# Some existence theorems for fractional integro-differential equations and inclusions with initial and non-separated boundary conditions

## Abstract

In this paper, we study the existence of solutions for a new class of boundary value problems of nonlinear fractional integro-differential equations and inclusions of arbitrary order with initial and non-separated boundary conditions. In the case of inclusion, the existence results are obtained for convex as well as non-convex multifunctions. Our results rely on the standard tools of fixed point theory and are well illustrated with the aid of examples.

## 1 Introduction

The subject of fractional calculus has recently been investigated in an extensive manner. The publication of several books, special issues, and a huge number of articles in journals of international repute, exploring numerous aspects of this branch of mathematics, clearly indicates the popularity of the topic. One of the key factors accounting for the utility of the subject is that fractional-order operators are nonlocal in nature in contrast to the integer-order operators and can describe the hereditary properties of many underlying phenomena and processes. Owing to this characteristic, the principles of fractional calculus have played a significant role in improving the modeling techniques for several real world problems [1]–[4].

Many researchers have focused their attention on fractional differential equations and inclusions, and a variety of interesting and important results concerning existence and uniqueness of solutions, stability properties of solutions, analytic and numerical methods of solutions of these equations have been obtained and the surge for investigating more and more results is still under way. For details and examples, we refer the reader to a series of papers [5]–[29] and the references therein. Anti-periodic boundary value problems occur in the mathematical modeling of a variety of physical processes and some works have been published in this area, for instance; see [30]–[34] and the references therein.

In this paper, for $\alpha \in \left(n-1,n\right]$, $n\ge 5$, $n\in \mathbb{N}$, $t\in I=\left[0,T\right]$, $T>0$, we investigate the fractional integro-differential equation

$\begin{array}{rl}D_{\alpha }^{c}x\left(t\right)=& f\left(t,x\left(t\right),{x}^{\prime }\left(t\right),{x}^{″}\left(t\right),{x}^{‴}\left(t\right),\varphi x\left(t\right),\psi x\left(t\right),{}^{c}{D}^{{\mu }_{1}}x\left(t\right),{}^{c}{D}^{{\mu }_{2}}x\left(t\right),\dots ,{}^{c}{D}^{{\mu }_{m}}x\left(t\right),\\ D_{{\nu }_{1}}^{c}x\left(t\right),{}^{c}{D}^{{\nu }_{2}}x\left(t\right),\dots ,{}^{c}{D}^{{\nu }_{{m}^{\prime }}}x\left(t\right),{}^{c}{D}^{{\xi }_{1}}x\left(t\right),{}^{c}{D}^{{\xi }_{2}}x\left(t\right),\dots ,{}^{c}{D}^{{\xi }_{{m}^{″}}}x\left(t\right)\right),\end{array}$
(1.1)

and related fractional integro-differential inclusion

$\begin{array}{rl}D_{\alpha }^{c}x\left(t\right)\in & F\left(t,x\left(t\right),{x}^{\prime }\left(t\right),{x}^{″}\left(t\right),{x}^{‴}\left(t\right),\varphi x\left(t\right),\psi x\left(t\right),{}^{c}{D}^{{\mu }_{1}}x\left(t\right),{}^{c}{D}^{{\mu }_{2}}x\left(t\right),\dots ,{}^{c}{D}^{{\mu }_{m}}x\left(t\right),\\ D_{{\nu }_{1}}^{c}x\left(t\right),{}^{c}{D}^{{\nu }_{2}}x\left(t\right),\dots ,{}^{c}{D}^{{\nu }_{{m}^{\prime }}}x\left(t\right),{}^{c}{D}^{{\xi }_{1}}x\left(t\right),{}^{c}{D}^{{\xi }_{2}}x\left(t\right),\dots ,{}^{c}{D}^{{\xi }_{{m}^{″}}}x\left(t\right)\right),\end{array}$
(1.2)

supplemented with initial boundary conditions

$\begin{array}{r}{x}^{\left(4\right)}\left(0\right)=\cdots ={x}^{\left(n-1\right)}\left(0\right)=0,\phantom{\rule{2em}{0ex}}ax\left(0\right)+bx\left(T\right)=0,\\ D_{p}^{c}x\left(0\right)=-{}^{c}{D}^{p}x\left(T\right),\phantom{\rule{2em}{0ex}}{}^{c}{D}^{q}x\left(0\right)=-{}^{c}{D}^{q}x\left(T\right),\\ D_{\gamma }^{c}x\left(0\right)=-{}^{c}{D}^{\gamma }x\left(T\right),\phantom{\rule{1em}{0ex}}0
(1.3)

where cD denotes the Caputo fractional derivative, $f:\left[0,T\right]×{\mathbb{R}}^{6+m+{m}^{\prime }+{m}^{″}}\to \mathbb{R}$ is a continuous function, $F:\left[0,1\right]×{\mathbb{R}}^{6+m+{m}^{\prime }+{m}^{″}}\to P\left(\mathbb{R}\right)$ is a multifunction, $P\left(\mathbb{R}\right)$ is the family of all non-empty subsets of $\mathbb{R},$ and the maps ϕ and ψ are defined by

$\varphi x\left(t\right)={\int }_{0}^{t}\gamma \left(t,s\right){h}_{1}\left(t,s,x\left(s\right),{x}^{\prime }\left(s\right),{x}^{″}\left(s\right),{x}^{‴}\left(s\right),{}^{c}{D}^{{\delta }_{1}}x\left(s\right),{}^{c}{D}^{{\beta }_{1}}x\left(s\right),{}^{c}{D}^{{\theta }_{1}}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds$

and

$\psi x\left(t\right)={\int }_{0}^{t}\lambda \left(t,s\right){h}_{2}\left(t,s,x\left(s\right),{x}^{\prime }\left(s\right),{x}^{″}\left(s\right),{x}^{‴}\left(s\right),{}^{c}{D}^{{\delta }_{2}}x\left(s\right),{}^{c}{D}^{{\beta }_{2}}x\left(s\right),{}^{c}{D}^{{\theta }_{2}}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,$

where $\gamma ,\lambda :\left[0,T\right]×\left[0,T\right]\to \mathbb{R}$ and ${h}_{1},{h}_{2}:\left[0,T\right]×\left[0,T\right]×{\mathbb{R}}^{7}\to \mathbb{R}$ are continuous maps, $0<{\mu }_{i}<1$ ($1\le i\le m$), $1<{\nu }_{j}<2$ ($1\le j\le {m}^{\prime }$), $2<{\xi }_{k}<3$ ($1\le k\le {m}^{″}$), $0<{\delta }_{i}<1$, $1<{\beta }_{i}<2$, and $2<{\theta }_{i}<3$ for $i=1,2$.

The paper is organized as follows. In Section 2, we recall some preliminary facts that we used in the sequel. Section 3 deals with the existence result for single-valued initial boundary value problem, while the results for multivalued problem are presented in Section 4. We present some examples illustrating the main results in Section 5.

## 2 Preliminaries

Let $\left(X,\parallel \cdot \parallel \right)$ be a normed space, $P\left(X\right)$ the set of all non-empty subsets of X, ${P}_{cl}\left(X\right)$ the set of all non-empty closed subsets of X, ${P}_{b}\left(X\right)$ the set of all non-empty bounded subsets of X, ${P}_{cp}\left(X\right)$ the set of all non-empty compact subsets of X and ${P}_{cp,c}\left(X\right)$ the set of all non-empty compact and convex subsets of X[35]. A multivalued map $G:X\to P\left(X\right)$ is said to be convex (closed) valued whenever $G\left(x\right)$ is convex (closed) for all $x\in X$[35]. The multifunction G is called bounded on bounded sets whenever $G\left(B\right)={\bigcup }_{x\in B}G\left(x\right)$ is bounded subset of X for all $B\in {P}_{b}\left(X\right)$, that is, ${sup}_{x\in B}\left\{sup\left\{|y|:y\in G\left(x\right)\right\}\right\}<\mathrm{\infty }$ for all $B\in {P}_{b}\left(X\right)$[35]. Also, the multifunction $G:X\to P\left(X\right)$ is called upper semi-continuous whenever for each ${x}_{0}\in X$ the set $G\left({x}_{0}\right)$ is a non-empty closed subset of X, and for every open set N of X containing $G\left({x}_{0}\right)$, there exists an open neighborhood ${N}_{0}$ of ${x}_{0}$ such that $G\left({N}_{0}\right)\subseteq N$[36], [37]. The multifunction $G:X\to P\left(X\right)$ is called compact whenever $G\left(B\right)$ is relatively compact for all $B\in {P}_{b}\left(X\right)$ and also is called completely continuous whenever G is upper semi-continuous and compact [38], [39]. It is well known that a compact multifunction G with non-empty compact valued is upper semi-continuous if and only if G has a closed graph, that is, ${x}_{n}\to {x}_{\ast }$, ${y}_{n}\in G\left({x}_{n}\right)$ for all n, and ${y}_{n}\to {y}_{\ast }$ imply ${y}_{\ast }\in G\left({x}_{\ast }\right)$[37]. We say that ${x}_{0}\in X$ is a fixed point of a multifunction G whenever ${x}_{0}\in G\left({x}_{0}\right)$[40]. Let $T>0$ and $G:\left[0,T\right]\to {P}_{cl}\left(\mathbb{R}\right)$ a multifunction. We say that G is measurable whenever the function $t↦d\left(y,G\left(t\right)\right)=inf\left\{|y-z|:z\in G\left(t\right)\right\}$ is measurable for all $y\in \mathbb{R}$[38], [39].

One can find basic notions of fractional calculus in [1] and [2]. We recall two necessary ones here.

The Riemann-Liouville fractional integral of order $q>0$ with the lower limit zero for a function $f:\left[0,\mathrm{\infty }\right)\to \mathbb{R}$ is defined by ${I}^{q}f\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(q\right)}{\int }_{0}^{t}\frac{f\left(s\right)}{{\left(t-s\right)}^{1-q}}\phantom{\rule{0.2em}{0ex}}ds$ for $t>0$ provided the integral exists.

The Caputo fractional derivative of order $q>0$ for a function $f\in {C}^{n}\left(\left[0,\mathrm{\infty }\right),\mathbb{R}\right)$ can be written as

$D_{q}^{c}f\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(n-q\right)}{\int }_{0}^{t}\frac{{f}^{\left(n\right)}\left(s\right)}{{\left(t-s\right)}^{q+1-n}}\phantom{\rule{0.2em}{0ex}}ds={I}^{n-q}{f}^{\left(n\right)}\left(t\right),$

where $n-1 and $t>0$.

To define the solution for problems (1.1)-(1.3) and (1.2)-(1.3), we establish the following lemma.

### Lemma 2.1

Let$y\in {L}^{1}\left(\left[0,T\right],\mathbb{R}\right)$. Then the integral solution of the linear equation

$D_{\alpha }^{c}x\left(t\right)=y\left(t\right)$
(2.1)

subject to the initial boundary conditions (1.3) is given by

$x\left(t\right)={\int }_{0}^{T}G\left(t,s\right)y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,$
(2.2)

where

and

$\begin{array}{rcl}{G}_{1}\left(t,s\right)& =& -\frac{b{\left(T-s\right)}^{\alpha -1}}{\left(a+b\right)\mathrm{\Gamma }\left(\alpha \right)}+\frac{\left[bT-\left(a+b\right)t\right]\mathrm{\Gamma }\left(2-p\right){\left(T-s\right)}^{\alpha -p-1}}{\left(a+b\right)\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}\\ -\frac{\left[bp{T}^{2}-\left(a+b\right)\left(2Tt-\left(2-p\right){t}^{2}\right)\right]\mathrm{\Gamma }\left(3-q\right){\left(T-s\right)}^{\alpha -q-1}}{2\left(a+b\right)\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}\\ -\frac{\mathrm{\Gamma }\left(4-\gamma \right){\left(T-s\right)}^{\alpha -\gamma -1}}{6\left(a+b\right)\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ ×\left(b\left(-6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right){T}^{3}\\ +\left(a+b\right)\left(6\left(p-q\right){T}^{2}t+\left(2-p\right)\left(3-p\right)\left(-3T{t}^{2}+\left(3-q\right){t}^{3}\right)\right)\right).\end{array}$

### Proof

It is well known that the solution of (2.1) can be written as

$\begin{array}{rcl}x\left(t\right)& =& {I}^{\alpha }y\left(t\right)-{b}_{0}-{b}_{1}t-{b}_{2}{t}^{2}-{b}_{3}{t}^{3}-{b}_{4}{t}^{4}-\cdots -{b}_{n-1}{t}^{n-1}\\ =& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{b}_{0}-{b}_{1}t-{b}_{2}{t}^{2}-{b}_{3}{t}^{3}-{b}_{4}{t}^{4}-\cdots -{b}_{n-1}{t}^{n-1},\end{array}$
(2.3)

where ${b}_{0},{b}_{1},{b}_{2},{b}_{3},{b}_{4},\dots ,{b}_{n-1}\in \mathbb{R}$ are arbitrary constants. Using the initial conditions ${x}^{\left(4\right)}\left(0\right)=\cdots ={x}^{\left(n-1\right)}\left(0\right)=0$, we find that ${b}_{4}=\cdots ={b}_{n-1}=0$. Since $D_{p}^{c}c=0$ for all constant c, $D_{p}^{c}t=\frac{{t}^{1-p}}{\mathrm{\Gamma }\left(2-p\right)}$, $D_{p}^{c}{t}^{2}=\frac{2{t}^{2-p}}{\mathrm{\Gamma }\left(3-p\right)}$, $D_{p}^{c}{t}^{3}=\frac{6{t}^{3-p}}{\mathrm{\Gamma }\left(4-p\right)}$, $D_{q}^{c}t=0$, $D_{q}^{c}{t}^{2}=\frac{2{t}^{2-q}}{\mathrm{\Gamma }\left(3-q\right)}$, $D_{q}^{c}{t}^{3}=\frac{6{t}^{3-q}}{\mathrm{\Gamma }\left(4-q\right)}$, $D_{\gamma }^{c}t=0$, $D_{\gamma }^{c}{t}^{2}=0$, $D_{\gamma }^{c}{t}^{3}=\frac{6{t}^{3-\gamma }}{\mathrm{\Gamma }\left(4-\gamma \right)}$, $D_{p}^{c}{I}^{\alpha }y\left(t\right)={I}^{\alpha -p}y\left(t\right)$, $D_{q}^{c}{I}^{\alpha }y\left(t\right)={I}^{\alpha -q}y\left(t\right)$, and $D_{\gamma }^{c}{I}^{\alpha }y\left(t\right)={I}^{\alpha -\gamma }y\left(t\right)$, therefore

$\begin{array}{r}D_{p}^{c}x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -p\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -p-1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{b}_{1}\frac{{t}^{1-p}}{\mathrm{\Gamma }\left(2-p\right)}-{b}_{2}\frac{2{t}^{2-p}}{\mathrm{\Gamma }\left(3-p\right)}-{b}_{3}\frac{6{t}^{3-p}}{\mathrm{\Gamma }\left(4-p\right)},\\ D_{q}^{c}x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -q\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -q-1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{b}_{2}\frac{2{t}^{2-q}}{\mathrm{\Gamma }\left(3-q\right)}-{b}_{3}\frac{6{t}^{3-q}}{\mathrm{\Gamma }\left(4-q\right)},\\ D_{\gamma }^{c}x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -\gamma \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -\gamma -1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{b}_{3}\frac{6{t}^{3-\gamma }}{\mathrm{\Gamma }\left(4-\gamma \right)}.\end{array}$

Now using the conditions $ax\left(0\right)+bx\left(T\right)=0$, $D_{p}^{c}x\left(0\right)=-{}^{c}{D}^{p}x\left(T\right)$, $D_{q}^{c}x\left(0\right)=-{}^{c}{D}^{q}x\left(T\right)$, and $D_{\gamma }^{c}x\left(0\right)=-{}^{c}{D}^{\gamma }x\left(T\right)$, we obtain

$\begin{array}{c}\begin{array}{rl}{b}_{0}=& \frac{b}{\left(a+b\right)}\left[\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{\mathrm{\Gamma }\left(2-p\right){T}^{p}}{\mathrm{\Gamma }\left(\alpha -p\right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{p\mathrm{\Gamma }\left(3-q\right){T}^{q}}{2\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left[-6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right]\mathrm{\Gamma }\left(4-\gamma \right){T}^{\gamma }}{6\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right],\end{array}\hfill \\ \begin{array}{rl}{b}_{1}=& \frac{\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{\mathrm{\Gamma }\left(3-q\right)}{\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{1-q}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left(q-p\right)\mathrm{\Gamma }\left(4-\gamma \right)}{\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{1-\gamma }}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\end{array}\hfill \\ \begin{array}{rl}{b}_{2}=& \frac{\mathrm{\Gamma }\left(3-q\right)}{2\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{\mathrm{\Gamma }\left(4-\gamma \right)}{2\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{2-\gamma }}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\end{array}\hfill \\ {b}_{3}=\frac{\mathrm{\Gamma }\left(4-\gamma \right)}{6\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.\hfill \end{array}$

Substituting the values of ${b}_{0},{b}_{1},{b}_{2},{b}_{3},{b}_{4},\dots ,{b}_{n-1}$ in (2.3), we get the solution (2.2). □

## 3 Existence results for problem (1.1)-(1.3)

Consider the space $X=\left\{u:u\in {C}^{3}\left(I\right)\right\}$ endowed with the norm

$\parallel u\parallel =\underset{t\in I}{sup}|u\left(t\right)|+\underset{t\in I}{sup}|{u}^{\prime }\left(t\right)|+\underset{t\in I}{sup}|{u}^{″}\left(t\right)|+\underset{t\in I}{sup}|{u}^{‴}\left(t\right)|.$

Obviously $\left(X,\parallel \cdot \parallel \right)$ is a Banach space.

We need the following result [40] in the sequel.

### Theorem 3.1

Let E be a Banach space, $S:E\to E$a completely continuous operator and

$V=\left\{x\in E:x=\mu Sx,0\le \mu \le 1\right\}$

a bounded set. Then S has a fixed point in E.

Let us define the operator $T:X\to X$ by

$\begin{array}{rcl}\left(Tx\right)\left(t\right)& =& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{b}{\left(a+b\right)\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left[bT-\left(a+b\right)t\right]\mathrm{\Gamma }\left(2-p\right)}{\left(a+b\right)\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{\left[bp{T}^{2}-\left(a+b\right)\left(2Tt-\left(2-p\right){t}^{2}\right)\right]\mathrm{\Gamma }\left(3-q\right)}{2\left(a+b\right)\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\left(\frac{\left[b\left(-6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right){T}^{3}}{6\left(a+b\right)\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ +\frac{\left(a+b\right)\left(6\left(q-p\right){T}^{2}t+\left(2-p\right)\left(3-p\right)\left(-3T{t}^{2}+\left(3-q\right){t}^{3}\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)}{6\left(a+b\right)\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\right)\\ ×{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\end{array}$

where

$\begin{array}{rcl}\stackrel{˜}{f}\left(s,x\left(s\right)\right)& =& f\left(s,x\left(s\right),{x}^{\prime }\left(s\right),{x}^{″}\left(s\right),{x}^{‴}\left(s\right),\varphi x\left(s\right),\psi x\left(s\right),{}^{c}{D}^{{\mu }_{1}}x\left(s\right),{}^{c}{D}^{{\mu }_{2}}x\left(s\right),\dots ,{}^{c}{D}^{{\mu }_{m}}x\left(s\right),\\ D_{{\nu }_{1}}^{c}x\left(s\right),{}^{c}{D}^{{\nu }_{2}}x\left(s\right),\dots ,{}^{c}{D}^{{\nu }_{{m}^{\prime }}}x\left(s\right),{}^{c}{D}^{{\xi }_{1}}x\left(s\right),{}^{c}{D}^{{\xi }_{2}}x\left(s\right),\dots ,{}^{c}{D}^{{\xi }_{{m}^{″}}}x\left(s\right)\right).\end{array}$

For the sake of convenience, we set

$\begin{array}{rcl}{M}_{1}& =& \left[\frac{|a|+2|b|}{|a+b|\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{\left(|a|+2|b|\right)\mathrm{\Gamma }\left(2-p\right)}{|a+b|\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{\left(|b|p+|a+b|\left(4-p\right)\right)\mathrm{\Gamma }\left(3-q\right)}{2|a+b|\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\left(\frac{\left[|b|\left(6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\\ +\frac{|a+b|\left(6\left(q-p\right)+\left(2-p\right)\left(3-p\right)\left(6-q\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right)\right]{T}^{\alpha }\\ +\left[\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{\left(3-p\right)\mathrm{\Gamma }\left(3-q\right)}{\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\frac{\left[2\left(q-p\right)+\left(2-p\right)\left(3-p\right)\left(5-q\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)}{2\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]{T}^{\alpha -1}\\ +\left[\frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}+\frac{\mathrm{\Gamma }\left(3-q\right)}{\mathrm{\Gamma }\left(\alpha -q+1\right)}+\frac{\left(4-q\right)\mathrm{\Gamma }\left(4-\gamma \right)}{\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]{T}^{\alpha -2}\\ +\left[\frac{1}{\mathrm{\Gamma }\left(\alpha -2\right)}+\frac{\mathrm{\Gamma }\left(4-\gamma \right)}{\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]{T}^{\alpha -3}.\end{array}$

### Theorem 3.2

The operator$T:X\to X$is completely continuous.

### Proof

First, we show that the operator $T:X\to X$ is continuous. Let $\left\{{x}_{n}\right\}$ be a sequence in X with ${x}_{n}\to {x}_{0}$ and $0<{\mu }_{1},\dots ,{\mu }_{m}<1$. Then we have

$\begin{array}{r}\underset{t\in I}{sup}|{}^{c}{D}^{{\mu }_{i}}{x}_{n}\left(t\right)-{}^{c}{D}^{{\mu }_{i}}{x}_{0}\left(t\right)|\\ \phantom{\rule{1em}{0ex}}=\underset{t\in I}{sup}|\frac{1}{\mathrm{\Gamma }\left(1-{\mu }_{i}\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-{\mu }_{i}}{x}_{n}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{\mathrm{\Gamma }\left(1-{\mu }_{i}\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-{\mu }_{i}}{x}_{0}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\\ \phantom{\rule{1em}{0ex}}=\underset{t\in I}{sup}|\frac{1}{\mathrm{\Gamma }\left(1-{\mu }_{i}\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-{\mu }_{i}}\left[{x}_{n}^{\prime }\left(s\right)-{x}_{0}^{\prime }\left(s\right)\right]\phantom{\rule{0.2em}{0ex}}ds|\\ \phantom{\rule{1em}{0ex}}\le \frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}\underset{t\in I}{sup}|{x}_{n}^{\prime }\left(t\right)-{x}_{0}^{\prime }\left(t\right)|\le \frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}\parallel {x}_{n}-{x}_{0}\parallel .\end{array}$

Since $\parallel {x}_{n}-x\parallel \to 0$, $lim_{n\to \mathrm{\infty }}^{c}{D}^{{\mu }_{i}}{x}_{n}\left(t\right)={}^{c}{D}^{{\mu }_{i}}{x}_{0}\left(t\right)$ uniformly on I. Similarly, $lim_{n\to \mathrm{\infty }}^{c}{D}^{{\nu }_{j}}{x}_{n}\left(t\right)={}^{c}{D}^{{\nu }_{j}}{x}_{0}\left(t\right)$ uniformly on I for $1\le j\le {m}^{\prime }$, $lim_{n\to \mathrm{\infty }}^{c}{D}^{{\xi }_{k}}{x}_{n}\left(t\right)={}^{c}{D}^{{\xi }_{k}}{x}_{0}\left(t\right)$ uniformly on I for $1\le k\le {m}^{″}$. Also, we get $lim_{n\to \mathrm{\infty }}^{c}{D}^{{\delta }_{i}}{x}_{n}\left(t\right)={}^{c}{D}^{{\delta }_{i}}{x}_{0}\left(t\right)$, $lim_{n\to \mathrm{\infty }}^{c}{D}^{{\beta }_{i}}{x}_{n}\left(t\right)={}^{c}{D}^{{\beta }_{i}}{x}_{0}\left(t\right)$, and $lim_{n\to \mathrm{\infty }}^{c}{D}^{{\theta }_{i}}{x}_{n}\left(t\right)={}^{c}{D}^{{\theta }_{i}}{x}_{0}\left(t\right)$ uniformly on I for $i=1,2$. Since

$\begin{array}{rcl}\parallel T{x}_{n}-T{x}_{0}\parallel & =& \underset{t\in I}{sup}|T{x}_{n}\left(t\right)-T{x}_{0}\left(t\right)|+\underset{t\in I}{sup}|{\left(T{x}_{n}\right)}^{\prime }\left(t\right)-{\left(T{x}_{0}\right)}^{\prime }\left(t\right)|\\ +\underset{t\in I}{sup}|{\left(T{x}_{n}\right)}^{″}\left(t\right)-{\left(T{x}_{0}\right)}^{″}\left(t\right)|+\underset{t\in I}{sup}|{\left(T{x}_{n}\right)}^{‴}\left(t\right)-{\left(T{x}_{0}\right)}^{‴}\left(t\right)|,\end{array}$

using the continuity of f, ${h}_{1}$, ${h}_{2}$, we get $\parallel T{x}_{n}-Tx\parallel \to 0$. Thus, T is continuous on X. Now, let $\mathrm{\Omega }\subseteq X$ be a bounded subset. Then there exists a positive constant $L>0$ such that $|\stackrel{˜}{f}\left(t,x\left(t\right)\right)|\le L$ for all $t\in I$ and $x\in \mathrm{\Omega }$. We show that T Ω is a bounded set. We have

$\begin{array}{rl}|\left(Tx\right)\left(t\right)|\le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds+\frac{|b|}{|a+b|\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|bT-\left(a+b\right)t|\mathrm{\Gamma }\left(2-p\right)}{|a+b|\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|bp{T}^{2}-\left(a+b\right)\left(2Tt-\left(2-p\right){t}^{2}\right)|\mathrm{\Gamma }\left(3-q\right)}{2|a+b|\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{|b\left(-6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right){T}^{3}}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ +\frac{\left(a+b\right)\left(6\left(q-p\right){T}^{2}t+\left(2-p\right)\left(3-p\right)\left(-3T{t}^{2}+\left(3-q\right){t}^{3}\right)\right)|\mathrm{\Gamma }\left(4-\gamma \right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\right)\\ ×{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{\left(|a|+2|b|\right)L{T}^{\alpha }}{|a+b|\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{\left(|a|+2|b|\right)\mathrm{\Gamma }\left(2-p\right)L{T}^{\alpha }}{|a+b|\mathrm{\Gamma }\left(\alpha -p+1\right)}\\ +\frac{\left(|b|p+|a+b|\left(4-p\right)\right)\mathrm{\Gamma }\left(3-q\right)L{T}^{\alpha }}{2|a+b|\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\left(\frac{\left[|b|\left(6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\\ +\frac{|a+b|\left(6\left(q-p\right)+\left(2-p\right)\left(3-p\right)\left(6-q\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)L{T}^{\alpha }}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right),\\ |{\left(Tx\right)}^{\prime }\left(t\right)|\le & \frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -2}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|T-\left(2-p\right)t|\mathrm{\Gamma }\left(3-q\right)}{\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|2\left(q-p\right){T}^{2}+\left(2-p\right)\left(3-p\right)\left(-2Tt+\left(3-q\right){t}^{2}\right)|\mathrm{\Gamma }\left(4-\gamma \right)}{2\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ ×{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{{T}^{\alpha -1}L}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{\mathrm{\Gamma }\left(2-p\right)L{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{\left(3-p\right)\mathrm{\Gamma }\left(3-q\right)L{T}^{\alpha -1}}{\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\frac{\left[2\left(q-p\right)+\left(2-p\right)\left(3-p\right)\left(5-q\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)L{T}^{\alpha -1}}{2\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)},\\ |{\left(Tx\right)}^{″}\left(t\right)|\le & \frac{1}{\mathrm{\Gamma }\left(\alpha -2\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -3}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(3-q\right)}{\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|-T+\left(3-q\right)t|\mathrm{\Gamma }\left(4-\gamma \right)}{\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{L{T}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}+\frac{\mathrm{\Gamma }\left(3-q\right)L{T}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -q+1\right)}+\frac{\left(4-q\right)\mathrm{\Gamma }\left(4-\gamma \right)L{T}^{\alpha -2}}{\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)},\end{array}$

and

$\begin{array}{rcl}|{\left(Tx\right)}^{‴}\left(t\right)|& \le & \frac{1}{\mathrm{\Gamma }\left(\alpha -3\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -4}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(4-\gamma \right)}{\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{L{T}^{\alpha -3}}{\mathrm{\Gamma }\left(\alpha -2\right)}+\frac{\mathrm{\Gamma }\left(4-\gamma \right)L{T}^{\alpha -3}}{\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\end{array}$

for all $x\in \mathrm{\Omega }$. Hence, we get

$\begin{array}{rcl}\parallel Tx\parallel & =& \underset{t\in I}{sup}|\left(Tx\right)\left(t\right)|+\underset{t\in I}{sup}|{\left(Tx\right)}^{\prime }\left(t\right)|+\underset{t\in I}{sup}|{\left(Tx\right)}^{″}\left(t\right)|+\underset{t\in I}{sup}|{\left(Tx\right)}^{‴}\left(t\right)|\\ \le & \left[\frac{|a|+2|b|}{|a+b|\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{\left(|a|+2|b|\right)\mathrm{\Gamma }\left(2-p\right)}{|a+b|\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{\left(|b|p+|a+b|\left(4-p\right)\right)\mathrm{\Gamma }\left(3-q\right)}{2|a+b|\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\left(\frac{\left[|b|\left(6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\\ +\frac{|a+b|\left(6\left(q-p\right)+\left(2-p\right)\left(3-p\right)\left(6-q\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right)\right]L{T}^{\alpha }\\ +\left[\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{\left(3-p\right)\mathrm{\Gamma }\left(3-q\right)}{\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\frac{\left[2\left(q-p\right)+\left(2-p\right)\left(3-p\right)\left(5-q\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)}{2\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]L{T}^{\alpha -1}\\ +\left[\frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}+\frac{\mathrm{\Gamma }\left(3-q\right)}{\mathrm{\Gamma }\left(\alpha -q+1\right)}+\frac{\left(4-q\right)\mathrm{\Gamma }\left(4-\gamma \right)}{\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]L{T}^{\alpha -2}\\ +\left[\frac{1}{\mathrm{\Gamma }\left(\alpha -2\right)}+\frac{\mathrm{\Gamma }\left(4-\gamma \right)}{\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]L{T}^{\alpha -3}={M}_{1}L.\end{array}$

This implies that the operator T maps bounded sets of X into bounded sets. Now, we prove that the sets $\left\{Tx:x\in \mathrm{\Omega }\right\}$, $\left\{{\left(Tx\right)}^{\prime }:x\in \mathrm{\Omega }\right\}$, $\left\{{\left(Tx\right)}^{″}:x\in \mathrm{\Omega }\right\}$, $\left\{{\left(Tx\right)}^{‴}:x\in \mathrm{\Omega }\right\}$ are equicontinuous on I. For $0\le {t}_{1}<{t}_{2}\le T$, we have

$\begin{array}{r}|\left(Tx\right)\left({t}_{2}\right)-\left(Tx\right)\left({t}_{1}\right)|\\ \phantom{\rule{1em}{0ex}}=|\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{{t}_{2}}{\left({t}_{2}-s\right)}^{\alpha -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{{t}_{1}}{\left({t}_{1}-s\right)}^{\alpha -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}-\frac{\left({t}_{2}-{t}_{1}\right)\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{\left[2T\left({t}_{2}-{t}_{1}\right)-\left(2-p\right)\left({t}_{2}^{2}-{t}_{1}^{2}\right)\right]\mathrm{\Gamma }\left(3-q\right)}{2\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}-\frac{\left[6\left(q-p\right){T}^{2}\left({t}_{2}-{t}_{1}\right)+\left(2-p\right)\left(3-p\right)\left(-3T\left({t}_{2}^{2}-{t}_{1}^{2}\right)+\left(3-q\right)\left({t}_{2}^{3}-{t}_{1}^{3}\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)}{6\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ \phantom{\rule{2em}{0ex}}×{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds|\\ \phantom{\rule{1em}{0ex}}\le \frac{L}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{{t}_{1}}\left[{\left({t}_{2}-s\right)}^{\alpha -1}-{\left({t}_{1}-s\right)}^{\alpha -1}\right]\phantom{\rule{0.2em}{0ex}}ds+\frac{L}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{{t}_{1}}^{{t}_{2}}{\left({t}_{2}-s\right)}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{\left({t}_{2}-{t}_{1}\right)\mathrm{\Gamma }\left(2-p\right)L{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{\left[2T\left({t}_{2}-{t}_{1}\right)+\left(2-p\right)\left({t}_{2}^{2}-{t}_{1}^{2}\right)\right]\mathrm{\Gamma }\left(3-q\right)L{T}^{\alpha -2}}{2\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ \phantom{\rule{2em}{0ex}}+\frac{\left[6\left(q-p\right){T}^{2}\left({t}_{2}-{t}_{1}\right)+\left(2-p\right)\left(3-p\right)\left(3T\left({t}_{2}^{2}-{t}_{1}^{2}\right)+\left(3-q\right)\left({t}_{2}^{3}-{t}_{1}^{3}\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)L{T}^{\alpha -3}}{6\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\\ \phantom{\rule{1em}{0ex}}\le \frac{L}{\mathrm{\Gamma }\left(\alpha +1\right)}\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right)\\ \phantom{\rule{2em}{0ex}}+\frac{\left({t}_{2}-{t}_{1}\right)\mathrm{\Gamma }\left(2-p\right)L{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{\left[2T\left({t}_{2}-{t}_{1}\right)+\left(2-p\right)\left({t}_{2}^{2}-{t}_{1}^{2}\right)\right]\mathrm{\Gamma }\left(3-q\right)L{T}^{\alpha -2}}{2\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ \phantom{\rule{2em}{0ex}}+\frac{\left[6\left(q-p\right){T}^{2}\left({t}_{2}-{t}_{1}\right)+\left(2-p\right)\left(3-p\right)\left(3T\left({t}_{2}^{2}-{t}_{1}^{2}\right)+\left(3-q\right)\left({t}_{2}^{3}-{t}_{1}^{3}\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)L{T}^{\alpha -3}}{6\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}.\end{array}$

In a similar manner, one can find that

$\begin{array}{c}\begin{array}{rl}|{\left(Tx\right)}^{\prime }\left({t}_{2}\right)-{\left(Tx\right)}^{\prime }\left({t}_{1}\right)|\le & \frac{L}{\mathrm{\Gamma }\left(\alpha \right)}\left({t}_{2}^{\alpha -1}-{t}_{1}^{\alpha -1}\right)+\frac{\left({t}_{2}-{t}_{1}\right)\mathrm{\Gamma }\left(3-q\right)L{T}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\frac{\left[2T\left({t}_{2}-{t}_{1}\right)+\left(3-q\right)\left({t}_{2}^{2}-{t}_{1}^{2}\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)L{T}^{\alpha -3}}{2\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)},\end{array}\hfill \\ |{\left(Tx\right)}^{″}\left({t}_{2}\right)-{\left(Tx\right)}^{″}\left({t}_{1}\right)|\le \frac{L}{\mathrm{\Gamma }\left(\alpha -1\right)}\left({t}_{2}^{\alpha -2}-{t}_{1}^{\alpha -2}\right)+\frac{\left({t}_{2}-{t}_{1}\right)\mathrm{\Gamma }\left(4-\gamma \right)L{T}^{\alpha -3}}{\mathrm{\Gamma }\left(\alpha -\gamma +1\right)},\hfill \end{array}$

and

$|{\left(Tx\right)}^{‴}\left({t}_{2}\right)-{\left(Tx\right)}^{‴}\left({t}_{1}\right)|\le \frac{L}{\mathrm{\Gamma }\left(\alpha -2\right)}\left({t}_{2}^{\alpha -3}-{t}_{1}^{\alpha -3}\right).$

Clearly the right-hand sides of the above inequalities tend to zero as ${t}_{2}\to {t}_{1}$. So T is completely continuous. This completes the proof. □

### Theorem 3.3

Assume that there exist positive constants${d}_{0}>0$, ${d}_{i}\ge 0$ ($1\le i\le 6$), ${\zeta }_{i}\ge 0$ ($1\le i\le m$), ${\eta }_{j}\ge 0$ ($1\le j\le {m}^{\prime }$), ${\tau }_{k}\ge 0$ ($1\le k\le {m}^{″}$), ${l}_{i1},{l}_{i2}\ge 0$ ($1\le i\le 7$), ${c}_{01},{c}_{02}>0$such that

$\begin{array}{r}|f\left(t,{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6},{y}_{1},{y}_{2},\dots ,{y}_{m},{z}_{1},{z}_{2},\dots ,{z}_{{m}^{\prime }},{w}_{1},{w}_{2},\dots ,{w}_{{m}^{″}}\right)|\\ \phantom{\rule{1em}{0ex}}\le {d}_{0}+{d}_{1}|{x}_{1}|+{d}_{2}|{x}_{2}|+{d}_{3}|{x}_{3}|+{d}_{4}|{x}_{4}|+{d}_{5}|{x}_{5}|+{d}_{6}|{x}_{6}|\\ \phantom{\rule{2em}{0ex}}+\sum _{i=1}^{m}{\zeta }_{i}|{y}_{i}|+\sum _{j=1}^{{m}^{\prime }}{\eta }_{j}|{z}_{j}|+\sum _{k=1}^{{m}^{″}}{\tau }_{k}|{w}_{k}|\end{array}$

for all $t\in I$ and ${x}_{1},\dots ,{x}_{6},{y}_{1},\dots ,{y}_{m},{z}_{1},\dots ,{z}_{{m}^{\prime }},{\tau }_{1},\dots ,{\tau }_{{m}^{″}}\in \mathbb{R}$ and

$|{h}_{j}\left(t,s,{u}_{1},{u}_{2},{u}_{3},{u}_{4},{u}_{5},{u}_{6},{u}_{7}\right)|\le {c}_{0j}+\sum _{i=1}^{7}{l}_{ij}|{u}_{i}|$

for$j=1,2$, all$t,s\in I$, and all${u}_{1},\dots ,{u}_{7}\in \mathbb{R}$. In addition, assume that

$\begin{array}{rcl}{M}_{1}^{\prime }& =& {M}_{1}\left[{d}_{1}+{d}_{2}+{d}_{3}+{d}_{4}+{d}_{5}{\gamma }_{0}\left({l}_{11}+{l}_{21}+{l}_{31}+{l}_{41}\\ +{l}_{51}\frac{{T}^{1-{\delta }_{1}}}{\mathrm{\Gamma }\left(2-{\delta }_{1}\right)}+{l}_{61}\frac{{T}^{2-{\beta }_{1}}}{\mathrm{\Gamma }\left(3-{\beta }_{1}\right)}+{l}_{71}\frac{{T}^{3-{\theta }_{1}}}{\mathrm{\Gamma }\left(4-{\theta }_{1}\right)}\right)\\ +{d}_{6}{\lambda }_{0}\left({l}_{12}+{l}_{22}+{l}_{32}+{l}_{42}+{l}_{52}\frac{{T}^{1-{\delta }_{2}}}{\mathrm{\Gamma }\left(2-{\delta }_{2}\right)}+{l}_{62}\frac{{T}^{2-{\beta }_{2}}}{\mathrm{\Gamma }\left(3-{\beta }_{2}\right)}+{l}_{72}\frac{{T}^{3-{\theta }_{2}}}{\mathrm{\Gamma }\left(4-{\theta }_{2}\right)}\right)\\ +\sum _{i=1}^{m}{\zeta }_{i}\frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}+\sum _{j=1}^{{m}^{\prime }}{\eta }_{j}\frac{{T}^{2-{\nu }_{j}}}{\mathrm{\Gamma }\left(3-{\nu }_{j}\right)}+\sum _{k=1}^{{m}^{″}}{\tau }_{k}\frac{{T}^{3-{\xi }_{k}}}{\mathrm{\Gamma }\left(4-{\xi }_{k}\right)}\right]<1,\end{array}$

where${\gamma }_{0}={sup}_{t\in I}{\int }_{0}^{t}|\gamma \left(t,s\right)|\phantom{\rule{0.2em}{0ex}}ds$and${\lambda }_{0}={sup}_{t\in I}{\int }_{0}^{t}|\lambda \left(t,s\right)|\phantom{\rule{0.2em}{0ex}}ds$. Then problem (1.1)-(1.3) has at least one solution.

### Proof

In view of Theorem 3.2, the operator $T:X\to X$ is completely continuous. Next we show that the set $V=\left\{x\in X:x=\mu Tx,0\le \mu \le 1\right\}$ is bounded. Let $x\in V$ and $t\in I$. Then we have

$\begin{array}{rl}x\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}\mu {\left(t-s\right)}^{\alpha -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{b}{\left(a+b\right)\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left[bT-\left(a+b\right)t\right]\mathrm{\Gamma }\left(2-p\right)}{\left(a+b\right)\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -p-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{\left[bp{T}^{2}-\left(a+b\right)\left(2Tt-\left(2-p\right){t}^{2}\right)\right]\mathrm{\Gamma }\left(3-q\right)}{2\left(a+b\right)\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -q-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\left(\frac{\left[b\left(-6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right){T}^{3}}{6\left(a+b\right)\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ +\frac{\left(a+b\right)\left(6\left(q-p\right){T}^{2}t+\left(2-p\right)\left(3-p\right)\left(-3T{t}^{2}+\left(3-q\right){t}^{3}\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)}{6\left(a+b\right)\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\right)\\ ×{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -\gamma -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\\ {x}^{\prime }\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}\mu {\left(t-s\right)}^{\alpha -2}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -p-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left[T-\left(2-p\right)t\right]\mathrm{\Gamma }\left(3-q\right)}{\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -q-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{\left[2\left(q-p\right){T}^{2}+\left(2-p\right)\left(3-p\right)\left(-2Tt+\left(3-q\right){t}^{2}\right)\right]\mathrm{\Gamma }\left(4-\gamma \right)}{2\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ ×{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -\gamma -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\\ {x}^{″}\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha -2\right)}{\int }_{0}^{t}\mu {\left(t-s\right)}^{\alpha -3}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{\mathrm{\Gamma }\left(3-q\right)}{\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -q-1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{\left[-T+\left(3-q\right)t\right]\mathrm{\Gamma }\left(4-\gamma \right)}{\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -\gamma -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\end{array}$

and ${x}^{‴}\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -3\right)}{\int }_{0}^{t}\mu {\left(t-s\right)}^{\alpha -4}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{\mathrm{\Gamma }\left(4-\gamma \right)}{\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}{\int }_{0}^{T}\mu {\left(T-s\right)}^{\alpha -\gamma -1}\stackrel{˜}{f}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds$. Thus, we get

$\begin{array}{rl}|x\left(t\right)|=& \mu |Tx\left(t\right)|\\ \le & \left[{d}_{0}+{d}_{1}\parallel x\parallel +{d}_{2}\parallel x\parallel +{d}_{3}\parallel x\parallel +{d}_{4}\parallel x\parallel \\ +{d}_{5}{\gamma }_{0}\left({c}_{01}+{l}_{11}\parallel x\parallel +{l}_{21}\parallel x\parallel +{l}_{31}\parallel x\parallel +{l}_{41}\parallel x\parallel \\ +{l}_{51}\frac{{T}^{1-{\delta }_{1}}}{\mathrm{\Gamma }\left(2-{\delta }_{1}\right)}\parallel x\parallel +{l}_{61}\frac{{T}^{2-{\beta }_{1}}}{\mathrm{\Gamma }\left(3-{\beta }_{1}\right)}\parallel x\parallel +{l}_{71}\frac{{T}^{3-{\theta }_{1}}}{\mathrm{\Gamma }\left(4-{\theta }_{1}\right)}\parallel x\parallel \right)\\ +{d}_{6}{\lambda }_{0}\left({c}_{02}+{l}_{12}\parallel x\parallel +{l}_{22}\parallel x\parallel +{l}_{32}\parallel x\parallel +{l}_{42}\parallel x\parallel \\ +{l}_{52}\frac{{T}^{1-{\delta }_{2}}}{\mathrm{\Gamma }\left(2-{\delta }_{2}\right)}\parallel x\parallel +{l}_{62}\frac{{T}^{2-{\beta }_{2}}}{\mathrm{\Gamma }\left(3-{\beta }_{2}\right)}\parallel x\parallel +{l}_{72}\frac{{T}^{3-{\theta }_{2}}}{\mathrm{\Gamma }\left(4-{\theta }_{2}\right)}\parallel x\parallel \right)\\ +\sum _{i=1}^{m}{\zeta }_{i}\frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}\parallel x\parallel +\sum _{j=1}^{{m}^{\prime }}{\eta }_{j}\frac{{T}^{2-{\nu }_{j}}}{\mathrm{\Gamma }\left(3-{\nu }_{j}\right)}\parallel x\parallel +\sum _{k=1}^{{m}^{″}}{\tau }_{k}\frac{{T}^{3-{\xi }_{k}}}{\mathrm{\Gamma }\left(4-{\xi }_{k}\right)}\parallel x\parallel \right]\\ ×\left[\frac{\left(|a|+2|b|\right){T}^{\alpha }}{|a+b|\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{\left(|a|+2|b|\right)\mathrm{\Gamma }\left(2-p\right){T}^{\alpha }}{|a+b|\mathrm{\Gamma }\left(\alpha -p+1\right)}\\ +\frac{\left(|b|p+|a+b|\left(4-p\right)\right)\mathrm{\Gamma }\left(3-q\right){T}^{\alpha }}{2|a+b|\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\left(\frac{\left[|b|\left(6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\\ +\frac{|a+b|\left(6\left(q-p\right)+\left(2-p\right)\left(3-p\right)\left(6-q\right)\right)\right]\mathrm{\Gamma }\left(4-\gamma \right){T}^{\alpha }}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right)\right],\\ |{x}^{\prime }\left(t\right)|=& \mu |{\left(Tx\right)}^{\prime }\left(t\right)|\\ \le & \left[{d}_{0}+{d}_{1}\parallel x\parallel +{d}_{2}\parallel x\parallel +{d}_{3}\parallel x\parallel +{d}_{4}\parallel x\parallel \\ +{d}_{5}{\gamma }_{0}\left({c}_{01}+{l}_{11}\parallel x\parallel +{l}_{21}\parallel x\parallel +{l}_{31}\parallel x\parallel +{l}_{41}\parallel x\parallel \\ +{l}_{51}\frac{{T}^{1-{\delta }_{1}}}{\mathrm{\Gamma }\left(2-{\delta }_{1}\right)}\parallel x\parallel +{l}_{61}\frac{{T}^{2-{\beta }_{1}}}{\mathrm{\Gamma }\left(3-{\beta }_{1}\right)}\parallel x\parallel +{l}_{71}\frac{{T}^{3-{\theta }_{1}}}{\mathrm{\Gamma }\left(4-{\theta }_{1}\right)}\parallel x\parallel \right)\\ +{d}_{6}{\lambda }_{0}\left({c}_{02}+{l}_{12}\parallel x\parallel +{l}_{22}\parallel x\parallel +{l}_{32}\parallel x\parallel +{l}_{42}\parallel x\parallel \\ +{l}_{52}\frac{{T}^{1-{\delta }_{2}}}{\mathrm{\Gamma }\left(2-{\delta }_{2}\right)}\parallel x\parallel +{l}_{62}\frac{{T}^{2-{\beta }_{2}}}{\mathrm{\Gamma }\left(3-{\beta }_{2}\right)}\parallel x\parallel +{l}_{72}\frac{{T}^{3-{\theta }_{2}}}{\mathrm{\Gamma }\left(4-{\theta }_{2}\right)}\parallel x\parallel \right)\\ +\sum _{i=1}^{m}{\zeta }_{i}\frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}\parallel x\parallel +\sum _{j=1}^{{m}^{\prime }}{\eta }_{j}\frac{{T}^{2-{\nu }_{j}}}{\mathrm{\Gamma }\left(3-{\nu }_{j}\right)}\parallel x\parallel +\sum _{k=1}^{{m}^{″}}{\tau }_{k}\frac{{T}^{3-{\xi }_{k}}}{\mathrm{\Gamma }\left(4-{\xi }_{k}\right)}\parallel x\parallel \right]\\ ×\left[\frac{{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{\left(3-p\right)\mathrm{\Gamma }\left(3-q\right){T}^{\alpha -1}}{\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\frac{\left[2\left(q-p\right)+\left(2-p\right)\left(3-p\right)\left(5-q\right)\right]\mathrm{\Gamma }\left(4-\gamma \right){T}^{\alpha -1}}{2\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right],\\ |{x}^{″}\left(t\right)|=& \mu |{\left(Tx\right)}^{″}\left(t\right)|\\ \le & \left[{d}_{0}+{d}_{1}\parallel x\parallel +{d}_{2}\parallel x\parallel +{d}_{3}\parallel x\parallel +{d}_{4}\parallel x\parallel \\ +{d}_{5}{\gamma }_{0}\left({c}_{01}+{l}_{11}\parallel x\parallel +{l}_{21}\parallel x\parallel +{l}_{31}\parallel x\parallel +{l}_{41}\parallel x\parallel \\ +{l}_{51}\frac{{T}^{1-{\delta }_{1}}}{\mathrm{\Gamma }\left(2-{\delta }_{1}\right)}\parallel x\parallel +{l}_{61}\frac{{T}^{2-{\beta }_{1}}}{\mathrm{\Gamma }\left(3-{\beta }_{1}\right)}\parallel x\parallel +{l}_{71}\frac{{T}^{3-{\theta }_{1}}}{\mathrm{\Gamma }\left(4-{\theta }_{1}\right)}\parallel x\parallel \right)\\ +{d}_{6}{\lambda }_{0}\left({c}_{02}+{l}_{12}\parallel x\parallel +{l}_{22}\parallel x\parallel +{l}_{32}\parallel x\parallel +{l}_{42}\parallel x\parallel \\ +{l}_{52}\frac{{T}^{1-{\delta }_{2}}}{\mathrm{\Gamma }\left(2-{\delta }_{2}\right)}\parallel x\parallel +{l}_{62}\frac{{T}^{2-{\beta }_{2}}}{\mathrm{\Gamma }\left(3-{\beta }_{2}\right)}\parallel x\parallel +{l}_{72}\frac{{T}^{3-{\theta }_{2}}}{\mathrm{\Gamma }\left(4-{\theta }_{2}\right)}\parallel x\parallel \right)\\ +\sum _{i=1}^{m}{\zeta }_{i}\frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}\parallel x\parallel +\sum _{j=1}^{{m}^{\prime }}{\eta }_{j}\frac{{T}^{2-{\nu }_{j}}}{\mathrm{\Gamma }\left(3-{\nu }_{j}\right)}\parallel x\parallel +\sum _{k=1}^{{m}^{″}}{\tau }_{k}\frac{{T}^{3-{\xi }_{k}}}{\mathrm{\Gamma }\left(4-{\xi }_{k}\right)}\parallel x\parallel \right]\\ ×\left[\frac{{T}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}+\frac{\mathrm{\Gamma }\left(3-q\right){T}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -q+1\right)}+\frac{\left(4-q\right)\mathrm{\Gamma }\left(4-\gamma \right){T}^{\alpha -2}}{\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right],\end{array}$

and

$\begin{array}{rcl}|{x}^{‴}\left(t\right)|& =& \mu |{\left(Tx\right)}^{‴}\left(t\right)|\\ \le & \left[{d}_{0}+{d}_{1}\parallel x\parallel +{d}_{2}\parallel x\parallel +{d}_{3}\parallel x\parallel +{d}_{4}\parallel x\parallel \\ +{d}_{5}{\gamma }_{0}\left({c}_{01}+{l}_{11}\parallel x\parallel +{l}_{21}\parallel x\parallel +{l}_{31}\parallel x\parallel +{l}_{41}\parallel x\parallel \\ +{l}_{51}\frac{{T}^{1-{\delta }_{1}}}{\mathrm{\Gamma }\left(2-{\delta }_{1}\right)}\parallel x\parallel +{l}_{61}\frac{{T}^{2-{\beta }_{1}}}{\mathrm{\Gamma }\left(3-{\beta }_{1}\right)}\parallel x\parallel +{l}_{71}\frac{{T}^{3-{\theta }_{1}}}{\mathrm{\Gamma }\left(4-{\theta }_{1}\right)}\parallel x\parallel \right)\\ +{d}_{6}{\lambda }_{0}\left({c}_{02}+{l}_{12}\parallel x\parallel +{l}_{22}\parallel x\parallel +{l}_{32}\parallel x\parallel +{l}_{42}\parallel x\parallel \\ +{l}_{52}\frac{{T}^{1-{\delta }_{2}}}{\mathrm{\Gamma }\left(2-{\delta }_{2}\right)}\parallel x\parallel +{l}_{62}\frac{{T}^{2-{\beta }_{2}}}{\mathrm{\Gamma }\left(3-{\beta }_{2}\right)}\parallel x\parallel +{l}_{72}\frac{{T}^{3-{\theta }_{2}}}{\mathrm{\Gamma }\left(4-{\theta }_{2}\right)}\parallel x\parallel \right)\\ +\sum _{i=1}^{m}{\zeta }_{i}\frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}\parallel x\parallel +\sum _{j=1}^{{m}^{\prime }}{\eta }_{j}\frac{{T}^{2-{\nu }_{j}}}{\mathrm{\Gamma }\left(3-{\nu }_{j}\right)}\parallel x\parallel +\sum _{k=1}^{{m}^{″}}{\tau }_{k}\frac{{T}^{3-{\xi }_{k}}}{\mathrm{\Gamma }\left(4-{\xi }_{k}\right)}\parallel x\parallel \right]\\ ×\left[\frac{{T}^{\alpha -3}}{\mathrm{\Gamma }\left(\alpha -2\right)}+\frac{\mathrm{\Gamma }\left(4-\gamma \right){T}^{\alpha -3}}{\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right].\end{array}$

This implies that

$\begin{array}{rcl}\parallel x\parallel & \le & {M}_{1}\left[{d}_{1}+{d}_{2}+{d}_{3}+{d}_{4}+{d}_{5}{\gamma }_{0}\left({l}_{11}+{l}_{21}+{l}_{31}+{l}_{41}\\ +{l}_{51}\frac{{T}^{1-{\delta }_{1}}}{\mathrm{\Gamma }\left(2-{\delta }_{1}\right)}+{l}_{61}\frac{{T}^{2-{\beta }_{1}}}{\mathrm{\Gamma }\left(3-{\beta }_{1}\right)}+{l}_{71}\frac{{T}^{3-{\theta }_{1}}}{\mathrm{\Gamma }\left(4-{\theta }_{1}\right)}\right)\\ +{d}_{6}{\lambda }_{0}\left({l}_{12}+{l}_{22}+{l}_{32}+{l}_{42}+{l}_{52}\frac{{T}^{1-{\delta }_{2}}}{\mathrm{\Gamma }\left(2-{\delta }_{2}\right)}+{l}_{62}\frac{{T}^{2-{\beta }_{2}}}{\mathrm{\Gamma }\left(3-{\beta }_{2}\right)}+{l}_{72}\frac{{T}^{3-{\theta }_{2}}}{\mathrm{\Gamma }\left(4-{\theta }_{2}\right)}\right)\\ +\sum _{i=1}^{m}{\zeta }_{i}\frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}+\sum _{j=1}^{{m}^{\prime }}{\eta }_{j}\frac{{T}^{2-{\nu }_{j}}}{\mathrm{\Gamma }\left(3-{\nu }_{j}\right)}+\sum _{k=1}^{{m}^{″}}{\tau }_{k}\frac{{T}^{3-{\xi }_{k}}}{\mathrm{\Gamma }\left(4-{\xi }_{k}\right)}\right]\parallel x\parallel \\ +{M}_{1}\left({d}_{0}+{d}_{5}{\gamma }_{0}{c}_{01}+{d}_{6}{\lambda }_{0}{c}_{02}\right)\end{array}$

and so $\parallel x\parallel \le \frac{{M}_{1}\left({d}_{0}+{d}_{5}{\gamma }_{0}{c}_{01}+{d}_{6}{\lambda }_{0}{c}_{02}\right)}{1-{M}_{1}^{\prime }}$. Thus, the set V is bounded. Hence it follows by Theorem 3.1 that the operator T has at least one fixed point, which in turn implies that problem (1.1)-(1.3) has a solution. □

### Theorem 3.4

Assume that$f:I×{\mathbb{R}}^{6+m+{m}^{\prime }+{m}^{″}}\to \mathbb{R}$and${h}_{1},{h}_{2}:I×I×{\mathbb{R}}^{7}\to \mathbb{R}$are continuous functions and there exist constants${n}_{i}\ge 0$ ($1\le i\le 6$), ${k}_{i}\ge 0$ ($1\le i\le m$), ${k}_{j}^{\prime }\ge 0$ ($1\le j\le {m}^{\prime }$), ${k}_{k}^{″}\ge 0$ ($1\le k\le {m}^{″}$), ${e}_{i1},{e}_{i2}\ge 0$ ($1\le i\le 7$) such that

$\begin{array}{r}|f\left(t,{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6},{y}_{1},{y}_{2},\dots ,{y}_{m},{z}_{1},{z}_{2},\dots ,{z}_{{m}^{\prime }},{w}_{1},{w}_{2},\dots ,{w}_{{m}^{″}}\right)\\ \phantom{\rule{2em}{0ex}}-f\left(t,{x}_{1}^{\prime },{x}_{2}^{\prime },{x}_{3}^{\prime },{x}_{4}^{\prime },{x}_{5}^{\prime },{x}_{6}^{\prime },{y}_{1}^{\prime },{y}_{2}^{\prime },\dots ,{y}_{m}^{\prime },{z}_{1}^{\prime },{z}_{2}^{\prime },\dots ,{z}_{{m}^{\prime }}^{\prime },{w}_{1}^{\prime },{w}_{2}^{\prime },\dots ,{w}_{{m}^{″}}^{\prime }\right)|\\ \phantom{\rule{1em}{0ex}}\le {n}_{1}|{x}_{1}-{x}_{1}^{\prime }|+{n}_{2}|{x}_{2}-{x}_{2}^{\prime }|+{n}_{3}|{x}_{3}-{x}_{3}^{\prime }|\\ \phantom{\rule{2em}{0ex}}+{n}_{4}|{x}_{4}-{x}_{4}^{\prime }|+{n}_{5}|{x}_{5}-{x}_{5}^{\prime }|+{n}_{6}|{x}_{6}-{x}_{6}^{\prime }|\\ \phantom{\rule{2em}{0ex}}+\sum _{i=1}^{m}{k}_{i}|{y}_{i}-{y}_{i}^{\prime }|+\sum _{j=1}^{{m}^{\prime }}{k}_{j}^{\prime }|{z}_{j}-{z}_{j}^{\prime }|+\sum _{k=1}^{{m}^{″}}{k}_{k}^{″}|{w}_{k}-{w}_{k}^{\prime }|\end{array}$

for all${x}_{1},\dots ,{x}_{6},{x}_{1}^{\prime },\dots ,{x}_{6}^{\prime },{y}_{1},\dots ,{y}_{m},{y}_{1}^{\prime },\dots ,{y}_{m}^{\prime },{z}_{1},\dots ,{z}_{{m}^{\prime }},{z}_{1}^{\prime },\dots ,{z}_{{m}^{\prime }}^{\prime },{w}_{1},\dots ,{w}_{{m}^{″}},{w}_{1}^{\prime },\dots ,{w}_{{m}^{″}}^{\prime }\in \mathbb{R}$, and$t\in I$, and also

$|{h}_{j}\left(t,s,{u}_{1},{u}_{2},{u}_{3},{u}_{4},{u}_{5},{u}_{6},{u}_{7}\right)-{h}_{j}\left(t,s,{u}_{1}^{\prime },{u}_{2}^{\prime },{u}_{3}^{\prime },{u}_{4}^{\prime },{u}_{5}^{\prime },{u}_{6}^{\prime },{u}_{7}^{\prime }\right)|\le \sum _{i=1}^{7}{e}_{ij}|{u}_{i}-{u}_{i}^{\prime }|$

for$j=1,2$, $t,s\in I$, and${u}_{1},\dots ,{u}_{7},{u}_{1}^{\prime },\dots ,{u}_{7}^{\prime }\in \mathbb{R}$. In addition, suppose that

$\begin{array}{rcl}\mathrm{\Delta }& =& {M}_{1}\left[{n}_{1}+{n}_{2}+{n}_{3}+{n}_{4}+{n}_{5}{\gamma }_{0}\left({e}_{11}+{e}_{21}+{e}_{31}+{e}_{41}\\ +{e}_{51}\frac{{T}^{1-{\delta }_{1}}}{\mathrm{\Gamma }\left(2-{\delta }_{1}\right)}+{e}_{61}\frac{{T}^{2-{\beta }_{1}}}{\mathrm{\Gamma }\left(3-{\beta }_{1}\right)}+{e}_{71}\frac{{T}^{3-{\theta }_{1}}}{\mathrm{\Gamma }\left(4-{\theta }_{1}\right)}\right)\\ +{n}_{6}{\lambda }_{0}\left({e}_{12}+{e}_{22}+{e}_{32}+{e}_{42}+{e}_{52}\frac{{T}^{1-{\delta }_{2}}}{\mathrm{\Gamma }\left(2-{\delta }_{2}\right)}+{e}_{62}\frac{{T}^{2-{\beta }_{2}}}{\mathrm{\Gamma }\left(3-{\beta }_{2}\right)}+{e}_{72}\frac{{T}^{3-{\theta }_{2}}}{\mathrm{\Gamma }\left(4-{\theta }_{2}\right)}\right)\\ +\sum _{i=1}^{m}{k}_{i}\frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}+\sum _{j=1}^{{m}^{\prime }}{k}_{j}^{\prime }\frac{{T}^{2-{\nu }_{j}}}{\mathrm{\Gamma }\left(3-{\nu }_{j}\right)}+\sum _{k=1}^{{m}^{″}}{k}_{k}^{″}\frac{{T}^{3-{\xi }_{k}}}{\mathrm{\Gamma }\left(4-{\xi }_{k}\right)}\right]<1.\end{array}$

Then problem (1.1)-(1.3) has a unique solution.

### Proof

Set $N={sup}_{0\le t\le T}|f\left(t,0,0,\dots ,0\right)|<\mathrm{\infty }$, ${\kappa }_{j}={sup}_{0\le t,s\le T}|{h}_{j}\left(t,s,0,0,\dots ,0\right)|<\mathrm{\infty }$ for $j=1,2$, and choose $r\ge \frac{\left(N+{n}_{5}{\gamma }_{0}{\kappa }_{1}+{n}_{6}{\lambda }_{0}{\kappa }_{2}\right){M}_{1}}{1-\mathrm{\Delta }}$. We show that $T\left({B}_{r}\right)\subseteq {B}_{r}$, where ${B}_{r}=\left\{x\in X:\parallel x\parallel \le r\right\}$. Let $x\in {B}_{r}$. Then we have

$\begin{array}{rcl}|\left(Tx\right)\left(t\right)|& \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds+\frac{|b|}{|a+b|\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|bT-\left(a+b\right)t|\mathrm{\Gamma }\left(2-p\right)}{|a+b|\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|bp{T}^{2}-\left(a+b\right)\left(2Tt-\left(2-p\right){t}^{2}\right)|\mathrm{\Gamma }\left(3-q\right)}{2|a+b|\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{|b\left(-6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right){T}^{3}}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ +\frac{\left(a+b\right)\left(6\left(q-p\right){T}^{2}t+\left(2-p\right)\left(3-p\right)\left(-3T{t}^{2}+\left(3-q\right){t}^{3}\right)\right)|\mathrm{\Gamma }\left(4-\gamma \right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\right)\\ ×{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}|\stackrel{˜}{f}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\left[|\stackrel{˜}{f}\left(s,x\left(s\right)\right)-f\left(s,0,0,\dots ,0\right)|+|f\left(s,0,0,\dots ,0\right)|\right]\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|b|}{|a+b|\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -1}\left[|\stackrel{˜}{f}\left(s,x\left(s\right)\right)-f\left(s,0,0,\dots ,0\right)|\\ +|f\left(s,0,0,\dots ,0\right)|\right]\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|bT-\left(a+b\right)t|\mathrm{\Gamma }\left(2-p\right)}{|a+b|\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}\left[|\stackrel{˜}{f}\left(s,x\left(s\right)\right)-f\left(s,0,0,\dots ,0\right)|\\ +|f\left(s,0,0,\dots ,0\right)|\right]\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{|bp{T}^{2}-\left(a+b\right)\left(2Tt-\left(2-p\right){t}^{2}\right)|\mathrm{\Gamma }\left(3-q\right)}{2|a+b|\left(2-p\right)\mathrm{\Gamma }\left(\alpha -q\right){T}^{2-q}}\\ ×{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -q-1}\left[|\stackrel{˜}{f}\left(s,x\left(s\right)\right)-f\left(s,0,0,\dots ,0\right)|+|f\left(s,0,0,\dots ,0\right)|\right]\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{|b\left(-6\left(q-p\right)+\left(2-p\right)\left(3-p\right)q\right){T}^{3}}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\\ +\frac{\left(a+b\right)\left(6\left(q-p\right){T}^{2}t+\left(2-p\right)\left(3-p\right)\left(-3T{t}^{2}+\left(3-q\right){t}^{3}\right)\right)|\mathrm{\Gamma }\left(4-\gamma \right)}{6|a+b|\left(2-p\right)\left(3-p\right)\left(3-q\right)\mathrm{\Gamma }\left(\alpha -\gamma \right){T}^{3-\gamma }}\right)\\ ×{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -\gamma -1}\left[|\stackrel{˜}{f}\left(s,x\left(s\right)\right)-f\left(s,0,0,\dots ,0\right)|+|f\left(s,0,0,\dots ,0\right)|\right]\phantom{\rule{0.2em}{0ex}}ds\\ \le & \left[\left({n}_{1}+{n}_{2}+{n}_{3}+{n}_{4}+{n}_{5}{\gamma }_{0}\left({e}_{11}+{e}_{21}+{e}_{31}+{e}_{41}\\ +{e}_{51}\frac{{T}^{1-{\delta }_{1}}}{\mathrm{\Gamma }\left(2-{\delta }_{1}\right)}+{e}_{61}\frac{{T}^{2-{\beta }_{1}}}{\mathrm{\Gamma }\left(3-{\beta }_{1}\right)}+{e}_{71}\frac{{T}^{3-{\theta }_{1}}}{\mathrm{\Gamma }\left(4-{\theta }_{1}\right)}\right)\\ +{n}_{6}{\lambda }_{0}\left({e}_{12}+{e}_{22}+{e}_{32}+{e}_{42}+{e}_{52}\frac{{T}^{1-{\delta }_{2}}}{\mathrm{\Gamma }\left(2-{\delta }_{2}\right)}+{e}_{62}\frac{{T}^{2-{\beta }_{2}}}{\mathrm{\Gamma }\left(3-{\beta }_{2}\right)}+{e}_{72}\frac{{T}^{3-{\theta }_{2}}}{\mathrm{\Gamma }\left(4-{\theta }_{2}\right)}\right)\\ +\sum _{i=1}^{m}{k}_{i}\frac{{T}^{1-{\mu }_{i}}}{\mathrm{\Gamma }\left(2-{\mu }_{i}\right)}+\sum _{j=1}^{{m}^{\prime }}{k}_{j}^{\prime }\frac{{T}^{2-{\nu }_{j}}}{\mathrm{\Gamma }\left(3-{\nu }_{j}\right)}+\sum _{k=1}^{{m}^{″}}{k}_{k}^{″}\frac{{T}^{3-{\xi }_{k}}}{\mathrm{\Gamma }\left(4-{\xi }_{k}\right)}\right)r\\ +N+{}_{}\end{array}$