Existence Results for Boundary Value Problem of Nonlinear Fractional Differential Equation

Abstract

In this chapter, we investigate the existence and uniqueness of solutions for class of nonlinear fractional differential equations with nonlocal boundary conditions. The existence results are obtained by using Leray-Schauder nonlinear alternative and Banach contraction principle. An illustrative example is presented at the end to illustrated the validity of our results.

1. Introduction

In this chapter, we are interested in the existence of solutions for nonlinear fractional difference equations

$${}^c{D}_{0^{+}}^{\alpha }u\left(t\right)-A^c{D}_{0^{+}}^{\beta }u\left(t\right)=f\left(t,u\left(t\right){,}^c{D}_{0^{+}}^{\beta }u\left(t\right){,}^c{D}_{0^{+}}^{\alpha }u\left(t\right)\right),t\in J=\left[0,T\right],$$E1

subject to the three-point boundary conditions

$$\left\{\begin{array}{c}\mathit{\lambda u}\left(0\right)-\mathit{\mu u}\left(T\right)-\mathit{\gamma u}\left(\eta \right)=d,\\ \lambda u^{\prime }\left(0\right)-\mu u^{\prime }\left(T\right)-\gamma u^{\prime }\left(\eta \right)=l,\end{array}\right.$$E2

where $T>0,0\le \eta \le T,\lambda \ne \mu +\gamma ,d,l,\lambda ,\mu ,\gamma \in \mathbb{R},\beta +1<\alpha$, $A$ is an ${\mathbb{R}}^{n\times n}$ matrix and ${}^c{D}_{0^{+}}^{\alpha },{}^c{D}_{0^{+}}^{\beta }$ are the Caputo fractional derivatives of order $1<\alpha \le 2,0<\beta \le 1$, respectively.

The first definition of fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus, was mentioned already in 1695 by Leibniz and L’Hospital. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electro-analytical chemistry, biology, control theory, fitting of experimental data, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For more details we refer the reader to [1, 2, 3, 4, 5] and the references cited therein.

Fractional differential equation theory have recieved increasing attention. This theory has been developed very quickly and attracted a considerable interest from researches in this field, which revealed the flexibility of fractional calculus theory in designing various mathematical models. The main methods conducted in this articles are by terms of fixed point techniques [6]. Boundary value problems for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory. A point of central importance in the study of nonlinear boundary value problems is to understand how the properties of nonlinearity in a problem influence the nature of the solutions to the boundary value problems. The multi-point boundary conditions are important in various physical problems of applied science when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the sensors located, at intermediate points, see [7, 8] and the references therein. We quote also that realistic problems arising from economics, optimal control, stochastic analysis can be modeled as differential inclusion. The study of fractional differential inclusions was initiated by EL-Sayad and Ibrahim [9]. Also, recently, several qualitative results for fractional differential inclusion were obtained in [10, 11, 12, 13] and the references therein.

The techniques of nonlinear analysis, as the main method to deal with the problems of nonlinear differential equations (DEs), nonlinear fractional differential equations (FDEs), nonlinear partial differential equations (PDEs), nonlinear fractional partial differential equations (FPDEs), nonlinear stochastic fractional partial differential equations (SFPDEs), plays an essential role in the research of this field, such as establishing the existence, uniqueness and multiplicity of solutions (or positive solutions) and mild solutions for nonlinear of different kinds of FPDEs, FPDEs, SFPDEs, inclusion differential equations and inclusion fractional differential equations with various boundary conditions, by using different techniques (approaches). For more details, see [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] and the references therein. For example, iterative method is an important tool for solving linear and nonlinear boundary value problems. It has been used in the research areas of mathematics and several branches of science and other fields. However, many authors showed the existence of positive solutions for a class of boundary value problem at resonance case. Some recent devolopment for resonant case can be found in [37, 38]. Let us cited few papers. In [39], the authors studied the boundary value problems of the fractional order differential equation:

$$\left\{\begin{array}{c}\begin{array}{c}{D}_{0+}^{\alpha }u\left(t\right)=f\left(t,u\left(t\right)\right)=0,t\in \left(0,1\right),\\ u\left(0\right)=0,D_{0+}^{\beta }u\left(1\right)={\mathit{aD}}_{0+}^{\beta }u\left(\eta \right),\end{array}\end{array}\right.$$

where $1<\alpha \le 2,0<\eta <1,0<a,\beta <1,$ $f\in C\left(\left[0,1\right]\times {\mathbb{R}}^2,\mathbb{R}\right)$ and $D_{0+}^{\alpha },D_{0+}^{\beta }$ are the standard Riemann-Liouville fractional derivative of order $\alpha$. They obtained the multiple positive solutions by the Leray-Schauder nonlinear alternative and the fixed point theorem on cones.

In 2017, Resapour et al. [40] investigated a Caputo fractional inclusion with integral boundary condition for the following problem

$$\left\{\begin{array}{c}\begin{array}{c}{}^c{D}^{\alpha }u\left(t\right)\in F\left(t,u\left(t\right){,}^c{D}^{\beta }u\left(t\right),u^{\prime }\left(t\right)\right),\\ u\left(0\right)+u^{\prime }\left(0\right){+}^c{D}^{\beta }u\left(0\right)={\int }_0^{\eta }u\left(s\right)\mathit{ds},\\ u\left(1\right)+u^{\prime }\left(1\right){+}^c{D}^{\beta }u\left(1\right)={\int }_0^{\nu }u\left(s\right)\mathit{ds},\end{array}\end{array}\right.$$

where $1<\alpha \le 2,\eta ,\nu ,\beta \in \left(0,1\right),$ $F:\left[0,1\right]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to 2^{\mathbb{R}}$ is a compact valued multifunction and ${}^c{D}^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha$.

In 2017, Sheng and Jiang [41] studied the existence and uniqueness of the solutions for fractional damped dynamical systems

$$\left\{\begin{array}{c}\begin{array}{c}{}^c{D}_{0^{+}}^{\alpha }u\left(t\right)-A^c{D}_{0^{+}}^{\beta }u\left(t\right)=f\left(t,u\left(t\right)\right),t\in \left[0,T\right],\\ u\left(0\right)=u_0,u^{\prime }\left(0\right)=u_0^{\text{'}},\end{array}\end{array}\right.$$

where $0<\beta \le 1<\alpha \le 2,0<T<\infty ,u\in {\mathbb{R}}^n,A$ is an ${\mathbb{R}}^{n\times n}$ matrix, $f:\left[0,1\right]\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ jointly continuous function and ${}^c{D}_{0^{+}}^{\alpha },{}^c{D}_{0^{+}}^{\beta }$ are the Caputo derivatives of order $\alpha ,\beta$, respectively.

In 2018, Abbes et al. [42] studied the existence and uniqueness of the solutions for fractional damped dynamical systems

$$\left\{\begin{array}{c}\begin{array}{c}{}^c{D}_{0^{+}}^{\alpha }u\left(t\right)-A^c{D}_{0^{+}}^{\beta }u\left(t\right)=f\left(t,u\left(t\right){,}^c{D}_{0^{+}}^{\beta }u\left(t\right){,}^c{D}_{0^{+}}^{\alpha }u\left(t\right)\right),t\in \left[0,T\right],\\ u\left(0\right)=u\left(T\right),u^{\prime }\left(0\right)=u^{\prime }\left(T\right),\end{array}\end{array}\right.$$

where $0<\beta \le 1<\alpha \le 2,0<T<\infty ,u\in {\mathbb{R}}^n,A$ is an ${\mathbb{R}}^{n\times n}$ matrix and $f:\left[0,1\right]\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ jointly continuous.

In 2019, Tao Zhu [43] studied the existence and uniqueness of positive solutions of the following fractional differential equations

$$\left\{\begin{array}{c}\begin{array}{c}{}^{}{D}_{0^{+}}^{\alpha }u\left(t\right)-A^c{D}_{0^{+}}^{\beta }u\left(t\right)=f\left(t,u\left(t\right)\right),t\in \left[0,T\right),0<\beta <\alpha <1,\\ u\left(0\right)=u_0.\end{array}\end{array}\right.$$

Inspired and motivated by the works mentioned above, we establish the existence results for the nonlocal boundary value problem (1.1)–(1.2) by using Leray-Schauder nonlinear alternative and the Banach fixed point theorem. Note that our work generalized the three works cited above [41, 42, 43]. The chapter is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel. In Section 3, deals with main results and we give an example to illustrate our results.

2. Existence and uniqueness results for our problem

2.1 Preliminaries

Let as introduce notations, definitions and preliminary facts that will be need in the sequel. For more details, see for example [44, 45, 46].

Definition 2.1. The Caputo fractional derivative of order $\alpha$ for the function $u\in C^n\left(\left[0,\infty \right),\mathbb{R}\right)$ is defined by

$${}^c{D}_{0^{+}}^{\alpha }u\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{n-\alpha -1}{u}^{\left(n\right)}\left(s\right)\mathit{ds}.$$

where $\mathrm{\Gamma }\left(\cdot \right)$ is the Eleur gamma function and $\alpha >0,n=\left[\alpha \right]+1,\left[\alpha \right]$ denotes the integer part of the real number $\alpha$.

Definition 2.2. The Riemann-Liouville fractional integral of order $\alpha >0$ of a function $u:\left(0,\infty \right)\to \mathbb{R}$ is given by

$$I_{0^{+}}^{\alpha }u\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}u\left(s\right)\mathit{ds},t>0.$$

where $\mathrm{\Gamma }\left(\cdot \right)$ is the Eleur gamma function, provided that the right side is pointwise defined on $\left(0,\infty \right)$.

Lemma 2.1. Let $u\in {\mathit{AC}}^n\left[0,T\right],n\in \mathbb{N}$ and $u\left(\cdot \right)\in C\left[0,T\right]$. Then, we have

$${}^c{D}_{0^{+}}^{\beta }\left(I_{0^{+}}^{\alpha }u\left(t\right)\right)=I_{0^{+}}^{\alpha -\beta }u\left(t\right),$$

$$I_{0^{+}}^{\alpha }\left({}^c{D}_{0^{+}}^{\alpha }u\left(t\right)\right)=u\left(t\right)-\stackrel{n-1}{\sum _{k=0}}\frac{t^k}{k!}{u}^{\left(k\right)}\left(0\right),t>0,n-1<\alpha <n,$$

Especially, when $1<\alpha <2$, then we have

$$I_{0^{+}}^{\alpha }\left({}^c{D}_{0^{+}}^{\alpha }u\left(t\right)\right)=u\left(t\right)-u\left(0\right)-t{u}^{\prime }\left(0\right).$$

Lemma 2.2. $\left(\left[10\right]\right)$ Let $0<\beta <1<\alpha <2$, then we have

$$I_{0^{+}}^{\alpha }\left({}^c{D}_{0^{+}}^{\beta }u\left(t\right)\right)=I_{0^{+}}^{\alpha -\beta }u\left(t\right)-\frac{u\left(0\right)t^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}.$$

2.2 Existence results

Let $C\left(J,{\mathbb{R}}^n\right)$ be the Banach space for all continuous function from $J$ into ${\mathbb{R}}^n$ equipped with the norm

$${‖u‖}_{\infty }=sup\left\{‖u\left(t\right)‖:t\in J\right\},$$

where $‖\cdot ‖$ denotes a suitable complete norm on ${\mathbb{R}}^n$. Denote $L^1\left(J,{\mathbb{R}}^n\right)$ the Banach space of the measurable functions $u:J\to {\mathbb{R}}^n$ that are Lebesgue integrable with norm

$${‖u‖}_{L^1}={\int }_0^T‖u\left(t\right)‖\mathit{dt}.$$

Let $\mathit{AC}\left(J,{\mathbb{R}}^n\right)$ be the Banach space of absolutely continuous valued functions on $J$ and set

$${\mathit{AC}}^n\left(J\right)=\left\{u:J\to {\mathbb{R}}^n:u,u^{\prime },{u^{\prime }}^{\prime },\dots ,u^{\left(n-1\right)}\in C\left(J,{\mathbb{R}}^n\right)\right\}\mathit{\text{and}}{u}^{\left(n-1\right)}\in \mathit{AC}\left(J,{\mathbb{R}}^n\right).$$

By

$$C^1\left(J\right)=\left\{u:J\to {\mathbb{R}}^n\mathit{\text{where}}{u}^{\prime }\in C\left(J,{\mathbb{R}}^n\right)\right\},$$

we denote the Banach space equipped with the norm

$${‖u‖}_1=max\left\{{‖u‖}_{\infty },{‖u^{\prime }‖}_{\infty }\right\}.$$

For the sake of brevity, we set

$$\begin{array}{l}\delta =\left(\lambda -\mu -\gamma \right)\mathrm{\Gamma }\left(\alpha -\beta +1\right)+A\left(\mu T^{\alpha -\beta }+{\mathit{\gamma \eta }}^{\alpha -\beta }\right),\mathrm{\Delta }=\frac{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{\delta }\\ \sigma =A\left(\alpha -\beta \right)\left(\mu T^{\alpha -\beta -1}+{\mathit{\gamma \eta }}^{\alpha -\beta -1}\right),\mathrm{\Lambda }=\left(\lambda -\mu -\gamma \right)-\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\left(\frac{\sigma }{\delta }\right),\\ R_1=\left(1+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)M_1+{\mathit{TM}}_2+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\\ R_2=M_2+\frac{\left(\alpha -\beta \right)‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{M}_1+\frac{‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\\ M_1=\mathrm{\Delta }\left\{\left[{\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right]\mathrm{\Phi }+{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\mathrm{\Theta }\right\}\end{array},$$

and

$$M_2={\mathrm{\Lambda }}^{-1}\left\{\left(\frac{\sigma }{\delta }\right)\mathrm{\Phi }+\mathrm{\Theta }\right\},$$

with

$$\mathrm{\Phi }=\frac{‖A‖}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\left(\mu T^{\alpha -\beta }+{\mathit{\gamma \eta }}^{\alpha -\beta }\right)+\frac{\left(\mu T^{\alpha }+{\mathit{\gamma \eta }}^{\alpha }\right)\left(L_1\mathrm{\Gamma }\left(2-\beta \right)+T^{1-\beta }\left(L_3‖A‖+L_2\right)\right)}{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(2-\beta \right)\left(1-L_3\right)},$$

$$\mathrm{\Theta }=\frac{‖A‖}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\left(\mu T^{\alpha -\beta -1}+{\mathit{\gamma \eta }}^{\alpha -\beta -1}\right)+\frac{\left(\mu T^{\alpha -1}+{\mathit{\gamma \eta }}^{\alpha -1}\right)\left(L_1\mathrm{\Gamma }\left(2-\beta \right)+T^{1-\beta }\left(L_3‖A‖+L_2\right)\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(2-\beta \right)\left(1-L_3\right)}.$$

Lemma 2.3. Let $y\left(\cdot \right)\in C\left(J,{\mathbb{R}}^n\right)$. The function $u\left(\cdot \right)\in C^1\left(J,{\mathbb{R}}^n\right)$ is a solution of the fractional differential problem

$$\left\{\begin{array}{c}\begin{array}{c}{}^c{D}_{0^{+}}^{\alpha }u\left(t\right)-A^c{D}_{0^{+}}^{\beta }u\left(t\right)=y\left(t\right),t\in J=\left[0,T\right],\\ \mathit{\lambda u}\left(0\right)-\mathit{\mu u}\left(T\right)-\mathit{\gamma u}\left(\eta \right)=d,\\ \lambda u^{\prime }\left(0\right)-\mu u^{\prime }\left(T\right)-\gamma u^{\prime }\left(\eta \right)=l,\end{array}\end{array}\right.$$E3

if and only if, $u$ is a solution of the fractional integral equation

$$\begin{array}{c}u\left(t\right)=\left(1-\frac{{\mathit{At}}^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)u\left(0\right)+t{u}^{\prime }\left(0\right)+\frac{A}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -\beta -1}u\left(s\right)\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}y\left(s\right)\mathit{ds},\end{array}$$E4

with

$$\begin{array}{l}u\left(0\right)=\mathrm{\Delta }\left\{\left[{\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right]\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta }\left(\mathit{\mu u}\left(T\right)+\mathit{\gamma u}\left(\eta \right)\right)+I_{0^{+}}^{\alpha }\left(\mathit{\mu y}\left(T\right)+\mathit{\gamma y}\left(\eta \right)\right)+d\right]\right.\\ \left.+{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta -1}\left(\mathit{\mu u}\left(T\right)+\mathit{\gamma u}\left(\eta \right)\right)+I_{0^{+}}^{\alpha -1}\left(\mathit{\mu y}\left(T\right)+\mathit{\gamma y}\left(\eta \right)\right)+l\right]\right\},\end{array}$$E5

and

$$\begin{array}{l}{u}^{\prime }\left(0\right)={\mathrm{\Lambda }}^{-1}\left\{\left(\frac{\sigma }{\delta }\right)\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta }\left(\mathit{\mu u}\left(T\right)+\mathit{\gamma u}\left(\eta \right)\right)+I_{0^{+}}^{\alpha }\left(\mathit{\mu y}\left(T\right)+\mathit{\gamma y}\left(\eta \right)\right)+d\right]\right.\\ \left. +\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta -1}\left(\mathit{\mu u}\left(T\right)+\mathit{\gamma u}\left(\eta \right)\right)+I_{0^{+}}^{\alpha -1}\left(\mathit{\mu y}\left(T\right)+\mathit{\gamma y}\left(\eta \right)\right)\right]+l\right\},\end{array}$$E6

where

$$\begin{array}{l}{I}_{0^{+}}^{\alpha }u\left(T\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -1}u\left(s\right)\mathit{ds},\\ I_{0^{+}}^{\alpha }u\left(\eta \right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{\eta }{\int }}{\left(\eta -s\right)}^{\alpha -1}u\left(s\right)\mathit{ds},\\ I_{0^{+}}^{\alpha -\beta }u\left(T\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -\beta -1}u\left(s\right)\mathit{ds},\\ I_{0^{+}}^{\alpha -\beta }u\left(\eta \right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{0}{\overset{\eta }{\int }}{\left(\eta -s\right)}^{\alpha -\beta -1}u\left(s\right)\mathit{ds}.\end{array}$$

Proof. From Lemmas 2.1 and 2.2, we have

$$\begin{array}{c}u\left(t\right)=u\left(0\right)+t{u}^{\prime }\left(0\right)-\frac{{\mathit{At}}^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}u\left(0\right)+\frac{A}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -\beta -1}u\left(s\right)\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}y\left(s\right)\mathit{ds}\end{array}$$

Applying conditions (2), we obtain (5) and (6).

Conversely, assume that $u$ satisfies the fractional integral (4), and using the facts that ${}^c{D}_{0^{+}}^{\alpha }$ is the left inverse of $I_{0^{+}}^{\alpha }$ and the fact that ${}^c{D}_{0^{+}}^{\alpha }C=0$, where $C$ is a constant, we get

$${}^c{D}_{0^{+}}^{\alpha }u\left(t\right)-A^c{D}_{0^{+}}^{\beta }u\left(t\right)=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),t\in J=\left[0,T\right].$$

Also, we can easily show that

$$\left\{\begin{array}{c}\begin{array}{c}\mathit{\lambda u}\left(0\right)-\mathit{\mu u}\left(T\right)-\mathit{\gamma u}\left(\eta \right)=d,\\ \lambda u^{\prime }\left(0\right)-\mu u^{\prime }\left(T\right)-\gamma u^{\prime }\left(\eta \right)=l.\end{array}\end{array}\right.$$

The proof is complete.

To simplify the proofs in the forthcoming theorem, we etablish the bounds for the integrals and the bounds for the term arising in the sequel.

Lemma 2.4. For $y\left(\cdot \right)\in C\left(J,{\mathbb{R}}^n\right)$, we have

$$\left|I_{0^{+}}^{\alpha }y\left(\eta \right)\right|=\left|\underset{0}{\overset{\eta }{\int }}\frac{{\left(\eta -\tau \right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}y\left(\tau \right)\mathit{d\tau }\right|\le \frac{{\eta }^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}‖y‖.$$

Proof. Obviously,

$$\underset{0}{\overset{\eta }{\int }}\frac{{\left(\eta -\tau \right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}y\left(\tau \right)\mathit{d\tau }={\left[-\frac{{\left(\eta -\tau \right)}^{\alpha }}{\alpha \mathrm{\Gamma }\left(\alpha \right)}\right]}_0^{\eta }=\frac{{\eta }^{\alpha }}{\alpha \mathrm{\Gamma }\left(\alpha \right)}=\frac{s^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}.$$

Hence

$$\left|\underset{0}{\overset{\eta }{\int }}\frac{{\left(s-\tau \right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}y\left(\tau \right)\mathit{d\tau }\right|\le \frac{{\eta }^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}‖y‖.$$

Lemma 2.5. For $u\left(\cdot \right)\in C^1\left(J,{\mathbb{R}}^n\right)$ and $0<\beta \le 1$, we have

$${‖{}^c{D}_{0^{+}}^{\beta }u\left(t\right)‖}_{\infty }\le \frac{T^{1-\beta }}{\mathrm{\Gamma }\left(2-\beta \right)}{‖u^{\prime }‖}_{\infty },$$

and, so

$${‖{}^c{D}_{0^{+}}^{\beta }u\left(t\right)‖}_{\infty }\le \frac{T^{1-\beta }}{\mathrm{\Gamma }\left(2-\beta \right)}{‖u‖}_1.$$

Proof.

Clearly, when $\beta =1$, the conclusion are true. So, consider the case $0<\beta <1$. By Definition 2.1, for each $u\left(\cdot \right)\in C^1\left(J,{\mathbb{R}}^n\right)$ and $t\in J$, we have

$$\begin{array}{l}\left|D_{0^{+}}^{\beta }u\left(t\right)\right|=\frac{1}{\mathrm{\Gamma }\left(1-\beta \right)}\left|\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{-\beta }{u}^{\prime }\left(s\right)\mathit{ds}\right|\\ \le {‖u^{\prime }‖}_{\infty }\frac{1}{\mathrm{\Gamma }\left(1-\beta \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{-\beta }\mathit{ds}\\ ={‖u^{\prime }‖}_{\infty }\frac{t^{1-\beta }}{\mathrm{\Gamma }\left(1-\beta \right)}\\ \le \frac{T^{1-\beta }}{\mathrm{\Gamma }\left(1-\beta \right)}{‖u^{\prime }‖}_{\infty }\\ \le \frac{T^{1-\beta }}{\mathrm{\Gamma }\left(1-\beta \right)}{‖u^{\prime }‖}_1.\end{array}$$

We need to give the following hypothesis:

$\left(H_1\right)$ there existe a constants $L_1,L_2>0$ and $0<L_3<1$ such that

$$\left|f\left(t,u,v,w\right)-f\left(t,\overline{u},\overline{v},\overline{w}\right)\right|\le L_1‖u-\overline{u}‖+L_2‖v-\overline{v}‖+L_3‖w-\overline{w}‖,$$

for any $u,v,\overline{u},\overline{v},\overline{w}\in {\mathbb{R}}^n$ and $t\in J$.

Now we are in a position to present the first main result of this paper. The existence results is based on Banach contraction principle.

Theorem 1.1. ([47] Banach’s fixed point theorem). Let $C$ be a non-empty closed subset of a Banach space $E$, then any contraction mapping $T$ of $C$ into itself has a unique fixed point.

Theorem 1.2. Assume that $\left(H_1\right)$ holds. If

$$max\left(R_1,R_2\right)<1,$$E7

then the boundary value problem (1.1)–(1.2) has a unique solution on $J$.

Proof. We transform the problem (1.1)–(1.2) into fixed point problem. Let $N:C^1\left(J,{\mathbb{R}}^n\right)\to C^1\left(J,{\mathbb{R}}^n\right)$ the operator defined by

$$\begin{array}{l}\left(\mathit{Nu}\right)\left(t\right)=\left(1-\frac{{\mathit{At}}^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)B+\mathit{tD}+\frac{A}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -\beta -1}u\left(s\right)\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g\left(s\right)\mathit{ds},\end{array}$$E8

with

$$\begin{array}{l}B=\mathrm{\Delta }\left\{\left[{\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right]\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta }\left(\mathit{\mu u}\left(T\right)+\mathit{\gamma u}\left(\eta \right)\right)+I_{0^{+}}^{\alpha }\left(\mathit{\mu y}\left(T\right)+\mathit{\gamma y}\left(\eta \right)\right)+d\right]\right.\\ \left.+{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta -1}\left(\mathit{\mu u}\left(T\right)+\mathit{\gamma u}\left(\eta \right)\right)+I_{0^{+}}^{\alpha -1}\left(\mathit{\mu y}\left(T\right)+\mathit{\gamma y}\left(\eta \right)\right)+l\right]\right\},\end{array}$$

and

$$\begin{array}{l}D={\mathrm{\Lambda }}^{-1}\left\{\left(\frac{\sigma }{\delta }\right)\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta }\left(\mathit{\mu u}\left(T\right)+\mathit{\gamma u}\left(\eta \right)\right)+I_{0^{+}}^{\alpha }\left(\mathit{\mu g}\left(T\right)+\mathit{\gamma g}\left(\eta \right)\right)+d\right]\right.\\ \left.+\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta -1}\left(\mathit{\mu u}\left(T\right)+\mathit{\gamma u}\left(\eta \right)\right)+I_{0^{+}}^{\alpha -1}\left(\mathit{\mu g}\left(T\right)+\mathit{\gamma g}\left(\eta \right)\right)\right]+l\right\},\end{array}$$

where $g\in C\left(J,{\mathbb{R}}^n\right)$ be such that

$$g\left(t\right)=f\left(t,u\left(t\right){,}^c{D}_{0^{+}}^{\beta }u\left(t\right),g\left(t\right)+A^c{D}_{0^{+}}^{\beta }u\left(t\right)\right)$$

For every $u\in C^1\left(J,{\mathbb{R}}^n\right)$ and any $t\in J$, we have

$$\begin{array}{c}\left(\mathit{Nu}\right)\left(t\right)=D-\frac{\left(\alpha -\beta \right){\mathit{At}}^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}B+\frac{A}{\mathrm{\Gamma }\left(\alpha -\beta -1\right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -\beta -2}u\left(s\right)\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -2}g\left(s\right)\mathit{ds}.\end{array}$$E9

Clearly, the fixed points of operator $N$ are solutions of problem (1.1)–(1.2).

It is clear that $\left(\mathit{Nu}\right)\in C\left(J,{\mathbb{R}}^n\right)$, consequently, $N$ is well defined.

Let $u,v\in C\left(J,{\mathbb{R}}^n\right)$. Then for $t\in J$, we have

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t\right)-\left(\mathit{Nv}\right)\left(t\right)‖\le \left(1+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)‖B-B_1‖+T‖D-D_1‖\\ +\frac{‖A‖}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -\beta -1}‖u\left(s\right)-v\left(s\right)‖\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -1}‖g\left(s\right)-h\left(s\right)‖\mathit{ds},\end{array}$$

with

$$\begin{array}{l}{B}_1=\mathrm{\Delta }\left\{\left[{\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right]\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta }\left(\mathit{\mu v}\left(T\right)+\mathit{\gamma v}\left(\eta \right)\right)+I_{0^{+}}^{\alpha }\left(\mathit{\mu h}\left(T\right)+\mathit{\gamma h}\left(\eta \right)\right)+d\right]\right.\\ \left. +{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta -1}\left(\mathit{\mu v}\left(T\right)+\mathit{\gamma v}\left(\eta \right)\right)+I_{0^{+}}^{\alpha -1}\left(\mathit{\mu h}\left(T\right)+\mathit{\gamma h}\left(\eta \right)\right)+l\right]\right\},\end{array}$$

and

$$\begin{array}{l}{D}_1={\mathrm{\Lambda }}^{-1}\left\{\left(\frac{\sigma }{\delta }\right)\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta }\left(\mathit{\mu v}\left(T\right)+\mathit{\gamma v}\left(\eta \right)\right)+I_{0^{+}}^{\alpha }\left(\mathit{\mu h}\left(T\right)+\mathit{\gamma h}\left(\eta \right)\right)+d\right]\right.\\ \left.+\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta -1}\left(\mathit{\mu v}\left(T\right)+\mathit{\gamma v}\left(\eta \right)\right)+I_{0^{+}}^{\alpha -1}\left(\mathit{\mu h}\left(T\right)+\mathit{\gamma h}\left(\eta \right)\right)\right]+l\right\},\end{array}$$

From $\left(H\right)$, for any $t\in J$, we have

$$\begin{array}{c}‖g\left(t\right)-h\left(t\right)‖=L_1‖u\left(t\right)-v\left(t\right)‖+L_2‖{}^c{D}_{0^{+}}^{\beta }u\left(t\right){-}^c{D}_{0^{+}}^{\beta }v\left(t\right)‖\\ +L_3‖g\left(t\right)+A^c{D}_{0^{+}}^{\beta }u\left(t\right)-h\left(t\right)-A^c{D}_{0^{+}}^{\beta }v\left(t\right)‖\\ \le L_1‖u\left(t\right)-v\left(t\right)‖+L_2‖{}^c{D}_{0^{+}}^{\beta }u\left(t\right){-}^c{D}_{0^{+}}^{\beta }v\left(t\right)‖\\ +L_3‖g\left(t\right)-h\left(t\right)‖+L_3‖A‖‖{}^c{D}_{0^{+}}^{\beta }u\left(t\right){-}^c{D}_{0^{+}}^{\beta }v\left(t\right)‖\\ \le L_1‖u\left(t\right)-v\left(t\right)‖+L_3‖g\left(t\right)-h\left(t\right)‖+\left(L_3‖A‖+L_2\right)‖{}^c{D}_{0^{+}}^{\beta }\left(u\left(t\right)-v\left(t\right)\right)‖.\end{array}$$

Thus

$$\begin{array}{c}‖g\left(t\right)-h\left(t\right)‖\le \frac{L_1}{1-L_3}‖u\left(t\right)-v\left(t\right)‖+\frac{L_3‖A‖+L_2}{1-L_3}‖{}^c{D}_{0^{+}}^{\beta }\left(u\left(t\right)-v\left(t\right)\right)‖\\ \le \frac{L_1}{1-L_3}{‖u-v‖}_{\infty }+\frac{L_3‖A‖+L_2}{1-L_3}{‖{}^c{D}_{0^{+}}^{\beta }\left(u-v\right)‖}_{\infty }.\end{array}$$

Then, according to the Lemma 3.2, we get

$$\begin{array}{c}‖g\left(t\right)-h\left(t\right)‖\le \frac{L_1}{1-L_3}{‖u-v‖}_1+\frac{T^{1-\beta }\left(L_3‖A‖+L_2\right)}{\mathrm{\Gamma }\left(2-\beta \right)\left(1-L_3\right)}{‖u-v‖}_1\\ =\frac{L_1\mathrm{\Gamma }\left(2-\beta \right)+T^{1-\beta }\left(L_3‖A‖+L_2\right)}{\mathrm{\Gamma }\left(2-\beta \right)\left(1-L_3\right)}{‖u-v‖}_1.\end{array}$$E10

By employing (10) and Lemma 3.1, we get

$$\begin{array}{l}‖B_1-B_2‖\le \mathrm{\Delta }\left\{\left[{\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right]\mathrm{\Phi }+{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\mathrm{\Theta }\right\}{‖u-v‖}_1\\ =M_1{‖u-v‖}_1.\end{array}$$

and

$$\begin{array}{c}‖D_1-D_2‖\le {\mathrm{\Lambda }}^{-1}\left\{\left(\frac{\sigma }{\delta }\right)\mathrm{\Phi }+\mathrm{\Theta }\right\}{‖u-v‖}_1\\ =M_2{‖u-v‖}_1,\end{array}$$

where $\mathrm{\Phi }$ and $\mathrm{\Theta }$ defined above.

Thus, for $t\in J$, we have

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t\right)-\left(\mathit{Nv}\right)\left(t\right)‖\le \left[\left(1+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)M_1+{\mathit{TM}}_2+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right.\\ +\left.\frac{T^{\alpha }{L}_1\mathrm{\Gamma }\left(2-\beta \right)+T^{1-\beta +\alpha }\left(L_3‖A‖+L_2\right)}{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(2-\beta \right)\left(1-L_2\right)}\right]{‖u-v‖}_1\\ =R_1{‖u-v‖}_1.\end{array}$$

Also

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t\right)-\left(\mathit{Nv}\right)\left(t\right)‖\le ‖D_2-D_1‖+\frac{\left(\alpha -\beta \right)‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}‖B_1-B_2‖\\ +\frac{‖A‖}{\mathrm{\Gamma }\left(\alpha -\beta -1\right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -\beta -2}‖u\left(s\right)-v\left(s\right)‖\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -2}‖g\left(s\right)-h\left(s\right)‖\mathit{ds}.\end{array}$$

By employing (10) and Lemma 3.2, we get

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t\right)-\left(\mathit{Nv}\right)\left(t\right)‖\le \left[M_2+\frac{\left(\alpha -\beta \right)‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{M}_1+\frac{‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\right.\\ +\left.\frac{T^{\alpha -1}{L}_1\mathrm{\Gamma }\left(2-\beta \right)+T^{\alpha -\beta }\left(L_3‖A‖+L_2\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(2-\beta \right)\left(1-L_2\right)}\right]{‖u-v‖}_1\\ =R_2{‖u-v‖}_1.\end{array}$$

Therefore

$$‖\left(\mathit{Nu}\right)\left(t\right)-\left(\mathit{Nv}\right)\left(t\right)‖\le max\left\{R_1,R_2\right\}{‖u-v‖}_1.$$

Thus, by (10) the operator $N$ is a contraction. Hence it follows by Banach’s contraction principle that the boundary value problem (1)–(12) has a unique solution on $J$.

Now we are in a position to present the second main result of this paper. The existence results is based on Leray-Schauder nonlinear alternative.

Theorem 1.3. ([6] Nonlinear alternative for single valued maps). Let $E$ be a Banach space, $C$ a closed, convex subset of $E$ and $U$ an open subset of $C$ with $0\in U$. Suppose that $F:\overline{U}\to C$ is a continuous and compact (that is $F\left(\overline{U}\right)$ is relatively compact subset of $C$) map. Then either

1. $F$ has a fixed point in $\overline{U}$, or

2. there is a $u\in \partial U$ (the boundary of $U$ in $C$) and $\lambda \in \left(0,1\right)$ with $u=\mathit{\lambda F}\left(u\right)$.

Set

$$l_1=M_3+{\mathit{TM}}_4+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{M}_3+{\mathit{TM}}_4+\frac{‖A‖{\mathit{rT}}^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}+\frac{T^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}M,$$

and

$$l_2=M_4+\frac{\left(\alpha -\beta \right)‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{M}_3+\frac{‖A‖{\mathit{rT}}^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta \right)}+\frac{T^{\alpha }M}{\mathrm{\Gamma }\left(\alpha \right)}.$$

Theorem 1.4. Assume that $\left(H_1\right)$ holds and there exists a positive constant $M>0$ such that $max\left\{l_1,l_2\right\}=l<M$. Then the boundary value problem (1.1)–(1.2) has at least one solution on $J$.

Proof. Let $N$ be the operator defined in (8).

$N$ is continuous. Let $\left(u_n\right)$ be a sequence such that $u_n\to u$ in $C\left(J,{\mathbb{R}}^n\right)$. Then for $t\in J$, we have

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t\right)-\left({\mathit{Nu}}_n\right)\left(t\right)‖\le \left(1+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)‖B_1-B_{n2}‖+T‖D_1-D_{n2}‖\\ +\frac{‖A‖}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -\beta -1}‖u\left(s\right)-u_n\left(s\right)‖\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -1}‖g\left(s\right)-g_n\left(s\right)‖\mathit{ds},\end{array}$$

where $B_{n2},D_{n2}\in {\mathbb{R}}^n$, with

$$\begin{array}{c}{B}_{n2}=\mathrm{\Delta }\left\{\left[{\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right]\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta }\left(\mu u_n\left(T\right)+\gamma u_n\left(\eta \right)\right)+I_{0^{+}}^{\alpha }\left(\mu g_n\left(T\right)+\gamma g_n\left(\eta \right)\right)+d\right]\right.\\ \left.+{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta -1}\left(\mu u_n\left(T\right)+\gamma u_n\left(\eta \right)\right)+I_{0^{+}}^{\alpha -1}\left(\mu g_n\left(T\right)+\gamma g_n\left(\eta \right)\right)+l\right]\right\},\\ D_{n2}={\mathrm{\Lambda }}^{-1}\left\{\left(\frac{\sigma }{\delta }\right)\left[{\mathit{AI}}_{0^{+}}^{\alpha -\beta }\left(\mu u_n\left(T\right)+\gamma u_n\left(\eta \right)\right)+I_{0^{+}}^{\alpha }\left(\mu g_n\left(T\right)+\gamma g_n\left(\eta \right)\right)+d\right]\right.\\ \left.+{\mathit{AI}}_{0^{+}}^{\alpha -\beta -1}\left(\mu u_n\left(T\right)+\gamma u_n\left(\eta \right)\right)+I_{0^{+}}^{\alpha -1}\left(\mu g_n\left(T\right)+\gamma g_n\left(\eta \right)\right)+l\right\},\end{array}$$

and

$$g_n\left(t\right)=f\left(t,u_n\left(t\right){,}^c{D}_{0^{+}}^{\beta }{u}_n\left(t\right),g_n\left(t\right)+A^c{D}_{0^{+}}^{\beta }{u}_n\left(t\right)\right).$$

From $\left(H\right)$, for any $t\in J$, we have

$$\begin{array}{c}‖g\left(t\right)-g_n\left(t\right)‖=L_1‖u\left(t\right)-u_n\left(t\right)‖+L_3‖g\left(t\right)+A^c{D}_{0^{+}}^{\beta }u\left(t\right)-g_n\left(t\right)-A^c{D}_{0^{+}}^{\beta }{u}_n\left(t\right)‖\\ +L_2‖{}^c{D}_{0^{+}}^{\beta }u\left(t\right){-}^c{D}_{0^{+}}^{\beta }{u}_n\left(t\right)‖\\ \le L_1‖u\left(t\right)-u_n\left(t\right)‖+L_2‖{}^c{D}_{0^{+}}^{\beta }u\left(t\right){-}^c{D}_{0^{+}}^{\beta }{u}_n\left(t\right)‖\\ +L_3‖g\left(t\right)-g_n\left(t\right)‖+L_3‖A‖‖{}^c{D}_{0^{+}}^{\beta }u\left(t\right){-}^c{D}_{0^{+}}^{\beta }{u}_n\left(t\right)‖\\ \le L_1‖u\left(t\right)-u_n\left(t\right)‖+L_3‖g\left(t\right)-g_n\left(t\right)‖+\left(L_3‖A‖+L_2\right)‖{}^c{D}_{0^{+}}^{\beta }\left(u\left(t\right)-u_n\left(t\right)\right)‖.\end{array}$$

Thus

$$\begin{array}{c}‖g\left(t\right)-g_n\left(t\right)‖\le \frac{L_1}{1-L_3}‖u\left(t\right)-u_n\left(t\right)‖+\frac{L_3‖A‖+L_2}{1-L_3}‖{}^c{D}_{0^{+}}^{\beta }\left(u\left(t\right)-u_n\left(t\right)\right)‖\\ \le \frac{L_1}{1-L_3}{‖u-u_n‖}_{\infty }+\frac{L_3‖A‖+L_2}{1-L_3}{‖{}^c{D}_{0^{+}}^{\beta }\left(u-u_n\right)‖}_{\infty }.\end{array}$$

Then, according to the Lemma 3.2, we get

$$\begin{array}{c}‖g\left(t\right)-g_n\left(t\right)‖\le \frac{L_1}{1-L_3}{‖u-u_n‖}_1+\frac{T^{1-\beta }\left(L_3‖A‖+L_2\right)}{\left(1-L_3\right)\mathrm{\Gamma }\left(2-\beta \right)}{‖u-u_n‖}_1\\ =\frac{L_1\mathrm{\Gamma }\left(2-\beta \right)+T^{1-\beta }\left(L_3‖A‖+L_2\right)}{\left(1-L_3\right)\mathrm{\Gamma }\left(2-\beta \right)}{‖u-u_n‖}_1.\end{array}$$

By employing (10) and Lemma 3.1, we get

$$\begin{array}{c}‖B_1-B_{n2}‖\le \mathrm{\Delta }\left\{\left[{\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right]\mathrm{\Phi }+{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\mathrm{\Theta }\right\}{‖u-u_n‖}_1,\\ =M_1{‖u-u_n‖}_1.\end{array}$$

and

$$\begin{array}{c}‖D_1-D_{n2}‖\le {\mathrm{\Lambda }}^{-1}\left\{\left(\frac{\sigma }{\delta }\right)\mathrm{\Phi }+\mathrm{\Theta }\right\}{‖u-u_n‖}_1\\ =M_2{‖u-u_n‖}_1,\end{array}$$

Thus, for $t\in J$, we have

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t\right)-\left({\mathit{Nu}}_n\right)\left(t\right)‖\le \left[\left(1+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)M_1+{\mathit{TM}}_2+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right.\\ +\left.\frac{T^{\alpha }{L}_1\mathrm{\Gamma }\left(2-\beta \right)+T^{1-\beta +\alpha }\left(L_3‖A‖+L_2\right)}{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(2-\beta \right)\left(1-L_2\right)}\right]{‖u-u_n‖}_1\\ =R_1{‖u-u_n‖}_1.\end{array}$$

Also

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t\right)-\left({\mathit{Nu}}_n\right)\left(t\right)‖\le ‖D_{n2}-D_1‖+\frac{\left(\alpha -\beta \right)‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}‖B_1-B_{n2}‖\\ +\frac{‖A‖}{\mathrm{\Gamma }\left(\alpha -\beta -1\right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -\beta -2}‖u\left(s\right)-u_n\left(s\right)‖\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}\underset{0}{\overset{T}{\int }}{\left(T-s\right)}^{\alpha -2}‖g\left(s\right)-g_n\left(s\right)‖\mathit{ds}.\end{array}$$

By employing (10), we get

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t\right)-\left({\mathit{Nu}}_n\right)\left(t\right)‖\le \left[M_2+\frac{\left(\alpha -\beta \right)‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{M}_1+\frac{‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\right.\\ +\left.\frac{T^{\alpha -1}{L}_1\mathrm{\Gamma }\left(2-\beta \right)\left(L_3‖A‖+L_2\right)T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{M}_1+\frac{‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\right]{‖u-u_n‖}_1.\\ =R_2{‖u-u_n‖}_1.\end{array}$$

Thus ${‖\mathit{Nu}-{\mathit{Nu}}_n‖}_1\to 0$ as $n\to \infty$, which implies that the operator $N$ is continuous.

Now, we show $N$ maps bounded sets into bounded sets in $C\left(J,{\mathbb{R}}^n\right)$. For a positive number $r$, let $B_r=\left\{u\in C^1\left(J,{\mathbb{R}}^n\right):{‖u‖}_1\le r\right\}$ be a bounded set in $C\left(J,{\mathbb{R}}^n\right)$. Then we have

$$\begin{array}{c}‖g\left(t\right)‖\le ‖f\left(t,u\left(t\right),g\left(t\right)+A^c{D}_{0^{+}}^{\beta }u\left(t\right),D_{0^{+}}^{\beta }u\left(t\right)\right)-f\left(t\mathrm{,0,0,0}\right)‖+‖f\left(t\mathrm{,0,0,0}\right)‖\\ \le L_1‖u\left(t\right)‖+L_3‖g\left(t\right)+A^c{D}_{0^{+}}^{\beta }u\left(t\right)‖+L_2‖D_{0^{+}}^{\beta }u\left(t\right)‖+‖f\left(t\mathrm{,0,0,0}\right)‖\\ \le L_1{‖u‖}_{\infty }+L_3‖g\left(t\right)‖+\left(L_3‖A‖+L_2\right){‖D_{0^{+}}^{\beta }u‖}_{\infty }+f^{\ast },\end{array}$$

where $\underset{t\in J}{sup}\left|f\left(t\mathrm{,0,0,0}\right)\right|=f^{\ast }<\infty$. Thus

$$‖g\left(t\right)‖\le \frac{L_1}{1-L_3}{‖u‖}_{\infty }+\frac{L_3‖A‖+L_2}{1-L_3}{‖D_{0^{+}}^{\beta }u‖}_{\infty }+\frac{f^{\ast }}{1-L_3}.$$

Then, By Lemma 3.2, we have

$$\begin{array}{c}‖g\left(t\right)‖\le \frac{L_1}{1-L_3}{‖u‖}_{\infty }+\frac{\left(L_3‖A‖+L_2\right)T^{1-\beta }}{\left(1-L_3\right)\mathrm{\Gamma }\left(2-\beta \right)}{‖u^{\prime }‖}_{\infty }+\frac{f^{\ast }}{1-L_3}\\ \le \frac{L_1}{1-L_3}{‖u‖}_1+\frac{\left(L_3‖A‖+L_2\right)T^{1-\beta }}{\left(1-L_3\right)\mathrm{\Gamma }\left(2-\beta \right)}{‖u‖}_1+\frac{f^{\ast }}{1-L_3}\\ \le \frac{L_{1}r}{1-L_3}+\frac{\left(L_3‖A‖+L_2\right){\mathit{rT}}^{1-\beta }}{\left(1-L_3\right)\mathrm{\Gamma }\left(2-\beta \right)}+\frac{f^{\ast }}{1-L_3}=M,\end{array}$$E11

which implies that

$$\begin{array}{l}‖B‖\le r‖A‖\mathrm{\Delta }\left[\left({\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right)\left(\frac{\mu T^{\alpha -\beta }+{\mathit{\gamma \eta }}^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)\right.\\ \left.+{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\left(\frac{\mu T^{\alpha -\beta -1}+{\mathit{\gamma \eta }}^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\right)\right]\\ +M\mathrm{\Delta }\left[\left({\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)+1\right)\left(\frac{\mu T^{\alpha }+{\mathit{\gamma \eta }}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)+{\mathrm{\Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\left(\frac{\mu T^{\alpha -1}+{\mathit{\gamma \eta }}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\right)\right]\\ +{\mathrm{\Delta \Lambda }}^{-1}\left(\mathit{\mu T}+\mathit{\gamma \eta }\right)\left[l+d\left(\left(\frac{\sigma }{\delta }\right)+1\right)\right]=M_3,\end{array}$$

and

$$\begin{array}{c}‖D‖\le r‖A‖\left[{\mathrm{\Lambda }}^{-1}\left(\frac{\sigma }{\delta }\right)\frac{\mu T^{\alpha -\beta }+{\mathit{\gamma \eta }}^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}+\frac{\mu T^{\alpha -\beta -1}+{\mathit{\gamma \eta }}^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\right]\\ +M{\mathrm{\Lambda }}^{-1}\left[\left(\frac{\sigma }{\delta }\right)\left(\frac{\mu T^{\alpha }+{\mathit{\gamma \eta }}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)+\left(\frac{\mu T^{\alpha -1}+{\mathit{\gamma \eta }}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\right)\right]+{\mathrm{\Lambda }}^{-1}\left[\left(\frac{\sigma }{\delta }\right)d+l\right]=M_4.\end{array}$$

Thus (8) implies

$$‖\left(\mathit{Nu}\right)\left(t\right)‖\le M_3+\frac{‖A‖T^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{M}_3+{\mathit{TM}}_4+\frac{‖A‖{\mathit{rT}}^{\alpha -\beta }}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}+\frac{T^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}M=l_1,$$

and

$$‖\left(\mathit{Nu}\right)\left(t\right)‖\le M_4+\frac{\left(\alpha -\beta \right)‖A‖T^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}{M}_3+\frac{‖A‖{\mathit{rT}}^{\alpha -\beta -1}}{\mathrm{\Gamma }\left(\alpha -\beta \right)}+\frac{T^{\alpha }M}{\mathrm{\Gamma }\left(\alpha \right)}=l_2.$$

Therefore

$${‖\left(\mathit{Nu}\right)‖}_1\le max\left\{l_1,l_2\right\}=l.$$E12

Now, we show that $N$ maps bounded sets into equicontinuous sets of $C^1\left(J,{\mathbb{R}}^n\right)$. Let $t_1,t_2\in \left[0,1\right]$ with $t_1<t_2$ and $u\in B_r$ is bounded sets of $C^1\left(J,{\mathbb{R}}^n\right)$. Then

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t_2\right)-\left(\mathit{Nu}\right)\left(t_1\right)‖\le M_4\left(t_2-t_1\right)+\left(1+\frac{‖A‖M_3}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\right)\left(t_2^{\alpha -\beta }-t_1^{\alpha -\beta }\right)\\ +\frac{‖A‖r}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{t_1}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\alpha -\beta -1}\mathit{ds}+\frac{‖A‖r}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\underset{0}{\overset{t_1}{\int }}\left[{\left(t_2-s\right)}^{\alpha -\beta -1}-{\left(t_1-s\right)}^{\alpha -\beta -1}\right]\mathit{ds}\\ +\frac{M_1}{\mathrm{\Gamma }\left(\alpha \right)}\left[\underset{t_1}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\alpha -1}\mathit{ds}+\underset{0}{\overset{t_1}{\int }}\left[{\left(t_2-s\right)}^{\alpha -1}-{\left(t_1-s\right)}^{\alpha -1}\right]\mathit{ds}\right]\end{array}$$

Obviously, the right-hand side of the above inequality tends to zero as $t_2\to t_1$.

Similarly, we have

$$\begin{array}{c}‖\left(\mathit{Nu}\right)\left(t_2\right)-\left(\mathit{Nu}\right)\left(t_1\right)‖\le \frac{\left(\alpha -\beta \right)‖A‖M_3}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\left(t_2^{\alpha -\beta -1}-t_1^{\alpha -\beta -1}\right)\\ +\frac{‖A‖r}{\mathrm{\Gamma }\left(\alpha -\beta -1\right)}\underset{t_1}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\alpha -\beta -2}\mathit{ds}+\frac{‖A‖r}{\mathrm{\Gamma }\left(\alpha -\beta -2\right)}\underset{0}{\overset{t_1}{\int }}\left[{\left(t_2-s\right)}^{\alpha -\beta -2}-{\left(t_1-s\right)}^{\alpha -\beta -2}\right]\mathit{ds}\\ +\frac{M}{\mathrm{\Gamma }\left(\alpha -1\right)}\left[\underset{t_1}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\alpha -2}\mathit{ds}+\underset{0}{\overset{t_1}{\int }}\left[{\left(t_2-s\right)}^{\alpha -2}-{\left(t_1-s\right)}^{\alpha -2}\right]\mathit{ds}\right]\end{array}$$

Again, it is seen that the right-hand side of the above inequality tends to zero as $t_2\to t_1$. Thus, $‖\left(\mathit{Nu}\right)\left(t_2\right)-\left(\mathit{Nu}\right)\left(t_1\right)‖\to 0$, as $t_2\to t_1$. This shows that the operator $N$ is completely continuous, by the Ascoli-Arzela theorem. Thus, the operator $N$ satisfies all the conditions of Theorem 3.4, and hence by its conclusion, either condition (i) or condition (ii) holds. We show that the condition (ii) is not possible.

Let $U=\left\{u\in C^1\left(J,{\mathbb{R}}^n\right):‖u‖<M\right\}$ with $max\left\{l_1,l_2\right\}=l<M$. In view of condition $l<M$ and by (12), we have

$$‖\mathit{Nu}‖\le max\left\{l_1,l_2\right\}<M.$$

Now, suppose there exists $u\in \partial U$ and $\lambda \in \left(0,1\right)$ such that $u=\mathit{\lambda Nu}$. Then for such a choice of $u$ and the constant $\lambda$, we have

$$M=‖u‖=\lambda ‖\mathit{Nu}‖<max\left\{l_1,l_2\right\}<M,$$

which is a contradiction. Consequently, by the Leray-Schauder alternative, we deduce that $F$ has a fixed point $u\in \overline{U}$ which is a solution of the boundary value problem (1)–(12). The proof is completed.

We construct an example to illustrate the applicability of the results presented.

Example 2.1. Consider the following fractional differential equation

$${}^c{D}_{0^{+}}^{\alpha }u\left(t\right)-A^c{D}_{0^{+}}^{\beta }u\left(t\right)=f\left(t,u\left(t\right){,}^c{D}_{0^{+}}^{\beta }u\left(t\right){,}^c{D}_{0^{+}}^{\alpha }u\left(t\right)\right),t\in J=\left[0,1\right],$$E13

subject to the three-point boundary conditions

$$\left\{\begin{array}{c}\begin{array}{c}u\left(0\right)-u\left(1\right)-u\left(\frac{1}{2}\right)=1,\\ u^{\prime }\left(0\right)-u^{\prime }\left(1\right)-u^{\prime }\left(\frac{1}{2}\right)=1,\end{array}\end{array}\right.$$E14

where $\alpha =2,\beta =1,\lambda =\mu =d=l=1,A=\left(\begin{array}{cc}2& 1\\ 0& 2\end{array}\right)$ and

$$f_i\left(t,u,v,w\right)=\frac{c_{i}t}{8}\mathit{arctan}\left(\left|u\right|+\left|v\right|+\left|w\right|\right),i=1,2,$$

such that $f=\left(f_1,f_2\right)$ with $0<c_i<1,i=1,2$.

For every $u_i,v_i\in {\mathbb{R}}^2,i=1,2,3$, we have

$$\left|f_i\left(t,u_1,u_2,u_3\right)-f_i\left(t,v_1,v_2,v_3\right)\right|\le \frac{c_i}{8}\left(\left|u_1-v_1\right|+\left|u_2-v_2\right|+\left|u_3-v_3\right|\right),i=1,2,$$

where $L_1=L_2=L_3=\frac{c_i}{8}$ for appropriate choice of constants $c_i,i=1,2$. we check the condition of Theorem 2.2. Clearly, assumption $\left(H_1\right)$ holds. A simple computations of $R_1,R_2,l_1$ and $l_2$ shows tha the second condition of Theorems 3.3 and 3.5 is satisfied. Thus the conclusion of Theorems 3.3 and 3.5 applies, and hence the problem (13)–(14) has a unique solution and at least one solution on $\left[0,1\right]$.

3. Conclusions

This chapter concerns the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal boundary conditions. By using Leray-Schauder nonlinear alternative and the Banach fixed point theorem, we shows the existence and uniqueness of positive solutions of our problem. In addition, an example is provided to demonstrate the effectiveness of the main results. The results of the present chapter are significantly contribute to the existing literature on the topic.

Acknowledgments

The authors want to thank the anonymous referee for the thorough reading of the manuscript and several suggestions that help us improve the presentation of the chapter.

Conflict of interest

The authors declare no conflict of interest.