Existence and Uniqueness of Solutions to Four-Point Impulsive Fractional Differential Equations with p-Laplacian Operator

Abstract

In this paper, by using fixed-point theorems, the existence and uniqueness of positive solutions to a class of four-point impulsive fractional differential equations with p-Laplacian operators are studied. In addition, three examples are given to justify the conclusion. The interest of this paper is to study impulsive fractional differential equations with p-Laplacian operators.

1. Introduction

It is well known that fractional differential equations describe a number of phenomena in natural life, especially in describing natural, physical and chemical phenomena more accurately and generally [1,2,3,4,5]. In the past few years, fractional differential equations described problems in greater depth and specificity, especially in the study of the liver, mumps virus, glucose and other problems [6,7,8,9].

Impulsive differential equations, which offer a depiction of surveyed evolving processes, were called mathematical instruments for the comprehension of several actual problems in science [10,11,12,13]. However, the theory of impulsive equations is more expressive than the corresponding theory of equations without impulse impacts since straightforward impulsive differential equations emerged for several fresh phenomena, such as rhythmical beating and the merging of solutions [14,15,16,17]. For the theory and applications of impulsive differential equations, we refer to the references [18,19,20,21].

In recent years, fractional differential equations with p-Laplacian operators are becoming significant, they can be used to depict a number of proliferate phenomena, which have been comprehensively applied in the fields of fluid mechanics, material memory and chemistry [22,23,24,25,26,27].

For convenience, among this paper, we mark ${\phi }_p\left(u\right)$ as a p-Laplacian operator, i.e.,

$${\phi }_p\left(u\right)={\left|u\right|}^{p-2}u,{\left({\phi }_p\right)}^{-1}={\phi }_q,$$

where $p>1$, $1/p+1/q=1.$

In [24], Li et al. discussed the existence of positive solutions to four-point boundary value problems,

$$\left\{\begin{array}{c}{}^c{D}_{0^{+}}^{\beta }\left({\phi }_p{(}^c{D}_{0^{+}}^{\alpha }u\left(t\right))\right)+f(t,u\left(t\right))=0,t\in (0,1),\hfill \\ {{(}^c{D}_{0^{+}}^{\alpha }u\left(0\right))}^{\left(i\right)}=0,(i=1,2,\cdots ,n-1),\hfill \\ {\left({\phi }_p{(}^c{D}_{0^{+}}^{\alpha }u\left(0\right))\right)}^{\left(j\right)}=0,(j=1,2,\cdots ,m-1),\hfill \\ u\left(0\right)-a^c{D}_{0^{+}}^{\alpha }u\left(\xi \right)=0,u\left(1\right)+b^c{D}_{0^{+}}^{\alpha }u\left(\eta \right)=0,\hfill \end{array}\right.$$

(1)

where ${\phi }_p$ is p-Laplacian operator, ${}^c{D}_{0^{+}}^{\alpha }u\left(t\right)$ and ${}^c{D}_{0^{+}}^{\beta }u\left(t\right)$ are Caputo fractional derivatives, $0<n-1<\alpha \le n,0<m-1<\beta \le m,m+n-1<\alpha +\beta <m+n,0<\xi <\eta <1.$ By using the monotone iterative technique, they obtained the existence of positive solutions to the fractional differential equation.

In [20], Tian et al. discussed the existence of solutions to a class of three-point impulse boundary value problems.

$$\left\{\begin{array}{c}{}^c{D}^{q}u\left(t\right)=f(t,u\left(t\right)),0<t<1,t\ne t_k,k=1,2,\cdots ,p,\hfill \\ \Delta u\left(t_k\right)=I_k\left(u\left(t_k\right)\right),\Delta u^{\prime }\left(t_k\right)=\overline{I_k}\left(u\left(t_k\right)\right),k=1,2,\cdots ,p,\hfill \\ u\left(0\right)+u^{\prime }\left(0\right)=0,u\left(1\right)+u^{\prime }\left(\xi \right)=0,\hfill \end{array}\right.$$

(2)

where ${}^c{D}^q$ is a Caputo fractional derivative, and $q\in \mathbb{R},1<q\le 2,f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function. By using a fixed-point theorem, the existence of solutions was found.

In [24], the authors considered a four-point fractional differential Equation $\left(1\right)$ with p-Laplacian operators; however, the boundary value conditions of this differential equations were impulsive. As we all know, the impulsive differential equations with p-Laplacian not only contain general dynamic equations but also have profound engineering physical significance—for instance, a p-Laplacian equation is needed to study the turbulence of non-Newtonian fluida, gas in porous media and the spontaneous combustion theory of chemically active gas.

Thus, in this paper, we studied the existence of solutions to four-point impulsive fractional differential equations with p-Laplacian operators. The studying of the existence of solutions to impulsive fractional differential equations with p-Laplacian is extremely important. In this paper, we discuss the existence and uniqueness of positive solutions to four-point impulsive fractional differential equations given by

$$\left\{\begin{array}{c}{{(}^c{D}_{0^{+}}^{\beta }\left({\phi }_p{(}^c{D}_{0^{+}}^{\alpha }u\left(t\right))\right))}^{\prime }=\lambda f(t,u\left(t\right),u^{\prime }\left(t\right)),2<\alpha ,\beta \le 3,t\in J^{\prime },\hfill \\ \Delta u\left(t_k\right)=I_k\left(u\left(t_k\right)\right),\Delta u^{\prime }\left(t_k\right)=I_k^{\prime }\left(u\left(t_k\right)\right),\Delta u^{″}\left(t_k\right)=I_k^{″}\left(u\left(t_k\right)\right),k=1,2,3,\cdots ,m,\hfill \\ au\left(0\right)-b{u}^{\prime }\left(\xi \right)=0,cu\left(1\right)+d{u}^{\prime }\left(\eta \right)=0,\hfill \\ {{(}^c{D}_{0^{+}}^{\alpha }u\left(t\right))}^{\left(j\right)}=0,(j=0,1,2),u^{″}\left(0\right)=0,\hfill \end{array}\right.$$

(3)

where ${}^{c}D_{0^{+}}^{\alpha }u\left(t\right)$ and ${}^{c}D_{0^{+}}^{\beta }u\left(t\right)$ are Caputo fractional differentials, $\lambda$ is a parameter, ${\phi }_p$ is a p-Laplacian operator, and ${\displaystyle {\phi }_p\left(s\right)={\left|s\right|}^{p-2}s,p>1,{\left({\phi }_p\right)}^{-1}\left(s\right)={\phi }_q\left(s\right),\frac{1}{p}+\frac{1}{q}=1,f\in C(J\times \mathbb{R}\times \mathbb{R},\mathbb{R}),I_k,I_k^{\prime },I_k^{″}\in C(\mathbb{R},\mathbb{R}),a,b,c,d\in [0,+\infty ),ac+bc+da=0,J=[0,1],0=t_0<t_1<\cdots <t_k<\cdots <t_m<t_{m+1}=1,J^{\prime }=J\setminus \{t_1,t_2,\cdots ,t_m\},c_i\in \mathbb{R},(i=1,2,\cdots ,9),0<\xi <\eta <1,\xi \ne t_k,\eta \ne t_k(k=1,2,\cdots ,m),\Delta u\left(t_k\right)=u\left(t_k^{+}\right)-u\left(t_k^{-}\right),\Delta u^{\prime }\left(t_k\right)=u^{\prime }\left(t_k^{+}\right)-u^{\prime }\left(t_k^{-}\right),\Delta u^{″}\left(t_k\right)=u^{″}\left(t_k^{+}\right)-u^{″}\left(t_k^{-}\right),}$ where $u\left(t_k^{+}\right)$ and $u\left(t_k^{-}\right)$ indicate the right and the left limits of $u\left(t\right)$ at $t=t_k(k=1,2,\cdots ,m),$ respectively, $u^{\prime }\left(t_k\right)$ and $u^{″}\left(t_k\right)$ have a kinsmanship implication for $u^{\prime }\left(t\right)$ and $u^{″}\left(t\right).$

The objective of this paper is to establish the existence and uniqueness of positive solutions to a class of four-point impulsive fractional differential equations (Equation (3)) by using a fixed-point theorem and contraction mapping principle. In addition, three examples are given to justify the conclusion. The main features of this paper are as follows. First, the nonlinear term involved the fractional order derivative. Next, the studied Equation (3) was used to find the first derivative on the outside of ${}^c{D}_{0^{+}}^{\beta }\left({\phi }_p{(}^c{D}_{0^{+}}^{\alpha }u\left(t\right))\right)$.

Compared with [20,24], our nonlinear terms are more general. Finally, our main results are optimal. This paper contributes to the further development of impulsive fractional differential equations. The rest of the paper is as follows: In Section 2, some basic definitions and related lemmas are given. The existence of positive solutions and the uniqueness of positive solutions for the four-point impulsive fractional differential Equation (3) with p-Laplacian operators under certain assumptions are proven in Section 3. Three examples are given in the Section 4.

2. Preliminaries

In this section, we present some basic definitions, related lemmas and Green functions.

Let

$$J_0=[0,t_1],J_1=(t_1,t_2],\cdots ,J_{m-1}=(t_{m-1},t_m],J_m=(t_m,1],$$

we define the spaces as follows. Let

$$PC(J,\mathbb{R})=\{u:J\to \mathbb{R}\mid u\in C\left(J_k\right),k=1,2,\cdots ,m\},$$

$$P{C}^1(J,\mathbb{R})=\{u:J\to \mathbb{R}\mid u\in C^1\left(J_k\right),k=1,2,\cdots ,m\},$$

$$P{C}^2(J,\mathbb{R})=\{u:J\to \mathbb{R}\mid u\in C^2\left(J_k\right),k=1,2,\cdots ,m\},$$

and endowed with norm

$$\parallel u\left(t\right)\parallel =\underset{t\in J}{sup}\left|u\left(t\right)\right|,$$

$${\parallel u\left(t\right)\parallel }_{P{C}^1}=max\{\parallel u\left(t\right)\parallel ,\parallel u^{\prime }\left(t\right)\parallel \},$$

$${\parallel u\left(t\right)\parallel }_{P{C}^2}=max\{\parallel u\left(t\right)\parallel ,\parallel u^{\prime }\left(t\right)\parallel ,\parallel u^{″}\left(t\right)\parallel \}.$$

where $u\left(t_k^{+}\right)$ indicate the right limit of $u\left(t\right)$ at $t=t_k(k=1,2,\cdots ,m),u^{\prime }\left(t_k\right)$ and $u^{″}\left(t_k\right)$ have a kinsmanship implication for $u^{\prime }\left(t\right)$ and $u^{″}\left(t\right).$

Clearly,

$$PC(J,\mathbb{R}),P{C}^1(J,\mathbb{R}),P{C}^2(J,\mathbb{R})$$

is a complete linear norm space. Thus,

$$PC(J,\mathbb{R}),P{C}^1(J,\mathbb{R}),P{C}^2(J,\mathbb{R})$$

are Banach spaces.

Definition 1 

([18]). Let $f\left(t\right)\in (0,1).$ The Caputo fractional derivative is defined as

$${}^{c}D_{0+}^{\alpha }f\left(t\right)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_0^t{(t-s)}^{n-\alpha -1}{f}^{\left(n\right)}\left(s\right)ds,n=\left[\alpha \right]+1,$$

where $\left[\alpha \right]$ indicates the integer part of real number α.

Definition 2 

([18]). Let $f\left(t\right)\in (0,1).$ The Riemann–Liouville fractional integral of order is defined as

$$I_{0+}^{\alpha }f\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}f\left(s\right)ds,\alpha >0,$$

provided the integral exists.

Lemma 1 

([18]). For $\alpha >0,u\in C[0,1],$the general solution of the fractional differential equation ${}^{c}D_{0+}^{\alpha }\left(u\left(t\right)\right)=0$ is stated by

$$u\left(t\right)=C_0+C_1+\cdots +C_{N-1}{t}^{N-1},C_i\in \mathbb{R}(i=1,2,\cdots ,N),N=\left[\alpha \right]+1,$$

where $\left[\alpha \right]$ indicates the integer part of real number α.

Lemma 2 

([18]). Let $u\left(t\right)\in C[0,1]\cap L[0,1],\alpha >0$and ${}^{c}D_{0+}^{\alpha }\left(u\left(t\right)\right)\in C[0,1]\cap L[0,1],$

$$I_{0+}^{\alpha }{}^{c}D_{0+}^{\alpha }u\left(t\right)=u\left(t\right)+C_0+C_1+\cdots +C_{N-1}{t}^{N-1},$$

(4)

$C_i\in \mathbb{R}(i=1,2,\cdots ,N),N=\left[\alpha \right]+1.$

Lemma 3 

([18]). [Contraction Mapping Principle] $(X,P)$ is a complete distance space, and T is a compressed map from $(X,P)$ to itself, and thus T has a unique fixed point on X.

Lemma 4 

([18]). [Arzela–Ascoli Theorem] Let $M:\zeta \cap (\overline{{\Omega }_2}\setminus {\Omega }_1)\to \zeta ;$ we say M is a compact operator if it is uniformly bounded and equicontinuous.

Lemma 5 

([18]). Let E be a Banach space. Assume that $T:E\to E$ is a completely continuous operator and that the set

$$V=\{u\in E\mid u=\mu Tu,0<\mu <1\}$$

is bounded, then T has a fixed point in E.

Lemma 6 

([18]). Let E be a Banach space. Assume that Ω is an open bounded subset of E with $\theta \in \Omega$ and let $T:\overline{\Omega }\to E$ be a completely continuous operator such that

$$\Vert Tu\Vert \le \Vert u\Vert ,\forall u\in \partial \Omega ,$$

then T has a fixed point in $\overline{\Omega }$, where θ is a fixed point.

Lemma 7. 

Let$y\in C[0,1],\xi ,\eta \in (t_l,t_l+1),l$is a nonnegative integer,$0\le l\le m.$then the solution of the fractional differential equations

$$\left\{\begin{array}{c}{}^{c}D_{0+}^{\beta }\left({\phi }_p{}^{c}D_{0+}^{\alpha }u\left(t\right)\right)\left)\right)\prime =\lambda y\left(t\right),2<\alpha ,\beta \le 3,t\in J^{\prime },\hfill \\ \Delta u\left(t_k\right)=I_k\left(u\left(t_k\right)\right),\Delta u^{\prime }\left(t_k\right)=I_k^{\prime }\left(u\left(t_k\right)\right),\Delta u^{″}\left(t_k\right)=I_k^{″}\left(u\left(t_k\right)\right),k=1,2,3,\cdots ,m,\hfill \\ au\left(0\right)-b{u}^{\prime }\left(\xi \right)=0,cu\left(1\right)+d{u}^{\prime }\left(\eta \right)=0,\hfill \\ {{}^{c}D_{0+}^{\alpha }u\left(t\right))}^{\left(j\right)}=0,(j=0,1,2),u^{″}\left(0\right)=0,\hfill \end{array}\right.$$

(5)

has the solution

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \left\{\begin{array}{c}{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_4+c_{5}t,t\in J_0,}\hfill \\ {\displaystyle {\int }_{t_k}^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ {\displaystyle +\sum _{i=1}^k{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ {\displaystyle +\sum _{i=1}^{k-1}(t_k-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ {\displaystyle +\sum _{i=1}^{k-1}\frac{{(t_k-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ {\displaystyle +\sum _{i=1}^k(t-t_k){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ {\displaystyle +\sum _{i=1}^k\frac{{(t-t_k)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ {\displaystyle +\sum _{i=1}^{k-1}(t-t_k)(t_k-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ {\displaystyle +\sum _{i=1}^k{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}(t_k-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}\frac{{(t_k-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^k(t-t_k)I_i^{\prime }\left(u\left(t_i\right)\right)}\hfill \\ {\displaystyle +\sum _{i=1}^k\frac{{(t-t_k)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}(t-t_k)(t_k-t_i)I_i^{″}\left(u\left(t_i\right)\right)+c_4+c_{5}t,t\in J_k,k=1,2,\cdots ,m,}\hfill \end{array}\right.\hfill \end{array}$$

(6)

where

$$\begin{array}{ccc}\hfill c_4& =& \frac{1}{ac+ad+bc}\left\{bc\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\right.\hfill \\ & & +\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}(t_m-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m(1-t_m)(t_m-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^m(1-t_m)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +bd\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\eta -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & -b(c+d)\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.\left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\xi -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill c_5& =& \frac{1}{ac+ad+bc}\left\{ac\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\right.\hfill \\ & & +\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}(t_m-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m(1-t_m)(t_m-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^m(1-t_m)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +ad\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\eta -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & +bc\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.\left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\xi -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\right\}.\hfill \end{array}$$

Proof. 

Let u be a solution of (5). By (4), we find

$$\begin{array}{c}\hfill {\phi }_p\left({}^{c}D_{0^{+}}^{\alpha }u\left(t\right)\right)=\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}\left({\int }_0^{s}y\left(r\right)dr\right)ds+c_1+c_{2}t+c_3{t}^2,\end{array}$$

according to conditions

$${\left({}^{c}D_{0^{+}}^{\alpha }u\left(t\right)\right)}^{\left(0\right)}=0,(j=0,1,2),$$

we find $c_1=c_2=c_3=0,$ and

$$\begin{array}{c}\hfill {}^c{D}_{0^{+}}^{\alpha }u\left(t\right)={\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}\left({\int }_0^{s}y\left(r\right)dr\right)ds\right).\end{array}$$

In addition,

$$u\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_4+c_{5}t+c_6{t}^2,$$

(7)

$$\begin{array}{c}\hfill u^{\prime }\left(t\right)=\frac{1}{\mathrm{\Gamma }(\alpha -1)}{\int }_0^t{(t-s)}^{\alpha -2}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_5+2c_{6}t,\end{array}$$

$$\begin{array}{c}\hfill u^{″}\left(t\right)=\frac{1}{\mathrm{\Gamma }(\alpha -2)}{\int }_0^t{(t-s)}^{\alpha -3}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+2c_6.\end{array}$$

If $t\in J_1,$

$$\begin{array}{c}\hfill u\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{t_1}^t{(t-s)}^{\alpha -1}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_7+c_8(t-t_1)+c_9{(t-t_1)}^2,\end{array}$$

$$\begin{array}{c}\hfill u^{\prime }\left(t\right)=\frac{1}{\mathrm{\Gamma }(\alpha -1)}{\int }_{t_1}^t{(t-s)}^{\alpha -2}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_8+2c_9(t-t_1),\end{array}$$

$$\begin{array}{c}\hfill u^{″}\left(t\right)=\frac{1}{\mathrm{\Gamma }(\alpha -2)}{\int }_{t_1}^t{(t-s)}^{\alpha -3}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+2c_9.\end{array}$$

Thus

$$\begin{array}{ccc}\hfill u\left(t_1^{-}\right)& =& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -1}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_4+c_5{t}_1+c_6{t}_1^2,\hfill \\ \hfill u\left(t_1^{+}\right)& =& e_1,\hfill \\ \hfill u^{\prime }\left(t_1^{-}\right)& =& \frac{1}{\mathrm{\Gamma }(\alpha -1)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -2}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_5+2c_6{t}_1,\hfill \\ \hfill u^{\prime }\left(t_1^{+}\right)& =& e_2,\hfill \\ \hfill u^{″}\left(t_1^{-}\right)& =& \frac{1}{\mathrm{\Gamma }(\alpha -2)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -3}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+2c_6,\hfill \\ \hfill u^{″}\left(t_1^{+}\right)& =& 2e_3.\hfill \end{array}$$

Due to

$$\Delta u\left(t_1\right)=u\left(t_1^{+}\right)-u\left(t_1^{-}\right)=I_1\left(u\left(t_1\right)\right),$$

$$\Delta u^{\prime }\left(t_1\right)=u^{\prime }\left(t_1^{+}\right)-u^{\prime }\left(t_1^{-}\right)=I_1^{\prime }\left(u\left(t_1\right)\right),$$

$$\Delta u^{″}\left(t_1\right)=u^{″}\left(t_1^{+}\right)-u^{″}\left(t_1^{-}\right)=I_1^{″}\left(u\left(t_1\right)\right),$$

we find

$$\begin{array}{ccc}\hfill c_7& =& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -1}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_4+c_5{t}_1+c_6{t}_1^2+I_1\left(u\left(t_1\right)\right),\hfill \\ \hfill c_8& =& \frac{1}{\mathrm{\Gamma }(\alpha -1)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -2}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+c_5+2c_6{t}_1+I_1^{\prime }\left(u\left(t_1\right)\right),\hfill \\ \hfill 2c_9& =& \frac{1}{\mathrm{\Gamma }(\alpha -2)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -3}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+2c_6+I_1^{″}\left(u\left(t_1\right)\right).\hfill \end{array}$$

Consequently,

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{t_1}^t{(t-s)}^{\alpha -1}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -1}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+I_1\left(u\left(t_1\right)\right)\hfill \\ & & +\frac{t-t_1}{\mathrm{\Gamma }(\alpha -1)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -2}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+(t-t_1)I_1^{\prime }\left(u\left(t_1\right)\right)\hfill \\ & & +\frac{{(t-t_1)}^2}{2\mathrm{\Gamma }(\alpha -2)}{\int }_0^{t_1}{(t_1-s)}^{\alpha -3}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds+\frac{{(t-t_1)}^2}{2}{I}_1^{″}\left(u\left(t_1\right)\right)\hfill \\ & & +c_4+c_{5}t+c_6{t}^2,t\in J_1.\hfill \end{array}$$

Thus, $t\in J_k,k=1,2,\cdots ,m$, and we get

$$\begin{array}{ccc}\hfill u\left(t\right)& =& {\displaystyle {\int }_{t_k}^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^k{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^{k-1}(t_k-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^{k-1}\frac{{(t_k-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^k(t-t_k){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^k\frac{{(t-t_k)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^{k-1}(t-t_k)(t_k-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^k{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}(t_k-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}\frac{{(t_k-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^k(t-t_k)I_i^{\prime }\left(u\left(t_i\right)\right)}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^k\frac{{(t-t_k)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}(t-t_k)(t_k-t_i)I_i^{″}\left(u\left(t_i\right)\right)+c_4+c_{5}t+c_6{t}^2.}\hfill \end{array}$$

(8)

Due to

$$au\left(0\right)-b{u}^{\prime }\left(\xi \right)=0,cu\left(1\right)+d{u}^{\prime }\left(\eta \right)=0,u^{″}\left(0\right)=0,$$

we find

$$\begin{array}{ccc}\hfill c_4& =& \frac{1}{ac+ad+bc}\left\{bc\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\right.\hfill \\ & & +\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}(t_m-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m(1-t_m)(t_m-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^m(1-t_m)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & +bd\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\eta -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & -b(c+d)\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.\left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\xi -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill c_5& =& \frac{1}{ac+ad+bc}\left\{ac\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\right.\hfill \\ & & +\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}(t_m-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m(1-t_m)(t_m-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^m(1-t_m)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & +ad\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\eta -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & +bc\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.\left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\xi -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\right\}\hfill \end{array}$$

and

$$c_6=0.$$

Substituting the value of $c_4,c_5,c_6$ in (7) and (8), we can find (6). This completes the proof. □

3. Main Results

In this section, the existence of positive solutions for the four-point impulsive fractional differential Equation (3) with p-Laplacian operators is proven.

We set

$$\begin{array}{c}\hfill {\sigma }_1=\frac{ad+2ac+2bc}{ad+ac+bc},{\sigma }_2=\frac{ad+2bc+2bd}{ad+ac+bc},{\sigma }_3=ac+2bc+2ad.\end{array}$$

Define the operator $T:PC(J,\mathbb{R})\to PC(J,\mathbb{R})$ as

$$\begin{array}{ccc}\hfill \left(Tu\right)\left(t\right)& =& {\displaystyle {\int }_{t_k}^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^k{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^{k-1}(t_k-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^{k-1}\frac{{(t_k-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^k(t-t_k){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^k\frac{{(t-t_k)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^{k-1}(t-t_k)(t_k-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^k{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}(t_k-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}\frac{{(t_k-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^k(t-t_k)I_i^{\prime }\left(u\left(t_i\right)\right)}\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^k\frac{{(t-t_k)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{k-1}(t-t_k)(t_k-t_i)I_i^{″}\left(u\left(t_i\right)\right)+g_1+g_{2}t.}\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill g_1& =& \frac{1}{ac+ad+bc}\left\{bc\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\right.\hfill \\ & & +\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}(t_m-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m(1-t_m)(t_m-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^m(1-t_m)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & +bd\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\eta -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & -b(c+d)\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.\left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\xi -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\right\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill g_2& =& \frac{1}{ac+ad+bc}\left\{ac\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\right.\hfill \\ & & +\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m{I}_i\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}(t_m-t_i)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m(1-t_m)(t_m-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^m(1-t_m)I_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{I}_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & +ad\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\eta -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\hfill \\ & & +bc\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\right.\hfill \\ & & +\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds}\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }y\left(r\right)dr\right)d\tau \right)ds\hfill \\ & & \left.\left.{\displaystyle +\sum _{i=1}^l{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^l(\xi -t_l)I_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{l-1}(t_l-t_i)I_i^{″}\left(u\left(t_i\right)\right)}\right)\right\}.\hfill \end{array}$$

Using Lemma 7 with $y\left(t\right)=f(t,u\left(t\right),u^{\prime }\left(t\right))$, problem (3) reduces to a fixed-point problem $u=Tu$. Thus, fractional differential Equation (3) has a solution if and only if the operator T has a fixed point.

Theorem 1. 

The operator$T:PC(J,\mathbb{R})\to PC(J,\mathbb{R})$is completely continuous.

Proof. 

It is clear that T is continuous by the continuity of $f,I_k,I_k^{\prime }and{I}_k^{″}.$ Let $\Omega \in PC(J,\mathbb{R})$ be bounded. Afterward, there exist positive constants $M_i>0(i=1,2,3,4),$ such that $\mid \left(f(t,u\left(t\right),u^{\prime }\left(t\right))\right)\mid \le {\phi }_p\left(M_1\right),\mid I_k\left(u\right)\mid \le M_2,\mid I_k^{\prime }\left(u\right)\mid \le M_3,\mid I_k^{″}\left(u\right)\mid \le M_4,\forall u\in \Omega .$ Thus, $\Omega \in PC(J,\mathbb{R}),$ set ${\displaystyle A=|{\phi }_q\left(\frac{1}{\mathrm{\Gamma }(\beta +1)}\right)|,\lambda =1,}$ we find

$$\begin{array}{ccc}\hfill |g_1|& \le & \frac{1}{ac+ad+bc}\left\{bc\left(A{M}_1{\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}ds+\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}ds\right.\right.\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^{m-1}\frac{1}{2}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^m\frac{1}{2}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds+\sum _{i=1}^m{M}_2+\sum _{i=1}^{m-1}{M}_3+\sum _{i=1}^{m-1}\frac{M_4}{2}}\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^m{M}_3+\sum _{i=1}^m\frac{M_4}{2}}\right)+bd\left(A{M}_1{\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds\right.\hfill \\ & & {\displaystyle +\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & \left.{\displaystyle +\sum _{i=1}^{m-1}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds+\sum _{i=1}^m{M}_3+\sum _{i=1}^{m-1}{M}_4+\sum _{i=1}^m{M}_4}\right)\hfill \\ & & -b(c+d)\left(A{M}_1{\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds\right.\hfill \\ & & {\displaystyle +\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds+\sum _{i=1}^{m-1}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & \left.\left.{\displaystyle +\sum _{i=1}^m{M}_3+\sum _{i=1}^{m-1}{M}_4+\sum _{i=1}^m{M}_4}\right)\right\}\hfill \\ & \le & \frac{1}{ac+ad+bc}\left(\frac{bc(1+m)A{M}_1}{\mathrm{\Gamma }(\alpha +1)}\right.\hfill \\ & & +\frac{(2b(c+d)m+b(2d+cm))A{M}_1}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & +\frac{(4b(c+d)m-b(\frac{5}{2}c+2d))A{M}_1}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & +bcm{M}_2+(bc(2m-1)+b(c+2d)m)M_3\hfill \\ & & \left.+(bc(2m-\frac{3}{2})+b(c+2d)(2m-1)M_4)\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill |g_2|& \le & \frac{1}{ac+ad+bc}\left(\frac{ac(1+m)A{M}_1}{\mathrm{\Gamma }(\alpha +1)}\right.\hfill \\ & & +\frac{(a(2c+d)m+bc(1+m)-a(c-d))A{M}_1}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & +\frac{(ac(2m-\frac{3}{2})+(ad+bc)(2m-1))A{M}_1}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & +acm{M}_2+(ac(2m-1)+(ad+bc)m)M_3\hfill \\ & & \left.+(ac(2m-\frac{3}{2})+(ad+bc)(2m-1))M_4\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \left|\right(Tu\left)\right(t\left)\right|& \le & {\displaystyle A{M}_1{\int }_{t_k}^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}ds+\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^{m-1}\frac{1}{2}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^m\frac{1}{2}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds+\sum _{i=1}^m{I}_i\left(u\left(t_i\right)\right)}\hfill \\ & & +\sum _{i=1}^{m-1}{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}\frac{1}{2}{I}_i^{″}\left(u\left(t_i\right)\right)\hfill \\ & & {\displaystyle +\sum _{i=1}^m{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^m\frac{1}{2}{I}_i^{″}\left(u\left(t_i\right)\right)}\hfill \\ & & +\sum _{i=1}^{m-1}{I}_i^{″}\left(u\left(t_i\right)\right)+|g_1|+|g_2|\hfill \\ & \le & \frac{{\sigma }_1(1+m)A{M}_1}{\mathrm{\Gamma }(\alpha +1)}+{\sigma }_{1}m{M}_2\hfill \\ & & +\frac{({\sigma }_1(2m-1)+{\sigma }_2(1+m))A{M}_1}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & +\frac{({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1))A{M}_1}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & +({\sigma }_1(2m-1)+{\sigma }_{2}m)M_3\hfill \\ & & +({\sigma }_1(2m-\frac{3}{2})+{\sigma }_1(2m-1))M_4:=M.\hfill \end{array}$$

Since $t\in [0,1],$ there exists a positive constant $M,$ such that $\Vert Tu\Vert \le M,$ which implies that the operator T is uniformly bounded.

However, for any $t\in J_k,0\le k\le m,$ we find

$$\begin{array}{ccc}\hfill {|\left(Tu\right)}^{\prime }\left(t\right)|& \le & {\displaystyle A{M}_1{\int }_{t_k}^t\frac{{(t-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds+\sum _{i=1}^{m}A{M}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^m{I}_i^{″}\left(u\left(t_i\right)\right)+\left|g_2\right|}\hfill \\ & \le & \frac{1}{ac+ad+bc}\left(\frac{acA{M}_1(m+1)}{\mathrm{\Gamma }(\alpha +1)}\right.\hfill \\ & & +\frac{acA{M}_1(2m-1)+A{M}_1{\sigma }_3(1+m)}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & +\frac{acA{M}_1(2m-\frac{3}{2})+A{M}_1{\sigma }_3(2m-1)}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & +acm{M}_2+(ac(2m-1)+m{\sigma }_3)M_3\hfill \\ & & \left.+(ac(2m-\frac{3}{2})+{\sigma }_3(2m-1))M_4\right):=\overline{M}.\hfill \end{array}$$

Thus, for $t_1,t_2\in J_k,t_1<t_2,0\le k\le m,$ we have

$$\mid \left(Tu\right)\left(t_2\right)-\left(Tu\right)\left(t_1\right)\mid \le {\int }_{t_1}^{t_2}\mid {\left(Tu\right)}^{\prime }\left(s\right)\mid ds\le \overline{M}(t_2-t_1),$$

which implies that T is equicontinuous on all $J_k,k=1,2,\cdots ,m.$ Therefore, by Lemma 4, the operator T is completely continuous. □

Theorem 2. 

Assume that there exist positive constants$H_i(i=1,2,3,4),$such that$\mid (f(t,u,u^{\prime })\mid \le {\phi }_p\left(H_1\right),\mid I_k\left(u\right)\mid \le H_2,\mid I_k^{\prime }\left(u\right)\mid \le H_3,\mid I_k^{″}\left(u\right)\mid \le H_4,$the fractional differential Equation (3) has a solution.

Proof. 

First, we prove that

$$B=\{u\in PC(J,\mathbb{R})\mid u=\lambda Tu,0<\lambda <1\}$$

is bounded. Let $u\in B,$ and then $u\in \lambda Tu,0<\lambda <1.$ For any $t\in J,$ we find

$$\begin{array}{ccc}\hfill \left|u\right(t\left)\right|=\lambda \left|\right(Tu\left)\right(t\left)\right|& \le & {\displaystyle A{H}_1{\int }_{t_k}^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}ds+\sum _{i=1}^{m}A{H}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}ds}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\displaystyle +\sum _{i=1}^{m-1}A{H}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^{m-1}\frac{1}{2}A{H}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m}A{H}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds+\sum _{i=1}^m\frac{1}{2}A{H}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}A{H}_1{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}ds+\sum _{i=1}^m{I}_i\left(u\left(t_i\right)\right)}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}{I}_i^{\prime }\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}\frac{1}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^m{I}_i^{\prime }\left(u\left(t_i\right)\right)}\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{1}{2}{I}_i^{″}\left(u\left(t_i\right)\right)+\sum _{i=1}^{m-1}{I}_i^{″}\left(u\left(t_i\right)\right)+|g_1|+|g_2|}\hfill \\ & \le & \frac{{\sigma }_1(1+m)A{H}_1}{\mathrm{\Gamma }(\alpha +1)}+{\sigma }_{1}m{H}_2\hfill \\ & & +\frac{({\sigma }_1(2m-1)+{\sigma }_2(1+m))A{H}_1}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & +\frac{({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1))A{H}_1}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & +({\sigma }_1(2m-1)+{\sigma }_{2}m)H_3\hfill \\ & & +({\sigma }_1(2m-\frac{3}{2})+{\sigma }_1(2m-1))H_4:=H,\hfill \end{array}$$

which illustrates that

$$\parallel u\parallel \le H.$$

For any $t\in J,$ the set B is bounded. Hence, by Lemma 5, the fractional differential Equation (3) has a solution. □

Theorem 3. 

Let${\displaystyle \underset{u\to 0}{lim}\frac{f(t,u,u^{\prime })}{{\phi }_p\left(u\right)}=0,\underset{u\to 0}{lim}\frac{I_k\left(u\right)}{u}=0,\underset{u\to 0}{lim}\frac{I_k^{\prime }\left(u\right)}{u}=0,\underset{u\to 0}{lim}\frac{I_k^{″}\left(u\right)}{u}=0,}$and

$$\begin{array}{ccc}& \sup & \left\{\frac{{\sigma }_1(m+1)A}{\mathrm{\Gamma }(\alpha +1)}{\delta }_1\right.\hfill \\ & & +\frac{({\sigma }_1(2m-1)+{\sigma }_2(m+1))A}{\mathrm{\Gamma }\left(\alpha \right)}{\delta }_1\hfill \\ & & +\frac{({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1))A}{\mathrm{\Gamma }(\alpha -1)}{\delta }_1\hfill \\ & & +{\sigma }_{1}m{\delta }_2+({\sigma }_1(2m-1)+{\sigma }_{2}m){\delta }_3\hfill \\ & & +\left.({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1)){\delta }_4\right\}\le 1.\hfill \end{array}$$

for some${\delta }_i>0(i=1,2,3,4),$then the fractional differential Equation (3) has a solution.

Proof. 

Since ${\displaystyle \underset{u\to 0}{lim}\frac{f(t,u,u^{\prime })}{{\phi }_p\left(u\right)}=0,\underset{u\to 0}{lim}\frac{I_k\left(u\right)}{u}=0,\underset{u\to 0}{lim}\frac{I_k^{\prime }\left(u\right)}{u}=0}$ and ${\displaystyle \underset{u\to 0}{lim}\frac{I_k^{″}\left(u\right)}{u}=0;}$ therefore, there exists a constant $s>0,$ such that $|f(t,u,u^{\prime })|\le {\phi }_q\left({\delta }_1\right)|{\phi }_p\left(u\right)|,|I_k\left(u\right)|\le {\delta }_2\left|u\right|,|I_k^{\prime }\left(u\right)|\le {\delta }_3\left|u\right|$ and $|I_k^{″}\left(u\right)|\le {\delta }_4\left|u\right|,$ for $0<\left|u\right|<s,$

$$\begin{array}{ccc}& \sup & \left\{\frac{{\sigma }_1(m+1)A}{\mathrm{\Gamma }(\alpha +1)}{\delta }_1\right.\hfill \end{array}$$

$$\begin{array}{ccc}& & +\frac{({\sigma }_1(2m-1)+{\sigma }_2(m+1))A}{\mathrm{\Gamma }\left(\alpha \right)}{\delta }_1\hfill \\ & & +\frac{({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1))A}{\mathrm{\Gamma }(\alpha -1)}{\delta }_1\hfill \\ & & +{\sigma }_{1}m{\delta }_2+({\sigma }_1(2m-1)+{\sigma }_{2}m){\delta }_3\hfill \\ & & +\left.({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1)){\delta }_4\right\}\le 1.\hfill \end{array}$$

Hence, it follows that $\Vert Tu\Vert \le \Vert u\Vert ,u\in \partial \Omega .$ Thus, by Lemma 6, the operator T has one fixed point. Thus, the fractional differential Equation (3) has a solution $u\in \overline{\Omega }.$ □

Theorem 4. 

There exist positive constants$L_i>0(i=1,2,3)$such that

$$\begin{array}{c}\hfill |{\phi }_q\left(f(t,u,u^{\prime })\right)-{\phi }_q\left(f(t,v,v^{\prime })\right)|\le L_1|u-v|,|I_k\left(u\right)-I_k\left(v\right)|\le L_2|u-v|,\\ \hfill |I_k^{\prime }\left(u\right)-I_k^{\prime }\left(v\right)|\le L_3|u-v|,|I_k^{″}\left(u\right)-I_k^{″}\left(v\right)|\le L_4|u-v|,\end{array}$$

for$t\in J,u,v\in C\left(\right[0,1],\mathbb{R})$ and $k=1,2,\cdots ,m.$ If

$$\begin{array}{ccc}\hfill K& =& \underset{t\in J}{max}\left\{\frac{{\sigma }_1(m+1)A}{\mathrm{\Gamma }(\alpha +1)}{L}_1\right.\hfill \\ & & +\frac{({\sigma }_1(2m-1)+{\sigma }_2(m+1))A}{\mathrm{\Gamma }\left(\alpha \right)}{L}_1\hfill \\ & & +\frac{({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1))A}{\mathrm{\Gamma }(\alpha -1)}{L}_1\hfill \\ & & +{\sigma }_{1}m{L}_2+({\sigma }_1(2m-1)+{\sigma }_{2}m)L_3\hfill \\ & & +\left.({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1))L_4\right\}<1,\hfill \end{array}$$

then the fractional differential Equation (3) has a unique solution.

Proof. 

For $u,v\in PC(J,\mathbb{R}),$ we obtain

$$\begin{array}{ccc}\hfill \left|Tu\right(t)-Tv(t\left)\right|& =& {\displaystyle {\int }_{t_k}^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))}\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^k{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ \hfill & & {\displaystyle +\sum _{i=1}^{k-1}(t_k-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))}\hfill \\ \hfill & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{k-1}\frac{{(t_k-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ & & {\displaystyle +\sum _{i=1}^k(t-t_k){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))}\hfill \\ \hfill & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^k\frac{{(t-t_k)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ \hfill & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{k-1}(t-t_k)(t_k-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ \hfill & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^k|I_i\left(u\left(t_i\right)\right)-I_i\left(v\left(t_i\right)\right)|}\hfill \\ & & +\sum _{i=1}^{k-1}(t_k-t_i)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|\hfill \\ \hfill & & +\sum _{i=1}^{k-1}\frac{{(t_k-t_i)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^k(t-t_k)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|\hfill \\ & & {\displaystyle +\sum _{i=1}^k\frac{{(t-t_k)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{k-1}(t-t_k)(t_k-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|}\hfill \\ & & +\frac{1}{ac+ad+bc}\left\{bc\right({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ & & {\displaystyle +\sum _{i=1}^m|I_i\left(u\left(t_i\right)\right)-I_i\left(v\left(t_i\right)\right)|+\sum _{i=1}^{m-1}(t_m-t_i)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & +\sum _{i=1}^m(1-t_m)(t_m-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^m(1-t_m)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)}\hfill \\ & & +bd\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\right|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\right({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{l-1}(t_l-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)}\hfill \\ & & -b(c+d)\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{l-1}(t_l-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|\left)\right\}}\hfill \\ & & +t\left(\frac{1}{ac+ad+bc}\left\{ac\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\right.\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ & & {\displaystyle +\sum _{i=1}^m|I_i\left(u\left(t_i\right)\right)-I_i\left(v\left(t_i\right)\right)|+\sum _{i=1}^{m-1}(t_m-t_i)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & +\sum _{i=1}^m(1-t_m)(t_m-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^m(1-t_m)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)}\hfill \\ & & +ad\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{l-1}(t_l-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)}\hfill \\ & & +bc\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ & & +\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{l-1}(t_l-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)\left\}\right)}\hfill \\ & \le & {\displaystyle A{\int }_{t_k}^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left|{\phi }_q(f(t,u\left(s\right),u^{\prime }\left(s\right))-f(t,v\left(s\right),v^{\prime }\left(s\right)))\right|ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m}A{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left|{\phi }_q(f(t,u\left(s\right),u^{\prime }\left(s\right))-f(t,v\left(s\right),v^{\prime }\left(s\right)))\right|ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}A{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q(f(t,u\left(s\right),u^{\prime }\left(s\right))-f(t,v\left(s\right),v^{\prime }\left(s\right)))\right|ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}\frac{1}{2}A{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\left|{\phi }_q(f(t,u\left(s\right),u^{\prime }\left(s\right))-f(t,v\left(s\right),v^{\prime }\left(s\right)))\right|ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m}A{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q(f(t,u\left(s\right),u^{\prime }\left(s\right))-f(t,v\left(s\right),v^{\prime }\left(s\right)))\right|ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{1}{2}A{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\left|{\phi }_q(f(t,u\left(s\right),u^{\prime }\left(s\right))-f(t,v\left(s\right),v^{\prime }\left(s\right)))\right|ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}A{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\left|{\phi }_q(f(t,u\left(s\right),u^{\prime }\left(s\right))-f(t,v\left(s\right),v^{\prime }\left(s\right)))\right|ds}\hfill \\ & & {\displaystyle +\sum _{i=1}^m|I_i\left(u\left(t_i\right)\right)-I_i\left(v\left(t_i\right)\right)|+\sum _{i=1}^{m-1}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|}\hfill \\ & & +\sum _{i=1}^{m-1}|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|+\sum _{i=1}^{m-1}\frac{1}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|\hfill \\ & & {\displaystyle +\sum _{i=1}^m|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|+\sum _{i=1}^m\frac{1}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|}\hfill \\ & & +\frac{1}{ac+ad+bc}\left\{bc\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ & & {\displaystyle +\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))}\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m|I_i\left(u\left(t_i\right)\right)-I_i\left(v\left(t_i\right)\right)|}\hfill \\ & & +\sum _{i=1}^{m-1}(t_m-t_i)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|+\sum _{i=1}^m(1-t_m)(t_m-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|\hfill \\ & & {\displaystyle +\sum _{i=1}^m(1-t_m)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|}\hfill \\ & & +\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)\hfill \\ & & +bd\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{l-1}(t_l-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)}\hfill \\ & & -b(c+d)\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{l-1}(t_l-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|\left)\right\}}\hfill \\ & & +t\left(\frac{1}{ac+ad+bc}\left\{ac\left({\int }_{t_m}^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\right.\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m(1-t_m){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{m-1}(1-t_m)(t_m-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ & & {\displaystyle +\sum _{i=1}^m|I_i\left(u\left(t_i\right)\right)-I_i\left(v\left(t_i\right)\right)|+\sum _{i=1}^{m-1}(t_m-t_i)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & +\sum _{i=1}^m(1-t_m)(t_m-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^m(1-t_m)|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|\hfill \\ & & {\displaystyle +\sum _{i=1}^m\frac{{(1-t_m)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{m-1}\frac{{(t_m-t_i)}^2}{2}|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)}\hfill \\ & & +ad\left({\int }_{t_l}^{\eta }\frac{{(\eta -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l(\eta -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\eta -t_l)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{l-1}(t_l-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)}\hfill \\ & & +bc\left({\int }_{t_l}^{\xi }\frac{{(\xi -s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\right.\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l{\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ & & \left|{\phi }_q\left(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}\left({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))-f(r,v\left(r\right),v^{\prime }\left(r\right))dr\right)d\tau \right)\right|ds\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))}\hfill \\ & & -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^{l-1}(t_l-t_i){\int }_{t_{i-1}}^{t_i}\frac{{(t_i-s)}^{\alpha -3}}{\mathrm{\Gamma }(\alpha -2)}\hfill \\ & & \left|{\phi }_q\right(\frac{\lambda }{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^s{(s-\tau )}^{\beta -1}({\int }_0^{\tau }f(r,u\left(r\right),u^{\prime }\left(r\right))\hfill \\ & & {\displaystyle -f(r,v\left(r\right),v^{\prime }\left(r\right))dr\left)d\tau \right)|ds+\sum _{i=1}^l|I_i^{\prime }\left(u\left(t_i\right)\right)-I_i^{\prime }\left(v\left(t_i\right)\right)|}\hfill \\ & & {\displaystyle +\sum _{i=1}^l(\xi -t_l)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|+\sum _{i=1}^{l-1}(t_l-t_i)|I_i^{″}\left(u\left(t_i\right)\right)-I_i^{″}\left(v\left(t_i\right)\right)|)\left\}\right)}\hfill \\ & \le & \left\{\frac{{\sigma }_1(m+1)A}{\mathrm{\Gamma }(\alpha +1)}{L}_1\right.\hfill \\ & & +\frac{({\sigma }_1(2m-1)+{\sigma }_2(m+1))A}{\mathrm{\Gamma }\left(\alpha \right)}{L}_1\hfill \\ & & +\frac{({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1))A}{\mathrm{\Gamma }(\alpha -1)}{L}_1\hfill \\ & & +{\sigma }_{1}m{L}_2+({\sigma }_1(2m-1)+{\sigma }_{2}m)L_3\hfill \\ & & +\left.({\sigma }_1(2m-\frac{3}{2})+{\sigma }_2(2m-1))L_4\right\}\parallel u-v\parallel .\hfill \end{array}$$

Consequently, we find $\parallel Tu-Tv\parallel \le K\parallel u-v\parallel .$ Hence, T is a contraction. Accordingly, the conjecture of the theorem follows by Lemma 3. This completes the proof. □

4. Example

In this section, three examples are given to prove our conclusions.

Example 1. 

Consider the following impulsive fractional differential equation

$$\left\{\begin{array}{c}{\displaystyle {\left({}^{c}D_{0^{+}}^{\beta }\left({\phi }_p{}^{c}D_{0^{+}}^{\alpha }u\left(t\right))\right)\right)}^{\prime }=\frac{cosu\left(t\right)}{3+u^3\left(t\right)},1<\alpha ,\beta \le 2,0<t<1,t\ne \frac{2}{3},}\hfill \\ {\displaystyle \Delta u\left(\frac{2}{3}\right)=e^{-u^2\left(\frac{2}{3}\right)}+5sin(1+e^u\left(\frac{2}{3}\right)),}\hfill \\ {\displaystyle \Delta u^{\prime }\left(\frac{2}{3}\right)=\frac{7+3u^2\left(\frac{2}{3}\right)}{2+5u^2\left(\frac{2}{3}\right)},}\hfill \\ {\displaystyle \Delta u^{″}\left(\frac{2}{3}\right)=\frac{e^{\frac{2}{3}}cosu\left(\frac{2}{3}\right)}{5+u^3\left(\frac{2}{3}\right)},}\hfill \\ {\displaystyle u\left(0\right)-2u^{\prime }\left(\frac{1}{5}\right)=0,3u\left(1\right)+5u^{\prime }\left(\frac{1}{7}\right)=0,}\hfill \\ {\displaystyle {}^{c}D_{0^{+}}^{\alpha }u\left(0\right)=0,u^{″}\left(0\right)=0,}\hfill \end{array}\right.$$

(9)

where$m=1and\lambda =1.$

Clearly,${\displaystyle H_1=\frac{1}{2},H_2=6,H_3=\frac{3}{5}}$and${\displaystyle H_4=\frac{4}{5},}$and the condition of Theorem 2 holds. Hence, by Theorem 2, the fractional differential Equation (11) has at least one solution.

Example 2. 

Consider the following impulsive fractional differential equation

$$\left\{\begin{array}{c}{\left({}^{c}D_{0^{+}}^{\beta }\left({\phi }_p{}^{c}D_{0^{+}}^{\alpha }u\left(t\right))\right)\right)}^{\prime }=t^2{u}^5\left(t\right)+u^3\left(t\right)arcta{n}^{4}u\left(t\right),1<\alpha ,\beta \le 2,0<t<1,t\ne \frac{1}{5},\hfill \\ {\displaystyle \Delta u\left(\frac{1}{5}\right)=e^{u^2\left(\frac{1}{5}\right)}{u}^2\left(\frac{1}{5}\right),}\hfill \\ {\displaystyle \Delta u^{\prime }\left(\frac{1}{5}\right)=\frac{2u^5\left(\frac{1}{5}\right)}{3+2u^3\left(\frac{1}{5}\right)},}\hfill \\ {\displaystyle \Delta u^{″}\left(\frac{1}{5}\right)=u^2\left(\frac{1}{5}\right)+sin(1-cosu\left(\frac{1}{5}\right)),}\hfill \\ {\displaystyle 4u\left(0\right)-3u^{\prime }\left(\frac{2}{7}\right)=0,2u\left(1\right)+5u^{\prime }\left(\frac{4}{5}\right)=0,}\hfill \\ {\displaystyle {}^{c}D_{0^{+}}^{\alpha }u\left(0\right)=0,u^{″}\left(0\right)=0,}\hfill \end{array}\right.$$

(10)

where$m=1and\lambda =1.$

$$\begin{array}{c}\hfill \underset{u\to 0}{lim}\left(\frac{f(t,u,u^{\prime })}{{\phi }_p\left(u\right)}\right)=\underset{u\to 0}{lim}\frac{t^2{u}^5\left(t\right)+u^3\left(t\right)arcta{n}^{4}u\left(t\right)}{u\left|u\right|}=0,\end{array}$$

$$\begin{array}{c}\hfill \underset{u\to 0}{lim}\frac{e^{u\left(t\right)}{u}^2\left(t\right)}{u}=0,\underset{u\to 0}{lim}\frac{\frac{2u^5\left(t\right)}{3+2u^3\left(t\right)}}{u}=0,\\ \hfill \underset{u\to 0}{lim}\frac{u^2\left(t\right)+sin(1-cosu\left(t\right))}{u}=0.\end{array}$$

Clearly, the condition of Theorem 3 holds, and we can find that the fractional differential Equation (10) has a solution.

Example 3. 

Consider the following impulsive fractional differential equation

$$\left\{\begin{array}{c}{\displaystyle {\left({}^{c}D_{0^{+}}^{\beta }\left({\phi }_p{(}^c{D}_{0^{+}}^{\alpha }u\left(t\right))\right)\right)}^{\prime }=\frac{sinu\left(t\right)}{5+u^2\left(t\right)},1<\alpha ,\beta \le 2,0<t<1,t\ne \frac{3}{7},}\hfill \\ {\displaystyle \Delta u\left(\frac{3}{7}\right)=e^{-u^2\left(\frac{3}{7}\right)}+3sin(1+e^u\left(\frac{3}{7}\right)),}\hfill \\ {\displaystyle \Delta u^{\prime }\left(\frac{3}{7}\right)=\frac{5+3u^2\left(\frac{3}{7}\right)}{6+3u^2\left(\frac{3}{7}\right)},}\hfill \\ {\displaystyle \Delta u^{″}\left(\frac{3}{7}\right)=\frac{e^{\frac{3}{7}}cosu\left(\frac{2}{3}\right)}{5+u^3\left(\frac{3}{7}\right)},}\hfill \\ {\displaystyle 2u\left(0\right)-u^{\prime }\left(\frac{1}{5}\right)=0,u\left(1\right)+3u^{\prime }\left(\frac{1}{7}\right)=0,}\hfill \\ {\displaystyle {}^{c}D_{0^{+}}^{\alpha }u\left(0\right)=0,u^{″}\left(0\right)=0,}\hfill \end{array}\right.$$

(11)

where$m=1and\lambda =1.$

Clearly,

$${\displaystyle L_1=\frac{1}{3},L_2=6,L_3=\frac{4}{5},L_4=\frac{3}{5},}$$

and we gain the condition of Theorem 4. Hence, by Theorem 4, the fractional differential Equation (11) has a unique solution.

5. Conclusions

In this paper, by using fixed-point theorems, the existence and uniqueness of positive solutions to a class of four-point impulsive fractional differential equations with p-Laplacian operators are studied. Three examples are provided at the end to sustain and validate the potentiality of all our acquired epilogs. As a result of our investigation into this research topic, our epilogs is novel. Besides, with the sustain of our novel epilogs in this work, further investigative tasks can be studied on this research topic. Our topic can be potentially enlarged in multiple domains of biology and engineering, such as physics, polymer rheology, biophysics, blood flow phenomena, electrical circuits, biology, and control theory.