\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 60, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/60\hfil Boundary value problem] {Boundary value problem for a coupled system of fractional differential equations with $p$-Laplacian operator at resonance} \author[L. Cheng, W. Liu, Q. Ye\hfil EJDE-2014/60\hfilneg] {Lingling Cheng, Wenbin Liu, Qingqing Ye} % in alphabetical order \address{Lingling Cheng \newline College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China} \email{chenglingling2006@163.com} \address{Wenbin Liu (Corresponding author) \newline College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China} \email{wblium@163.com} \address{Qingqing Ye \newline School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, China} \email{yeqingzero@gmail.com} \thanks{Submitted March 10, 2013. Published February 28, 2014.} \thanks{Supported by grant 11271364 from the NNSF of China} \subjclass[2000]{34A08, 34B15} \keywords{Fractional differential equation; boundary value problem; \hfill\break\indent coincidence degree; $p$-Laplacian operator} \begin{abstract} In this article, we discuss the existence of solutions to boundary-value problems for a coupled system of fractional differential equations with $p$-Laplacian operator at resonance. We prove the existence of solutions when $\dim \ker L\geq 2$, using the coincidence degree theory by Mawhin. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Along with the development of sciences and technology, the subject of fractional differential equations (FDEs for short) has emerged as an important area of investigation. Indeed, we can find a large number of applications in physics, electrochemistry, control, biology, etc. (see \cite{H1,P1}). Recently, many results on FDEs have been obtained; see for example \cite{A1,B1,B2,B3,K1,L1,M1}. Many authors have studied boundary value problems (BVPs for short) of FDEs; see \cite{A2,B4,B5,L2,W,Z1,Z2,Z3}. The papers \cite{C1,C2,L3,L4} considered the BVPs of FDEs with $p$-Laplacian operator. In 2012, Chen et al.~\cite{C2} showed the existence solutions by coincidence degree for the Caputo fractional $p$-Laplacian equations \begin{gather*} D_{0^{+}}^{\beta}\phi_{p}(D^{\alpha}_{0^{+}}x(t)) =f(t,x(t),D^{\alpha}_{0^{+}}x(t)),\quad 01$, $f:[0,1]\times R^{2}\to \mathbb{R}$ is continuous, $D^{\alpha}_{0^{+}}$ and $D_{0^{+}}^{\beta}$ are Caputo fractional derivatives. They used the operator $Lu=D_{0^{+}}^{\beta}\phi_{p}(D^{\alpha}_{0^{+}}x(t))$ with $D^{\alpha}_{0^{+}}x(0)=D^{\alpha}_{0^{+}}x(1)=0$ and obtained $\dim\ker L=1$. Articles \cite{J1,S2} considered BVPs for a coupled system of FDEs. In 2009, Su \cite{S2} showed the existence result by Schauder fix-point theorem for the coupled system of FDEs: \begin{gather*} D^{\alpha}u(t)=f(t,v(t),D^{\mu }v(t)),\quad 00$, $\alpha -\nu \geq 1$, $\beta - \mu \geq 1$, $f,g:[0,1]\times R^{2}\to \mathbb{R}$ are given functions and $D$ is the standard Riemann-Liouville dervative. In 2012 Jiang \cite{J1} considered the existence results for a coupled system of FDEs: \begin{gather*} D^{\alpha}u(t)=f(t,u(t),v(t)),\quad u(0)=0,\quad D^{\gamma}u(t)|_{t=1}=\sum ^{n}_{i=1}a_iD^{\gamma}u(t)|_{t=\xi _i},\\ D^{\beta}v(t)=g(t,u(t),v(t)),\quad v(0)=0,\quad D^{\delta}v(t)|_{t=1}=\sum ^{m}_{i=1}b_iD^{\delta}v(t)|_{t=\eta _i}, \end{gather*} where $t\in [0,1]$, $1<\alpha$, $\beta \leq 2$, $0<\gamma \leq \alpha -1$, $0<\delta \leq \beta -1$, $0<\xi _1<\xi _2<\dots <\xi _{n}<1$, $0<\eta _1<\eta _2<\dots <\eta _{m}<1$, and proved that $\dim\ker L=1$. As we know, there are only a few papers devoted to investigate the BVPs for a coupled system of FDEs with $p$-Laplacian operator at resonance. What is more, the case of $\dim\ker L\geq 2$ have not been studied. In this paper we will study the BVPs for higher order FDEs as follows: \begin{equation} \label{e1.1} \begin{gathered} D_{0^{+}}^{\gamma}\phi_{p}(D^{\alpha}_{0^{+}}u(t))=f(t,v(t)),\\ D_{0^{+}}^{\gamma}\phi_{p}(D^{\beta}_{0^{+}}v(t))=g(t,u(t)),\\ D^{\alpha}_{0^{+}}u(0)=D^{\alpha}_{0^{+}}u(1) =D^{\beta}_{0^{+}}v(0)=D^{\beta}_{0^{+}}v(1)=0, \end{gathered} \end{equation} where $t\in[0,1]$, $n-1<\alpha,\beta\leq n$, $0<\gamma\leq1$, $f,g:[0,1] \times R \to \mathbb{R}$ are continuous functions, $D_{0^{+}}^{\alpha},D_{0^{+}}^{\beta}$ and $D_{0^{+}}^{\gamma}$ are Caputo derivatives, and $ \phi_{p}(s)= \begin{cases} |s|^{p-2}s & s\neq 0,\\ 0 & s=0 \end{cases} $ is a $p$-Laplacian operator with $p>1$. Hence, if $L(u,v)=(D_{0^{+}}^{\gamma}\phi _{p}(D_{0^{+}}^{\alpha}u), D_{0^{+}}^{\gamma}\phi _{p}(D_{0^{+}}^{\beta}v))$ and \begin{align*} \operatorname{dom}L =\big\{&(u,v)\in X|(D_{0^{+}}^{\gamma}\phi_{p} (D^{\alpha}_{0^{+}}u),D_{0^{+}}^{\gamma}\phi_{p}(D^{\beta}_{0^{+}}v)) \in Y,\\ &D^{\alpha}_{0^{+}}u(0)=D^{\alpha}_{0^{+}}u(1) =D^{\beta}_{0^{+}}v(0)=D^{\beta}_{0^{+}}v(1)=0\big\}, \end{align*} then $\dim\ker L=n,n\geq 2$. \section{Preliminaries} For convenience, we present here some necessary basic knowledge and a theorem, which can be found in \cite{M2}. Let $X$ and $Y$ be real Banach spaces and $L:\operatorname{dom}L \subset X \to Y$ be a Fredholm operator with index zero, $P:X \to X$, $Q:Y\to Y$ be projectors such that \[ \operatorname{Im}P=\ker L, \quad \ker Q=\operatorname{Im}L, \quad X=\ker L \oplus \ker P, \quad Y=\operatorname{Im}L \oplus \operatorname{Im}Q. \] It follows that \[ L|_{\operatorname{dom}L \cap \ker P}:\operatorname{dom}L \cap \ker P \to \operatorname{Im}L, \] is invertible. We denote the inverse by $K_{p}$. If $\Omega$ is an open bounded subset of $X$, $\operatorname{dom}L \cap \overline{\Omega} \neq \emptyset$, the map $N:X \to Y$ will be called $L$-compact on $\overline{\Omega}$ if $QN(\overline{\Omega})$ is bounded and $K_{p}(I-Q)N:\overline{\Omega} \to X$ is compact. \begin{theorem}[\cite{M2}] \label{thm2.1} Let $L:\operatorname{dom} \subset X \to Y$ be a Fredholm operator of index zero and $N:X\to Y$ be called $L$-compact on $\overline{\Omega}$. Assume that the following conditions are satisfied: \begin{itemize} \item[(1)] $Lx \neq \lambda Nx$ for every $(x,\lambda)\in [(\operatorname{dom}L\backslash \ker L)\cap \partial \Omega]\times (0,1)$; \item[(2)] $Nx\notin \operatorname{Im}L$ for every $x\in \ker L\cap \partial \Omega$; \item[(3)] $\deg(QN|_{\ker L},\Omega \cap \ker L,0)\neq 0$, where $Q:Y\to Y$ is a projection such that $\operatorname{Im}L=\ker Q$. \end{itemize} Then the equation $Lx=Nx$ has at least one solution in $\operatorname{dom}L\cap \overline{\Omega }$. \end{theorem} In this article, we take $ X=C^{\alpha -1}[0,1] \times C^{\beta -1}[0,1]$ with norm \[ \|(u,v)\|=\max\{\|u\|_{\infty},\|v\|_{\infty}, \|D^{\alpha-1}_{0^{+}}u\|_{\infty},\|D^{\beta-1}_{0^{+}}v\|_{\infty}\}, \] and $Y=C[0,1] \times C[0,1]$ with norm \[ \|(f,g)\|=\max\{\|f(x)\|_{\infty},\|g(x)\|_{\infty}\}, \] where $C^{\alpha -1}[0,1]=\{u|u,D^{\alpha}_{0^{+}}u \in C[0,1]\}$, $C^{\beta }[0,1]=\{v|v,D^{\beta}_{0^{+}}v \in C[0,1]\}$. Define the operator $L:\operatorname{dom}L\cap X \to Y$ ,by \begin{equation} \label{e2.1} L(u(t),v(t))=(D_{0^{+}}^{\gamma}\phi_{p}(D^{\alpha}_{0^{+}}u(t)), D_{0^{+}}^{\gamma}\phi_{p}(D^{\beta}_{0^{+}}v(t))), \end{equation} where \begin{align*} \operatorname{dom}L=\big\{&(u,v)\in X|(D_{0^{+}}^{\gamma}\phi_{p}(D^{\alpha}_{0^{+}} u(t)), D_{0^{+}}^{\gamma}\phi_{p}(D^{\beta}_{0^{+}}v(t)))\in Y,\\ & D^{\alpha}_{0^{+}}u(0)=D^{\alpha}_{0^{+}}u(1)=D^{\beta}_{0^{+}}v(0) =D^{\beta}_{0^{+}}v(1)=0\}. \end{align*} Define the operator $N: X \to Y$, by \[ N(u(t),v(t))=(N_1u(t),N_2v(t)),t\in [0,1], \] where $N_1u(t)=f(t,v(t)),N_2v(t)=g(t,u(t))$. It is easy to see that $X$ is a Banach space, and problem \eqref{e1.1} is equivalent to the operator equation \[ L(u,v)=N(u,v), (u,v)\in \operatorname{dom}L. \] The following definitions can be found in \cite{P1,S3}. \begin{definition} \label{def2.1}\rm The Riemann-Liouville fractional integral of order $\alpha >0$ of a function $u:(0,1)\to \mathbb{R}$ is given by \[ I^{\alpha}_{0^{+}}u(t)=\frac{1}{\Gamma (\alpha)} \int ^{t}_{0}(t-s)^{\alpha -1}u(s)ds, \] provided that the right side integral is pointwise defined on $(0,+\infty)$. \end{definition} \begin{definition} \label{def2.2}\rm The Caputo fractional derivative of order $\alpha >0$ of a continuous function $u:(0,1)\to \mathbb{R}$ is given by \[ D^{\alpha}_{0^{+}}u(t)=I^{n-\alpha}_{0^{+}}\frac{d^{n}u(t)}{dt^{n}} =\frac{1}{\Gamma (n-\alpha)}\int ^{t}_{0}(t-s)^{n-\alpha -1}u^{n}(s)ds, \] where $n$ is the smallest integer greater than or equal to $\alpha$, provided that the right side integral is pointwise defined on $(0,+\infty)$. \end{definition} \begin{lemma}[\cite{L5}] \label{lem2.1}\rm Let $\alpha >0$.The fractional differential equation $D^{\alpha}_{0^{+}}u(t)=0$ has solution \[ u(t)=C_1+C_2t+C_3t^{2}+\dots +C_{n}t^{n-1}. \] \end{lemma} \begin{lemma}[\cite{K1}] \label{lem2.2}\rm Assume that $u(t)$ with a fractional derivative of order $\alpha >0$. Then \[ I^{\alpha}_{0^{+}}D^{\alpha}_{0^{+}}u(t)=u(t)+C_1+C_2t+C_3t^{2}+\dots +C_{n}t^{n-1},\quad C_i\in R,i=1,2,\dots ,n, \] where $n$ is the smallest integer greater than or equal to $\alpha$. \end{lemma} \section{Main result} In this section, a theorem on existence of solutions for problem \eqref{e1.1} will be given. Define the operators $T_1$ and $T_2$ as follows: \[ T_1y_1(s)=\int ^{1}_{0}(1-s)^{\alpha -1}y_1(s)ds, \quad T_2y_2(s)=\int ^{1}_{0}(1-s)^{\beta -1}y_2(s)ds. \] \begin{theorem} \label{thm3.1} Let $f,g:[0,1]\times R \to \mathbb{R}$ be continuous and assume that \begin{itemize} \item[(H1)] there exist nonnegative functions $a(t),b(t),c(t),d(t) \in C[0,1]$, such that \[ |f(t,v)| \leq a(t)+b(t)|v|^{p-1}; \quad |g(t,u)| \leq c(t)+d(t)|u|^{p-1}; \] \item[(H2)] for $(u,v) \in \operatorname{dom}L$, there exist constants $M_i>0$, $i=1,2$, such that, if either $|u(t)|>M_1,t \in [\xi ,1]$, or $|v(t)|>M_2,t \in [\eta ,1]$, then either \[ T_1N_1u\neq 0,\quad\text{or}\quad T_2N_2v\neq 0; \] \item[(H3)] there exist a positive constant $B$, such that for each $(u,v) \in \ker L$, if $\min\{|\pi _i|,|\pi '_i|\}>B$, $i=1,2,\dots n$. \end{itemize} Then either (1) \begin{itemize} \item[(i)] $(\sum _{i=1}^{n}\pi '_i)T_1N_1u>0,(\sum _{i=1}^{n} \pi _i)T_2N_2v>0$, \item[(ii)] $(\sum _{i=1}^{n}\pi '_i)T_1N_1u>0, (\sum _{i=1}^{n}\pi _i)T_2N_2v<0$; \end{itemize} or (2) \begin{itemize} \item[(i)] $(\sum _{i=1}^{n}\pi '_i)T_1N_1u<0$, $(\sum _{i=1}^{n}\pi _i) T_2N_2v<0$, \item[(ii)] $(\sum _{i=1}^{n}\pi '_i)T_1N_1u<0$, $(\sum _{i=1}^{n} \pi _i)T_2N_2v>0$, where $b(t), d(t)$ satisfy \[ \|b\|_{\infty}\|d\|_{\infty}<\frac{(\Gamma (\gamma +1))^{2}}{4} (\frac{\xi \eta \Gamma (\alpha +1) \Gamma (\beta +1)}{(1+\xi)(1+\eta)})^{1-q}. \] \end{itemize} \end{theorem} \begin{lemma} \label{lem3.1} Let $L$ be defined by (2), then \begin{gather} \label{e3.1} \begin{aligned} \ker L=\big\{&(u,v)\in X: (u,v)=(\sum _{i=1}^{n}\pi _it^{i-1},\sum _{i=1}^{n}\pi '_it^{i-1}),\\ &\pi _i,\pi '_i\in R,i=1,2,\dots ,n,t\in [0,1]\big\}, \end{aligned}\\ \operatorname{Im}L=\{(y_1,y_2)\in Y|T_1y_1=0,T_2y_2=0\}. \label{e3.2} \end{gather} \end{lemma} \begin{proof} By Lemmas \ref{lem2.1} and \ref{lem2.2}, and $\phi _{p}^{-1}(s)=\phi _{q}(s)$, $1/p+1/q=1$, the equation $D_{0^{+}}^{\gamma}\phi_{p}(D^{\alpha}_{0^{+}}u(t))=0$ has solution \[ u(t)=I^{\alpha}_{0^{+}}\phi _{q}(c)+\sum _{i=1}^{n}\pi _it^{i-1}, \quad \pi _i\in R,\;i=1,2,\dots ,n, \] which satisfies $D^{\alpha}_{0^{+}}u(t)=\phi _{q}(c)$, combining with the boundary value condition $D^{\alpha}_{0^{+}}u(0)=0$, we can get $u(t)=\sum _{i=1}^{n}\pi _it^{i-1}$, similarly $v(t)=\sum _{i=1}^{n}\pi '_it^{i-1}$. So, it has \eqref{e3.1} holds. On the one hand, if $(y_1,y_2)\in \operatorname{Im}L$, then there exist two functions $u,v\in \operatorname{dom}L$ such that \[ y_1=D_{0^{+}}^{\gamma}\phi_{p}(D^{\alpha}_{0^{+}}u(t)),y_2 =D_{0^{+}}^{\gamma}\phi_{p}(D^{\beta}_{0^{+}}v(t)). \] Based on Lemma \ref{lem2.2} and $D^{\alpha}_{0^{+}}u(0)=D^{\alpha}_{0^{+}}v(0)=0$, \[ D^{\alpha}_{0^{+}}u(t)=\phi _{q}I^{\gamma}_{0^{+}}y_1,D^{\beta}_{0^{+}}u(t) =\phi _{q}I^{\gamma}_{0^{+}}y_2. \] From condition the $D^{\alpha}_{0^{+}}u(1)=D^{\beta}_{0^{+}}v(1)=0$, we obtain that \[ T_1y_1=\int ^{1}_{0}(1-s)^{\alpha -1}y_1(s)ds=0,T_2y_2 =\int ^{1}_{0}(1-s)^{\beta -1}y_2(s)ds=0. \] On the other hand, for each $(y_1,y_2)\in Y$ satisfying $T_iy_i=0$, $i=1,2$. Let \[ u(t)=I^{\alpha}_{0^{+}}\phi _{q}(I^{\gamma}_{0^{+}}y_1(t)),\quad v(t)=I^{\beta}_{0^{+}}\phi _{q}(I^{\gamma}_{0^{+}}y_2(t)), \] then $(u,v)\in \operatorname{dom}L$ and \[ L(u(t),v(t))=(D_{0^{+}}^{\gamma}\phi_{p}(D^{\alpha}_{0^{+}}u(t)), D_{0^{+}}^{\gamma}\phi_{p}(D^{\beta}_{0^{+}}v(t))), \] so that $(y_1,y_2)\in \operatorname{Im}L$. Therefore, \eqref{e3.2} holds. The proof is complete. \end{proof} \begin{lemma} \label{lem3.2} Let $L$ be defined by \eqref{e2.1}, then $L$ is a Fredholm operator of index zero, and the linear continuous projector operators $P:X\to X,Q:Y\to Y$ can be defined as \begin{gather} \label{e3.3} P(u(t),v(t))=(P_1u(t),P_2v(t)), \\ Q(y_1(t),y_2(t))=(Q_1y_1(t),Q_2y_2(t)), \label{e3.4} \end{gather} where \begin{gather*} P_1u(t)=u(0)+\sum _{i=1}^{n-1}u^{(i)}t^{i},P_2v(t) =v(0)+\sum _{i=1}^{n-1}v^{(i)}t^{i},\\ Q_1y_1(t)=\Lambda (\sum _{i=1}^{n}\Lambda _it^{i-1})T_1y_1(t),Q_2y_2(t) =\Lambda '(\sum _{i=1}^{n}\Lambda '_it^{i-1})T_2y_2(t),\\ \frac{1}{\Lambda } =\sum _{i=1}^{n} \frac{\Lambda _i\Gamma (\alpha )\Gamma (i)}{\Gamma (\alpha +i)}, \frac{1}{\Lambda '} =\sum _{i=1}^{n}\frac{\Lambda '_i\Gamma (\beta )\Gamma (i)}{\Gamma (\beta +i)}. \end{gather*} Furthermore, the operator $K_{p}:\operatorname{Im}L\to \operatorname{dom}L\cap \ker P$ can be written as \begin{align*} K_{P}(y_1(t),y_2(t)) &=(K_{P_1}y_1(t),K_{P_2}y_2(t))\\ &=(I^{\alpha}_{0^{+}}\phi _{q}(I^{\gamma}_{0^{+}}y_1(t)), I^{\beta}_{0^{+}}\phi _{q}(I^{\gamma}_{0^{+}}y_2(t))),\quad \forall t\in [0,1]. \end{align*} \end{lemma} \begin{proof} For each $(y_1,y_2)\in Y$ and \eqref{e3.4}, we have \begin{align*} Q_1^{2}y_1 &=Q_1[\Lambda \Big(\sum _{i=1}^{n}\Lambda _it^{i-1}\Big)T_1y_1(t)]\\ &=\Lambda \Big(\sum _{i=1}^{n}\Lambda _it^{i-1}\Big)T_1\Lambda \Big(\sum _{i=1}^{n}\Lambda _it^{i-1}\Big)T_1y_1(t)\\ &=\Lambda \Big(\sum _{i=1}^{n}\Lambda _it^{i-1}\Big) \sum _{i=1}^{n}\frac{\Lambda _i\Gamma (\alpha )\Gamma (i)}{\Gamma (\alpha +i)} T_1y_1(t)\\ &=\Lambda \sum _{i=1}^{n}\frac{\Lambda _i\Gamma (\alpha )\Gamma (i)}{\Gamma (\alpha +i)}Q_1y_1. \end{align*} From $\frac{1}{\Lambda } =\sum _{i=1}^{n} \frac{\Lambda _i\Gamma (\alpha )\Gamma (i)}{\Gamma (\alpha +i)}$, we obtain \begin{equation} \label{e3.5} Q_1^{2}y_1=Q_1y_1. \end{equation} Similarly, we can derive \begin{equation} \label{e3.6} Q_2^{2}y_2=Q_1y_1. \end{equation} So, for each $(y_1,y_2)\in Y$ and $t\in [0,1]$ , it follows from \eqref{e3.5} \eqref{e3.6} that \[ Q^{2}(y_1,y_2)=Q(Q_1y_1,Q_1y_1)=(Q_1^{2}y_1,Q_2^{2}y_2)=(Q_1y_1,Q_1y_1)=Q(y_1,y_2). \] Obviously, \[ \ker Q=\{(y_1,y_2)\in Y|T_1y_1=T_2y_2=0\}=\operatorname{Im}L. \] Let $(y_1,y_2)=[(y_1,y_2)-Q(y_1,y_2)]+(y_1,y_2)$, then $(y_1,y_2)-Q(y_1,y_2)\in \ker Q =\operatorname{Im}L,Q(y_1,y_2)\in \operatorname{Im}Q$. For $(y_1,y_2)\in \operatorname{Im}L\cap \operatorname{Im}Q$, we can get $(y_1,y_2)=(0,0)$, then we have \[ Y=\operatorname{Im}L\oplus \operatorname{Im}Q. \] For each $(u,v)\in X$ by \eqref{e3.3}, we have \begin{align*} P_1^{2}u(t) &=P_1(u(0)+\sum _{i=1}^{n-1}u^{(i)}t^{i})\\ &=u(0)+\sum _{i=1}^{n-1}(u(0)+\sum _{i=1}^{n-1}u^{(i)}t^{i})^{(i)}|_{t=0}t^{i}\\ &=u(0)+\sum _{i=1}^{n-1}u^{(i)}t^{i}\\ &=P_1u(t); \end{align*} that is, \begin{equation} \label{e3.7} P_1^{2}u(t)=P_1u(t). \end{equation} Similarly, we can derive that \begin{equation} \label{e3.8} P_2^{2}u(t)=P_2u(t). \end{equation} So, for each $(u,v)\in X$ and $t\in [0,1]$, it follows from \eqref{e3.7} \eqref{e3.8} that \[ P^{2}(u(t),v(t))=P(u(t),v(t)). \] Obviously, $\operatorname{Im}P=\ker L$, \[ \ker P=\{(u,v)\in X: u(0)=v(0)=u^{(i)}(0)=v^{(i)}(0)=0,i=1,2,\dots ,n-1\}. \] Let $(u,v)=[(u,v)-P(u,v)]+P(u,v)$, we can get $(u,v)-P(u,v)\in \ker P$, $P(u,v)\in \operatorname{Im}P$, so $X=\ker P+\ker L$. By simple calculation, we can get $\ker L\cap \ker P=(0,0)$, then \[ X=\ker L\oplus \ker P. \] Thus \[ \dim\ker L=\dim\operatorname{Im}Q=\operatorname{codim}\operatorname{Im}L=n,\quad n\geq 2. \] This means that $L$ is a Fredholm operator of index zero. From the definitions of $P,K_{p}$, it is easy to see that the generalized inverse of $L$ is $K_{P}$. In fact, for $(y_1,y_2)\in \operatorname{Im}L$, we have \begin{equation} \label{e3.9} LK_{P}(y_1,y_2)=L(I^{\alpha}_{0^{+}}\phi _{q}(I^{\gamma}_{0^{+}}y_1(t)), I^{\beta}_{0^{+}}\phi _{q}(I^{\gamma}_{0^{+}}y_2(t)))=(y_1,y_2). \end{equation} Moreover, for $(u,v)\in \operatorname{dom}L\cap \ker P$, we get $u(0)=v(0)=u^{(i)}(0)=v^{(i)}(0)=0$, $i=1,2,\dots ,n-1$. Hence \begin{equation} \label{e3.10} K_{P}L(u,v)=K_{P}(D_{0^{+}}^{\gamma}\phi_{p} (D^{\alpha}_{0^{+}}u(t)),D_{0^{+}}^{\gamma}\phi_{p}(D^{\beta}_{0^{+}}v(t)))=(u,v). \end{equation} Combining \eqref{e3.9} and \eqref{e3.10}, we know that $K_{P}=(L|_{\operatorname{dom}L\cap \ker P})^{-1}$. The proof is complete. \end{proof} \begin{lemma} \label{lem3.3} Assume $\Omega \subset X$ is an open boundary subset such that $\operatorname{dom}L\cap \overline{\Omega}\neq \emptyset$, then $N$ is $L$-compact on $\overline{\Omega}$. \end{lemma} \begin{proof} By the continuity of $f,g$, we can get that $QN(\overline{\Omega})$ and $K_{P}(I-Q)N(\overline{\Omega})$ are bounded. So, in view of the Arzela-Ascoli theorem, we need only prove that $K_{P}(I-Q)(\overline{\Omega})\subset X$ is equicontinuous. From the continuity of $f,g$, there exists a constant $M>0$ such that \[ |(I-Q_i)N_i(u,v)|\leq M,\quad \forall t\in [0,1],\;(u,v)\in \overline{\Omega},\; i=1,2, \] where $I:C[0,1]\to C[0,1]$ is the indentity mapping. Furthermore, denote $K_{P,Q}=K_{P}(I-Q)N$ and for $0\leq t_1< t_2\leq 1$,$(u,v)\in \overline{\Omega}$, we have \begin{align*} &K_{P,Q}(u(t_2),v(t_2))-K_{P,Q}(u(t_1),v(t_1))\\ &=(K_{P_1}(I-Q_1)N_1u(t_2)-K_{P_1}(I-Q_1)N_1u(t_1),\\ &\quad K_{P_2}(I-Q_2)N_2u(t_2)-K_{P_2}(I-Q_2)N_2u(t_1)), \end{align*} From \begin{align*} &|K_{P_1}(I-Q_1)N_1u(t_2)-K_{P_1}(I-Q_1)N_1u(t_1)|\\ &=\frac{1}{\Gamma (\alpha)}|\int ^{t_2}_{0}(t_2-s)^{\alpha -1}\phi _{q} (\frac{1}{\Gamma (\gamma)}\int ^{s}_{0}(s-\tau)^{\gamma -1}I-Q_1)N_1u(\tau)d\tau)ds\\ &\quad -\int ^{t_1}_{0}(t_1-s)^{\alpha -1}\phi _{q} (\frac{1}{\Gamma (\gamma)}\int ^{s}_{0}(s-\tau)^{\gamma -1} I-Q_1)N_1u(\tau)d\tau)ds|\\ &\leq \frac{\phi _{q}(M)}{\Gamma (\alpha)}|\int ^{t_1}_{0}[(t_2-s)^{\alpha -1} -(t_1-s)^{\alpha -1}]ds+\int ^{t_2}_{t_1}(t_2-s)^{\alpha -1}ds|\\ &\leq \frac{\phi _{q}(M)}{\Gamma (\alpha)}(t_2^{\alpha}-t_1^{\alpha}), \end{align*} and \begin{align*} &|D_{0^{+}}^{\alpha -1}K_{P_1}(I-Q_1)N_1u(t_2)-D_{0^{+}}^{\alpha -1} K_{P_1}(I-Q_1)N_1u(t_1)|\\ &=|\int ^{t_2}_{0}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{s}_{0}(s-\tau)^{\gamma-1}(I-Q_1)N_1u(\tau)d\tau)ds\\ &\quad -\int ^{t_1}_{0}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{s}_{0}(s-\tau)^{\gamma -1}(I-Q_1)N_1u(\tau)d\tau)ds|\\ &=|\int ^{t_2}_{t_1}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{s}_{0}(s-\tau)^{\gamma-1}(I-Q_1)N_1u(\tau)d\tau)ds|\\ &\leq \phi _{q}(M)(t_2-t_1). \end{align*} Similarly, \begin{gather*} |K_{P_2}(I-Q_2)N_1u(t_2)-K_{P_2}(I-Q_2)N_1u(t_1)| \leq \frac{\phi _{q}(M)}{\Gamma (\beta)}(t_2^{\beta}-t_1^{\beta}), \\ |D_{0^{+}}^{\beta -1}K_{P_2}(I-Q_2)N_1u(t_2)-D_{0^{+}}^{\beta -1}K_{P_2} (I-Q_1)N_2u(t_1)| \leq \phi _{q}(M)(t_2-t_1), \end{gather*} and since $t^{\alpha},t^{\beta}$ are uniformly continuous on $[0,1]$, we can get that $K_{P}(I-Q)N(\overline{\Omega})\subset X$ is equicontinuous. Thus, we get that $K_{P}(I-Q)N:\overline{\Omega}\to X$ is compact. The proof is complete. \end{proof} \begin{lemma} \label{lem3.4} Suppose {\rm (H1)--(H2)} hold. Then the set \[ \Omega _1=\{(u,v)|(u,v)\in \operatorname{dom}L\backslash \ker L,L(u,v) =\lambda N(u,v),\lambda \in (0,1)\} \] is bounded. \end{lemma} \begin{proof} Take $(u,v)\in \Omega _1$, then $N(u,v)\in \operatorname{Im}L$. By \eqref{e3.2},we have \[ T_1N_1u=0,\quad T_2N_2v=0. \] By $L(u,v)=\lambda N(u,v)$ and $D_{0^{+}}^{\alpha}u(0)=D_{0^{+}}^{\beta}v(0)=0$, we have \begin{equation} \label{e3.11} \begin{aligned} &(u(t),v(t))\\ &=\lambda \Big(\frac{1}{\Gamma (\alpha)}\int _{0} ^{t}(t-s)^{\alpha -1} \phi _{q}(\frac{1}{\Gamma (\gamma)}\int _{0}^{s}(s-\tau)^{\gamma -1} f(\tau ,v(\tau))d\tau)ds+\sum _{i=0}^{n-1}c_it^{i},\\ &\quad \frac{1}{\Gamma (\beta)}\int _{0} ^{t}(t-s)^{\beta -1}\phi _{q} (\frac{1}{\Gamma (\gamma)}\int _{0}^{s}(s-\tau)^{\gamma -1} g(\tau ,u(\tau))d\tau)ds+\sum _{i=0}^{n-1}c'_it^{i}\Big). \end{aligned} \end{equation} Together with (H2) means that there exist constants $t_{0}\in [\xi ,1]$, $t_1\in [\eta ,1]$ such that $|u(t_{0})|\leq M_1$, $|v(t_1)|\leq M_2$. By \eqref{e3.11}, we have \begin{gather} \label{e3.12} \sum _{i=0}^{n-1}|c_i|t_{0}^{i} \leq M_1+\frac{1}{\Gamma (\alpha)}\int _{0} ^{1}(1-s)^{\alpha -1} \phi _{q}(\frac{1}{\Gamma (\gamma)}\int _{0}^{1}(1-\tau)^{\gamma -1} f(\tau ,v(\tau))d\tau)ds, \\ \label{e3.13} \sum _{i=0}^{n-1}|c'_i|t_1^{i}\leq M_2+\frac{1}{\Gamma (\beta)} \int _{0} ^{1}(1-s)^{\beta -1}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int _{0}^{1}(1-\tau)^{\gamma -1}g(\tau ,u(\tau))d\tau)ds. \end{gather} It follows from (H1) and \eqref{e3.11} \eqref{e3.12} that \begin{align*} &\vert u(t)\vert \\ &\leq \frac{1}{\Gamma (\alpha)}\int ^{1}_{0}(1-s)^{\alpha -1}\phi _{q} (\frac{1}{\Gamma (\gamma)}\int ^{1}_{0}(1-\tau)^{\gamma -1} |f(\tau ,v(\tau))|d\tau )ds+|c_{0}|+\frac{1}{\xi } (\sum _{i=1}^{n-1}|c_i|t_{0}^{i})\\ &\leq \frac{M_1}{\xi}+\frac{1+\xi }{\xi \Gamma (\alpha)} \int ^{1}_{0}(1-s)^{\alpha -1}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{1}_{0}(1-\tau)^{\gamma -1}(a(t)+b(t)|v(t)|^{p-1})d\tau)ds\\ &\leq \frac{M_1}{\xi}+\frac{1+\xi }{\xi \Gamma (\alpha)} \int ^{1}_{0}(1-s)^{\alpha -1}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{1}_{0}(1-\tau)^{\gamma -1}(\|a\|_{\infty} +\|b\|_{\infty}\|v\|_{\infty}^{p-1})d\tau)ds\\ &= \frac{M_1}{\xi}+\frac{1+\xi }{\xi \Gamma (\alpha +1)} \phi _{q}(\frac{1}{\Gamma (\gamma +1)}(\|a\|_{\infty} +\|b\|_{\infty}\|v\|_{\infty}^{p-1}))\\ &\leq \frac{M_1}{\xi}+\frac{2^{q-1}(1+\xi )}{\xi \Gamma (\alpha +1)} (\phi _{q}(\frac{\|a\|_{\infty}}{\Gamma (\gamma +1)}) +(\phi_{q}\frac{\|b\|_{\infty}\|v\|_{\infty}^{p-1}}{\Gamma (\gamma +1)}))\\ &\leq \frac{M_1}{\xi}+\frac{2^{q-1}(1+\xi )}{\xi \Gamma (\alpha +1)} ((\frac{\|a\|_{\infty}}{\Gamma (\gamma +1)})^{q-1} +(\frac{\|b\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}\|v\|_{\infty}); \end{align*} that is, \[ \Vert u(t)\Vert _{\infty}\leq \frac{M_1}{\xi} +\frac{2^{q-1}(1+\xi )}{\xi \Gamma (\alpha +1)} ((\frac{\|a\|_{\infty}}{\Gamma (\gamma +1)})^{q-1} +(\frac{\|b\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}\|v\|_{\infty}). \] Similarly, from (H1), \eqref{e3.11}, \eqref{e3.13} and $\phi _{p}(s+t)\leq 2^{p}(\phi _{p}(s)+\phi _{p}(t))$, $s,t>0$, we obtain \[ \Vert v(t)\Vert _{\infty}\leq \frac{M_2}{\eta}+\frac{2^{q-1}(1+\eta )} {\xi \Gamma (\beta +1)}((\frac{\|c\|_{\infty}}{\Gamma (\gamma +1)})^{q-1} +(\frac{\|d\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}\|u\|_{\infty}). \] Let \begin{gather*} \frac{M_1}{\xi}+\frac{2^{q-1}(1+\xi )}{\xi \Gamma (\alpha +1)} (\frac{\|a\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}=A,\quad \frac{2^{q-1}(1+\xi )}{\xi \Gamma (\alpha +1)} (\frac{\|b\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}=B,\\ \frac{M_2}{\eta}+\frac{2^{q-1}(1+\eta )}{\eta \Gamma (\beta +1)} (\frac{\|c\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}=A',\quad \frac{2^{q-1}(1+\eta )}{\eta \Gamma (\beta +1)} (\frac{\|d\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}=B', \end{gather*} then, the condition \[ \|b\|_{\infty}\|d\|_{\infty} <\frac{(\Gamma (\gamma +1))^{2}}{4}(\frac{\xi \eta \Gamma (\alpha +1) \Gamma (\beta +1)}{(1+\xi)(1+\eta)})^{1-q}, \] which by Theorem \ref{thm3.1} could written as $BB'<1$, so, we obtain \[ \Vert u(t)\Vert _{\infty}\leq \frac{A+A'B}{1-BB'},\quad \Vert v(t)\Vert _{\infty}\leq \frac{A'+AB'}{1-BB'}. \] By \eqref{e3.12} and \eqref{e3.13} we have \begin{equation} \label{e3.14} \begin{aligned} \vert c_{n-1}\vert &\leq \frac{M_1}{\xi}+\frac{1}{\xi \Gamma (\alpha)} \int ^{1}_{0}(1-s)^{\alpha -1}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{1}_{0}(1-\tau)^{\gamma -1}|f(\tau ,v(\tau))|d\tau )ds\\ &\leq \frac{M_1}{\xi}+\frac{2^{q-1}}{\xi \Gamma (\alpha +1)} ((\frac{\|a\|_{\infty}}{\Gamma (\gamma +1)})^{q-1} +(\frac{\|b\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}\|v\|_{\infty}), \end{aligned} \end{equation} \begin{equation} \label{e3.15} \begin{aligned} \vert c'_{n-1}\vert &\leq \frac{M_2}{\eta}+\frac{1}{\eta \Gamma (\beta)} \int ^{1}_{0}(1-s)^{\beta -1}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{1}_{0}(1-\tau)^{\gamma -1}|f(\tau ,u(\tau))|d\tau )ds\\ &\leq \frac{M_2}{\eta}+\frac{2^{q-1}}{\xi \Gamma (\beta +1)} ((\frac{\|c\|_{\infty}}{\Gamma (\gamma +1)})^{q-1} +(\frac{\|d\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}\|u\|_{\infty}). \end{aligned} \end{equation} Then, by \eqref{e3.11}, \eqref{e3.12} and \eqref{e3.13} we obtain \begin{align*} \vert D_{0}^{\alpha -1}u(t)\vert &\leq \int ^{1}_{0}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{1}_{0}(1-\tau)^{\gamma -1}|f(\tau ,v(\tau))|d\tau )ds +\frac{|c_{n-1}|t^{n-\alpha}}{\Gamma (n+1-\alpha)}\\ &\leq \frac{M_1}{\xi}+\frac{2^{q-1}(1+\xi \Gamma (\alpha +1))} {\xi \Gamma (\alpha +1)} ((\frac{\|a\|_{\infty}}{\Gamma (\gamma +1)})^{q-1} +(\frac{\|b\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}\|v\|_{\infty}), \end{align*} \begin{align*} \vert D_{0}^{\beta -1}u(t)\vert &\leq \int ^{1}_{0}\phi _{q}(\frac{1}{\Gamma (\gamma)} \int ^{1}_{0}(1-\tau)^{\gamma -1}|f(\tau ,u(\tau))|d\tau )ds +\frac{|c'_{n-1}|t^{n-\beta}}{\Gamma (n+1-\beta)}\\ &\leq \frac{M_2}{\eta}+\frac{2^{q-1}(1+\eta \Gamma (\beta +1))} {\xi \Gamma (\beta +1)}((\frac{\|a\|_{\infty}}{\Gamma (\gamma +1)})^{q-1} +(\frac{\|d\|_{\infty}}{\Gamma (\gamma +1)})^{q-1}\|u\|_{\infty}). \end{align*} Hence the $\Omega _1$ is bounded in $X$. The proof is complete. \end{proof} \begin{lemma} \label{lem3.5} Suppose that {\rm (H3)} hold. Then the set \[ \Omega _2=\{(u,v)|(u,v)\in \ker L,N(u,v)\in \operatorname{Im}L\} \] is bounded in $X$. \end{lemma} \begin{proof} For $(u,v)\in \Omega _2$, we have $(u(t),v(t))=(\sum _1^{n}\pi _it^{i-1},\sum _1^{n}\pi '_it^{i-1}), \pi _i,\pi '_i\in R,i=1,2,\dots ,n$ and $T_1N_1(\sum _1^{n}\pi _it^{i-1})=T_2N_2(\sum _1^{n}\pi '_it^{i-1})=0$. By (H3), we obtain that $\max\{|\pi _i|,|\pi '_i|\}\leq B,i=1,2,\dots ,n$, so $\max\{\|u\|_{\infty},\|v\|_{\infty}\}\leq 2B$. Furthermore, \begin{gather*} \vert D_{0^{+}}^{\alpha -1}u(t)| = \frac{1}{\Gamma (n-\alpha)}\int _{0}^{t}(t-s)^{n-1-\alpha }|\pi _{n}|ds \leq \frac{|\pi|_{n}}{\Gamma (n+1-\alpha )}\leq \frac{B}{\Gamma (n+1-\alpha )}, \\ \vert D_{0^{+}}^{\beta -1}v(t)|\leq \frac{B}{\Gamma (n+1-\beta )}. \end{gather*} Hence, $\Omega _2$ is bounded in $X$. The proof is complete. \end{proof} \begin{lemma} \label{lem3.6} Suppose that {\rm (H3)(1)} holds. Then the set \[ \Omega _3=\{(u,v)\in \ker L|\lambda J(u,v)+(1-\lambda ) Q(N_1u,\theta N_2v)=(0,0),\lambda \in [0,1]\} \] is bounded in $X$. If {\rm (H3)(1)(i)} holds, then $\theta =1$, if {\rm (H3)(1)(ii)} hold, then $\theta =-1$, where, $J:\ker L\to \operatorname{Im}Q$ is a linear isomorphism given by \[ J(\sum _1^{n}\pi _it^{i-1},\sum _1^{n}\pi '_it^{i-1}) =(\Lambda (\sum _1^{n}\Lambda _i)(\sum _1^{n}\pi '_it^{i-1}), \Lambda '(\sum _1^{n}\Lambda '_i)(\sum _1^{n}\pi _it^{i-1})), \] where $\Lambda (\sum _1^{n}\Lambda _i)\neq 0,\Lambda '(\sum _1^{n}\Lambda '_i)\neq 0$. \end{lemma} \begin{proof} For $(u,v)\in \Omega _3$, we have $(u(t),v(t))=(\sum _1^{n}\pi _it^{i-1}, \sum _1^{n}\pi '_it^{i-1}),\pi _i,\pi '_i\in R,i=1,2,\dots ,n$, by (H3)(1)(i), there exists $\lambda \in [0,1]$ such that \begin{equation} \label{e3.16} \begin{aligned} &\lambda J(\sum _1^{n}\pi _it^{i-1},\sum _1^{n}\pi '_it^{i-1})+(1-\lambda ) (\Lambda (\sum _1^{n}\Lambda _i)T_1N_1(\sum _1^{n}\pi _it^{i-1}),\\ &\Lambda '(\sum _1^{n}\Lambda '_i)T_2N_2(\sum _1^{n}\pi '_it^{i-1})))=(0,0). \end{aligned} \end{equation} If $\lambda =0$, we can get that $\max\{|\pi _i|,|\pi '_i|\}\leq B$, $i=1,2$, then $\max\{\|u\|_{\infty},\|v\|_{\infty}\}\leq 2B$. Hence, $\Omega _3$ is bounded. If $\lambda =1$, then $u=v=0$. For $\lambda (0,1)$, let $\Lambda _i=\pi '_i$, $\Lambda '_i=\pi _i$, $i=1,2,\dots ,n$, if $\min\{|\pi _i|,|\pi '_i|\}> B$, $i=1,2,\dots ,n$, we have the following inequalities: \begin{gather*} \lambda (\sum _1^{n}\pi '_i)^{2}+(1-\lambda )(\sum _1^{n} \pi '_i)T_1N_1(\sum _1^{n}\pi _i)>0, \\ \lambda (\sum _1^{n}\pi _i)^{2}+(1-\lambda )(\sum _1^{n}\pi _i) T_2N_2(\sum _1^{n}\pi '_i)>0, \end{gather*} this contradicts \eqref{e3.16}, so, $\Omega _3$ is bounded in $X$. Similarly, if (H3)(1)(ii) holds, we have $\Omega _3$ is bounded in $X$. The proof is complete. \end{proof} \begin{lemma} \label{lem3.7} If {\rm (H3)(2)} hold, then the set \[ \Omega _3=\{(u,v)\in \ker L|-\lambda J(u,v)+(1-\lambda ) Q(N_1u,\theta N_2v)=(0,0),\lambda \in [0,1]\} \] is bounded in $X$. \end{lemma} The proof of the above lemma is similarly with Lemma \ref{lem3.6}, and it is omitted. Now with Lemmas \ref{lem3.1}--\ref{lem3.7} in hand, we prove our main result. \begin{proof}[Proof the Theorem \ref{thm3.1}] Let $\Omega$ is a bounded open set of $X$ with $\cup ^{3}_{i=1}\subset \Omega$. By Lemma \ref{lem3.3}, we can get that $N$ is $L$-compact on $\overline{\Omega }$. Then by Lemmas \ref{lem3.4} and \ref{lem3.5}, we have (1) $Lx \neq \lambda Nx$ for every $(x,\lambda)\in [(\operatorname{dom}L\backslash \ker L)\cap \partial \Omega]\times (0,1)$; (2) $Nx\notin \operatorname{Im}L$ for every $x\in \ker L\cap \partial \Omega$; we need to prove only (3) $\deg(QN|_{\ker L},\Omega \cap \ker L,0)\neq 0$. Take \[ H(u,v,\lambda )=\pm \lambda J(u,v)+(1-\lambda )Q(N_1u,\theta N_2v), \] according to Lemma \ref{lem3.6}, we have $H(u,v,\lambda )\neq 0$ for $(u,v)\in \partial \Omega \cap \ker L$. By the homotopy property of degree, we can get \begin{align*} \deg(QN\vert _{\ker L},\Omega \cap \ker L,(0,0)) &=\deg(H(\cdot ,0),\Omega \cap \ker L,(0,0))\\ &=\deg(H(\cdot ,1),\Omega \cap \ker L,(0,0))\\ &=\deg(\pm J,\Omega \cap \ker L,(0,0)) \neq 0. \end{align*} By Theorem \ref{thm2.1}, we obtain that $L(u,v)=N(u,v)$ has at least one solution in $\operatorname{dom}L\cap \overline{\Omega }$; i.e, problem \eqref{e1.1} has at least one solution in $X$, The proof is complete. \end{proof} \begin{thebibliography}{99} \bibitem{A1} Agarwal, R. P.; O'Regan, D.; Stanek, S.; \emph{Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations}, J. Math. Anal. 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