Theory and Modern Applications

# Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations

## Abstract

In this manuscript, by using the fixed point theorems, the existence and the uniqueness of solutions for multi-term nonlinear fractional integro-differential equations are reported. Two examples are presented to illustrate our results.

## 1 Introduction

The study of fractional differential equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. Fractional differential equations appear naturally in a number of fields such as physics, polymer rheology, regular variational in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, etc. An excellent account in the study of fractional differential equations can be found in [1, 2] and . For more details and examples, one can study  and . It is considerable that there are many works about fractional integro-differential equations (see, for example,  and ).

In 2007, Xinwei and Landong reviewed the existence of solutions for the nonlinear fractional differential equation

${}^{c}D^{\alpha }u\left(t\right)=f\left(t,u\left(t\right){,}^{c}{D}^{\beta }u\left(t\right)\right)\phantom{\rule{1em}{0ex}}\left(0

with boundary values $u\left(0\right)={u}^{\prime }\left(1\right)=0$ or ${u}^{\prime }\left(0\right)=u\left(1\right)=0$ or $u\left(0\right)=u\left(1\right)=0$, where $1<\alpha \le 2$, $0<\beta \le 1$, and f is continuous on $\left[0,1\right]×\mathbb{R}×\mathbb{R}$ . In 2009, Su and Zhang studied the existence and uniqueness of solutions for the following nonlinear two-point fractional boundary value problem

${}^{c}D^{\alpha }u\left(t\right)=f\left(t,u\left(t\right){,}^{c}{D}^{\beta }u\left(t\right)\right)\phantom{\rule{1em}{0ex}}\left(0

with boundary values ${a}_{1}u\left(0\right)-{a}_{2}{u}^{\prime }\left(0\right)=A$ and ${b}_{1}u\left(1\right)+{b}_{2}{u}^{\prime }\left(1\right)=B$, where α, β, ${a}_{i}$, ${b}_{i}$ ($i=1,2$) satisfy certain conditions . In 2010, Ahmad and Sivasundaram studied the existence of solutions for the nonlinear fractional integro-differential equation

with boundary values ${u}^{\prime }\left(0\right)+au\left({\eta }_{1}\right)=0$, $b{u}^{\prime }\left(1\right)+u\left({\eta }_{2}\right)=0$ and $0<{\eta }_{1}\le {\eta }_{2}<1$, where ${}^{c}D^{q}$ is the Caputo fractional derivative, $a,b\in \left(0,1\right)$, $f:\left[0,1\right]×X×X×X\to X$ is continuous and for the mappings $\gamma ,\lambda :\left[0,1\right]×\left[0,1\right]\to \left[0,\mathrm{\infty }\right)$ with the property ${sup}_{t\in \left[0,1\right]}|{\int }_{0}^{t}\lambda \left(t,s\right)\phantom{\rule{0.2em}{0ex}}ds|<\mathrm{\infty }$ and ${sup}_{t\in \left[0,1\right]}|{\int }_{0}^{t}\gamma \left(t,s\right)\phantom{\rule{0.2em}{0ex}}ds|<\mathrm{\infty }$, the maps ϕ and ψ are defined by $\left(\varphi u\right)\left(t\right)={\int }_{0}^{t}\gamma \left(t,s\right)u\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$ and $\left(\psi u\right)\left(t\right)={\int }_{0}^{t}\lambda \left(t,s\right)u\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$. Here, X is a Banach space (see ).

## 2 Main results

### 2.1 The basic problem

In this paper, we study the existence and uniqueness of solutions for the multi-term nonlinear fractional integro-differential equation

${}^{c}D^{\alpha }u\left(t\right)=f\left(t,u\left(t\right),\left(\varphi u\right)\left(t\right),\left(\psi u\right)\left(t\right){,}^{c}{D}^{{\beta }_{1}}u\left(t\right){,}^{c}{D}^{{\beta }_{2}}u\left(t\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(t\right)\right)\phantom{\rule{1em}{0ex}}\left(0
(1)

with boundary values $u\left(0\right)+au\left(1\right)=0$ and ${u}^{\prime }\left(0\right)+b{u}^{\prime }\left(1\right)=0$, where $1<\alpha <2$, $0<{\beta }_{i}<1$, $\alpha -{\beta }_{i}\ge 1$, $a,b\ne -1$, $f:\left[0,1\right]×{\mathbb{R}}^{n+3}\to \mathbb{R}$ is continuous, and for the mappings

$\gamma ,\lambda :\left[0,1\right]×\left[0,1\right]\to \left[0,\mathrm{\infty }\right)$

with the property ${sup}_{t\in \left[0,1\right]}|{\int }_{0}^{t}\lambda \left(t,s\right)\phantom{\rule{0.2em}{0ex}}ds|<\mathrm{\infty }$ and ${sup}_{t\in \left[0,1\right]}|{\int }_{0}^{t}\gamma \left(t,s\right)\phantom{\rule{0.2em}{0ex}}ds|<\mathrm{\infty }$, the maps ϕ and ψ are defined by $\left(\varphi u\right)\left(t\right)={\int }_{0}^{t}\gamma \left(t,s\right)u\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$ and $\left(\psi u\right)\left(t\right)={\int }_{0}^{t}\lambda \left(t,s\right)u\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$. In this way, we need the following result, which has been proved in .

Lemma 2.1 Let $\alpha >0$ and $n=\left[\alpha \right]+1$. Then

$I^{\alpha }{}^{c}{D}^{\alpha }u\left(t\right)=u\left(t\right)+{c}_{0}+{c}_{1}t+{c}_{2}t+\cdots +{c}_{n-1}{t}^{n-1},$

where ${c}_{0},{c}_{1},\dots ,{c}_{n-1}$ are some real numbers.

The proof of the following result by using Lemma 2.1 is straightforward.

Lemma 2.2 Let $y\in C\left[0,1\right]$, $a,b\ne -1$ and $1<\alpha <2$. Then the problem ${}^{c}D^{\alpha }u\left(t\right)=y\left(t\right)$ with boundary values $u\left(0\right)+au\left(1\right)=0$ and ${u}^{\prime }\left(0\right)+b{u}^{\prime }\left(1\right)=0$ has the unique solution

$\begin{array}{rcl}u\left(t\right)& =& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{a}{\left(1+a\right)\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{ab-b\left(1+a\right)t}{\left(1+a\right)\left(1+b\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

### 2.2 Some results on solving the problem

Let $C\left(I\right)$ be the space of all continuous real-valued functions on $I=\left[0,1\right]$ and

endowed with the norm $\parallel u\parallel ={max}_{t\in I}|u\left(t\right){|+{\sum }_{i=1}^{n}{max}_{t\in I}|}^{c}{D}^{{\beta }_{i}}u\left(t\right)|$. It is known that $\left(X,\parallel \cdot \parallel \right)$ is a Banach space.

Theorem 2.3 Assume that there exist $\kappa \in \left(0,\alpha -1\right)$ and $\mu \left(t\right)\in {L}^{\frac{1}{\kappa }}\left(\left[0,1\right],\left(0,\mathrm{\infty }\right)\right)$ such that

$\begin{array}{c}|f\left(t,x,y,w,{u}_{1},{u}_{2},\dots ,{u}_{n}\right)-f\left(t,{x}^{\prime },{y}^{\prime },{w}^{\prime },{v}_{1},{v}_{2},\dots ,{v}_{n}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \mu \left(t\right)\left(|x-{x}^{\prime }|+|y-{y}^{\prime }|+|w-{w}^{\prime }|+|{u}_{1}-{v}_{1}|+|{u}_{2}-{v}_{2}|+\cdots +|{u}_{n}-{v}_{n}|\right)\hfill \end{array}$

for all $t\in \left[0,1\right]$ and $x,y,w,{x}^{\prime },{y}^{\prime },{w}^{\prime },{u}_{1},{u}_{2},\dots ,{u}_{n},{v}_{1},{v}_{2},\dots ,{v}_{n}\in \mathbb{R}$. Then problem (1) has a unique solution whenever

$\begin{array}{rcl}\mathrm{\Delta }& =& \left(1+{\gamma }_{0}+{\lambda }_{0}\right)\left[\frac{\left(1+2|a|\right){\mu }^{\ast }}{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }+\frac{|b|\left(1+2|a|\right){\mu }^{\ast }}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\\ +\sum _{i=1}^{n}\left(\frac{\mathrm{\Gamma }\left(\alpha -\kappa \right){\mu }^{\ast }}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -{\beta }_{i}-\kappa +1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\\ +\frac{|b|{\mu }^{\ast }}{|1+b|\mathrm{\Gamma }\left(2-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\right)\right]<1,\end{array}$

where ${\gamma }_{0}={sup}_{t\in I}|{\int }_{0}^{t}\gamma \left(t,s\right)\phantom{\rule{0.2em}{0ex}}ds|$, ${\lambda }_{0}={sup}_{t\in I}|{\int }_{0}^{t}\lambda \left(t,s\right)\phantom{\rule{0.2em}{0ex}}ds|$, ${\mu }^{\ast }={\left({\int }_{0}^{1}{\left(\mu \left(s\right)\right)}^{\frac{1}{\kappa }}\phantom{\rule{0.2em}{0ex}}ds\right)}^{\kappa }$.

Proof Define the mapping $F:X\to X$ by

$\begin{array}{rcl}\left(Fu\right)\left(t\right)& =& {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{a}{\left(1+a\right)}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\\ ×f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{ab-b\left(1+a\right)t}{\left(1+a\right)\left(1+b\right)}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\\ ×f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

For each $u,v\in X$ and $t\in \left[0,1\right]$, by using the Hölder inequality, we have

$\begin{array}{c}|\left(Fu\right)\left(t\right)-\left(Fv\right)\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left(f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\hfill \\ \phantom{\rule{2em}{0ex}}-f\left(s,v\left(s\right),\left(\varphi v\right)\left(s\right),\left(\psi v\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}v\left(s\right){,}^{c}{D}^{{\beta }_{2}}v\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(s\right)\right)\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}-\frac{a}{\left(1+a\right)}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left(f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\hfill \\ \phantom{\rule{2em}{0ex}}-f\left(s,v\left(s\right),\left(\varphi v\right)\left(s\right),\left(\psi v\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}v\left(s\right){,}^{c}{D}^{{\beta }_{2}}v\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(s\right)\right)\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{ab-b\left(1+a\right)t}{\left(1+a\right)\left(1+b\right)}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\left(f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\hfill \\ \phantom{\rule{2em}{0ex}}\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)-f\left(s,v\left(s\right),\left(\varphi v\right)\left(s\right),\left(\psi v\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}v\left(s\right){,}^{c}{D}^{{\beta }_{2}}v\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(s\right)\right)\right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\hfill \\ \phantom{\rule{2em}{0ex}}-f\left(s,v\left(s\right),\left(\varphi v\right)\left(s\right),\left(\psi v\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}v\left(s\right){,}^{c}{D}^{{\beta }_{2}}v\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|a|}{|1+a|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\hfill \\ \phantom{\rule{2em}{0ex}}-f\left(s,v\left(s\right),\left(\varphi v\right)\left(s\right),\left(\psi v\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}v\left(s\right){,}^{c}{D}^{{\beta }_{2}}v\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|ab-b\left(1+a\right)t|}{|1+a||1+b|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\hfill \\ \phantom{\rule{2em}{0ex}}\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)-f\left(s,v\left(s\right),\left(\varphi v\right)\left(s\right),\left(\psi v\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}v\left(s\right){,}^{c}{D}^{{\beta }_{2}}v\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\mu \left(s\right)\left(|u\left(s\right)-v\left(s\right)|+|\left(\varphi u\right)\left(s\right)-\left(\varphi v\right)\left(s\right)|+|\left(\psi u\right)\left(s\right)-\left(\psi v\right)\left(s\right)|\hfill \\ \phantom{\rule{2em}{0ex}}+{|}^{c}{D}^{{\beta }_{1}}u\left(s\right){-}^{c}{D}^{{\beta }_{1}}v\left(s\right){|+|}^{c}{D}^{{\beta }_{2}}u\left(s\right){-}^{c}{D}^{{\beta }_{2}}v\left(s\right){|+\cdots +|}^{c}{D}^{{\beta }_{n}}u\left(s\right){-}^{c}{D}^{{\beta }_{n}}v\left(s\right)|\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|a|}{|1+a|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\mu \left(s\right)\left(|u\left(s\right)-v\left(s\right)|+|\left(\varphi u\right)\left(s\right)-\left(\varphi v\right)\left(s\right)|+|\left(\psi u\right)\left(s\right)-\left(\psi v\right)\left(s\right)|\hfill \\ \phantom{\rule{2em}{0ex}}+{|}^{c}{D}^{{\beta }_{1}}u\left(s\right){-}^{c}{D}^{{\beta }_{1}}v\left(s\right){|+|}^{c}{D}^{{\beta }_{2}}u\left(s\right){-}^{c}{D}^{{\beta }_{2}}v\left(s\right){|+\cdots +|}^{c}{D}^{{\beta }_{n}}u\left(s\right){-}^{c}{D}^{{\beta }_{n}}v\left(s\right)|\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|\left(1+2|a|\right)}{|1+a||1+b|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\mu \left(s\right)\left(|u\left(s\right)-v\left(s\right)|+|\left(\varphi u\right)\left(s\right)-\left(\varphi v\right)\left(s\right)|\hfill \\ \phantom{\rule{2em}{0ex}}+|\left(\psi u\right)\left(s\right)-\left(\psi v\right)\left(s\right)|+{|}^{c}{D}^{{\beta }_{1}}u\left(s\right){-}^{c}{D}^{{\beta }_{1}}v\left(s\right)|\hfill \\ \phantom{\rule{2em}{0ex}}+{|}^{c}{D}^{{\beta }_{2}}u\left(s\right){-}^{c}{D}^{{\beta }_{2}}v\left(s\right){|+\cdots +|}^{c}{D}^{{\beta }_{n}}u\left(s\right){-}^{c}{D}^{{\beta }_{n}}v\left(s\right)|\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\mu \left(s\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|a|\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}\mu \left(s\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|\left(1+2|a|\right)\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}\mu \left(s\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{\mathrm{\Gamma }\left(\alpha \right)}{\left({\int }_{0}^{t}{\left({\left(t-s\right)}^{\alpha -1}\right)}^{\frac{1}{1-\kappa }}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-\kappa }{\left({\int }_{0}^{t}{\left(\mu \left(s\right)\right)}^{\frac{1}{\kappa }}\phantom{\rule{0.2em}{0ex}}ds\right)}^{\kappa }\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|a|\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left({\int }_{0}^{1}{\left({\left(1-s\right)}^{\alpha -1}\right)}^{\frac{1}{1-\kappa }}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-\kappa }{\left({\int }_{0}^{1}{\left(\mu \left(s\right)\right)}^{\frac{1}{\kappa }}\phantom{\rule{0.2em}{0ex}}ds\right)}^{\kappa }\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|\left(1+2|a|\right)\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{0}^{1}{\left({\left(1-s\right)}^{\alpha -2}\right)}^{\frac{1}{1-\kappa }}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-\kappa }{\left({\int }_{0}^{1}{\left(\mu \left(s\right)\right)}^{\frac{1}{\kappa }}\phantom{\rule{0.2em}{0ex}}ds\right)}^{\kappa }\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{{\mu }^{\ast }\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }+\frac{|a|{\mu }^{\ast }\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|\left(1+2|a|\right){\mu }^{\ast }\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(1+{\gamma }_{0}+{\lambda }_{0}\right)\left[\frac{\left(1+2|a|\right){\mu }^{\ast }}{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|\left(1+2|a|\right){\mu }^{\ast }}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\right]\parallel u-v\parallel .\hfill \end{array}$

Also, we have

$\begin{array}{c}{|}^{c}{D}^{{\beta }_{i}}\left(Fu\right)\left(t\right){-}^{c}{D}^{{\beta }_{i}}\left(Fv\right)\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\left(Fu\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\left(Fv\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,v\left(\tau \right),\left(\varphi v\right)\left(\tau \right),\left(\psi v\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}v\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}v\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,v\left(\tau \right),\left(\varphi v\right)\left(\tau \right),\left(\psi v\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}v\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}v\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×|f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\hfill \\ \phantom{\rule{2em}{0ex}}-f\left(\tau ,v\left(\tau \right),\left(\varphi v\right)\left(\tau \right),\left(\psi v\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}v\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}v\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|}{|1+b|}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×\left({\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}|f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\hfill \\ \phantom{\rule{2em}{0ex}}-f\left(\tau ,v\left(\tau \right),\left(\varphi v\right)\left(\tau \right),\left(\psi v\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}v\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}v\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}v\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}\left({\int }_{0}^{s}{\left(s-\tau \right)}^{\alpha -2}\mu \left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+b|\mathrm{\Gamma }\left(1-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}\left({\int }_{0}^{1}{\left(1-\tau \right)}^{\alpha -2}\mu \left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{{\mu }^{\ast }\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}{s}^{\alpha -\kappa -1}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|{\mu }^{\ast }\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+b|\mathrm{\Gamma }\left(1-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{{\mu }^{\ast }\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }{\int }_{0}^{1}{\left(1-\xi \right)}^{-{\beta }_{i}}{\xi }^{\alpha -\kappa -1}\phantom{\rule{0.2em}{0ex}}d\xi \hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|{\mu }^{\ast }\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\parallel u-v\parallel }{|1+b|\mathrm{\Gamma }\left(2-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }.\hfill \end{array}$

Since $B\left(\alpha -\kappa ,1-{\beta }_{i}\right)={\int }_{0}^{1}{\left(1-\xi \right)}^{-{\beta }_{i}}{\xi }^{\alpha -\kappa -1}\phantom{\rule{0.2em}{0ex}}d\xi =\frac{\mathrm{\Gamma }\left(\alpha -\kappa \right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}-\kappa +1\right)}$, we obtain

$\begin{array}{rcl}{|}^{c}{D}^{{\beta }_{i}}\left(Fu\right)\left(t\right){-}^{c}{D}^{{\beta }_{i}}\left(Fv\right)\left(t\right)|& \le & \left(1+{\gamma }_{0}+{\lambda }_{0}\right)\left[\frac{\mathrm{\Gamma }\left(\alpha -\kappa \right){\mu }^{\ast }}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -{\beta }_{i}-\kappa +1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\\ +\frac{|b|{\mu }^{\ast }}{|1+b|\mathrm{\Gamma }\left(2-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\right]\parallel u-v\parallel \end{array}$

for all $i=1,2,\dots ,n$. Hence, we get

$\begin{array}{c}\parallel Fu-Fv\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \left(1+{\gamma }_{0}+{\lambda }_{0}\right)\left[\frac{\left(1+2|a|\right){\mu }^{\ast }}{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }+\frac{|b|\left(1+2|a|\right){\mu }^{\ast }}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{i=1}^{n}\left(\frac{\mathrm{\Gamma }\left(\alpha -\kappa \right){\mu }^{\ast }}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -{\beta }_{i}-\kappa +1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|{\mu }^{\ast }}{|1+b|\mathrm{\Gamma }\left(2-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\right)\right]\parallel u-v\parallel =\mathrm{\Delta }\parallel u-v\parallel .\hfill \end{array}$

Since $\mathrm{\Delta }<1$, F is a contraction mapping, therefore, by using the Banach contraction principle, F has a unique fixed point, which is the unique solution of problem (1) by using Lemma 2.2. □

Corollary 2.4 Assume that there exists $L>0$ such that

$\begin{array}{c}|f\left(t,x,y,w,{u}_{1},{u}_{2},\dots ,{u}_{n}\right)-f\left(t,{x}^{\prime },{y}^{\prime },{w}^{\prime },{v}_{1},{v}_{2},\dots ,{v}_{n}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le L\left(|x-{x}^{\prime }|+|y-{y}^{\prime }|+|w-{w}^{\prime }|+|{u}_{1}-{v}_{1}|+|{u}_{2}-{v}_{2}|+\cdots +|{u}_{n}-{v}_{n}|\right)\hfill \end{array}$

for all $t\in \left[0,1\right]$ and $x,y,w,{x}^{\prime },{y}^{\prime },{w}^{\prime },{u}_{1},{u}_{2},\dots ,{u}_{n},{v}_{1},{v}_{2},\dots ,{v}_{n}\in \mathbb{R}$. Then problem (1) has a unique solution whenever

$\begin{array}{r}\left(1+{\gamma }_{0}+{\lambda }_{0}\right)\left[\frac{\left(1+2|a|\right)\left(1+\left(\alpha +1\right)|b|\right)L}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha +1\right)}\\ \phantom{\rule{1em}{0ex}}+\sum _{i=1}^{n}\left(\frac{L}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}+1\right)}+\frac{|b|L}{|1+b|\mathrm{\Gamma }\left(2-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha \right)}\right)\right]<1,\end{array}$

where ${\gamma }_{0}={sup}_{t\in I}|{\int }_{0}^{t}\gamma \left(t,s\right)\phantom{\rule{0.2em}{0ex}}ds|$, ${\lambda }_{0}={sup}_{t\in I}|{\int }_{0}^{t}\lambda \left(t,s\right)\phantom{\rule{0.2em}{0ex}}ds|$.

Now, we restate the Schauder’s fixed point theorem, which is needed to prove next result (see Theorem 1.10.16 in ).

Theorem 2.5 Let E be a closed, convex and bounded subset of a Banach space X, and let $F:E\to E$ be a continuous mapping such that $F\left(E\right)$ is a relatively compact subset of X. Then F has a fixed point in E.

Theorem 2.6 Let $f:\left[0,1\right]×{\mathbb{R}}^{n+3}\to \mathbb{R}$ be a continuous function such that there exists a constant $l\in \left(0,\alpha -1\right)$ and a real-valued function $m\left(t\right)\in {L}^{\frac{1}{l}}\left(\left[0,1\right],\left(0,\mathrm{\infty }\right)\right)$ such that

$|f\left(t,x,y,w,{u}_{1},{u}_{2},\dots ,{u}_{n}\right)|\le m\left(t\right)+d{|x|}^{\rho }+{d}^{\prime }{|y|}^{\rho \prime }+{d}^{\prime \prime }{|w|}^{\rho \prime \prime }+{d}_{1}{|{u}_{1}|}^{{\rho }_{1}}+{d}_{2}{|{u}_{2}|}^{{\rho }_{2}}+\dots {d}_{n}{|{u}_{n}|}^{\rho n},$
(*)

where $d,{d}^{\prime },{d}^{″},{d}_{i}\ge 0$ and $0<\rho ,{\rho }^{\prime },{\rho }^{″},{\rho }_{i}<1$ for $i=1,2,\dots ,n$, or

$\begin{array}{c}|f\left(t,x,y,w,{u}_{1},{u}_{2},\dots ,{u}_{n}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le d{|x|}^{\rho }+{d}^{\prime }{|y|}^{{\rho }^{\prime }}+{d}^{″}{|w|}^{{\rho }^{″}}+{d}_{1}{|{u}_{1}|}^{{\rho }_{1}}+{d}_{2}{|{u}_{2}|}^{{\rho }_{2}}+\cdots +{d}_{n}{|{u}_{n}|}^{{\rho }_{n}},\hfill \end{array}$

where $d,{d}^{\prime },{d}^{″},{d}_{i}>0$ and $\rho ,{\rho }^{\prime },{\rho }^{″},{\rho }_{i}>1$ for $i=1,2,\dots ,n$. Then problem (1) has a solution.

Proof First, suppose that f satisfy condition (). Define ${B}_{r}=\left\{u\in X,\parallel u\parallel \le r\right\}$, where

$\begin{array}{r}\begin{array}{rl}r\ge & max\left\{{\left(\left(n+4\right)Ad\right)}^{\frac{1}{1-\rho }},{\left(\left(n+4\right)A{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}\right)}^{\frac{1}{1-{\rho }^{\prime }}},{\left(\left(n+4\right)A{d}^{″}{\lambda }_{0}^{{p}^{″}}\right)}^{\frac{1}{1-{\rho }^{″}}},{\left(\left(n+4\right)A{d}_{1}\right)}^{\frac{1}{1-{\rho }_{1}}},\\ {\left(\left(n+4\right)A{d}_{2}\right)}^{\frac{1}{1-{\rho }_{2}}},\dots ,{\left(\left(n+4\right)A{d}_{n}\right)}^{\frac{1}{1-{\rho }_{n}}},\left(n+4\right)K\right\},\end{array}\\ \begin{array}{rl}K=& \frac{\left(1+2|a|\right)M}{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{|b|\left(1+2|a|\right)M}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\sum _{i=1}^{n}\left(\frac{\mathrm{\Gamma }\left(\alpha -l\right)M}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -{\beta }_{i}-l+1\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }\left(2-{\beta }_{i}\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\right),\end{array}\\ A=\frac{\left(1+2|a|\right)\left(1+\left(1+\alpha \right)|b|\right)}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha +1\right)}+\sum _{i=1}^{n}\left(\frac{1}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}+1\right)}+\frac{|b|}{|1+b|\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}\right)\end{array}$

and $M={\left({\int }_{0}^{1}{\left(m\left(t\right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{l}$. Note that ${B}_{r}$ is a closed, bounded and convex subset of the Banach space X. For each $u\in {B}_{r}$, we have

$\begin{array}{rcl}|\left(Fu\right)\left(t\right)|& =& |{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{a}{\left(1+a\right)}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\\ ×f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{ab-b\left(1+a\right)t}{\left(1+a\right)\left(1+b\right)}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\\ ×f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds|\\ \le & {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|a|}{|1+a|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\\ ×|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|\left(1+2|a|\right)}{|1+a||1+b|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\\ ×|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}m\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right){\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|a|}{|1+a|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}m\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|a|}{|1+a|}\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)\\ ×{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\phantom{\rule{0.2em}{0ex}}ds+\frac{|b|\left(1+2|a|\right)}{|1+a||1+b|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}m\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|\left(1+2|a|\right)}{|1+a||1+b|}\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)\\ ×{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left({\int }_{0}^{t}{\left({\left(t-s\right)}^{\alpha -1}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-l}{\left({\int }_{0}^{t}{\left(m\left(s\right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{l}\\ +\frac{|a|}{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left({\int }_{0}^{1}{\left({\left(1-s\right)}^{\alpha -1}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-l}{\left({\int }_{0}^{1}{\left(m\left(s\right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{l}\\ +\frac{|b|\left(1+2|a|\right)}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\left({\int }_{0}^{1}{\left({\left(1-s\right)}^{\alpha -2}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-l}{\left({\int }_{0}^{1}{\left(m\left(s\right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{l}\\ +\frac{\left(1+2|a|\right)\left(1+\left(1+\alpha \right)|b|\right)}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha +1\right)}\\ ×\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)\\ \le & \frac{\left(1+2|a|\right)M}{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{|b|\left(1+2|a|\right)M}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{\left(1+2|a|\right)\left(1+\left(1+\alpha \right)|b|\right)}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha +1\right)}\\ ×\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right).\end{array}$

Also, we have

$\begin{array}{rcl}{|}^{c}{D}^{{\beta }_{i}}\left(Fu\right)\left(t\right)|& =& |{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\left(Fu\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\\ =& |{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\\ ×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \\ -\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\\ ×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right),\\ {}^{c}D^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds|\\ \le & {\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}|f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right),\\ {}^{c}D^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|}{|1+b|}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}|f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right),\\ {}^{c}D^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ \le & {\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}m\left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ +\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)\\ ×{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|}{|1+b|}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}m\left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|}{|1+b|}\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)\\ ×{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}\\ ×\left[{\left({\int }_{0}^{s}{\left({\left(s-\tau \right)}^{\alpha -2}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}d\tau \right)}^{1-l}{\left({\int }_{0}^{s}{\left(m\left(\tau \right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}d\tau \right)}^{l}\right]\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\\ ×{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}{s}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds+\frac{|b|}{|1+b|\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\\ ×{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}\left[{\left({\int }_{0}^{1}{\left({\left(1-\tau \right)}^{\alpha -2}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}d\tau \right)}^{1-l}{\left({\int }_{0}^{1}{\left(m\left(\tau \right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}d\tau \right)}^{l}\right]\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)}{|1+b|\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}\\ \le & \frac{M}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}{s}^{\alpha -l-1}\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\\ ×{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}{s}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}{\int }_{0}^{t}{\left(t-s\right)}^{-{\beta }_{i}}\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)}{|1+b|\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}\\ \le & \frac{M}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}{\int }_{0}^{1}{\left(1-\xi \right)}^{-{\beta }_{i}}{\xi }^{\alpha -l-1}\phantom{\rule{0.2em}{0ex}}d\xi \\ +\frac{\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\\ ×{\int }_{0}^{1}{\left(1-\xi \right)}^{-{\beta }_{i}}{\xi }^{\alpha -1}\phantom{\rule{0.2em}{0ex}}d\xi \\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{|b|\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)}{|1+b|\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}.\end{array}$

Since $B\left(\alpha -l,1-{\beta }_{i}\right)={\int }_{0}^{1}{\left(1-\xi \right)}^{-{\beta }_{i}}{\xi }^{\alpha -l-1}\phantom{\rule{0.2em}{0ex}}d\xi =\frac{\mathrm{\Gamma }\left(\alpha -l\right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}-l+1\right)}$ and, on the other hand, $B\left(\alpha ,1-{\beta }_{i}\right)={\int }_{0}^{1}{\left(1-\xi \right)}^{-{\beta }_{i}}{\xi }^{\alpha -1}\phantom{\rule{0.2em}{0ex}}d\xi =\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}+1\right)}$, we conclude that

$\begin{array}{rcl}{|}^{c}{D}^{{\beta }_{i}}\left(Fu\right)\left(t\right)|& \le & \frac{\mathrm{\Gamma }\left(\alpha -l\right)M}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -{\beta }_{i}-l+1\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}+1\right)}\\ +\frac{|b|\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)}{|1+b|\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}\end{array}$

for all $i=1,2,\dots ,n$. Thus,

$\begin{array}{rcl}\parallel Fu\parallel & \le & \frac{\left(1+2|a|\right)M}{|1+a|\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{|b|\left(1+2|a|\right)M}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha -1\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\sum _{i=1}^{n}\left[\frac{\mathrm{\Gamma }\left(\alpha -l\right)M}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -{\beta }_{i}-l+1\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\right]\\ +\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)\\ ×\left(\frac{\left(1+2|a|\right)\left(1+\left(1+\alpha \right)|b|\right)}{|1+a||1+b|\mathrm{\Gamma }\left(\alpha +1\right)}+\sum _{i=1}^{n}\left[\frac{1}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}+1\right)}+\frac{|b|}{|1+b|\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}\right]\right)\\ =& K+\left(d{r}^{\rho }+{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}{r}^{{\rho }^{\prime }}+{d}^{″}{\lambda }_{0}^{{p}^{″}}{r}^{{\rho }^{″}}+{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}+\cdots +{d}_{n}{r}^{{\rho }_{n}}\right)A\\ \le & \frac{r}{n+4}×\left(n+4\right)=r.\end{array}$

Hence, F maps ${B}_{r}$ into ${B}_{r}$. Now, suppose that f satisfy the second condition. In this case, choose

$\begin{array}{rcl}0& <& r\\ \le & min\left\{{\left(\frac{1}{\left(n+3\right)Ad}\right)}^{\frac{1}{\rho -1}},{\left(\frac{1}{\left(n+3\right)A{d}^{\prime }{\gamma }_{0}^{{p}^{\prime }}}\right)}^{\frac{1}{{\rho }^{\prime }-1}},{\left(\frac{1}{\left(n+3\right)A{d}^{″}{\lambda }_{0}^{{p}^{″}}}\right)}^{\frac{1}{{\rho }^{″}-1}},\\ {\left(\frac{1}{\left(n+3\right)A{d}_{1}}\right)}^{\frac{1}{{\rho }_{1}-1}},{\left(\frac{1}{\left(n+3\right)A{d}_{2}}\right)}^{\frac{1}{{\rho }_{2}-1}},\dots ,{\left(\frac{1}{\left(n+3\right)A{d}_{n}}\right)}^{\frac{1}{{\rho }_{n}-1}}\right\}.\end{array}$

By using similar arguments, one can show that $\parallel Fu\parallel \le \frac{r}{n+3}×\left(n+3\right)=r$, and so F maps ${B}_{r}$ into ${B}_{r}$. Since f is continuous, it is easy to get that F is also continuous. Now, we show that F is completely continuous operator on ${B}_{r}$. For each $u\in {B}_{r}$, put

$N=\underset{t\in I}{max}\left\{f\left(t,u\left(t\right),\left(\varphi u\right)\left(t\right),\left(\psi u\right)\left(t\right){,}^{c}{D}^{{\beta }_{1}}u\left(t\right){,}^{c}{D}^{{\beta }_{2}}u\left(t\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(t\right)\right)\right\}+1.$

For each ${t}_{1},{t}_{2}\in I$ with ${t}_{1}<{t}_{2}$, we have

$\begin{array}{c}|\left(Fu\right)\left({t}_{2}\right)-\left(Fu\right)\left({t}_{1}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{b\left({t}_{1}-{t}_{2}\right)}{1+b}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{b\left({t}_{1}-{t}_{2}\right)}{1+b}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}-{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \\ \phantom{\rule{2em}{0ex}}×|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{|b|\left({t}_{2}-{t}_{1}\right)}{|1+b|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×|f\left(s,u\left(s\right),\left(\varphi u\right)\left(s\right),\left(\psi u\right)\left(s\right){,}^{c}{D}^{{\beta }_{1}}u\left(s\right){,}^{c}{D}^{{\beta }_{2}}u\left(s\right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le N{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}-{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\phantom{\rule{0.2em}{0ex}}ds+N{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{N|b|\left({t}_{2}-{t}_{1}\right)}{|1+b|}{\int }_{0}^{1}\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{N}{\mathrm{\Gamma }\left(\alpha +1\right)}\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right)+\frac{N|b|}{|1+b|\mathrm{\Gamma }\left(\alpha \right)}\left({t}_{2}-{t}_{1}\right).\hfill \end{array}$

On the other hand, for each $i\in \left\{1,2,\dots ,n\right\}$, we have

$\begin{array}{c}{|}^{c}{D}^{{\beta }_{i}}\left(Fu\right)\left({t}_{2}\right){-}^{c}{D}^{{\beta }_{i}}\left(Fu\right)\left({t}_{1}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\left(Fu\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{1}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}{\left(Fu\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{1}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{1}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-{\beta }_{i}}-{\left({t}_{1}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right),\hfill \\ {\phantom{\rule{2em}{0ex}}}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-{\beta }_{i}}}{\mathrm{\Gamma }\left(1-{\beta }_{i}\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}u\left(\tau \right){,}^{c}{D}^{{\beta }_{2}}u\left(\tau \right),\dots {,}^{c}{D}^{{\beta }_{n}}u\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{b}{1+b}{\int }_{0}^{1}\frac{{\left(1-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×f\left(\tau ,u\left(\tau \right),\left(\varphi u\right)\left(\tau \right),\left(\psi u\right)\left(\tau \right){,}^{c}{D}^{{\beta }_{1}}\hfill \end{array}$