ABSTRACT

In this article, we consider the three-dimensional fractional differential system of the form

$\begin{array}{c}{D}_{0^{+}}^{\alpha }u\left(t\right)=f_1\left(t,v\left(t\right),v^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left(0,1\right),\\ D_{0^{+}}^{\beta }v\left(t\right)=f_2\left(t,w\left(t\right),w^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left(0,1\right),\\ D_{0^{+}}^{\gamma }w\left(t\right)=f_3\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left(0,1\right),\end{array}$

together with the Neumann boundary conditions,

$u^{\prime }\left(0\right)=u^{\prime }\left(1\right)=0,\mathrm{<mi>\; </mi>}{v}^{\prime }\left(0\right)=v^{\prime }\left(1\right)=0,\mathrm{<mi>\; </mi>}{w}^{\prime }\left(0\right)=w^{\prime }\left(1\right)=\mathrm{0,}$

where $D_{0^{+}}^{\alpha },D_{0^{+}}^{\beta },D_{0^{+}}^{\gamma }$ are the standard Caputo fractional derivatives, $1<\alpha ,\beta ,\gamma \le 2$ . A new result on the existence of solutions for a class of fractional differential system is obtained by using Mawhin’s coincidence degree theory. Suitable examples are given to illustrate the main results.

Keywords:

Fractional Differential Equations, Boundary Value Problem, Coincidence Degree Theory

1. Introduction

Fractional calculus is a very effective tool in the modeling of many phenomena like control of dynamical systems, porous media, electro chemistry, viscoelasticity, electromagnetic and so on. The fractional theory and its applications are mentioned by many papers and monographs, we refer [1] - [9] . For nonlinear fractional boundary value problem, many fixed point theorems were applied to investigate the existence of solutions as in references [10] [11] [12] [13] . On the other hand, there is another effective approach, Mawhin’s coincidence theory, which proves to be very useful for determining the existence of solutions for fractional order differential equations. In recent years, boundary value problems for fractional differential equations at resonance have been studied in many papers (see [14] - [21] ). The main motivation for investigating the fractional boundary value problem arises from fractional advection-dispersion equation.

Hu et al. [22] investigated the two-point boundary value problem for fractional differential equations of the following form

$\begin{array}{l}{D}_{0^{+}}^{\alpha }x\left(t\right)=f\left(t,x\left(t\right),x^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left[0,1\right],\\ x\left(0\right)=0,\mathrm{<mi>\; </mi>}{x}^{\prime }\left(0\right)=x^{\prime }\left(1\right),\end{array}$

where $D_{0^{+}}^{\alpha }$ is the Caputo fractional differential operator, $1<\alpha \le 2$ , and $f\mathrm{<mo>\; :\; </mo>}\left[\mathrm{0,1}\right]\times R^2\to R$ is continuous.

In [23] , Hu et al. extended the above boundary value problem to the existence of solutions for the following coupled system of fractional differential equations of the form

$\begin{array}{l}{D}_{0^{+}}^{\alpha }u\left(t\right)=f\left(t,v\left(t\right),v^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left(0,1\right),\\ D_{0^{+}}^{\beta }v\left(t\right)=g\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left(0,1\right),\\ u\left(0\right)=v\left(0\right)=0,\mathrm{<mi>\; </mi>}{u}^{\prime }\left(0\right)=u^{\prime }\left(1\right),\mathrm{<mi>\; </mi>}{v}^{\prime }\left(0\right)=v^{\prime }\left(1\right)\mathrm{.}\end{array}$

where $D_{0^{+}}^{\alpha }\mathrm{,\; <mi>\; </mi>}{D}_{0^{+}}^{\beta }$ are the Caputo fractional derivatives, $1<\alpha \le 2,$ $1<\beta \le 2$ , and $f\mathrm{,}g\mathrm{<mo>\; :\; </mo>}\left[\mathrm{0,1}\right]\times R^2\to R$ is continuous.

It seems that there has been no work done on the boundary value problem of system involving three nonlinear fractional differential equations. Motivated by the above observation, we investigate the following three-dimensional fractional differential system of the form

$\begin{array}{l}{D}_{0^{+}}^{\alpha }u\left(t\right)=f_1\left(t,v\left(t\right),v^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left(0,1\right),\\ D_{0^{+}}^{\beta }v\left(t\right)=f_2\left(t,w\left(t\right),w^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left(0,1\right),\\ D_{0^{+}}^{\gamma }w\left(t\right)=f_3\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),\mathrm{<mi>\; </mi>}t\in \left(0,1\right),\end{array}$ (1)

together with the Neumann boundary conditions,

$u^{\prime }\left(0\right)=u^{\prime }\left(1\right)=0,\mathrm{<mi>\; </mi>}{v}^{\prime }\left(0\right)=v^{\prime }\left(1\right)=0,\mathrm{<mi>\; </mi>}{w}^{\prime }\left(0\right)=w^{\prime }\left(1\right)=0,$

where $D_{0^{+}}^{\alpha }\mathrm{,}{D}_{0^{+}}^{\beta }\mathrm{,}{D}_{0^{+}}^{\gamma }$ are the standard Caputo fractional derivatives, $1<\alpha ,\beta ,\gamma$ Math_19#, and $f_1,f_2,f_3:\left[0,1\right]\times R\times R\to R$ is continuous.

The main goal of this paper is to establish some new criteria for the existence of solutions of (1). The method is based on Mawhin’s coincidence degree theory. The results in this paper are generalized of the existing ones.

2. Preliminaries

In this section, we give the definitions of fractional derivatives and integrals and some notations which are useful throughout this paper. There are several kinds of definitions of fractional derivatives and integrals. In this paper, we use the Riemann-Liouville left sided definition on the half-axis $R_{+}$ and the Caputo fractional derivative.

Let X and Y be real Banach spaces and let $L:\text{dom}L\subset X\to Y$ be a Fred- holm operator with index zero if $\dim \text{Ker}L=\text{codim}\mathrm{Im}L<\infty$ and ImL is closed in Y and there exist continuous projectors $P:X\to X,\mathrm{<mi>\; </mi>}Q:Y\to Y$ such that

$\mathrm{Im}P=\text{Ker}L,\mathrm{<mi>\; </mi>}\text{Ker}Q=\mathrm{Im}L,\mathrm{<mi>\; </mi>}X=\text{Ker}L\oplus \text{Ker}P,\mathrm{<mi>\; </mi>}Y=\mathrm{Im}L\oplus \mathrm{Im}Q\mathrm{.}$

It follows that

${L|}_{\text{dom}L\cap \text{Ker}P}:\mathrm{dom}L\cap \text{Ker}P\to \mathrm{Im}L$

is invertible. Here $K_P$ denotes the inverse of ${L|}_{\text{dom}L\cap \text{Ker}P}$ .

If $\Omega$ is an open bounded subset of X, and $\text{dom}L\cap \bar{\Omega }\ne \varphi$ , then the map $N\mathrm{<mo>\; :\; </mo>}X\to Y$ will be called L-compact on $\bar{\Omega }$ , if $QN\left(\bar{\Omega }\right)$ is bounded and $K_P\left(I-Q\right)N\mathrm{<mo>\; :\; </mo>}\bar{\Omega }\to X$ is compact, where I is the identity operator.

Lemma 1. [14] Let $L\mathrm{<mo>\; :\; </mo>}\text{dom}L\subset X\to Y$ be a Fredholm operator with index zero and $N\mathrm{<mo>\; :\; </mo>}X\to Y$ be L-compact on $\bar{\Omega }$ . Assume that the following condi- tions are satisfied.

1. $Lx\ne \lambda Nx$ for every $\left(x,\lambda \right)\in \left[\left(\text{dom}L\backslash \text{Ker}L\right)\cap \partial \Omega \right]\times \left(0,1\right)$ ;

2. $Nx\notin \mathrm{Im}L$ for every $x\in \text{Ker}L\cap \partial \Omega$ ;

3. $\mathrm{deg}\left({QN|}_{\text{Ker}L},\text{Ker}L\cap \Omega ,0\right)\ne 0$ , where $Q\mathrm{<mo>\; :\; </mo>}Y\to Y$ is a projection such that $\mathrm{Im}L=\text{Ker}Q$ .

Then the operator equation $Lx=Nx$ has at least one solution in $\text{dom}L\cap \bar{\Omega }$ .

Definition 1. [6] The Riemann-Liouville fractional integral of order $\alpha >0$ of a function $y\mathrm{<mo>\; :\; </mo>}{R}_{+}\to R$ on the half-axis $R_{+}$ is given by

$\left(I_{0^{+}}^{\alpha }y\right)\left(t\right):=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t}{\left(t-v\right)}^{\alpha -1}y\left(v\right)\text{d}v\text{for}t>0$

provided the right hand side is pointwise defined on $R_{+}\mathrm{.}$

Definition 2. [6] Assume that $x\left(t\right)$ is $\left(n-1\right)$ -times absolutely continuous function, the Caputo fractional derivative of order $\alpha >0$ of x is given by

$\left(D_{0^{+}}^{\alpha }x\right)\left(t\right):=I_{0^{+}}^{n-\alpha }\frac{{\text{d}}^{n}x\left(t\right)}{\text{d}{t}^n}=\frac{1}{\Gamma \left(n-\alpha \right)}{\displaystyle {\int }_0^t}{\left(t-v\right)}^{n-\alpha -1}{x}^{\left(n\right)}\left(v\right)\text{d}v\text{for}t>0$

where n is the smallest integer greater than or equal to $\alpha$ , provided that the right side integral is pointwise defined on $\left(\mathrm{0,}+\infty \right)$ .

Lemma 2. [6] Let $\alpha >0$ and $n=-\left[-\alpha \right]$ . If $x^{\left(n-1\right)}\in AC\left[\mathrm{0,1}\right]$ , then

$I_{0^{+}}^{\alpha }{D}_{0^{+}}^{\alpha }x\left(t\right):=x\left(t\right)+c_0+c_{1}t+c_2{t}^2+\cdots +c_{n-1}{t}^{n-1}$

where $c_i=-\frac{x^{\left(i\right)}\left(0\right)}{i!}\in R,\mathrm{<mi>\; </mi>}i=0,1,2,\cdots ,n-1$ , here n is the smallest integer greater

than or equal to $\alpha$ .

In this paper, let us take $X=C^1\left[0,1\right]$ with the norm ${\Vert x\Vert }_X=\max \left\{{\Vert x\Vert }_{\infty },{\Vert x^{\prime }\Vert }_{\infty }\right\}$ and $Y=C\left[0,1\right]$ with the norm ${\Vert y\Vert }_Y={\Vert y\Vert }_{\infty }$ , where ${\Vert x\Vert }_{\infty }={{\displaystyle \max }}_{t\in \left[0,1\right]}\left|x\left(t\right)\right|$ . Then we denote $\bar{X}=X\times X\times X$ with the norm ${\Vert \left(u,v,w\right)\Vert }_{\bar{X}}=\max \left\{{\Vert u\Vert }_X,{\Vert v\Vert }_X,{\Vert w\Vert }_X\right\}$ and $\bar{Y}=Y\times Y\times Y$ with the norm ${\Vert \left(x,y,z\right)\Vert }_{\bar{Y}}=\max \left\{{\Vert x\Vert }_Y,{\Vert y\Vert }_Y,{\Vert z\Vert }_Y\right\}$ . Clearly, both $\bar{X}$ and $\bar{Y}$ are Banach spaces.

Define the operators $L_i:\text{dom}L\subset X\to Y,\left(i=1,2,3\right)$ by

$L_{1}u=D_{0^{+}}^{\alpha }u,\mathrm{<mi>\; </mi>}{L}_{2}v=D_{0^{+}}^{\beta }v\mathrm{<mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi>}{L}_{3}w=D_{0^{+}}^{\gamma }w\mathrm{,}$

where

$\text{dom}{L}_1=\left\{u\in X|D_{0^{+}}^{\alpha }u\left(t\right)\in Y,\mathrm{<mi>\; </mi>}{u}^{\prime }\left(0\right)=u^{\prime }\left(1\right)=0\right\},$

$\text{dom}{L}_2=\left\{v\in X|D_{0^{+}}^{\beta }v\left(t\right)\in Y,\mathrm{<mi>\; </mi>}{v}^{\prime }\left(0\right)=v^{\prime }\left(1\right)=0\right\},$

and

$\text{dom}{L}_3=\left\{w\in X|D_{0^{+}}^{\gamma }w\left(t\right)\in Y,\mathrm{<mi>\; </mi>}{w}^{\prime }\left(0\right)=w^{\prime }\left(1\right)=0\right\}\mathrm{.}$

Define the operator $L:\text{dom}L\subset \bar{X}\to \bar{Y}$ by

$L\left(u,v,w\right)=\left(L_{1}u,L_{2}v,L_{3}w\right)\mathrm{,}$ (2)

where $\text{dom}L=\left\{\left(u,v,w\right)\in \bar{X}|u\in \text{dom}{L}_1,v\in \text{dom}{L}_2,w\in \text{dom}{L}_3\right\}\mathrm{.}$

Let the Nemytski operator $N\mathrm{<mo>\; :\; </mo>}\bar{X}\to \bar{Y}$ be defined as

$N\left(u,v,w\right)=\left(N_{1}v,N_{2}w,N_{3}u\right),$

where $N_1\mathrm{<mo>\; :\; </mo>}X\to Y$ is defined by

$N_{1}v\left(t\right)=f_1\left(t,v\left(t\right),v^{\prime }\left(t\right)\right)\mathrm{,}$

$N_2\mathrm{<mo>\; :\; </mo>}X\to Y$ is defined by

$N_{2}w\left(t\right)=f_2\left(t,w\left(t\right),w^{\prime }\left(t\right)\right),$

and $N_3\mathrm{<mo>\; :\; </mo>}X\to Y$ is defined by

$N_{3}u\left(t\right)=f_3\left(t,u\left(t\right),u^{\prime }\left(t\right)\right).$

Then Neumann boundary value problem (1) is equivalent to the operator equation

$L\left(u,v,w\right)=N\left(u,v,w\right),\mathrm{<mi>\; </mi>}\left(u,v,w\right)\in \text{dom}L\mathrm{.}$

3. Main Results

In this section, we begin with the following theorem on existence of solutions for

Neumann boundary value problem (1).

Theorem 1. Let $f_1\mathrm{,}{f}_2\mathrm{,}{f}_3\mathrm{<mo>\; :\; </mo>}\left[\mathrm{0,1}\right]\times R\times R\to R$ be continuous. Assume that

(H1) there exist nonnegative functions $a_i,b_i,c_i\in C\left[0,1\right],\mathrm{<mi>\; </mi>}\left(i=1,2,3\right)$ with

$\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)\Gamma \left(\gamma \right)-\left(B_1+C_1\right)\left(B_2+C_2\right)\left(B_3+C_3\right)}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)\Gamma \left(\gamma \right)}>0$

such that for all $\left(u,v\right)\in R^2,\mathrm{<mi>\; </mi>}t\in \left[0,1\right],$

$\left|f_i\left(t,u,v\right)\right|\le a_i\left(t\right)+b_i\left(t\right)\left|u\right|+c_i\left(t\right)\left|v\right|\mathrm{<mi>\; </mi>}\text{for}\mathrm{<mi>\; </mi>}i=1,2,3,$

where $A_i={\Vert a_i\Vert }_{\infty },\mathrm{<mi>\; </mi>}{B}_i={\Vert b_i\Vert }_{\infty },\mathrm{<mi>\; </mi>}{C}_i={\Vert c_i\Vert }_{\infty },\mathrm{<mi>\; </mi>}\left(i=1,2,3\right)$ ;

(H2) there exists a constant $M>0$ such that for all $t\in \left[0,1\right],\mathrm{<mi>\; </mi>}\left|u\right|>M,v\in R$ either

$u{f}_1\left(t,u,v\right)>0,\mathrm{<mi>\; </mi>}u{f}_2\left(t,u,v\right)>0,\mathrm{<mi>\; </mi>}u{f}_3\left(t,u,v\right)>0$

or

$u{f}_1\left(t,u,v\right)<0,\mathrm{<mi>\; </mi>}u{f}_2\left(t,u,v\right)<0,\mathrm{<mi>\; </mi>}u{f}_3\left(t,u,v\right)<0;$

(H3) there exists a constant $M^{*}>0$ such that for every $m_1,m_2,m_3\in R$ satisfying $\min \left\{m_1,m_2,m_3\right\}>M^{*}$ either

$m_1{N}_1\left(m_2\right)>0,\mathrm{<mi>\; </mi>}{m}_2{N}_2\left(m_3\right)>0,\mathrm{<mi>\; </mi>}{m}_3{N}_3\left(m_1\right)>0$

or

$m_1{N}_1\left(m_2\right)<0,\mathrm{<mi>\; </mi>}{m}_2{N}_2\left(m_3\right)<0,\mathrm{<mi>\; </mi>}{m}_3{N}_3\left(m_1\right)<0.$

Then Neumann boundary value problem (1) has at least one solution.

Lemma 3. Let L be defined by (2). Then

$\begin{array}{c}\text{Ker}L=\left(\text{Ker}{L}_1,\text{Ker}{L}_2,\text{Ker}{L}_3\right)\\ =\left\{\left(u,v,w\right)\in \bar{X}|\left(u,v,w\right)=\left(u\left(0\right),v\left(0\right),w\left(0\right)\right)\right\},\end{array}$ (3)

and

$\begin{array}{c}\mathrm{Im}L=\left(\mathrm{Im}{L}_1,\mathrm{Im}{L}_2,\mathrm{Im}{L}_3\right)\\ =\{\left(x,y,z\right)\in \bar{Y}|{\displaystyle {\int }_0^1}{\left(1-s\right)}^{\alpha -2}x\left(s\right)\text{d}s\\ ={\displaystyle {\int }_0^1}{\left(1-s\right)}^{\beta -2}y\left(s\right)\text{d}s={\displaystyle {\int }_0^1}{\left(1-s\right)}^{\gamma -2}z\left(s\right)\text{d}s=0\}.\end{array}$ (4)

Proof. By Lemma 2, $L_{1}u=D_{0^{+}}^{\alpha }u\left(t\right)=0$ has the solution

$u\left(t\right)=u\left(0\right)+u^{\prime }\left(0\right)t.$

From the boundary conditions, we have

$\text{Ker}{L}_1=\left\{u\in X|u=u\left(0\right)\right\}\mathrm{.}$

For $x\in \mathrm{Im}{L}_1$ , there exists $u\in \text{dom}{L}_1$ such that $x=L_{1}u\in Y$ . By using the Lemma 2, we get

$u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\alpha -1}x\left(s\right)\text{d}s+u\left(0\right)+u^{\prime }\left(0\right)t.$

Then, we have

$u^{\prime }\left(t\right)=\frac{1}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\alpha -2}x\left(s\right)\text{d}s+u^{\prime }\left(0\right)\mathrm{.}$

By the boundary value conditions of (1), we can get that x satisfies

${\displaystyle {\int }_0^1}{\left(1-s\right)}^{\alpha -2}x\left(s\right)\text{d}s=0.$

On the other hand, suppose $x\in Y$ and satisfies ${\displaystyle {\int }_0^1}{\left(1-s\right)}^{\alpha -2}x\left(s\right)\text{d}s=0.$ Let $u\left(t\right)=I_{0^{+}}^{\alpha }x\left(t\right),$ then $u\in \text{dom}{L}_1$ and $D_{0^{+}}^{\alpha }u\left(t\right)=x\left(t\right)$ . Hence, $x\in \mathrm{Im}{L}_1$ . Then we get

$\mathrm{Im}{L}_1=\left\{x\in Y|{\displaystyle {\int }_0^1}{\left(1-s\right)}^{\alpha -2}x\left(s\right)\text{d}s=0\right\}\mathrm{.}$

Similarly, we have

$\text{Ker}{L}_2=\left\{v\in X|v=v\left(0\right)\right\},\mathrm{<mi>\; </mi>}\mathrm{Im}{L}_2=\left\{y\in Y|{\displaystyle {\int }_0^1}{\left(1-s\right)}^{\beta -2}y\left(s\right)\text{d}s=0\right\},$

and

$\text{Ker}{L}_3=\left\{w\in X|w=w\left(0\right)\right\},\mathrm{<mi>\; </mi>}\mathrm{Im}{L}_3=\left\{z\in Y|{\displaystyle {\int }_0^1}{\left(1-s\right)}^{\gamma -2}z\left(s\right)\text{d}s=0\right\}.$

$\square$

Lemma 4. Let L be defined by (2). Then L is a Fredholm operator of index zero, $P\mathrm{<mo>\; :\; </mo>}\bar{X}\to \bar{X}$ and $Q\mathrm{<mo>\; :\; </mo>}\bar{Y}\to \bar{Y}$ are the linear continuous projector opera- tors can be defined as

$P\left(u,v,w\right)=\left(P_{1}u,P_{2}v,P_{3}w\right)=\left(u\left(0\right),v\left(0\right),w\left(0\right)\right),$

$\begin{array}{c}Q\left(x,y,z\right)=\left(Q_{1}x,Q_{2}y,Q_{3}z\right)\\ =(\left(\alpha -1\right){\displaystyle {\int }_0^1}{\left(1-s\right)}^{\alpha -2}x\left(s\right)\text{d}s,\\ \left(\beta -1\right){\displaystyle {\int }_0^1}{\left(1-s\right)}^{\beta -2}y\left(s\right)\text{d}s,\left(\gamma -1\right){\displaystyle {\int }_0^1}{\left(1-s\right)}^{\gamma -2}z\left(s\right)\text{d}s)\mathrm{.}\end{array}$

Further more, the operator $K_P:\mathrm{Im}L\to \text{dom}L\cap \text{Ker}P$ can be written by

$K_P\left(x,y,z\right)=\left(I_{0^{+}}^{\alpha }x\left(t\right),I_{0^{+}}^{\beta }y\left(t\right),I_{0^{+}}^{\gamma }z\left(t\right)\right)\mathrm{.}$

Proof. Clearly, $\mathrm{Im}P=\text{Ker}L$ and $P^2\left(u,v,w\right)=P\left(u,v,w\right)$ . It follows that $\left(u,v,w\right)=\left(\left(u,v,w\right)-P\left(u,v,w\right)\right)+P\left(u,v,w\right)$ , we have $\bar{X}=\text{Ker}P+\text{Ker}L$ . By using simple calculation, we get that $\text{Ker}L\cap \text{Ker}P=\left\{\left(0,0,0\right)\right\}$ . Then we have

$\bar{X}=\text{Ker}P\oplus \text{Ker}L\mathrm{.}$

For $\left(x\mathrm{,}y\mathrm{,}z\right)\in \bar{Y}$ , we have

$Q^2\left(x,y,z\right)=Q\left(Q_{1}x,Q_{2}y,Q_{3}z\right)=\left(Q_1^{2}x,Q_2^{2}y,Q_3^{2}z\right)$

By the definition of $Q_1$ , we get

$Q_1^{2}x=Q_{1}x\cdot \left(\alpha -1\right){\displaystyle {\int }_0^1}{\left(1-s\right)}^{\alpha -2}x\left(s\right)\text{d}s=Q_{1}x.$

Similarly, we can show that $Q_2^{2}y=Q_{2}y$ and $Q_3^{2}z=Q_{3}z$ . Thus, we can get $Q^2\left(x,y,z\right)=Q\left(x,y,z\right)$ .

Let

$\left(x,y,z\right)=\left(\left(x,y,z\right)-Q\left(x,y,z\right)\right)+Q\left(x,y,z\right)$ ,

where $\left(x,y,z\right)-Q\left(x,y,z\right)\in \text{Ker}Q,$ $Q\left(x,y,z\right)\in \mathrm{Im}Q$ . It follows that $\text{Ker}Q=\mathrm{Im}L$ and $Q^2\left(x,y,z\right)=Q\left(x,y,z\right)$ , we get $\mathrm{Im}Q\cap \mathrm{Im}L=\left\{\left(0,0,0\right)\right\}$ . It is clear that

$\bar{Y}=\mathrm{Im}L\oplus \mathrm{Im}Q\mathrm{.}$

Thus

$\dim \text{Ker}L=\dim \mathrm{Im}Q=\text{codim}\mathrm{Im}L\mathrm{.}$

Hence L is a Fredholm operator of index zero.

From the definitions of P and $K_P$ , we will prove that $K_P$ is the inverse of ${L|}_{\text{dom}L\cap \text{Ker}P}$ . Infact, for $\left(x,y,z\right)\in \mathrm{Im}L$ , we have

$L{K}_P\left(x,y,z\right)=\left(D_{0^{+}}^{\alpha }\left(I_{0^{+}}^{\alpha }x\right),D_{0^{+}}^{\beta }\left(I_{0^{+}}^{\beta }y\right),D_{0^{+}}^{\gamma }\left(I_{0^{+}}^{\gamma }z\right)\right)=\left(x,y,z\right)\mathrm{.}$ (5)

Moreover, for $\left(u\mathrm{,}v\mathrm{,}w\right)\in \text{dom}L\cap \text{Ker}P$ , we have $u\left(0\right)=v\left(0\right)=w\left(0\right)=0$ and

$\begin{array}{c}{K}_{P}L\left(u,v,w\right)=\left(I_{0^{+}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(t\right)\right),I_{0^{+}}^{\beta }\left(D_{0^{+}}^{\beta }v\left(t\right)\right),I_{0^{+}}^{\gamma }\left(D_{0^{+}}^{\gamma }w\left(t\right)\right)\right)\\ =\left(u\left(t\right)-u\left(0\right)-u^{\prime }\left(0\right)t,v(t)-v\left(0\right)-v^{\prime }\left(0\right)t,w(t)-w\left(0\right)-w^{\prime }\left(0\right)t\right)\mathrm{,}\end{array}$

which together with the boundary condition $u^{\prime }\left(0\right)=v^{\prime }(0)=w^{\prime }(0)=0$ yields that

$K_{P}L\left(u,v,w\right)=\left(u,v,w\right)\mathrm{.}$ (6)

From (5) and (6), we get $K_P={\left({L|}_{\text{dom}L\cap \text{Ker}P}\right)}^{-1}$ . $\square$

Lemma 5. Assume $\Omega \subset \bar{X}$ is an open bounded subset such that $\text{dom}L\cap \bar{\Omega }\ne \varphi$ , then N is L-compact on $\bar{\Omega }$ .

Proof. By the continuity of f₁, f₂ and f₃, we can get $QN\left(\bar{\Omega }\right)$ and $K_P\left(I-Q\right)N\left(\bar{\Omega }\right)$ are bounded. By the Arzela-Ascoli theorem, we will prove that $K_P\left(I-Q\right)N\left(\bar{\Omega }\right)\subset \bar{X}$ is equicontinuous.

From the continuity of f₁, f₂ and f₃, there exist constants $M_i>0,\mathrm{<mi>\; </mi>}\left(i=1,2,3\right)$ such that for all $\left(u\mathrm{,}v\mathrm{,}w\right)\in \bar{\Omega }$ .

$\left|\left(I-Q_1\right)N_{1}v\right|\le M_1,\left|\left(I-Q_2\right)N_{2}w\right|\le M_2,\mathrm{<mi>\; </mi>}\left|\left(I-Q_3\right)N_{3}u\right|\le M_3\mathrm{.}$

Furthermore, for $0\le t_1<t_2\le 1,\mathrm{<mi>\; </mi>}\left(u,v,w\right)\in \bar{\Omega }$ , we have

$\begin{array}{l}\left|\left(K_P\left(I-Q\right)N\left(u\left(t_2\right),v\left(t_2\right),w\left(t_2\right)\right)\right)-\left(K_P\left(I-Q\right)N\left(u\left(t_1\right),v\left(t_1\right),w\left(t_1\right)\right)\right)\right|\\ =\left|K_P\left(I-Q\right)\left(N_{1}v\left(t_2\right),N_{2}w\left(t_2\right),N_{3}u\left(t_2\right)\right)-K_P\left(I-Q\right)\left(N_{1}v\left(t_1\right),N_{2}w\left(t_1\right),N_{3}u\left(t_1\right)\right)\right|\\ =|\left(I_{0^{+}}^{\alpha }\left(I-Q_1\right)N_{1}v\left(t_2\right),I_{0^{+}}^{\beta }\left(I-Q_2\right)N_{2}w\left(t_2\right),I_{0^{+}}^{\gamma }\left(I-Q_3\right)N_{3}u\left(t_2\right)\right)\\ -\left(I_{0^{+}}^{\alpha }\left(I-Q_1\right)N_{1}v\left(t_1\right),I_{0^{+}}^{\beta }\left(I-Q_2\right)N_{2}w\left(t_1\right),I_{0^{+}}^{\gamma }\left(I-Q_3\right)N_{3}u\left(t_1\right)\right)|\\ =|I_{0^{+}}^{\alpha }\left(I-Q_1\right)N_{1}v\left(t_2\right)-I_{0^{+}}^{\alpha }\left(I-Q_1\right)N_{1}v\left(t_1\right)\mathrm{,}\\ I_{0^{+}}^{\beta }\left(I-Q_2\right)N_{2}w\left(t_2\right)-I_{0^{+}}^{\beta }\left(I-Q_2\right)N_{2}w\left(t_1\right)\mathrm{,}\\ I_{0^{+}}^{\gamma }\left(I-Q_3\right)N_{3}u\left(t_2\right)-I_{0^{+}}^{\gamma }\left(I-Q_3\right)N_{3}u\left(t_1\right)|\mathrm{.}\end{array}$

By

$\begin{array}{l}\left|I_{0^{+}}^{\alpha }\left(I-Q_1\right)N_{1}v\left(t_2\right)-I_{0^{+}}^{\alpha }\left(I-Q_1\right)N_{1}v\left(t_1\right)\right|\\ \le \frac{1}{\Gamma \left(\alpha \right)}\left|{\displaystyle {\int }_0^{t_2}}{\left(t_2-s\right)}^{\alpha -1}\left(I-Q_1\right)N_{1}v\left(s\right)\text{d}s-{\displaystyle {\int }_0^{t_1}}{\left(t_1-s\right)}^{\alpha -1}\left(I-Q_1\right)N_{1}v\left(s\right)\text{d}s\right|\\ \le \frac{M_1}{\Gamma \left(\alpha \right)}\left[{\displaystyle {\int }_0^{t_1}}\left({\left(t_2-s\right)}^{\alpha -1}-{\left(t_1-s\right)}^{\alpha -1}\right)\text{d}s+{\displaystyle {\int }_{t_1}^{t_2}}{\left(t_2-s\right)}^{\alpha -1}\text{d}s\right]=\frac{M_1}{\Gamma \left(\alpha +1\right)}\left(t_2^{\alpha }-t_1^{\alpha }\right)\end{array}$

and

$\begin{array}{l}\left|{\left(I_{0^{+}}^{\alpha }\left(I-Q_1\right)N_{1}v\right)}^{\prime }\left(t_2\right)-{\left(I_{0^{+}}^{\alpha }\left(I-Q_1\right)N_{1}v\right)}^{\prime }\left(t_1\right)\right|\\ \le \frac{\alpha -1}{\Gamma \left(\alpha \right)}\left|{\displaystyle {\int }_0^{t_2}}{\left(t_2-s\right)}^{\alpha -2}\left(I-Q_1\right)N_{1}v\left(s\right)\text{d}s-{\displaystyle {\int }_0^{t_1}}{\left(t_1-s\right)}^{\alpha -2}\left(I-Q_1\right)N_{1}v\left(s\right)\text{d}s\right|\\ \le \frac{M_1}{\Gamma \left(\alpha -1\right)}\left[{\displaystyle {\int }_0^{t_1}}\left({\left(t_2-s\right)}^{\alpha -2}-{\left(t_1-s\right)}^{\alpha -2}\right)\text{d}s+{\displaystyle {\int }_{t_1}^{t_2}}{\left(t_2-s\right)}^{\alpha -2}\text{d}s\right]\\ =\frac{M_1}{\Gamma \left(\alpha \right)}\left(t_2^{\alpha -1}-t_1^{\alpha -1}\right).\end{array}$

Similarly, we can show that

$\left|I_{0^{+}}^{\beta }\left(I-Q_2\right)N_{2}w\left(t_2\right)-I_{0^{+}}^{\beta }\left(I-Q_2\right)N_{2}w\left(t_1\right)\right|\le \frac{M_2}{\Gamma \left(\beta +1\right)}\left(t_2^{\beta }-t_1^{\beta }\right)\mathrm{,}$

$\left|{\left(I_{0^{+}}^{\beta }\left(I-Q_2\right)N_{2}w\right)}^{\prime }\left(t_2\right)-{\left(I_{0^{+}}^{\beta }\left(I-Q_2\right)N_{2}w\right)}^{\prime }\left(t_1\right)\right|\le \frac{M_2}{\Gamma \left(\beta +1\right)}\left(t_2^{\beta -1}-t_1^{\beta -1}\right)\mathrm{,}$

$\left|I_{0^{+}}^{\gamma }\left(I-Q_3\right)N_{3}u\left(t_2\right)-I_{0^{+}}^{\gamma }\left(I-Q_3\right)N_{3}u\left(t_1\right)\right|\le \frac{M_3}{\Gamma \left(\gamma +1\right)}\left(t_2^{\gamma }-t_1^{\gamma }\right),$

$\left|{\left(I_{0^{+}}^{\gamma }\left(I-Q_3\right)N_{3}u\right)}^{\prime }\left(t_2\right)-{\left(I_{0^{+}}^{\gamma }\left(I-Q_3\right)N_{3}u\right)}^{\prime }\left(t_1\right)\right|\le \frac{M_3}{\Gamma \left(\gamma +1\right)}\left(t_2^{\gamma -1}-t_1^{\gamma -1}\right).$

Since $t^{\alpha },\mathrm{<mi>\; </mi>}{t}^{\alpha -1},\mathrm{<mi>\; </mi>}{t}^{\beta },\mathrm{<mi>\; </mi>}{t}^{\beta -1},\mathrm{<mi>\; </mi>}{t}^{\gamma }$ and $t^{\gamma -1}$ are uniformly continuous on [0, 1], we have $K_P\left(I-Q\right)N\left(\bar{\Omega }\right)\subset X$ is equicontinuous. Thus $K_P\left(I-Q\right)N\mathrm{<mo>\; :\; </mo>}\bar{\Omega }\to X$ is compact. $\square$

Lemma 6. Assume that $\left(H_1\right)\mathrm{,}\mathrm{<mi>\; </mi>}\left(H_2\right)$ hold, then the set

${\Omega }_1=\left\{\left(u,v,w\right)\in \text{dom}L\backslash \text{Ker}L|\mathrm{<mi>\; </mi>}L\left(u,v,w\right)=\lambda N\left(u,v,w\right),\mathrm{<mi>\; </mi>}\lambda \in \left(0,1\right)\right\}$

is bounded.

Proof. Let $\left(u\mathrm{,}v\mathrm{,}w\right)\in {\Omega }_1$ , then $N\left(u,v,w\right)\in \mathrm{Im}L$ . By (4), we get

${\displaystyle {\int }_0^1}{\left(1-s\right)}^{\alpha -2}{f}_1\left(s,v\left(s\right),v^{\prime }\left(s\right)\right)\text{d}s=\mathrm{0,}$

${\displaystyle {\int }_0^1}{\left(1-s\right)}^{\beta -2}{f}_2\left(s,w\left(s\right),w^{\prime }\left(s\right)\right)\text{d}s=0$

and

${\displaystyle {\int }_0^1}{\left(1-s\right)}^{\gamma -2}{f}_3\left(s,u\left(s\right),u^{\prime }\left(s\right)\right)\text{d}s=0.$

Then, by integral mean value theorem, there exist constants $\xi \mathrm{,\; <mi>\; </mi>}\eta \mathrm{,\; <mi>\; </mi>}\zeta \in \left(\mathrm{0,1}\right)$ such that $f_1\left(\xi ,v\left(\xi \right),v^{\prime }\left(\xi \right)\right)=0,\mathrm{<mi>\; </mi>}{f}_2\left(\eta ,w\left(\left(\eta \right),w^{\prime }\left(\eta \right)\right)\right)=0$ and $f_3\left(\zeta ,u\left(\zeta \right),u^{\prime }\left(\zeta \right)\right)=0.$ Then we get

$\begin{array}{l}v\left(\xi \right)f_1\left(\xi ,v\left(\xi \right),v^{\prime }\left(\xi \right)\right)=0,\mathrm{<mi>\; </mi>}w\left(\eta \right)f_2\left(\eta ,w\left(\eta \right),w^{\prime }\left(\eta \right)\right)=\mathrm{0\; <mi>\; </mi>}\\ \text{and}\mathrm{<mi>\; </mi>}u\left(\zeta \right)f_3\left(\zeta ,u\left(\zeta \right),u^{\prime }\left(\zeta \right)\right)=0.\end{array}$

From $\left({\text{H}}_{\text{2}}\right)$ , we get $\left|v\left(\xi \right)\right|\le M,\mathrm{<mi>\; </mi>}\left|w\left(\eta \right)\right|\le M$ and $\left|u\left(\zeta \right)\right|\le M$ . Hence we have

$\left|u\left(t\right)\right|=\left|u\left(\zeta \right)+{\displaystyle {\int }_{\zeta }^t}{u}^{\prime }\left(s\right)\text{d}s\right|\le M+{\Vert u^{\prime }\Vert }_{\infty }.$ (7)

We obtain

${\Vert u\Vert }_{\infty }\le M+{\Vert u^{\prime }\Vert }_{\infty }.$ (8)

Similarly, we can show that

${\Vert v\Vert }_{\infty }\le M+{\Vert v^{\prime }\Vert }_{\infty }$ (9)

and

${\Vert w\Vert }_{\infty }\le M+{\Vert w^{\prime }\Vert }_{\infty }.$ (10)

By $L\left(u,v,w\right)=\lambda N\left(u,v,w\right)$ , we get

$u\left(t\right)=\frac{\lambda }{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\alpha -1}{f}_1\left(s,v\left(s\right),v^{\prime }\left(s\right)\right)\text{d}s+u\left(0\right),$

$v\left(t\right)=\frac{\lambda }{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\beta -1}{f}_2\left(s,w\left(s\right),w^{\prime }\left(s\right)\right)\text{d}s+v\left(0\right)$

and

$w\left(t\right)=\frac{\lambda }{\Gamma \left(\gamma \right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\gamma -1}{f}_3\left(s,u\left(s\right),u^{\prime }\left(s\right)\right)\text{d}s+w\left(0\right)\mathrm{.}$

Then

$u^{\prime }\left(t\right)=\frac{\lambda }{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\alpha -2}{f}_1\left(s,v\left(s\right),v^{\prime }\left(s\right)\right)\text{d}s\mathrm{,}$

$v^{\prime }\left(t\right)=\frac{\lambda }{\Gamma \left(\beta -1\right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\beta -2}{f}_2\left(s,w\left(s\right),w^{\prime }\left(s\right)\right)\text{d}s$

and

$\mathrm{<mi>\; </mi>}{w}^{\prime }\left(t\right)=\frac{\lambda }{\Gamma \left(\gamma -1\right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\gamma -2}{f}_3\left(s,u\left(s\right),u^{\prime }\left(s\right)\right)\text{d}s\mathrm{.}$

So,

$\begin{array}{c}{\Vert u^{\prime }\Vert }_{\infty }=\frac{1}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\alpha -2}\left|f_1\left(s,v\left(s\right),v^{\prime }\left(s\right)\right)\right|\text{d}s\\ \le \frac{1}{\Gamma \left(\alpha -1\right)}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\alpha -2}\left[a_1\left(s\right)+b_1\left(s\right)\left|v\left(s\right)\right|+c_1\left(s\right)\left|v^{\prime }\left(s\right)\right|\right]\text{d}s\\ \le \frac{1}{\Gamma \left(\alpha -1\right)}\left[A_1+B_{1}M+\left(B_1+C_1\right){\Vert v^{\prime }\Vert }_{\infty }\right]{\displaystyle {\int }_0^t}{\left(t-s\right)}^{\alpha -2}\text{d}s\\ \le \frac{1}{\Gamma \left(\alpha \right)}\left[A_1+B_{1}M+\left(B_1+C_1\right){\Vert v^{\prime }\Vert }_{\infty }\right]\mathrm{.}\end{array}$ (12)