Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

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.

1 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.

2 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

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

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

• (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. □

3 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

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

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

• (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}$$

4 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.

5 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.