frac{\theta \Delta \lambda (t_{1}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ &{}+ \biggl(\Delta \frac{(t_{1}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{1}}(t_{1}-s)^{ \alpha +\beta -1}f(s) \,ds \\ &{}- \biggl(\Delta \frac{(t_{1}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{o}^{t_{1}}(t_{1}-s)^{\beta -1}x(s) \,ds \\ &{}- \biggl(\Delta \frac{(t_{1}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)g _{1}(t_{1}). \end{aligned}$$

Generally speaking, for \(t\in (s_{k-1},t_{k}]\), \(x(t_{k})=g_{k}(t)\).

Case 3. For \(t\in (t_{k},s_{k}]\), we consider

$$ {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)x(t)=f(t), \quad \textrm{with } x(t_{k})=g_{k}(t_{k}) \quad \textrm{and} \quad x(T)= \theta I^{p}x(\eta ). $$

By repeating again the same process, we have

$$\begin{aligned} x(t) =&\frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1}f(s)\,ds- \frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{ \beta -1}x(s)\,ds \\ &{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s)\,ds \\ &{}- \frac{ \lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ &{}-\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s)\,ds \\ &{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1}f(s) \,ds \\ &{}- \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{\beta -1}x(s) \,ds \\ &{}- \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)g _{k}(t_{k}), \end{aligned}$$

where

$$\begin{aligned} &\Delta =\frac{\varGamma (\beta +1)\varGamma (\beta +p+1)}{\varGamma (\beta +p+1) \eta ^{\beta }-\varGamma (\beta +1)\theta \eta ^{\beta +p}+ \varGamma (\beta +1)t_{k}^{\beta } ( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )} \\ &\quad \textrm{with } t_{1}^{\beta }=0 \textrm{ for } t\in [0,s _{0}] \textrm{ and }t_{k}^{\beta }\neq 0, \textrm{ for } t\in (t_{k},s_{k}], k=2,3,\dots . \end{aligned}$$

Conversely, one can verify the fact by proceeding the standard steps to complete the proof. □

Generalized Ulam–Hyers–Rassias stability

Using the ideas of stability in [24, 31], we can generate a generalized Ulam–Hyers–Rassias stability concept for Eq. (1.1).

Let \(\epsilon , \psi \geq 0\) and for a nondecreasing \(\varphi \in PC(J, \mathbb{R_{+}})\) consider

$$\begin{aligned} \textstyle\begin{cases} \vert {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(t)-f(t,x(t),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(t)) \vert \leq \varphi (t), \\ \quad t\in (t_{k},s_{k}], k=0,1,\dots ,m, 0< \alpha ,\beta < 1, \\ \vert x(t)-(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g_{k}(t,x(t)) \vert \leq \psi , \quad t\in (s_{k-1},t_{k}], k=0,1,\dots ,m. \end{cases}\displaystyle \end{aligned}$$
(3.1)

Remark 3.1

A function \(x\in PC(J,\mathbb{R})\) is a solution of the inequality (3.1) if and only if there is \(G\in PC(J,\mathbb{R})\) and a sequence \(G_{k}\), \(k=1,2,\dots ,m\) (which depends on x) such that

  1. (i)

    \(|G(t)|\leq \varphi (t)\), \(t\in J\) and \(|G_{k}|\leq \psi \), \(k=1,2,\dots ,m\),

  2. (ii)

    \({^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(t)=f(t,x(t), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(t))+G(t)\), \(t \in (t_{k},s_{k}]\), \(k=1,2,\dots ,m\),

  3. (iii)

    \(x(t)=g_{k}(t,x(t))+G_{k}\), \(t\in (s_{k-1},t_{k}]\), \(k=1, \dots ,m\).

Definition 3.1

Equation (1.1) is called generalized Ulam–Hyers–Rassias stable with respect to \((\varphi , \psi )\) if there exists \(c_{f,\alpha , \beta ,g_{i},\varphi }>0\) such that for each solution \(y\in PC(J, \mathbb{R})\) of inequality (3.1) there is a solution \(x\in PC(J,\mathbb{R})\) of Eq. (1.1) with

$$ \bigl\vert y(t)-x(t) \bigr\vert \leq c_{f,\alpha ,\beta ,G_{i},\varphi } \bigl( \varphi (t)+ \psi \bigr), \quad t\in J. $$

Remark 3.2

If \(x\in PC(J,\mathbb{R})\) is a solution of inequality (3.1), then x is a solution of the following integral inequality:

$$ \textstyle\begin{cases} \vert x(t)- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g_{k}(t,x(t)) \vert \leq \psi ,\\ \quad t\in (s_{k-1},t_{k}], k=1,2,\dots ,m; \\ \vert x(t)-x(0) -\frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{ \alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{ \beta }+\lambda )x(s))\,ds \\ \qquad {}+\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ \qquad {}+\frac{ \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )}\int _{0} ^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D _{0,t}^{\beta }+\lambda )x(s))\,ds \\ \qquad {}-\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma ( \beta +1)}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma (\beta +p)}\int _{0}^{ \eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ \qquad {}-\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(s))\,ds \vert \\ \quad \leq \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha + \beta -1}\varphi (s)\,ds +\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{ \beta -1}\varphi (s)\,ds \\ \qquad {} +\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1}\varphi (s)\,ds +\frac{ \lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)}\int _{0} ^{T}(T-s)^{\beta -1}x(s)\,ds \\ \qquad {}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1) \varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}\varphi (s)\,ds \\ \qquad{} +\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \varphi (s)\,ds, \quad t\in (0,s_{0}]; \\ \vert x(t)- (1-\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} ) )g_{k}(t_{k},x(t_{k})) \\ \qquad{} -\frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha + \beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+ \lambda )x(s))\,ds\\ \qquad{} +\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{ \beta -1}x(s)\,ds \\ \qquad{} -\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(s))\,ds \\ \qquad{} +\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta )}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds -\frac{ \theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s))\,ds \\ \qquad{} +\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+ \lambda )x(s))\,ds \\ \qquad{} - (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-\lambda )\\ \qquad{} \times \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t_{k}}(t _{k}-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D _{0,t}^{\beta }+\lambda )x(s))\,ds \\ \qquad{} + (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{ \beta -1}x(s)\,ds \vert \\ \quad \leq \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha + \beta -1}\varphi (s)\,ds+\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{ \beta -1}\varphi (s)\,ds \\ \qquad{} +\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1}\varphi (s)\,ds +\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )}\int _{0}^{T}(T-s)^{\beta -1}\varphi (s)\,ds \\ \qquad{} +\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1} \varphi (s)\,ds\\ \qquad{} +\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1}\varphi (s)\,ds \\ \qquad{} + (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-\lambda ) \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t _{k}}(t_{k}-s)^{\alpha +\beta -1}\varphi (s)\,ds \\ \qquad{} + (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{ \beta -1}\varphi (s)\,ds+\psi ,\\ \quad t\in (t_{k},s_{k}], k=1,2,\dots ,m. \end{cases} $$
(3.2)

In fact, by Remark 3.1, we get

$$ \textstyle\begin{cases} {^{c}}D^{\alpha }_{0,t}({^{c}}D^{\beta }_{0,t}+\lambda )x(t)=f(t,x(t), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(t))+G(t), \\ \quad t\in (t_{k},s_{k}], k=0,1,\dots ,m, 0< \alpha ,\beta < 1, \\ x(t)= (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g_{k}(t,x(t))+G_{k},\\\quad t\in (s_{k-1},t_{k}],k=1,2,\dots ,m. \end{cases} $$
(3.3)

Clearly, the solution of Eq. (3.3) is given by

$$ x(t)= \textstyle\begin{cases} \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1} (f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(s))+G(s) )\,ds \\ \quad{}-\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} (f(s,x(s),{^{c}}D_{0,t}^{ \alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(s))+G(s) )\,ds \\ \quad{}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds -\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ \quad{}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} (f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(s)) +G(s) )\,ds \\ \quad{}-\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds - (\frac{ \Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{\varGamma (p+1)\varGamma ( \beta +1)}-1 )x_{0}, \quad t\in (0,s_{0}], \\ \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1} (f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(s))+G(s) )\,ds \\ \quad{}-\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds-\frac{ \lambda \Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma ( \beta )}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1} (f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(s))+G(s) )\,ds \\ \quad{}-\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} (f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda )x(s)) +G(s) )\,ds \\ \quad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds + (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-\lambda ) \\ \quad{}\times \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} (f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t} ^{\beta }+\lambda )x(s))+G(s) )\,ds \\ \quad{}- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )\frac{ \lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{\beta -1}x(s)\,ds \\ \quad{}- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g _{k}(t_{k},x(t_{k}))+G_{k}, \quad t\in (t_{k},s_{k}], k=0,1,\dots ,m, \\ (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g _{k}(t,x(t)) +G_{k},\quad t\in (s_{k-1},t_{k}],k=1,2,\dots ,m. \end{cases} $$

For \(t\in (t_{k},s_{k}]\), \(k=0,1,\dots ,m\), we get

$$\begin{aligned} & \biggl\vert x(t)-\frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1}f \bigl(s,x(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)x(s)\bigr)\,ds \\ &\qquad {}+\frac{\lambda }{\varGamma {\beta }} \int _{o} ^{t}(t-s)^{\beta -1}x(s)\,ds \\ &\qquad {}-\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f\bigl(s,x(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)x(s)\bigr)\,ds \\ &\qquad {} +\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds\\ &\qquad {}- \frac{ \theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s))\,ds \\ &\qquad {} +\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f \bigl(s,x(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)x(s)\bigr)\,ds \\ &\qquad {} - \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)- \lambda \biggr) \\ &\qquad {} \times \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t _{k}-s)^{\alpha +\beta -1}f \bigl(s,x(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D _{0,t}^{\beta }+\lambda \bigr)x(s)\bigr)\,ds \\ &\qquad {} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{ \beta -1}x(s)) \,ds \\ &\qquad {} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)g_{k}\bigl(t_{k},x(t_{k})\bigr) \biggr\vert \\ &\quad \leq \biggl\vert \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1}G(s)\,ds \biggr\vert \, + \biggl\vert \frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \biggr\vert \\ &\qquad {} + \biggl\vert \frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}G(s)\,ds \biggr\vert \\ &\qquad {}+ \biggl\vert \frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \biggr\vert \\ &\qquad {} + \biggl\vert \frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1}G(s)\,ds \biggr\vert \\ &\qquad {} + \biggl\vert \frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \biggr\vert \\ &\qquad {} + \biggl\vert \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t _{k}}(t_{k}-s)^{\alpha +\beta -1}G(s) \,ds \biggr\vert \\ &\qquad {} + \biggl\vert \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{ \beta -1}x(s)) \,ds \biggr\vert + \vert G_{k} \vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1}\varphi (s)\,ds + \frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ &\qquad {} + \frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \varphi (s)\,ds\\ &\qquad {} + \frac{\lambda \Delta (t_{k}^{\beta }- t^{ \beta })}{\varGamma (\beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ &\qquad {} + \frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1}\varphi (s) \,ds \\ &\qquad {} + \frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s))\,ds \\ &\qquad {} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t _{k}}(t_{k}-s)^{\alpha +\beta -1} \varphi (s)\,ds \\ &\qquad {} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{ \beta -1}x(s)) \,ds +\psi . \end{aligned}$$

Proceeding as above, we derive

$$\begin{aligned} &\biggl\vert x(t)- \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)g_{k}\bigl(t,x(t)\bigr) \biggr\vert \leq \vert G_{k} \vert \leq \psi , \\ &\quad t\in (s _{k-1},t_{k}], k=0,1,\dots ,m, \end{aligned}$$

and

$$\begin{aligned} & \biggl\vert x(t)- \biggl(1-\frac{\Delta (\theta \eta ^{p}-\varGamma (p+1))t^{ \beta }}{\varGamma (p+1)\varGamma (\beta +1)} \biggr)x_{0}- \frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{ \beta -1}x(s)\,ds \\ &\qquad {} +\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds\\ &\qquad {} - \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1}f\bigl(s,x(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)x(s)\bigr)\,ds \\ &\qquad {} +\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f \bigl(s,x(s),{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)x(s)\bigr)\,ds \\ &\qquad {} -\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f \bigl(s,x(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)x(s)\bigr)\,ds \\ &\qquad {} +\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1) \varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \biggr\vert \\ &\quad \leq \biggl\vert \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1}G(s)\,ds \biggr\vert \\ &\qquad {} + \biggl\vert \frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}G(s)\,ds \biggr\vert \\ &\qquad {} + \biggl\vert \frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \biggr\vert \\ &\qquad {} + \biggl\vert \frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{ \beta -1}x(s)\,ds \biggr\vert + \biggl\vert \frac{\theta \Delta t^{\beta }}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}G(s)\,ds \biggr\vert \\ &\qquad {} + \biggl\vert \frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1) \varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \biggr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1}\varphi (s)\,ds \\ &\qquad {} + \frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}\varphi (s)\,ds \\ &\qquad {} + \frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma ( \beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ &\qquad {} + \frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{ \beta -1}x(s)\,ds + \frac{\theta \Delta t^{\beta }}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}\varphi (s) \,ds \\ &\qquad {} + \frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1) \varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds , \quad t\in (0,s_{0}]. \end{aligned}$$

Main results via fixed point methods

In order to apply a fixed point theorem of the alternative for contractions on a generalized complete metric space to achieve our main result, we want to collect the following facts.

Definition 4.1

For a nonempty set V, a function \(d:V\times V\rightarrow [0,\infty ]\) is called a generalized metric on V if and only if d satisfies

:

\(d(v_{1},v_{2})=0\) if and only if \(v_{1}=v_{2}\);

:

\(d(v_{1},v_{2})=d(v_{2},v_{1})\) for all \(v_{1},v _{2}\in V\);

:

\(d(v_{1},v_{3})\leq d(v_{1},v_{2})+d(v_{2},v_{3})\) for all \(v_{1}, v_{2}, v_{3}\in V\).

Lemma 4.1

([9])

Let \((V,d)\) be a generalized complete metric space. Assume that \(T:V\rightarrow V\) is a strictly contractive operator with the Lipschitz constant \(L<1\). If there exists a \(k\geq 0\) such that \(d(T^{k+1} v,T ^{k} v)<\infty \) for some v in V, then the followings statements are true:

(\(B_{1}\)):

The sequence \(\{T^{n} v\}\) converges to a fixed point \(v^{*}\) of T;

(\(B_{2}\)):

The unique fixed point of T is \(v^{*}\in V^{*}= \{u\in V \textrm{ such that } d(T^{k} v,u)<\infty \}\);

(\(B_{3}\)):

If \(u\in V^{*}\), then \(d(u,v^{*})\leq \frac{1}{1-L}d(Tu,u)\).

We can introduce some assumptions as follows:
\((H_{1})\) :

\(f\in C(J\times \mathbb{R}\times \mathbb{R},\mathbb{R})\).

\((H_{2})\) :

There exists a positive constant \(L_{f}\) such that

$$\bigl\vert f(t,u_{1},\bar{u_{1}})-f(t,u_{2}, \bar{u_{2}}) \bigr\vert \leq L_{f_{1}} \vert u_{1}-u_{2} \vert +\bar{L}_{f_{2}} \vert \bar{u_{1}}-\bar{u_{2}} \vert , \quad \textit{for each }t\in J\textit{ and all }u_{1}, u_{2}\in \mathbb{R}. $$
\((H_{3})\) :

\(g_{k}\in C((s_{k-1},t_{k}]\times \mathbb{R}, \mathbb{R})\) and there are positive constant \(L_{gk}\), \(k=1,2,\dots ,m\) such that

$$\bigl\vert g_{k} (t,u_{1})-g_{k} (t,u_{2}) \bigr\vert \leq L_{gk} \vert u_{1}-u_{2} \vert , \quad \textit{for each }t\in (s_{k-1},t_{k}],\textit{ and all }u_{1},u_{2} \in \mathbb{R}. $$
\((H_{4})\) :

Let \(\varphi \in C(J,\mathbb{R}_{+})\) be a nondecreasing function. There exists \(c_{\varphi }> 0\) such that

$$ \biggl( \int _{0}^{t}\bigl(\varphi (s)\bigr)^{\frac{1}{p}} \,ds \biggr)^{p}\leq C_{\varphi }\varphi (t)\quad \textit{for each } t\in J. $$
(4.1)

Theorem 4.2

Suppose that \((H_{1})\) and \((H_{2})\) are satisfied and also a function \(y\in PC(J,\mathbb{R})\) satisfies (3.1). Then there exists a unique solution x of Eq. (1.1) such that

$$ x(t)= \textstyle\begin{cases} \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {}-\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds +\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {} -\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }( {^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} +\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} -\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds - (\frac{ \Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{\varGamma (p+1)\varGamma ( \beta +1)}-1 )x, \\ \quad t\in (0,s_{0}], \\ \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds\\ \quad {} -\frac{ \lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ \quad {}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D _{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {}-\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {}-\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds + (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-\lambda ) \\ \quad {}\times \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{ \beta }+\lambda ))x(s)\,ds \\ \quad {}- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )\frac{ \lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{\beta -1}x(s)\,ds \\ \quad {}- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g _{k} (t_{k},x(t_{k}) ), \quad t\in (t_{k},s_{k}], k=1,2, \dots ,m, \\ g_{k}(t_{k},x(t_{k})), \quad t\in (s_{k-1},t_{k}], k=1,2,\dots ,m \end{cases} $$
(4.2)

and

$$\begin{aligned} \bigl\vert y(t)-x(t) \bigr\vert \leq & \biggl\{ \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r} \\ &{} +\frac{\Delta C_{\varphi }(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T ^{\alpha +\beta -r}\\ &{} +\frac{\lambda \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl( \frac{1-r}{\beta -r} \biggr)^{1-r}T^{\beta -r} \\ &{} +\frac{\theta \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl( \frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &{} +\frac{C_{\varphi }\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &{} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &{}\times\frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &{} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\\ &{}\times\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1 \biggr\} \\ &{}\times \biggl(\frac{\varphi (t)+\psi }{1-M} \biggr) \end{aligned}$$
(4.3)

for all \(t\in J\) if \(0<\alpha <\beta <1\), with

$$ M=\max \{M_{1},M_{2}\}< 1, $$
(4.4)

where

$$\begin{aligned} M_{1} =&\max \biggl\{ \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r} (L_{f_{1}}C_{\varphi }+\bar{L}_{f _{2}} C_{\varphi } )s_{k}^{\alpha +\beta -r}\\ &{} +\frac{\lambda C_{ \varphi }\varphi (t)}{\varGamma {\beta }} \biggl( \frac{1-r}{\beta -r} \biggr)^{1-r}s_{k}^{\beta -r} \\ &{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r} (L _{f_{1}}C_{\varphi }+\bar{L}_{f_{2}} C_{\varphi } )T^{\alpha + \beta -r} \\ &{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} (L_{f_{1}}C_{\varphi }+\bar{L}_{f_{2}} C_{\varphi } ) \eta ^{\alpha +\beta +p-r} \\ &{}+\frac{C_{\varphi }\varphi (t)\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{ \beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r}\\ &{} + \biggl(\Delta \frac{(t_{k} ^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}- \varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &{} \times \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha + \beta -r} \biggr)^{1-r} (L_{f_{1}}C_{\varphi }+\bar{L}_{f_{2}} C_{ \varphi } )t_{k}^{\alpha +\beta -r} \\ &{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })C_{\varphi } \varphi (t)}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{\beta -r}\\ &{} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{ \beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{ \varGamma (p+1)} \biggr)-1 \biggr) \\ &{} \times \biggl( \frac{\lambda C_{\varphi }\varphi (t)}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+L_{gk} \biggr) \textit{ such that } k=1,2,\dots ,m \biggr\} , \end{aligned}$$
$$\begin{aligned} M_{2} =&\max \biggl\{ \frac{L_{f_{1}}}{\varGamma (\alpha +\beta +1)}s_{k} ^{\alpha +\beta } +\frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta +1)}s _{k}^{\alpha +\beta } + \frac{\lambda }{\varGamma ({\beta }+1)}s_{k}^{ \beta }\\ &{} +\frac{\Delta (t_{k}^{\beta }-t^{\beta })L_{f_{1}}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +1)}T^{\alpha +\beta } \\ &{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +1)}T^{\alpha +\beta } +\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +1)}T ^{\beta }\\ &{} + \frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p+1)}\eta ^{\alpha +\beta +p} \\ &{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })\bar{L}_{f_{2}}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p+1)}\eta ^{\alpha +\beta +p} +\frac{ \theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta +p+1)}\eta ^{\beta +p} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &{}\times\biggl( \frac{L_{f_{1}}}{\varGamma (\alpha +\beta +1)}t_{k}^{\alpha +\beta }+ \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta +1)}t_{k}^{ \alpha +\beta } \biggr) \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \\ &{}\times\biggl(\frac{\lambda }{\beta \varGamma {\beta }} t_{k}^{\beta }+L_{gk} \biggr)\textit{ such that }k=0,1,\dots ,m \biggr\} . \end{aligned}$$

Proof

Consider the space of piecewise continuous functions

$$ V= \bigl\{ g:J\rightarrow \mathbb{R} \textrm{ such that } g\in PC(J, \mathbb{R}) \bigr\} , $$

endowed with the generalized metric on V, defined by

$$\begin{aligned} d(g,h) =&\inf \bigl\{ C_{1}+C_{2}\in [0,+\infty ] \\ &{} \text{such that } \bigl\vert g(t)-h(t) \bigr\vert \leq (C_{1}+ C_{2}) \bigl(\varphi (t)+ \psi \bigr) \textrm{ for all } t\in J \bigr\} , \end{aligned}$$
(4.5)

where

$$\begin{aligned} C_{1}\in \bigl\{ C\in [0,\infty ] \text{ such that } \bigl\vert g(t)-h(t) \bigr\vert \leq C \varphi (t)\text{ for all } t\in (t_{k},s_{k}], k=0,1, \dots ,m \bigr\} \end{aligned}$$

and

$$\begin{aligned} C_{2}\in \bigl\{ C\in [0,\infty ] \text{ such that } \bigl\vert g(t)-h(t) \bigr\vert \leq C\psi \text{ for all } t\in (s_{k-1},t_{k}], k=1,2, \dots ,m \bigr\} . \end{aligned}$$

It is easy to verify that \((V,d)\) is a complete generalized metric space [19].

Define an operator \(\varLambda :V\rightarrow V\) by

$$ (\varLambda x) (t)= \textstyle\begin{cases} \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {}-\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds +\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {} -\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }( {^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} +\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} -\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds - (\frac{ \Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{\varGamma (p+1)\varGamma ( \beta +1)}-1 )x_{0}, \\ \quad t\in (0,s_{0}], \\ \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} -\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds -\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {} +\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D _{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} -\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} +\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds + (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-\lambda ) \\ \quad {} \times \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{ \beta }+\lambda ))x(s)\,ds \\ \quad {} - (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1}x(s)\,ds \\ \quad {} - (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g _{k}(t_{k},x(t_{k})),\\ \quad t\in (t_{k},s_{k}], k=1,2,\dots ,m, \\ g_{k}(t_{k},x(t_{k})), \quad t\in (s_{k-1},t_{k}], k=1,2,\dots ,m \end{cases} $$
(4.6)

for all x belongs to V and \(t\in J\). Obviously, according to \((H_{1})\), Λ is a well-defined operator.

Next we shall verify that Λ is strictly contractive on V. Note that according to definition of \((V,d)\), for any \(g,h\in V\), it is possible to find \(C_{1},C_{2},C_{3},C_{4}\in [0,\infty ]\) such that

$$ \bigl\vert g(t)-h(t) \bigr\vert \leq \textstyle\begin{cases} C_{1}\varphi (t),\quad t\in (t_{k},s_{k}], k=0,\dots ,m, \\ C_{2}\psi , \quad t\in (s_{k-1},t_{k}],k=1,\dots ,m, \end{cases} $$
(4.7)

and

$$\begin{aligned} &\bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)- {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \\ &\quad \leq \textstyle\begin{cases} C_{3}\zeta (t)\leq C_{1}\varphi (t),\quad t\in (t_{k},s_{k}], k=0, \dots ,m, \\ C_{4}\zeta (t)\leq C_{2}\psi , \quad t\in (s_{k-1},t_{k}], k=1, \dots ,m. \end{cases}\displaystyle \end{aligned}$$

From the definition of Λ in Eq. (4.6), \((H_{2})\), \((H_{3})\) and (4.7), we obtain that

Case 1. For \(t\in [0,s_{0}]\),

$$\begin{aligned} &\bigl\vert (\varLambda g) (t)-(\varLambda h) (t) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1} \\ &\qquad{} \times \bigl[L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \bigr]\,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl[L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \bigr]\,ds \\ &\qquad{}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl[L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \bigr]\,ds \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\quad =\frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{ \bar{L}_{f_{2}}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \\ &\qquad{}\times \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{ \alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha }\bigl( {^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad {}+\frac{\bar{L}_{f_{2}}\theta \Delta t^{\beta }}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad{}\times \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert {^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\quad \leq \frac{L_{f_{1}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds +\frac{\lambda C_{1}}{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad {}+\frac{L_{f_{1}}C_{1}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda C_{1}\Delta t^{\beta }}{\varGamma (\beta )\varGamma ( \beta +1)} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad {}+\frac{ \bar{L}_{f_{2}} C_{1}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}} C_{1}\theta \Delta t^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}\theta \Delta t^{\beta }}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{C_{1}\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1) \varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0} ^{t}(t-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0} ^{t} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}}C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0} ^{t}(t-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0} ^{t} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\lambda C_{1}}{\varGamma {\beta }} \biggl( \int _{o}^{t}(t-s)^{\frac{ \beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{t} \bigl(\varphi (s) \bigr)^{ \frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{L_{f_{1}} C_{1}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \biggl( \int _{0}^{T}(T-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}\Delta t^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta )} \biggl( \int _{0}^{T}(T-s)^{\frac{\alpha + \beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{C_{1}\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma ( \beta +1)} \biggl( \int _{0}^{T}(T-s)^{\frac{\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{L_{f_{1}} C_{1}\theta \Delta t^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl( \int _{0}^{\eta }(\eta -s)^{\frac{ \alpha +\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{\eta }\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}\theta \Delta t^{\beta }}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +p)} \biggl( \int _{0}^{\eta }(\eta -s)^{\frac{ \alpha +\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \bigl(\varphi (s) \bigr)^{ \frac{1}{r}}\,ds )^{r} \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }C_{1}}{\varGamma (\beta +1) \varGamma (\beta +p)} \biggl( \int _{0}^{\eta }(\eta -s)^{ \frac{\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{\eta } \bigl( \varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\quad \leq \frac{L_{f_{1}} C_{1}C_{\varphi }\varphi (t)}{\varGamma (\alpha + \beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\bar{L}_{f_{2}} C_{1}C_{\varphi }\varphi (t)}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{ \alpha +\beta -r} \\ &\qquad{}+\frac{\lambda C_{1}C_{\varphi }\varphi (t)}{\varGamma {\beta }} \biggl(\frac{r-1}{ \beta -r} \biggr)^{1-r}t^{\beta -r} +\frac{L_{f_{1}} C_{1}C_{\varphi } \varphi (t)\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}C_{\varphi }\varphi (t)\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha + \beta -r} \biggr)^{1-r}T^{\alpha +\beta -r} +\frac{C_{1}C_{\varphi } \varphi (t)\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{\beta -r} \\ &\qquad{}+\frac{L_{f_{1}} C_{1}C_{\varphi }\varphi (t)\theta \Delta t^{ \beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{ \alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}C_{\varphi }\varphi (t)\theta \Delta t ^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{ \alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }C_{1}C_{\varphi }\varphi (t)}{ \varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &\quad \leq \biggl\{ \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r} (L_{f_{1}} +\bar{L}_{f_{2}} )s _{0}^{\alpha +\beta -r}+\frac{\lambda \Delta t^{\beta }}{\varGamma ( \beta )\varGamma (\beta +1)} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{ \beta -r} \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \biggl(\frac{r-1}{\beta -r} \biggr)^{1-r}s _{0}^{\beta -r} +\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r} (L _{f_{1}}+\bar{L}_{f_{2}} )T^{\alpha +\beta -r} \\ &\qquad{}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} (L_{f _{1}} +\bar{L}_{f_{2}} )\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \biggr\} C_{1} C_{\varphi }\varphi (t). \end{aligned}$$

Case 2. For \(t\in (s_{k-1},t_{k}]\), we have

$$ \bigl\vert (\varLambda g)t-(\varLambda h)t \bigr\vert = \bigl\vert g_{k}\bigl(t,g(t)\bigr)-g_{k}\bigl(t,h(t)\bigr) \bigr\vert \leq L_{gk} \bigl\vert g(t)-h(t) \bigr\vert \leq L_{gk}C_{2}\psi . $$

Case 3. For \(t\in (t_{k},s_{k}]\) and \(s\in (t_{k},s_{k}]\),

$$\begin{aligned} &\bigl\vert (\varLambda g) (t)-(\varLambda h) (t) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds+\frac{ \Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr))-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)h(s)\bigr)) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1} \\ &\qquad{} \times \bigl(L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl(L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl(L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} \\ &\qquad{} \times [L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad =\frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta L_{f_{1}} (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta \bar{L}_{f_{2}}(t_{k}^{\beta }-t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta )} \\ &\qquad{}\times \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{ \alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha }\bigl( {^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad{}\times \int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t _{k}}(t_{k}-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D _{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t} ^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t _{k}-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds +\frac{\bar{L}_{f_{2}}C_{1}}{ \varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda C_{1}}{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds +\frac{\Delta L_{f_{1}}C_{1} (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta \bar{L}_{f_{2}}C_{1}(t_{k}^{\beta }-t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha + \beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda C_{1}\Delta (t_{k} ^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{ \beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}C_{1}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}C_{1}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }( \eta -s)^{\alpha +\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta C_{1}\Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t _{k}-s)^{\alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{\bar{L}_{f_{2}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0} ^{t_{k}}(t_{k}-s)^{\alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda C_{1}}{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk} \bigl\vert g(t_{k})-h(t_{k})) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0} ^{t}(t-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0} ^{t}\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0}^{t}(t-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{t} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\lambda C_{1}}{\varGamma {\beta }} \biggl( \int _{o}^{t}(t-s)^{\frac{ \beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{t}\bigl(\varphi (s)\bigr)^{ \frac{1}{r}} \,ds \biggr)^{r} \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })L_{f_{1}} C_{1}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \biggl( \int _{0}^{T}(T-s)^{\frac{ \alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T}\bigl(\varphi (s)\bigr)^{ \frac{1}{r}} \,ds \biggr)^{r} \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}} C_{1}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl( \int _{0}^{T}(T-s)^{\frac{ \alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })C_{1}}{\varGamma ( \beta +1)\varGamma (\beta )} \biggl( \int _{0}^{T}(T-s)^{ \frac{\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T}\bigl(\varphi (s)\bigr)^{ \frac{1}{r}} \,ds \biggr)^{r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}} C_{1}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl( \int _{0}^{\eta }( \eta -s)^{\frac{\alpha +\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0} ^{\eta }\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })\bar{L}_{f_{2}} C _{1}}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl( \int _{0}^{ \eta }(\eta -s)^{\frac{\alpha +\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{\eta }\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{C_{1}\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)} \biggl( \int _{0}^{\eta }(\eta -s)^{\frac{ \beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{\eta }\bigl(\varphi (s) \bigr)^{ \frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times\frac{L_{f_{1}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0} ^{t_{k}}(t_{k}-s)^{\frac{\alpha +\beta -1}{1-r}} \,ds \biggr)^{1-r} \biggl( \int _{0}^{t_{k}}\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times\frac{\bar{L}_{f_{2}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0}^{t_{k}}(t_{k}-s)^{\frac{\alpha +\beta -1}{1-r}} \,ds \biggr)^{1-r} \biggl( \int _{0}^{t_{k}}\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \\ &\qquad{}\times \frac{\lambda C_{1}}{\varGamma {\beta }} \biggl( \int _{o}^{t_{k}}(t _{k}-s)^{\frac{\beta -1}{1-r}} \,ds \biggr)^{1-r} \biggl( \int _{0}^{t_{k}}\bigl( \varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk}C_{2}\psi \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}} C_{1} C_{\varphi }\varphi (t)}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\bar{L}_{f_{2}} C_{1} C_{\varphi }\varphi (t)}{ \varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t ^{\alpha +\beta -r} \\ &\qquad{}+\frac{\lambda C_{1}C_{\varphi }\varphi (t)}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r}+ \frac{\Delta (t_{k}^{\beta }-t^{ \beta })L_{f_{1}} C_{1}C_{\varphi }\varphi (t)}{\varGamma (\beta +1) \varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T ^{\alpha +\beta -r} \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}} C_{1}C_{ \varphi }\varphi (t)}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r} \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })C_{1}C_{\varphi } \varphi (t)}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{\beta -r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}} C_{1}C _{\varphi }\varphi (t)}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })\bar{L}_{f_{2}} C _{1}C_{\varphi }\varphi (t)}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{C_{1}C_{\varphi }\varphi (t)\theta \Delta \lambda (t_{k}^{ \beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{ \beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}} C_{1}C_{\varphi }\varphi (t)}{\varGamma (\alpha + \beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha + \beta -r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{\bar{L}_{f_{2}} C_{1}C_{\varphi }\varphi (t)}{\varGamma ( \alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t_{k} ^{\alpha +\beta -r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda C_{1}C_{\varphi }\varphi (t)}{\varGamma {\beta }} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}t_{k}^{\beta -r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk}C_{2}\psi \\ &\quad \leq \biggl\{ \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r} (L_{f_{1}}+\bar{L}_{f_{2}} )s_{0} ^{\alpha +\beta -r} +\frac{\lambda }{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}s_{0}^{\beta -r} \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r} (L _{f_{1}}+\bar{L}_{f_{2}} )T^{\alpha +\beta -r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} (L_{f_{1}}+\bar{L}_{f_{2}} )\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{ \varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{} \times \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha + \beta -r} \biggr)^{1-r} (L_{f_{1}}+\bar{L}_{f_{2}} )t_{k}^{\alpha +\beta -r} + \frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{ \beta -r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \biggl(\frac{\lambda }{\varGamma {\beta }} \biggl( \frac{1-r}{\beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+L_{gk} \biggr) \biggr\} C_{\varphi } \\ &\qquad{}\times (C_{1}+C_{2} ) \bigl( \varphi (t)+\psi \bigr). \end{aligned}$$

Also, for \(t\in (t_{k},s_{k}]\) and \(s\in (s_{k-1},t_{k}]\), we have

$$\begin{aligned} &\bigl\vert (\varLambda g) (t)-(\varLambda h) (t) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1} \\ &\qquad{} \times \bigl[L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \bigr]\,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times [L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times [L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} \\ &\qquad{} \times [L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad =\frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta L_{f_{1}} (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta \bar{L}_{f_{2}}(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \\ &\qquad{}\times \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)- {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad{}\times \int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t _{k}}(t_{k}-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D _{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t} ^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times \frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t _{k}-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}}C_{2}\psi }{\varGamma (\alpha +\beta )} \int _{0} ^{t}(t-s)^{\alpha +\beta -1}\,ds + \frac{\bar{L}_{f_{2}}C_{2}\psi }{ \varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1}\,ds \\ &\qquad{}+ \frac{ \lambda C_{2}\psi }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1}\,ds +\frac{\Delta L_{f_{1}}C_{2}\psi (t_{k}^{\beta }-t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha + \beta -1}\,ds \\ &\qquad{}+ \frac{\Delta \bar{L}_{f_{2}}C_{2}\psi (t_{k}^{\beta }-t ^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{ \alpha +\beta -1}\,ds \\ &\qquad{}+\frac{\lambda C_{2}\psi \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}\,ds \\ &\qquad{}+ \frac{L _{f_{1}}C_{2}\psi \theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}\,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}C_{2}\psi \theta \Delta (t_{k}^{\beta }- t ^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{ \eta }(\eta -s)^{\alpha +\beta +p-1}\,ds \\ &\qquad{}+ \frac{\theta C_{2}\psi \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}\,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}}C_{2}\psi }{\varGamma (\alpha +\beta )} \int _{0} ^{t_{k}}(t_{k}-s)^{\alpha +\beta -1} \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{\bar{L}_{f_{2}}C_{2}\psi }{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{\alpha +\beta -1} \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda C_{2}\psi }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{ \beta -1} \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk} \bigl\vert g(t_{k})-h(t_{k})) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}} C_{2}\psi }{\varGamma (\alpha +\beta )(\alpha + \beta )}t^{\alpha +\beta } +\frac{\bar{L}_{f_{2}} C_{2}\psi }{\varGamma (\alpha +\beta )(\alpha +\beta )}t^{\alpha +\beta } +\frac{\lambda C _{2}\psi }{\beta \varGamma {\beta }}t^{\beta } \\ &\qquad{}+\frac{\Delta (t_{k}^{ \beta }-t^{\beta })L_{f_{1}} C_{2}\psi }{\varGamma (\beta +1)(\alpha + \beta )\varGamma (\alpha +\beta )}T^{\alpha +\beta } +\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}} C_{2} \psi }{\varGamma (\beta +1)(\alpha +\beta )\varGamma (\alpha +\beta )}T^{ \alpha +\beta } \\ &\qquad{} +\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta }) C _{2}\psi }{\varGamma (\beta +1)\beta \varGamma (\beta )}T^{\beta } +\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}} C_{2} \psi }{\varGamma (\beta +1)(\alpha +\beta +p)\varGamma (\alpha +\beta +p)} \eta ^{\alpha +\beta +p} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{ \beta })\bar{L}_{f_{2}} C_{2}\psi }{\varGamma (\beta +1)(\alpha +\beta +p) \varGamma (\alpha +\beta +p)}\eta ^{\alpha +\beta +p} \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta }) C_{2} \psi }{\varGamma (\beta +1)(\beta +p)\varGamma (\beta +p)}\eta ^{\beta +p} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda C_{2}\psi }{\beta \varGamma {\beta }} t_{k}^{\beta } \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}} C_{2}\psi }{(\alpha +\beta )\varGamma (\alpha + \beta )}t_{k}^{\alpha +\beta } \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{\bar{L}_{f_{2}} C_{2}\psi }{(\alpha +\beta )\varGamma ( \alpha +\beta )}t_{k}^{\alpha +\beta } \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk} C_{2}\psi \\ &\quad \leq \biggl\{ \frac{L_{f_{1}}}{\varGamma (\alpha +\beta +1)}s_{k}^{ \alpha +\beta }+ \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta +1)}s _{k}^{\alpha +\beta } +\frac{\lambda }{\varGamma ({\beta }+1)}s_{k}^{ \beta } \\ &\qquad{}+ \frac{\Delta (t_{k}^{\beta }-t^{\beta })L_{f_{1}}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +1)}T^{\alpha +\beta } \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +1)}T^{\alpha +\beta }+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +1)}T ^{\beta } \\ &\qquad{}+ \frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p+1)}\eta ^{\alpha +\beta +p} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })\bar{L}_{f_{2}}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p+1)}\eta ^{\alpha +\beta +p} +\frac{ \theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta +p+1)}\eta ^{\beta +p} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times \biggl( \frac{L_{f_{1}}}{\varGamma (\alpha +\beta +1)}t_{k}^{\alpha +\beta }+ \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta +1)}t_{k}^{ \alpha +\beta } \biggr) \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \biggl(\frac{\lambda }{\beta \varGamma {\beta }} t_{k}^{\beta }+L_{gk} \biggr) \biggr\} \\ &\qquad{}\times(C_{1}+C_{2}) \bigl(\varphi (t)+\psi \bigr). \end{aligned}$$

From above, we have

$$ \bigl\vert (\varLambda g) (t)-(\varLambda h) (t) \bigr\vert \leq M(C_{1}+C_{2}) \bigl(\varphi (t)+\psi \bigr), \quad t\in [0, \tau ], $$

that is,

$$ d(\varLambda g,\varLambda h)\leq M(C_{1}+C_{2}) \bigl(\varphi (t)+\psi \bigr). $$

Hence, we conclude that

$$ d(\varLambda g,\varLambda h)\leq Md(g,h), \quad \text{for any } g,h\in V. $$

Since condition (4.4) is strictly contractive, continuity property is thus shown. Now we take \(g_{0}\in V\). From the piecewise continuity property of \(g_{0}\) and \(\varLambda g_{0}\), it follows that there exists a constant \(0< G_{1}<\infty \) such that

$$\begin{aligned} &\bigl\vert (\varLambda g_{0}) (t)-g_{0}(t) \bigr\vert \\ &\quad \leq \biggl\vert \frac{1}{\varGamma (\alpha + \beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,g_{0}(s),{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}-\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ &\qquad {}-\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,g_{0}(s),{^{c}}D_{0,t}^{ \alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g_{0}(s) \,ds \\ &\qquad {}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds+ \frac{\theta \Delta t^{\beta }}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad {} \times \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,g_{0}(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}-\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ &\qquad {}+ \biggl(\frac{\Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{ \varGamma (p+1)\varGamma (\beta +1)}+1 \biggr)x_{0}-g_{0}(t) \biggr\vert \\ &\quad \leq G_{1} \varphi (t)\leq G_{1}\bigl(\varphi (t)+\psi \bigr), \quad t\in (0,s_{0}]. \end{aligned}$$

There exists a constant \(0< G_{2}<\infty \) such that

$$\begin{aligned} &\bigl\vert (\varLambda g_{0}) (t)-g_{0}(t) \bigr\vert = \bigl\vert g_{k}\bigl(t,g_{0}(t)\bigr)-g_{0}(t) \bigr\vert \leq G_{2} \psi \leq G_{2}\bigl(\varphi (t)+\psi \bigr), \\ &\quad t\in (s_{k-1},t_{k}], k=1,2,\dots ,m. \end{aligned}$$

Also we can find a constant \(0< G_{3}<\infty \) such that

$$\begin{aligned} &\bigl\vert (\varLambda g_{0}) (t)-g_{0}(t) \bigr\vert \\ &\quad \leq \biggl\vert \frac{1}{\varGamma (\alpha + \beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,g_{0}(s),{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}-\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ &\qquad {}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,g_{0}(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}-\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds- \frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad {} \times \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,g_{0}(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ &\qquad {}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad {}\times \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1}f(s,g_{0}(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t} ^{\beta }+\lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}- \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{\beta -1}x(s) \,ds \\ &\qquad {}- \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)g _{k}(t_{k})-g_{0}(t) \biggr\vert , \quad t\in (t_{k},s_{k}], \\ &\bigl\vert (\varLambda g_{0}) (t)-g_{0}(t) \bigr\vert \leq G_{3}\varphi (t)\leq G_{3}\bigl(\varphi (t)+\psi \bigr), \quad t\in (t_{k},s_{k}], k=1,2,\dots ,m. \end{aligned}$$

Since f, \(g_{k}\) and \(g_{0}\) are bounded on J and \(\varphi (\cdot )>0\), Eq. (4.5) implies that \(d(\varLambda g_{0},g_{0})<\infty \).

By using Banach fixed point theorem, there exists a continuous function \(x:J\rightarrow \mathbb{R}\) such that \(\varLambda ^{n}g_{0}\rightarrow x\) in \((V,d)\) as \(n\rightarrow \infty \) and \(\varLambda x=y_{0}\), that is, x satisfies Eq. (4.2) for every \(t\in J\).

Now we show that \(\{g\in V \textrm{ such that } d(g_{0},g)<\infty \}=V\). For any \(g\in V\), since g and \(g_{0}\) are bounded on J and \(\min_{t\in J}\varphi (t)>0\), there exists a constant \(0< C_{g}<\infty \) such that \(|g_{0}(t)-g(t)|\leq C_{g}(\varphi (t)+\psi)\), for any \(t\in J\). Hence, we have \(d(g_{0},g)<\infty \) for all \(g\in V\), that is, \(\{g\in V \textrm{ such that } d(g_{0},g)<\infty \}=V\). Thus, we determine that x is the unique continuous function satisfying Eq. (4.2). Using (3.2) and \((H_{4})\), we can write

$$\begin{aligned} d(y,\varLambda y) \leq & \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha +\beta -r} +\frac{ \lambda C_{\varphi }}{\varGamma {\beta }} \biggl( \frac{1-r}{\beta -r} \biggr)^{1-r}t^{\beta -r} \\ &{}+\frac{\Delta C_{\varphi }(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r}\\ &{}+\frac{\lambda \Delta C_{\varphi }(t _{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}T^{\beta -r} \\ &{}+\frac{\theta \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha + \beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &{}+\frac{C_{\varphi }\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1. \end{aligned}$$

Summarizing, we have

$$\begin{aligned} d(y,x) \leq & \frac{d(\varLambda y,y)}{1-M} \\ \leq & \biggl(\frac{1}{1-M} \biggr) \biggl\{ \frac{C_{\varphi }}{\varGamma ( \alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{ \alpha +\beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r} \\ &{}+\frac{\Delta C_{\varphi }(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r}\\ &{}+\frac{\lambda \Delta C_{\varphi }(t _{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}T^{\beta -r} \\ &{}+\frac{\theta \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha + \beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &{}+\frac{C_{\varphi }\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1 \biggr\} . \end{aligned}$$

This shows that (4.3) is true for \(t\in J\). □

Here, we give an example to illustrate our main result.

Example 4.3

$$ \textstyle\begin{cases} {^{c}}D_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)\\ \quad = \frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t ^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)} \\ \qquad {} +\int _{0}^{t}\frac{(t-s)^{\frac{3}{2}}}{\varGamma {\frac{5}{2}}} (\frac{ \vert x(s) \vert }{8+e^{s}+s^{2}} )\,ds, \quad t\in (0,1]\cup (2,3], \\ x(t)=\frac{x(t)}{(3+t^{2})(1+ \vert x(t) \vert )}, \quad t\in (1,2], \\ x(0)=\frac{\sqrt{2}}{3}, \qquad x(1)=\frac{5}{6}\int _{0}^{\frac{1}{4}}\frac{(\frac{1}{4}-s)}{\varGamma \frac{4}{3}}\,ds \quad 0< \eta < 1 \end{cases} $$
(4.8)

and

$$ \textstyle\begin{cases} \vert {^{c}}D_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )y(t)-\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )y(t)}{8+e^{t}+t ^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )y(t)} \\ \quad {} -\int _{0}^{t}\frac{(t-s)^{\frac{3}{2}}}{\varGamma {\frac{5}{2}}} (\frac{ \vert y(s) \vert }{8+e^{s}+s^{2}} )\,ds \vert \leq e^{t}, \quad t\in (0,1]\cup (2,3], \\ \vert y(t)-\frac{y(t)}{(3+t^{2})(1+ \vert x(t) \vert )} \vert \leq 1, \quad t\in (1,2]. \end{cases} $$

Let \(J=[0,3]\), \(\alpha =\beta =\frac{1}{2}\), \(r=\frac{1}{3}\), \(\Delta =-2.70\), \(\theta =\frac{5}{6}\), \(p=\frac{4}{3}\), \(\eta = \frac{1}{4}\) and \(0=t_{0}< s_{0}=1< t_{1}=2< s_{1}=\tau =3\). Denote \(f(t,x(t))=\frac{|x(t)|}{8+e^{t}+t^{2}}\) with \(L_{f}=\frac{1}{9}\) for \(t\in (0,1]\cup (2,3]\) and \(g_{1}(t,x(t))= \frac{x(t)}{(3+t^{2})(1+|x(t)|)}\) with \(L_{g_{k}}=\frac{1}{4}\) for \(t\in (1,2]\). Putting \(\psi =1\), \(L_{f_{1}}=\bar{L}_{f_{2}}= \frac{1}{4}\) \(\varphi (t)=e^{t}\) and \(c_{\varphi }=1\), we have \((\int _{0}^{t}(e^{s})^{3}\,ds )^{\frac{1}{3}}\leq e^{t}\) and let \(M_{1}\approx -0.5900\), \(M_{2}\approx 0.9713\), so \(M=0.9713 < 1\).

By Theorem 4.2, there exists a unique solution \(x:[0,3]\rightarrow \mathbb{R}\) such that

$$ x(t)= \textstyle\begin{cases} \int _{0}^{t} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ( {^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)})\,ds -0.0846\int _{o}^{t}(t-s)^{\frac{-1}{2}}x(s)\,ds \\ \quad {} +0.0650 t^{\frac{1}{2}}\int _{0}^{1} (\frac{ \vert x(t) \vert + {{^{c}}D} _{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D _{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)} )\,ds \\ \quad {} -0.0901 t^{\frac{1}{2}}\int _{0}^{1}(1-s)^{\frac{-1}{2}}x(s)\,ds \\ \quad {} -0.7454 \sqrt{t}\int _{0}^{\frac{1}{4}}(\frac{1}{4}-s)^{ \frac{4}{3}} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ( {^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)} )\,ds \\ \quad {} +0.1415\sqrt{t}\int _{0}^{\frac{1}{4}}(\frac{1}{4}-s)^{\frac{5}{6}}x(s)\,ds + (0.9476\sqrt{t}+1 )x_{0}, \quad t\in [0,1], \\ \int _{0}^{t} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ( {^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)} )\,ds \\ \quad {} -0.0846\int _{o}^{t}(t-s)^{\frac{-1}{2}}x(s)\,ds \\ \quad {} -1.0650(\sqrt{2}-\sqrt{t})\int _{0}^{1} (\frac{ \vert x(t) \vert + {{^{c}}D} _{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D _{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)} )\,ds \\ \quad {} +0.0901(\sqrt{2}-\sqrt{t})\int _{0}^{1}(1-s)^{\frac{-1}{2}}x(s)\,ds \\ \quad {} +0.7454(\sqrt{2}-\sqrt{t})\int _{0}^{\frac{1}{4}}(\frac{1}{4}-s)^{ \frac{4}{3}} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ( {^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)} )\,ds \\ \quad {} -0.1415(\sqrt{2}-\sqrt{t})\int _{0}^{\frac{1}{4}}(\frac{1}{4}-s)^{ \frac{5}{6}}x(s))\,ds \\ \quad {} + (0.9476(\sqrt{2}-\sqrt{t})-\frac{3}{20} )\int _{0}^{2} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t} ^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t} ^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)} )\,ds \\ \quad {} - (0.9476(\sqrt{2}-\sqrt{t})-1 )0.846 \int _{o}^{2}(2-s)^{ \frac{-1}{2}}x(s)\,ds \\ \quad {} - (0.9476(\sqrt{2}-\sqrt{t})-1 ) \frac{x(t)}{(3+t^{2})(1+ \vert x(t) \vert )}, \quad t\in (2,3] \\ \frac{x(t)}{(3+t^{2})(1+ \vert x(t) \vert )}, \quad t\in (1,2]. \end{cases} $$

Then

$$\begin{aligned} &\bigl\vert y(t)-x(t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r} \\ &\qquad {}+\frac{\Delta C_{\varphi }(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r} +\frac{\lambda \Delta C_{\varphi }(t _{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}T^{\beta -r} \\ &\qquad {}+\frac{\theta \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha + \beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad {}+\frac{C_{\varphi }\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \\ &\qquad {}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &\qquad {}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} -\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1 \biggr\} \biggl(\frac{\varphi (t)+ \psi }{1-M} \biggr), \end{aligned}$$

which can further be reduced to

$$\begin{aligned} &\bigl\vert y(t)-x(t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r} \\ &\qquad {}-\frac{\Delta C_{\varphi }t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{ \alpha +\beta -r} -\frac{\lambda \Delta C_{\varphi }t^{\beta }}{ \varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T ^{\beta -r} \\ &\qquad {}-\frac{\theta \Delta C_{\varphi }t^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} \eta ^{\alpha +\beta +p-r} \\ &\qquad {}-\frac{C_{\varphi }\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &\qquad {}- \biggl(\frac{\Delta t^{\beta }}{\varGamma (\beta +1)} \biggl(\frac{ \theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C _{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &\qquad {}- \biggl(\frac{\Delta t^{\beta }}{\varGamma (\beta +1)} \biggl(\frac{ \theta \eta ^{p} -\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \frac{ \lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1 \biggr\} \biggl( \frac{\varphi (t)+\psi }{1-M} \biggr). \end{aligned}$$

This implies

$$\begin{aligned} &\bigl\vert y(t)-x(t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r} \tau ^{\alpha +\beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}\tau ^{\beta -r} \\ &\qquad {}-\frac{\Delta C_{\varphi }\tau ^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{ \alpha +\beta -r} -\frac{\lambda \Delta C_{\varphi }\tau ^{\beta }}{ \varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T ^{\beta -r} \\ &\qquad {}-\frac{\theta \Delta C_{\varphi }\tau ^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} \eta ^{\alpha +\beta +p-r} \\ &\qquad {}-\frac{C_{\varphi }\theta \Delta \lambda \tau ^{\beta }}{\varGamma ( \beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &\qquad {}- \biggl(\frac{\Delta \tau ^{\beta }}{\varGamma (\beta +1)} \biggl(\frac{ \theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C _{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}\tau ^{\alpha +\beta -r} \\ &\qquad {}- \biggl(\frac{\Delta \tau ^{\beta }}{\varGamma (\beta +1)} \biggl(\frac{ \theta \eta ^{p} -\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \frac{ \lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}\tau ^{\beta -r}+1 \biggr\} \biggl( \frac{\varphi (t)+\psi }{1-M} \biggr). \end{aligned}$$

Plugging-in the values, we have

$$\begin{aligned} &\bigl\vert y(t)-x(t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{1}{\varGamma (\frac{1}{2}+ \frac{1}{2})} \biggl( \frac{1-\frac{1}{3}}{\frac{1}{2}+\frac{1}{2}- \frac{1}{3}} \biggr)^{(1-\frac{1}{3})}3^{(\frac{1}{2}+\frac{1}{2}- \frac{1}{3})} +\frac{(0.15)}{\varGamma {\frac{1}{2}}} \biggl(\frac{1- \frac{1}{3}}{\frac{1}{2}-\frac{1}{3}} \biggr)^{(1-\frac{1}{3})}3^{( \frac{1}{2}-\frac{1}{3})} \\ &\qquad {}-\frac{(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)\varGamma ( \frac{1}{2}+\frac{1}{2})} \biggl(\frac{1-\frac{1}{3}}{\frac{1}{2}+ \frac{1}{2}-\frac{1}{3}} \biggr)^{(1-\frac{1}{3})}3^{(\frac{1}{2}+ \frac{1}{2}-\frac{1}{3})}\\ &\qquad {}-\frac{(0.15)(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)\varGamma (\frac{1}{2})} \biggl(\frac{1-\frac{1}{3}}{ \frac{1}{2}-\frac{1}{3}} \biggr)^{(1-\frac{1}{3})}3^{(\frac{1}{2}- \frac{1}{3})} \\ &\qquad {}-\frac{(0.833)(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)\varGamma (\frac{1}{2}+\frac{1}{2}+\frac{4}{3})} \biggl(\frac{1-\frac{1}{3}}{ \frac{1}{2}+\frac{1}{2}+\frac{4}{3}-\frac{1}{3}} \biggr)^{1-\frac{1}{3}}(0.25)^{ \frac{1}{2}+\frac{1}{2}+\frac{4}{3}-\frac{1}{3}} \\ &\qquad {}-\frac{(0.833)(-2.7)(0.15) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1) \varGamma (\frac{1}{2}+\frac{4}{3})} \biggl(\frac{1-\frac{1}{3}}{ \frac{1}{2}+\frac{4}{3}-\frac{1}{3}} \biggr)^{1-\frac{1}{3}}(0.25)^{ \frac{1}{2}+\frac{4}{3}-\frac{1}{3}} \\ &\qquad {}- \biggl(\frac{(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)} \biggl(\frac{(0.833)(0.25)^{ \frac{4}{3}}-\varGamma (\frac{4}{3}+1)}{\varGamma (\frac{4}{3}+1)} \biggr)-(0.15) \biggr)\\ &\qquad {}\times \frac{1}{\varGamma (\frac{1}{2}+\frac{1}{2})} \biggl(\frac{1- \frac{1}{3}}{\frac{1}{2}+\frac{1}{2}-\frac{1}{3}} \biggr)^{(1- \frac{1}{3})}3^{(\frac{1}{2}+\frac{1}{2}-\frac{1}{3})} \\ &\qquad {}- \biggl(\frac{(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)} \biggl(\frac{(0.833)(0.25)^{ \frac{4}{3}} -\varGamma (\frac{4}{3}+1)}{\varGamma (\frac{4}{3}+1)} \biggr)-1 \biggr) \frac{(0.15) }{\varGamma {\frac{1}{2}}} \biggl(\frac{1-\frac{1}{3}}{ \frac{1}{2}-\frac{1}{3}} \biggr)^{1-\frac{1}{3}}3^{(\frac{1}{2}- \frac{1}{3})}+1 \biggr\} \\ &\qquad {}\times \biggl(\frac{e^{t}+1}{1-0.9714} \biggr) \\ &\quad \leq 5.4846 \biggl(\frac{e^{t}+1}{0.0286} \biggr) \\ &\quad \leq 191.769 \bigl(e^{t}+1 \bigr), \end{aligned}$$

thus problem (4.8) is Ulam–Hyers–Rassias stable.

Conclusions

In this article, we considered a nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses. After introduction, we built a uniform structure for the solutions of our proposed model. We studied the concept of generalized Ulam–Hyers–Rassias stability to our proposed model. And, finally, we presented a particular example for the applicability of our main result.

References

  1. 1.

    Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)

  2. 2.

    Ahmad, B., Nieto, J.J., Alsaedi, A., El-Shahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 13(2), 599–602 (2012)

  3. 3.

    Alkhazzan, A., Jiang, P., Baleanu, D., Khan, H., Khan, A.: Stability and existence results for a class of nonlinear fractional differential equations with singularity. Math. Methods Appl. Sci. (2018). https://doi.org/10.1002/mma.5263

  4. 4.

    Bai, Z.: On positive solutions of a non-local fractional boundary value problem. Nonlinear Anal., Theory Methods Appl. 72(2), 916–924 (2010)

  5. 5.

    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012)

  6. 6.

    Benchohra, M., Graef, J.R., Hamani, S.: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 87(7), 851–863 (2008)

  7. 7.

    Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)

  8. 8.

    Benchohra, M., Seba, D.: Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2009, 8 (2009)

  9. 9.

    Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative, for contractions on a generalized complete matric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

  10. 10.

    Fa, K.S.: Generalized Langevin equation with fractional derivative and long-time correlation function. Phys. Rev. E 73(6), 061104 (2006)

  11. 11.

    Feckan, M., Zhou, Y., Wang, J.: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 3050–3060 (2012)

  12. 12.

    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. 27, 222–224 (1941)

  13. 13.

    Khan, A., Syam, M.I., Zada, A., Khan, H.: Stability analysis of nonlinear fractional differential equations with Caputo and Riemann–Liouville derivatives. Eur. Phys. J. Plus 133, 264 (2018)

  14. 14.

    Khan, H., Chen, W., Sun, H.: Analysis of positive solution and Hyers–Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space. Math. Methods Appl. Sci. 41(9), 3430–3440 (2018)

  15. 15.

    Khan, H., Li, Y., Chen, W., Baleanu, D., Khan, A.: Existence theorems and Hyers–Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator. Bound. Value Probl. 2017, 157 (2017)

  16. 16.

    Khan, H., Tunc, C., Chen, W., Khan, A.: Existence theorems and Hyers–Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator. J. Appl. Anal. Comput. 8(4), 1211–1226 (2018)

  17. 17.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equation. Elsevier, Amsterdam (2006)

  18. 18.

    Kosmatov, N.: Initial value problems of fractional order with fractional impulsive conditions. Results Math. 63, 1289–1310 (2013)

  19. 19.

    Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

  20. 20.

    Lim, S.C., Li, M., Teo, L.P.: Langevin equation with two fractional orders. Phys. Lett. A 372(42), 6309–6320 (2008)

  21. 21.

    Mainardi, F., Pironi, P.: The fractional Langevin equation: Brownian motion revisited. Extr. Math. 11(1), 140–154 (1996)

  22. 22.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

  23. 23.

    Rassias, Th.M.: On the stability of linear mappings in Banach spaces. Proc. Amer. Math. 72, 297–300 (1978)

  24. 24.

    Rus, I.A.: Ulam stability of ordinary differential equations. Stud. Univ. Babeş–Bolyai, Math. 54, 125–133 (2009)

  25. 25.

    Saad, K.M., Atangana, A., Baleanu, D.: Fractional derivatives with non-singular kernel applied to the Burger’s equation. Chaos 28(6), 063109 (2018)

  26. 26.

    Saad, K.M., Baleanu, D., Atangana, A.: Fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations. Comput. Appl. Math. 37(4), 5203–5216 (2018)

  27. 27.

    Saad, K.M., Sharif, Al.: Analytical study for time and time-space fractional Burgers’ equation solutions. Adv. Differ. Equ. 2017, 300 (2017)

  28. 28.

    Shah, R., Zada, A.: A fixed point approach to the stability of a nonlinear Volterra integrodiferential equation with delay. Hacet. J. Math. Stat. 47(3), 615–623 (2018)

  29. 29.

    Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin (2011)

  30. 30.

    Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1968)

  31. 31.

    Wang, J., Feckan, M., Zhou, Y.: Ulam stype stability of impulsive ordinary differential equation. J. Math. Anal. Appl. 35, 258–264 (2012)

  32. 32.

    Wang, J., Zada, A., Ali, W.: Ulam’s-type stability of first–order impulsive differential equations with variable delay in quasi-Banach spaces. Int. J. Nonlinear Sci. Numer. Simul. 19, 553–560 (2018)

  33. 33.

    Wang, J., Zhou, Y., Feckan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)

  34. 34.

    Wang, J., Zhou, Y., Lin, Z.: On a new class of impulsive fractional differential equations. Appl. Math. Comput. 242, 649–657 (2014)

  35. 35.

    Wu, G.C., Baleanu, D.: Stability analysis of impulsive fractional difference equations. Fract. Calc. Appl. Anal. 21, 354–375 (2018)

  36. 36.

    Wu, G.C., Baleanu, D., Huang, L.L.: Novel Mittag–Leffler stability of linear fractional delay difference equations with impulse. Appl. Math. Lett. 82, 71–78 (2018)

  37. 37.

    Zada, A., Ali, S.: Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses. Int. J. Nonlinear Sci. Numer. Simul. 19(7), 763–774 (2018)

  38. 38.

    Zada, A., Ali, S., Li, Y.: Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition. Adv. Differ. Equ. 2017, 317 (2017)

  39. 39.

    Zada, A., Ali, W., Farina, S.: Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses. Math. Methods Appl. Sci. 40, 5502–5514 (2017)

  40. 40.

    Zada, A., Shah, O., Shah, R.: Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl. Math. Comput. 271, 512–518 (2015)

  41. 41.

    Zada, A., Shah, S.O.: Hyers–Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses. Hacet. J. Math. Stat. 47(5), 1196–1205 (2018)

  42. 42.

    Zada, A., Wang, P., Lassoued, D., Li, T.X.: Connections between Hyers–Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems. Adv. Differ. Equ. 2017, 192 (2017)

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All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.

Correspondence to Xiaoming Wang.

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MSC

  • 26A33
  • 34A08
  • 34B27

Keywords

  • Implicit Langevin equation
  • Caputo derivative
  • noninstantaneous impulses
  • Ulam–Hyers–Rassias stability