Existence and uniqueness of solutions for a higher-order coupled fractional differential equations at resonance

Abstract

In this article, we present the existence of solutions for a higher-order coupled fractional differential equations with the Caputo fractional derivative. Our main approach is the coincidence degree theory due to Mawhin. The most interesting point is the proof of the uniqueness of the solution for the higher-order coupled fractional differential equations at resonance. We give an example to demonstrate our results.

1 Introduction

In this article, we study the higher-order coupled fractional differential equation

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)=f (t,v(t),v'(t),v''(t),\ldots,v^{(N-1)}(t) ),\quad 0< t< 1, \\ D_{0^{+}}^{\beta}v(t)=g (t,u(t),u'(t),u''(t),\ldots,u^{(N-1)}(t) ), \quad 0< t< 1, \end{cases} $$

(1.1)

with the coupled two-point boundary conditions

$$ \textstyle\begin{cases} u(0)=u'(0)=\cdots=u^{(N-2)}=0, \qquad u^{(N-1)}(0)=u^{(N-1)}(1), \\ v(0)=v'(0)=\cdots=v^{(N-2)}=0, \qquad v^{(N-1)}(0)=v^{(N-1)}(1), \end{cases} $$

(1.2)

where $N-1<\alpha,\beta<N$, $N\geq2$, $D_{0^{+}}^{\alpha}$ and $D_{0^{+}}^{\beta}$ denote the Caputo fractional derivative, and f, g are given continuous functions.

Fractional differential equations have been studied extensively. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications such as physics, chemistry, phenomena arising in engineering, economy and science; see e.g. [1–4].

Recently, more and more authors paid attention to the boundary value problems of fractional differential equations; see [5–19]. In [6], the author has investigated the existence of solutions to the coupled systems of fractional differential equations at nonresonance. Moreover, there have been many works related to the existence of solutions for boundary value problems at resonance; see [10–13, 15–18, 20, 21]. Some papers have dealt with the solutions of multipoint boundary value problems of a coupled fractional differential equations at resonance; see [12, 17].

In [17], Zhang et al. considered a three-point boundary value problem for a coupled system of nonlinear fractional differential equations at resonance given by

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)=f(t,v(t),D_{0^{+}}^{\beta-1}v(t)),\quad 0< t< 1, \\ D_{0^{+}}^{\beta}v(t)=g(t,u(t),D_{0^{+}}^{\alpha-1}u(t)), \quad 0< t< 1, \\ u(0)=v(0)=0, \qquad u(1)=\sigma_{1} u(\eta_{1}),\qquad v(1)=\sigma_{2} v(\eta_{2}), \end{cases} $$

where $1<\alpha,\beta\leq2$, $0<\eta_{1},\eta_{2} <1$, $\sigma_{1},\sigma_{2} >0$, $\sigma_{1}\eta_{1}^{\alpha-1}=\sigma_{2}\eta_{2}^{\beta-1}=1$, D is Riemann-Liouville fractional derivative, and $f,g:[0,1]\times \mathbb{R}^{2} \rightarrow\mathbb{R}$ are given functions.

In [12], the authors discussed a two-point boundary value problem for a coupled system of fractional differential equations at resonance:

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha}u(t)=f(t,v(t),v'(t)), \quad 0< t< 1, \\ D_{0^{+}}^{\beta}v(t)=g(t,u(t),u'(t)), \quad 0< t< 1, \\ u(0)=v(0)=0, \qquad u'(0)=u'(1),\qquad v'(0)=v'(1), \end{cases} $$

where $D_{0^{+}}^{\alpha}$, $D_{0^{+}}^{\beta}$ is Caputo fractional derivative, $1<\alpha,\beta\leq2$, and $f,g:[0,1]\times \mathbb{R}^{2} \rightarrow\mathbb{R}$ are given function.

From the above work, we see three facts. Firstly, although the two-point boundary value problems for coupled system of fractional differential equations have been studied by some authors, to the best of our knowledge, higher-order fractional differential equations with the Caputo fractional derivative are seldom considered. Secondly, the nonlinear terms in the equations of this paper satisfy a sublinear growth condition that is weaker than the previous ones (see [11, 15]), meanwhile, the present work generalizes and improves the available results (see [5, 12]). Thirdly, the uniqueness of the solution is useful for many applications. As far as we know, there are few contributions to the uniqueness of a solution for fractional differential equations. The objective of this paper is to fill the gap in the relevant literature.

The rest of this paper is organized as follows. In Section 2, we give some necessary notations, definitions and lemmas. In Section 3, we study the existence of solutions of (1.1) and (1.2) by the coincidence degree theory due to Mawhin [22]. Finally, an example is given to illustrate our results in Section 4.

2 Preliminaries

We present the necessary definitions and lemmas from fractional calculus theory that will be used to prove our main theorems.

Definition 2.1

([1])

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

$$I_{0+}^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s) \, ds, $$

provided that the right-hand side is pointwise defined on $(0,\infty)$.

Definition 2.2

([1])

The Riemann-Liouville fractional derivative of order $\alpha>0$ of a continuous function $f:(0,\infty)\to\mathbb{R}$ is given by

$$D_{0^{+}}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)} \int_{0}^{t} \frac {f^{(n)}(s)}{(t-s)^{\alpha-n+1}}\, ds, $$

where $n-1<\alpha\leq n$, provided that the right-hand side is pointwise defined on $(0,\infty)$.

Lemma 2.1

([1])

Let $n-1<\alpha\leq n$, $u\in C(0,1)\cap L^{1}(0,1)$, then

$$I_{0+}^{\alpha}D_{0+}^{\alpha}u(t)=u(t)+c_{0}+ c_{1}t+\cdots+c_{n-1}t^{n-1}, $$

where $c_{i}\in\mathbb{R}$, $i=0,1,\ldots, n-1$.

Lemma 2.2

([1])

If $\beta>0$, $\alpha+\beta>0$, then the equation

$$I_{0+}^{\alpha}I_{0+}^{\beta}f(x)=I_{0+}^{\alpha+\beta}f(x), $$

is satisfied for continuous function f.

Now let us recall some notation about the coincidence degree continuation theorem.

Let Y, Z be real Banach spaces, $L:\operatorname{dom} L \subset Y \to Z$ be a Fredholm map of index zero and $P:Y\to Y$, $Q:Z\to Z$ be continuous projectors such that $\ker L = \operatorname{Im}P$, $\operatorname{Im}L= \ker Q$, and $Y=\ker L \oplus\ker P$, $Z=\operatorname{Im}L \oplus\operatorname{Im}Q$. 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 of this map by $K_{P}$. If Ω is an open bounded subset of Y, the map N will be called L-compact on $\overline{\Omega}$ if $QN(\overline{\Omega})$ is bounded and $K_{P,Q}N=K_{P}(I-Q)N:\overline{\Omega}\to Y$ is compact.

Theorem 2.1

Let L be a Fredholm operator of index zero and N be L-compact on $\overline{\Omega}$. Suppose that the following conditions are satisfied:

• (1) $Lx \neq\lambda Nx$ for each $(x,\lambda) \in [(\operatorname{dom}L\setminus\ker L) \cap \partial\Omega ]\times (0,1)$;

• (2) $Nx \notin\operatorname{Im} L$ for each $x \in\ker L \cap\partial\Omega$;

• (3) $\deg(JQN|_{\ker L}, \Omega\cap\ker L, 0)\neq0$, where $Q:Z\to Z$ is a continuous projection as above with $\operatorname{Im}L= \ker Q$ and $J:\operatorname{Im}Q\to \ker L$ is any isomorphism.

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

3 Main results

In this section, we will prove the existence and uniqueness results for (1.1) and (1.2).

We use the Banach space $E=C[0,1]$ with the norm $\|u\|_{\infty}=\max_{0 \le t \le1}|u(t)|$. We define a linear space $X =\{u^{(i)}\in E:i=1,2,\ldots,N-1\}$. By means of the linear functional analysis theory, we can prove that X is a Banach space with the norm ${\Vert x \Vert }_{X}=\max \{\Vert u \Vert _{\infty}, \Vert u' \Vert _{\infty},\ldots, \Vert {{u}^{(N-1)}} \Vert _{\infty}\}$. Further we consider a Banach space $Y=X\times X$ endowed with the norm defined by $\|(u,v)\|_{Y}=\max\{\|u\|_{X},\|v\|_{X} \}$, and $Z=E\times E$ is a Banach space with the norm defined by $\|(x,y)\|_{Z}=\max \{\|x\|_{\infty},\|y\|_{\infty}\}$.

Define the linear operator $L_{1}$ from $\operatorname{dom}L_{1}\cap X$ to E by

$$L_{1}u=D_{0+}^{\alpha}u, $$

where $\operatorname{dom}L_{1}=\{u\in X|u^{(i)}(0)=0, u^{(N-1)}(0)=u^{(N-1)}(1), i=0,1,\ldots, N-2\}$.

Define the linear operator $L_{2}$ from $\operatorname{dom}L_{2}\cap X$ to E by

$$L_{2}v=D_{0+}^{\beta}v, $$

where $\operatorname{dom}L_{2}=\{v\in X|v^{(i)}(0)=0, v^{(N-1)}(0)=v^{(N-1)}(1), i=0,1,\ldots, N-2\}$.

Define the operator $L: \operatorname{dom}L\cap Y \rightarrow Z$ by

$$L(u,v)=(L_{1}u,L_{2}v), $$

where $\operatorname{dom}L=\{(u,v)\in Y|u\in\operatorname{dom}L_{1},v\in\operatorname{dom}L_{2}\}$, and we define $N:Y\to Z$ by setting

$$N(u,v)=(N_{1}v,N_{2}u), $$

where $N_{1}:X\to E$ is defined by

$$N_{1}v(t)=f\bigl(t,v(t),v'(t),v''(t), \ldots,v^{(N-1)}(t)\bigr), $$

and $N_{2}:X\to E$ is defined by

$$N_{2}u(t)=g\bigl(t,u(t),u'(t),u''(t), \ldots,u^{(N-1)}(t)\bigr). $$

Then the problem (1.1) and (1.2) can be written by $L(u,v)=N(u,v)$.

Lemma 3.1

The mapping $L:\operatorname{dom} L \subset Y\rightarrow Z$ is a Fredholm operator of index zero.

Proof

It is clear that $\operatorname{Ker}L=(c_{1}t^{N-1},c_{2}t^{N-1})\cong\mathbb{R}^{2}$.

Let $(x,y)\in\operatorname{Im}L$, so there exists $(u,v)\in\operatorname{dom}L$ which satisfies $L(u,v)=(x,y)$. By Lemma 2.1, we have

$$\begin{aligned}& u(t) =I_{0+}^{\alpha}x(t)+{{c}_{0}}+{{c}_{1}}t+ \cdots +{{c}_{N-1}} {{t}^{N-1}}, \\& v(t) =I_{0+}^{\alpha}y(t)+{{d}_{0}}+{{d}_{1}}t+ \cdots +{{d}_{N-1}} {{t}^{N-1}}. \end{aligned}$$

By the definition of domL, we have $c_{i}=d_{i}=0$, $i=0,1,\ldots, N-2$. Hence

$$ u(t)=I_{0+}^{\alpha }x(t)+c_{N-1}t^{N-1},\qquad v(t)=I_{0+}^{\beta}y(t)+d_{N-1}t^{N-1}. $$

According to Lemma 2.2, we get

$$\begin{aligned}& {{u}^{(N-1)}}(t)=D_{0+}^{N-1} \bigl(I_{0+}^{\alpha }x(t)+{{c}_{N-1}} {{t}^{N -1}} \bigr)=I_{0+}^{\alpha -N+1}x(t)+{{c}_{N-1}}(N-1)!, \\& {{v}^{(N-1)}}(t)=D_{0+}^{N-1} \bigl(I_{0+}^{\beta }y(t)+{{d}_{N-1}} {{t}^{N -1}} \bigr)=I_{0+}^{\beta-N+1}y(t)+{{d}_{N-1}}(N-1)!. \end{aligned}$$

Taking into account $u^{(N-1)}(0)=u^{(N-1)}(1)$ and $v^{(N-1)}(0)=v^{(N-1)}(1)$, we obtain

$$\int_{0}^{1}{{{(1-s)}^{\alpha -N}}x}(s)\, ds=0, \qquad \int_{0}^{1}{{{(1-s)}^{\beta-N}}y}(s) \, ds=0. $$

On the other hand, suppose $(x,y)$ satisfies the above equations. Let $u(t)=I_{0+}^{\alpha}x(t)$ and $v(t)=I_{0+}^{\beta}y(t)$, we can easily prove $(u(t),v(t) )\in\operatorname{dom} L$.

Thus, we conclude that

$$\operatorname{Im} L = \biggl\{ (x,y) \Big|\int_{0}^{1}{{{(1-s)}^{\alpha -N}}x}(s) \, ds=0; \int_{0}^{1}{{{(1-s)}^{\beta -N}}y}(s) \, ds=0 \biggr\} . $$

Consider the linear operators $Q_{1},Q_{2} :E\rightarrow E$ defined by

$$\begin{aligned}& Q_{1}x(t)=(\alpha-N+1)\int_{0}^{1}{{{(1-s)}^{\alpha-N}}x}(s) \, ds, \\& Q_{2}y(t)=(\beta-N+1)\int_{0}^{1}{{{(1-s)}^{\beta-N}}y}(s) \, ds. \end{aligned}$$

Obviously, $Q(x,y)= (Q_{1}x(t),Q_{2}y(t) )\cong\mathbb{R}^{2}$. Taking $x(t)\in E$, by a direct computation, we have

$$\begin{aligned} {{Q}_{1}}\bigl({{Q}_{1}}x(t)\bigr)&={{Q}_{1}}x(t) \cdot(\alpha -N+1)\int_{0}^{1}{{{(1-s)}^{\alpha-N}}} \, ds \\ &={{Q}_{1}}x(t). \end{aligned}$$

Similarly, $Q_{2}^{2}=Q_{2}$. This gives $Q^{2}(x,y)=Q(x,y)$. It is easy to check from $(x,y)=(x,y)-Q(x,y)+Q(x,y)$ that $Z= \operatorname{Im}L+\operatorname{Im}Q$. Moreover, we can see that $Z=\operatorname{Im}L\oplus \operatorname{Im}Q$.

Now, $\operatorname{Ind}L = \operatorname{dim} \ker L - \operatorname{codim} \operatorname{Im}L = 0$, and so L is a Fredholm mapping of index zero. □

We can define the operators $P_{1}:X\rightarrow X$, $P_{2}:X\rightarrow X$ and $P:(u,v)\rightarrow(P_{1}u,P_{2}v)$, where

$${{P}_{1}}u=\frac{{{u}^{(N-1)}}(0)}{(N-1)!}{{t}^{N-1}},\qquad {{P}_{2}}v=\frac{{{v}^{(N-1)}}(0)}{(N-1)!}{{t}^{N-1}}. $$

Obviously, $P_{1}^{2}=P_{1}$ and $P_{2}^{2}=P_{2}$.

Note that

$$\operatorname{Ker}P= \bigl\{ (u,v)|{{u}^{(N-1)}}(0)=0, {{v}^{(N-1)}}(0)=0 \bigr\} . $$

Since $(u,v)=(u,v)-P(u,v)+P(u,v)$, it is clear that $Y= \operatorname{Ker}P+\operatorname{Ker}L$. By a simple calculation, we get $\operatorname{Ker}L\cap\operatorname{Ker}P=\{(0,0)\}$. Thus, we get $Y= \operatorname{Ker}L\oplus\operatorname{Ker}P$.

For every $(u,v)\in Y$,

$$\begin{aligned} \bigl\Vert P(u,v)\bigr\Vert _{Y} &=\bigl\Vert (P_{1}u,P_{2}v)\bigr\Vert _{Y}=\max \bigl\{ \Vert P_{1}u\Vert _{X};\|P_{2}v\| _{X} \bigr\} \\ & =\max \biggl\{ \frac{\vert {{u}^{(N-1)}}(0) \vert }{(N-1)!}{{\bigl\Vert {{t}^{N-1}} \bigr\Vert }_{X}}; \frac{\vert {{v}^{(N-1)}}(0) \vert }{(N-1)!}{{\bigl\Vert {{t}^{N-1}} \bigr\Vert }_{X}} \biggr\} \\ &\leq\max \bigl\{ \bigl\vert {u}^{(N-1)}(0)\bigr\vert ; \bigl\vert {v}^{(N-1)}(0)\bigr\vert \bigr\} . \end{aligned}$$

(3.1)

We define $K_{P}:\operatorname{Im}L\to \operatorname{dom}L \cap\operatorname{Ker} P$ by $K_{P}(x,y)= (I_{0+}^{\alpha}x,I_{0+}^{\beta}y )$.

For $(x,y)\in\operatorname{Im} L$, we have

$$LK_{P}(x,y)=L\bigl(I_{0+}^{\alpha}x,I_{0+}^{\beta}y \bigr)=\bigl(D_{0+}^{\alpha }I_{0+}^{\alpha}x,D_{0+}^{\beta}I_{0+}^{\beta}y \bigr)=(x,y). $$

For $(u,v)\in\operatorname{dom} L \cap\operatorname{Ker} P$, we have ${u}^{(N-1)}(0)={v}^{(N-1)}(0)=0$, so the coefficients $c_{i}$ and $d_{i}$, $i=0,1,\ldots,N-1$, in the expressions

$$\begin{aligned}& u=I_{0+}^{\alpha}D_{0+}^{\alpha}u(t)+{{c}_{0}}+{{c}_{1}}t+ \cdots +{{c}_{N-1}} {{t}^{N-1}}, \\& v=I_{0+}^{\beta}D_{0+}^{\beta}u(t)+{{d}_{0}}+{{d}_{1}}t+ \cdots +{{d}_{N-1}} {{t}^{N-1}}, \end{aligned}$$

are all equal to zero. Thus, we obtain

$$K_{p}L(x,y)= \bigl(I_{0+}^{\alpha}D_{0+}^{\alpha}x,I_{0+}^{\beta }D_{0+}^{\beta}y \bigr)=(x,y). $$

That shows that $K_{P}=(L_{\operatorname{dom} L \cap\operatorname{Ker} P })^{-1}$.

Again, for each $(x,y)\in\operatorname{Im} L$,

$$\begin{aligned} \bigl\Vert K_{P}(x,y)\bigr\Vert _{Y} &= \bigl\Vert \bigl(I_{0+}^{\alpha}x,I_{0+}^{\beta}y \bigr)\bigr\Vert _{Y} =\max \bigl\{ \bigl\Vert I_{0+}^{\alpha}x\bigr\Vert _{X}; \bigl\Vert I_{0+}^{\beta}y \bigr\Vert _{X} \bigr\} \\ &\leq\max \biggl\{ \frac{1}{\Gamma(\alpha-N+2)}\|x\|_{\infty}; \frac {1}{\Gamma(\beta-N+2)}\|y \|_{\infty}\biggr\} \\ &=\max\bigl\{ a\Vert x\Vert _{\infty}; b\|y\|_{\infty}\bigr\} , \end{aligned}$$

(3.2)

where $a=\frac{1}{\Gamma(\alpha-N+2)}$, $b=\frac{1}{\Gamma(\beta-N+2)}$.

With similar arguments to [5], we obtain the following lemma.

Lemma 3.2

$K_{P}(I-Q)N:Y\rightarrow Y$ is completely continuous.

To obtain our main results, we need the following conditions.

(H₁):

There exist positive constants $a_{1}$, $a_{2}$, $b_{i}$, $c_{i}$, and ${\theta_{i}},{\lambda_{i}}\in[0,1]$, $i=1,2,\ldots,N$, such that for all $(x_{1},x_{2},\ldots,x_{N})\in\mathbb{R}^{N}$,

$$\begin{aligned}& \bigl\vert f(t,x_{1},x_{2},\ldots,x_{N})\bigr\vert \leq a_{1}+ b_{1}|x_{1}|^{\theta_{1}}+b_{2}|x_{2}|^{\theta _{2}}+ \cdots+b_{N}|x_{N}|^{\theta_{N}}, \quad \forall t\in[0,1], \\& \bigl\vert g(t,x_{1},x_{2},\ldots,x_{N})\bigr\vert \leq a_{2}+ c_{1}|x_{1}|^{\lambda _{1}}+c_{2}|x_{2}|^{\lambda_{2}}+ \cdots+c_{N}|x_{N}|^{\lambda_{N}},\quad \forall t\in[0,1]. \end{aligned}$$

(H₂):

There exists a constant $A>0$ such that for any $c_{1},c_{2}\in\mathbb{R}^{2}$, if $\min\{|c_{1}|,|c_{2}|\}>A$, one has either

$$c_{1}\cdot N_{1}\bigl(c_{2}t^{N-1}\bigr)> 0, \qquad c_{2}\cdot N_{2}\bigl(c_{1}t^{N-1} \bigr)>0, $$

or

$$c_{1}\cdot N_{1}\bigl(c_{2}t^{N-1}\bigr)< 0,\qquad c_{2}\cdot N_{2}\bigl(c_{1}t^{N-1} \bigr)< 0. $$

(H₃):

$\max \{2a\sum_{i=1}^{N}{{{b}_{i}}}, a\sum_{i=1}^{N}{{{b}_{i}}}+b\sum_{i=1}^{N}{{{c}_{i}}}, 2b\sum_{i=1}^{N}{{{c}_{i}}} \}<1$.

Lemma 3.3

$\Omega_{1}= \{(u,v)\in\operatorname{dom}L\setminus\operatorname{Ker}L: L(u,v)= \lambda N(u,v),\lambda\in[0,1] \}$ is bounded.

Proof

For $(u,v)\in\Omega_{1}$, thus $\lambda\neq0$. Also, $L(u,v)=\lambda N(u,v)\in\operatorname{Im}L=\operatorname{Ker}Q$, that is,

$$\begin{aligned}& \lambda(\alpha-N+1)\int_{0}^{1}{{{(1-s)}^{\alpha-N}}f \bigl(s,v(s),v'(s),v''(s ), \ldots,v^{(N-1)}(s) \bigr)}\, ds=0, \\& \lambda (\beta-N+1)\int_{0}^{1}{{{(1-s)}^{\beta -N}}g \bigl(s,u(s),u'(s),u''(s), \ldots,u^{(N-1)}(s) \bigr)}\, ds=0. \end{aligned}$$

By the integral mean value theorem, there exist $t_{0},t_{1}\in[0,1]$ such that

$$\begin{aligned}& f\bigl(t_{0},v(t_{0}),v'(t_{0}),v''(t_{0} ),\ldots,v^{(N-1)}(t_{0})\bigr) =0, \\& g\bigl(t_{1},u(t_{1}),u'(t_{1}),u''(t_{1} ),\ldots,u^{(N-1)}(t_{1})\bigr) =0. \end{aligned}$$

From (H₂), we get $|u^{(N-1)}(t_{1})|\leq A$ and $| v^{(N-1)}(t_{0})| \leq A$.

Again for $(u,v)\in\Omega_{1}$, $(u,v)\in\operatorname{dom}(L)\setminus \operatorname{Ker}(L)$, then $(I-P)(u,v)\in\operatorname{dom}L\cap\operatorname{Ker}P$ and $LP(u,v)=(0,0)$, thus from (3.2), we have

$$\begin{aligned} \bigl\Vert (I-P) (u,v)\bigr\Vert _{Y} &=\bigl\Vert K_{P}L(I-P) (u,v)\bigr\Vert _{Y}=\bigl\Vert K_{P}(L_{1}u,L_{2}v)\bigr\Vert _{Y} \\ &\leq\max\bigl\{ a\Vert N_{1}v\Vert _{\infty}; b \|N_{2}u\|_{\infty}\bigr\} . \end{aligned}$$

(3.3)

By $Lu=\lambda Nu$ and $u\in\operatorname{dom}L$, we have

$$\begin{aligned} u(t) =&\frac{1}{\Gamma ( \alpha )}\int_{0}^{t}{{{ ( t-s )}^{\alpha-1}} f \bigl( s,v(s),\ldots,{{v}^{(N-1)}}(s) \bigr)}\, ds \\ &{}-u(0)- u'(0)t-\cdots-\frac{{{u}^{(N-1)}}(0)}{ ( N-1 )!}{{t}^{N-1}}. \end{aligned}$$

Furthermore, we have

$${{u}^{(N-1)}}(t)=\frac{1}{\Gamma( \alpha-N+1 )}\int_{0}^{t}{{{ ( t-s )}^{\alpha-N}} f \bigl( s,v(s), \ldots,{{v}^{(N-1)}}(s) \bigr)}\, ds-{{u}^{(N-1)}}(0). $$

Substituting $t=t_{1}$ into the above equation, we get

$${{u}^{(N-1)}}(t_{1})=\frac{1}{\Gamma ( \alpha-N+1 )}\int _{0}^{ t_{1}}{{{ ( t_{1}-s )}^{\alpha-N}} f \bigl( s,v(s), \ldots,{{v}^{(N-1)}}(s) \bigr)}\, ds-{{u}^{(N-1)}}(0). $$

Together with $|u^{(N-1)}(t_{1})|\leq A$, we derive that

$$\begin{aligned} \bigl\vert {{u}^{(N-1)}}(0) \bigr\vert \le&\biggl\vert \frac{1}{\Gamma ( \alpha-N+1 )}\int_{0}^{{{t}_{1}}}{{{ ( {{t}_{1}}-s )}^{\alpha-N}}f \bigl( s,v(s),\ldots,{{v}^{(N-1)}}(s) \bigr)}\, ds \biggr\vert \\ &{}+ \bigl\vert {{u}^{(N-1)}}({{t}_{1}}) \bigr\vert \\ \le& A+\frac{1}{\Gamma ( \alpha-N+1 )}\int_{0}^{{t}_{1}} {{ ( {{t}_{1}}-s )}^{\alpha-N}}\bigl\vert f \bigl( s,v(s), \ldots,{{v}^{(N-1)}}(s) \bigr)\bigr\vert \, ds \\ \leq& A+\frac{1}{\Gamma ( \alpha-N+1 )}\int_{0}^{{{t}_{1}}}{{{ ( {{t}_{1}}-s )}^{\alpha -N}}} \Biggl( {a_{1}}+{\sum _{i=1}^{N}b_{i}\bigl\vert v^{(i-1)} \bigr\vert ^{\theta_{i}}} \Biggr)\, ds \\ \leq& A+\frac{1}{\Gamma ( \alpha-N+1 )} \Biggl( {a_{1}}+{\sum _{i=1}^{N}b_{i}\bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}} \Biggr)\cdot\int _{0}^{{{t}_{1}}} (t_{1}-s )^{\alpha -N}\, ds \\ \leq& A+a{a_{1}}+a{\sum_{i=1}^{N}b_{i} \bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}} . \end{aligned}$$

(3.4)

With similar arguments, we obtain

$$ \bigl\vert {{v}^{(N-1)}}(0) \bigr\vert \le A+{{a}_{2}b}+ b{\sum_{i=1}^{N}c_{i} \bigl\Vert u^{(i-1)} \bigr\Vert ^{\lambda_{i}}_{\infty}} . $$

(3.5)

From (3.1) and (3.3), we have

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{Y} =&\bigl\Vert P(u,v)+(I-P) (u,v)\bigr\Vert _{Y} \\ \leq&\bigl\Vert P(u,v)\bigr\Vert _{Y}+\bigl\Vert (I-P) (u,v)\bigr\Vert _{Y} \\ \leq&\max \bigl\{ \bigl\vert {{u}^{(N-1)}}(0)\bigr\vert +a \|N_{1}v\|_{\infty}, \bigl\vert {{u}^{(N-1)}}(0)\bigr\vert +b\|N_{2}u\|_{\infty}, \\ &\bigl\vert {{v}^{(N-1)}}(0)\bigr\vert +a\|N_{1}v \|_{\infty}, \bigl\vert {{v}^{(N-1)}}(0)\bigr\vert +b \|N_{2}u\|_{\infty}\bigr\} . \end{aligned}$$

In what follows, the proof can be divided into four cases.

Case 1. $\|(u,v)\|_{Y} \leq \vert {{u}^{(N-1)}}(0) \vert +a\|N_{1}v\|_{\infty}$.

By (3.4) and (H₁), we have

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{Y} &\leq \bigl\vert {{u}^{(N-1)}}(0) \bigr\vert +a\|N_{1}v\|_{\infty}\\ &\leq A+a{a_{1}}+a{\sum_{i=1}^{N}b_{i} \bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}}+a \|N_{1}v\|_{\infty}\\ &\leq A+a{a_{1}}+a{\sum_{i=1}^{N}b_{i} \bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}}+a \bigl\Vert f \bigl(t,v(t), v'(t),\ldots,v^{(N-1)}(t) \bigr)\bigr\Vert _{\infty}\\ &\leq A+2a{a_{1}}+a{\sum_{i=1}^{N}b_{i} \bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}}+a{\sum _{i=1}^{N}b_{i}\bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}} \\ &= A+2a{{a}_{1}}+2a\sum_{i=1}^{N}{{{b}_{i}}} \bigl\Vert v ^{(i-1)}\bigr\Vert ^{\theta_{i}}_{\infty}. \end{aligned}$$

According to (H₃) and the definition of $\|(u,v)\|_{Y}$, we can derive $\|v\|_{X}$ are bounded. Therefore $\Omega_{1}$ is bounded.

Case 2. $\|(u,v)\|_{Y} \leq | {{v}^{(N-1)}}(0) |+b\|N_{2}u\|_{\infty}$. The proof is similar to Case 1. Here, we omit it.

Case 3. $\|(u,v)\|_{Y} \leq | {{u}^{(N-1)}}(0) |+b\|N_{2}u\|_{\infty}$.

From (3.4) and (H₁), we obtain

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{Y} &\leq \bigl\vert {{u}^{(N-1)}}(0) \bigr\vert +b\|N_{2}u\|_{\infty}\\ &\leq A+a{a_{1}}+a{\sum_{i=1}^{N}b_{i} \bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}}+b \|N_{2}u\|_{\infty}\\ &\leq A+a{a_{1}}+a{\sum_{i=1}^{N}b_{i} \bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}}+b\bigl\Vert g \bigl(t,u(t), u'(t),\ldots ,u^{(N-1)}(t) \bigr)\bigr\Vert _{\infty}\\ &\leq A+a{a_{1}}+ba_{2}+a{\sum _{i=1}^{N}b_{i}\bigl\Vert v^{(i-1)} \bigr\Vert ^{\theta_{i}}_{\infty}} + b\sum _{i=1}^{N}c_{i}\bigl\Vert u^{(i-1)}\bigr\Vert ^{\lambda_{i}}_{\infty}. \end{aligned}$$

By (H₃), we easily conclude that $\|(u,v)\|_{Y}$ is bounded. Therefore $\Omega_{1}$ is bounded.

Case 4. $\|(u,v)\|_{\infty}\leq | {{v}^{(N-1)}}(0) |+a\|N_{1}v\|_{\infty}$. The proof is similar to the Case 2. Here, we omit it.

According to the above arguments, we prove that $\Omega_{1}$ is bounded. □

Lemma 3.4

$\Omega_{2}= \{(u,v)\in\operatorname{Ker}L:N(u,v)\in\operatorname{Im}L \}$ is bounded.

Proof

Let $(u,v)\in\operatorname{Ker}L$, so we have $u=c_{1}t^{N-1}$, $v=c_{2}t^{N-1}$, $c_{1},c_{2}\in\mathbb{R}$. In view of $N(u,v)=(N_{1}v,N_{2}u)\in\operatorname{Im}L=\operatorname{Ker}Q$, we have

$$\begin{aligned}& \int_{0}^{1}(1-s)^{\alpha-N}f \bigl(t,c_{2}t^{N-1},(N-1)c_{2}t^{N-2}, \ldots ,c_{2}(N-1)! \bigr)\, dt=0, \\& \int_{0}^{1}(1-s)^{\beta-N}g \bigl(t,c_{1}t^{N-1},(N-1)c_{1}t^{N-2}, \ldots ,c_{1}(N-1)! \bigr)\, dt=0. \end{aligned}$$

By the integral mean value theorem, there exist constants $t_{0}, t_{1}\in[0,1]$ such that

$$\begin{aligned}& f \bigl({{t}_{0}},{{c}_{2}}t_{0}^{N-1},(N-1){{c}_{2}}t_{0}^{N-2}, \ldots ,{{c}_{2}}(N-1)! \bigr)=0, \\& g \bigl({{t}_{1}},{{c}_{1}}t_{1}^{N-1},(N-1){{c}_{1}}t_{1}^{N-2}, \ldots ,{{c}_{1}}(N-1)! \bigr)=0, \end{aligned}$$

which together with (H₂) imply $|c_{i}|\leq\frac{A}{ (N-1)!}$, $i=1,2$. Hence, $\Omega_{2}$ is bounded. □

Lemma 3.5

$\Omega_{3}= \{(u,v)\in\operatorname{Ker}L:\lambda(u,v)+(1-\lambda) QN(u,v)=(0,0),\lambda\in[0,1] \}$ is bounded.

Proof

Let $(u,v)\in\operatorname{Ker}L$, so we have $u=c_{1}t^{N-1}$, $v=c_{2}t^{N-1}$, $c_{1},c_{2}\in\mathbb{R}$, and

$$ \lambda c_{1}t^{N-1}+(1-\lambda)Q_{1}N_{1}(v)=0, \qquad \lambda c_{2}t^{N-1}+(1-\lambda)Q_{2}N_{2}(u)=0, $$

that is to say,

$$\begin{aligned}& \lambda c_{1}t^{N-1}+(1-\lambda)\int_{0}^{1}f \bigl(t,c_{2}t^{N-1},c_{2}(N-1)t^{N-2}, \ldots ,c_{2}(N-1)! \bigr)\, dt=0, \end{aligned}$$

(3.6)

$$\begin{aligned}& \lambda c_{2}t^{N-1}+(1-\lambda)\int_{0}^{1}g \bigl(t,c_{1}t^{N-1},c_{1}(N-1)t^{N-2}, \ldots ,c_{1}(N-1)! \bigr)\, dt=0. \end{aligned}$$

(3.7)

If $\lambda=0$, then $|c_{i}|\leq\frac{A}{ (N-1)!}$, $i=1,2$. If $\lambda\in(0,1]$, then we can have $|c_{i}|\leq\frac{A}{ (N-1)!}$, $i=1,2$. Otherwise, if $|c_{i}|>\frac{A}{ (N-1)!}$, $i=1,2$, in view of the first part of (H₂), one has

$$\begin{aligned}& \lambda c_{1}^{2}t^{N-1} +(1-\lambda)\int _{0}^{1}c_{1} f \bigl(t,c_{2}t^{N-1},c_{2}(N-1)t^{N-2}, \ldots,c_{2}(N-1)! \bigr)\, dt>0, \\& \lambda c_{2}^{2}t^{N-1}+(1-\lambda)\int _{0}^{1}c_{2} g \bigl(t,c_{1}t^{N-1},c_{1}(N-1)t^{N-2}, \ldots,c_{1}(N-1)! \bigr)\, dt>0, \end{aligned}$$

which contradict (3.6) and (3.7). Thus, $\Omega_{3}$ is bounded. □

Remark 3.1

If the second part of (H₂) holds, then the set

$$\Omega_{3}'= \bigl\{ (u,v)\in\operatorname{Ker}L:- \lambda(u,v)+(1-\lambda) QN(u,v)=(0,0),\lambda\in[0,1] \bigr\} $$

is bounded.

Theorem 3.1

Suppose (H₁)-(H₃) hold, then the problem (1.1) and (1.2) has at least on solution in Y.

Proof

Let Ω to be a bounded open subset of Y, such that $\bigcup_{i=1}^{3}{\overline{\Omega}_{i}}\subset\Omega$. It follows from Lemma 3.2 that N is L-compact on Ω. By Lemma 3.3, Lemma 3.4, and Lemma 3.5, we get:

• (1) $Lu\neq\lambda Nu$, for every $(u,v,\lambda)\in[( \operatorname{dom}L\setminus\operatorname{Ker}L)\cap\partial\Omega]\times(0,1)$.

• (2) $Nu\notin\operatorname{Im}L$ for every $u\in\operatorname{Ker}L\cap\partial\Omega$.

• (3) Let $H((u,v),\lambda)=\pm\lambda I(u,v)+(1-\lambda)JQN(u,v)$, where I is the identical operator. Via the homotopy property of the degree, we obtain

  $$\begin{aligned} \deg (JQ N|_{\ker L},\Omega\cap\ker L,0 ) &= \deg \bigl(H(\cdot,0),\Omega \cap\ker L,0 \bigr) \\ &= \deg \bigl(H(\cdot,1),\Omega\cap\ker L,0 \bigr) \\ &= \deg (I,\Omega\cap\ker L,0 )=1\neq0. \end{aligned}$$

Applying Theorem 2.1, we conclude that $L(u,v)=N(u,v)$ has at least one solution in $\operatorname{dom}L\cap\overline{\Omega}$. □

Under the stronger conditions imposed on f, we can prove the uniqueness of the solutions to BVP (1.1) and (1.2).

Theorem 3.2

Suppose the condition (H₁) in Theorem  2.1 is replaced by the following conditions:

($\mathrm{H}'_{1}$):

There exist positive constants $a_{i}$, $b_{i}$, $i=0,1,\ldots ,N-1$, such that for all $(x_{1},x_{2},\ldots,x_{N}), (y_{1},y_{2},\ldots,y_{N})\in\mathbb{R}^{N}$ one has

$$\begin{aligned}& \bigl\vert f(t,x_{1},x_{2},\ldots,x_{N})-f(t,y_{1},y_{2}, \ldots,y_{N})\bigr\vert \leq a_{0}|x_{1}-y_{1}|+ \cdots+a_{N-1}|x_{N}-y_{N}|, \\& \bigl\vert g(t,x_{1},x_{2},\ldots,x_{N})-g(t,y_{1},y_{2}, \ldots,y_{N})\bigr\vert \leq b_{0}|x_{1}-y_{1}|+ \cdots+b_{N-1}|x_{N}-y_{N}|. \end{aligned}$$

($\mathrm{H}''_{1}$):

There exist positive constants $k_{i}$, $l_{i}$, $i=0,1,\ldots,N-1$, such that for all $(x_{1},x_{2},\ldots,x_{N}), (y_{1},y_{2},\ldots,y_{N})\in\mathbb{R}^{N}$, one has

$$\begin{aligned}& \bigl\vert f (t,x_{1},x_{2},\ldots,x_{N})-f(t,y_{1},y_{2}, \ldots,y_{N})\bigr\vert \\& \quad \geq l_{N-1}|x_{N}-y_{N}|-l_{0}|x_{1}-y_{1}|-l_{1}|x_{2}-y_{2}|- \cdots -l_{N-2}|x_{N-1}-y_{N-1}|, \\& \bigl\vert g(t,x_{1},x_{2},\ldots,x_{N})-g(t,y_{1},y_{2}, \ldots,y_{N})\bigr\vert \\& \quad \geq k_{N-1}|x_{N}-y_{N}|-k_{0}|x_{1}-y_{1}|-k_{1}|x_{2}-y_{2}|- \cdots-k_{N-2}|x_{N-1}-y_{N-1}|. \end{aligned}$$

Then BVP (1.1) and (1.2) has a unique solution, provided that

$$\begin{aligned}& \max \{p_{1}+q_{1}+2r_{2}, p_{2}+q_{2}+2r_{1}, \\& \quad p_{1}+q_{1}+r_{1}+r_{2}, p_{2}+q_{2}+r_{1}+r_{2} \}< 1, \end{aligned}$$

(3.8)

where $p_{1}=\frac{l_{0}}{l_{N-1}}$, $p_{2}=\frac{{{k}_{0}}}{{{k}_{N-1}}}$, $q_{1}=\sum_{i=1}^{N-2}\frac{l_{i}}{l_{N-1}}$, $q_{2}=\sum_{i=1}^{N-2}{\frac{{{k}_{i}}}{{{k}_{N-1}}}}$, $r_{1}=a\sum_{i=0}^{N-1}{{{a}_{i}}}$, $r_{2}=b\sum_{i=0}^{N-1}{{{b}_{i}}}$.

Proof

We let $y_{i}=0$, $i=1,2,\ldots,N$, $a_{1}=\max_{t\in[0,1]}|f(t,0,\ldots ,0)|$ and $a_{2}=\max_{t\in[0,1]}|g(t,0, \ldots,0)|$, then from (3.8) we can show that the condition (H₁) is satisfied. Therefore, the existence of a solution for the coupled system (1.1) and (1.2) follows from Theorem 3.1.

Suppose $(u_{i},v_{i})\in Y$, $i=1,2$ are two solutions of BVP (1.1) and (1.2), then

$$\begin{aligned}& D_{0^{+}}^{\alpha}u_{i}(t)=f \bigl(t,v_{i}(t),v_{i}'(t), \ldots,v^{(N-1)}_{i}(t) \bigr), \\& D_{0^{+}}^{\beta}v_{i}(t)=g \bigl(t,u_{i}(t),u_{i}'(t), \ldots,u^{(N-1)}_{i}(t) \bigr). \end{aligned}$$

Note $u=u_{1}-u_{2}$, $v=v_{1}-v_{2}$, thus we have the following equations:

$$ \begin{aligned} &D_{0^{+}}^{\alpha}u(t)=f \bigl(t,v_{1}(t),v_{1}'(t),\ldots,v^{(N-1)}_{1}(t) \bigr)- f \bigl(t,v_{2}(t),v_{2}'(t), \ldots,v^{(N-1)}_{2}(t) \bigr), \\ &D_{0^{+}}^{\beta}v(t)=g \bigl(t,u_{1}(t),u_{1}'(t), \ldots,u^{(N-1)}_{1}(t) \bigr)- g \bigl(t,u_{2}(t),u_{2}'(t), \ldots,u^{(N-1)}_{2}(t) \bigr). \end{aligned} $$

(3.9)

By $\operatorname{Im}L=\operatorname{Ker}Q$, we have

$$\begin{aligned}& \int_{0}^{1}(1-s)^{\alpha-N}f \bigl(t,v_{1}(t),v_{1}'(t),\ldots,v^{(N-1)}_{1}(t) \bigr)- f \bigl(t,v_{2}(t),v_{2}'(t), \ldots,v^{(N-1)}_{2}(t) \bigr)\, dt=0, \\& \int_{0}^{1}(1-s)^{\beta-N}g \bigl(t,u_{1}(t),u_{1}'(t),\ldots,u^{(N-1)}_{1}(t) \bigr)- g \bigl(t,u_{2}(t),u_{2}'(t), \ldots,u^{(N-1)}_{2}(t) \bigr)\, dt=0. \end{aligned}$$

By the integral mean value theorem, there exist $\xi,\eta\in[0,1]$, such that

$$\begin{aligned}& f \bigl(\xi,v_{1}(\xi),v_{1}'(\xi), \ldots,v_{1}^{(N-1)}(\xi) \bigr)- f \bigl(\xi,v_{2}( \xi),v_{2}'(\xi),\ldots,v_{2}^{(N-1)}(\xi) \bigr)=0, \\& g \bigl(\eta,u_{1}(\eta),u_{1}'(\eta), \ldots,u_{1}^{(N-1)}(\eta) \bigr)- g \bigl(\eta,u_{2}( \eta),u_{2}'(\eta),\ldots,u_{2}^{(N-1)}( \eta) \bigr)=0. \end{aligned}$$

By ($\mathrm{H}''_{1}$), we have

$$\begin{aligned}& 0=\bigl\vert f\bigl(\xi,v_{1}(\xi),v_{1}'( \xi),\ldots,v_{1}^{(N-1)}(\xi)\bigr)- f\bigl(\xi,v_{2}( \xi),v_{2}'(\xi),\ldots,v_{2}^{(N-1)}(\xi) \bigr)\bigr\vert \\& \hphantom{0}\geq l_{N-1}\bigl\vert v^{(N-1)}(\xi)\bigr\vert -l_{0}\bigl\vert v(\xi)\bigr\vert -l_{1}\bigl\vert v'(\xi)\bigr\vert -\cdots -l_{N-2}\bigl\vert v^{(N-2)}(\xi)\bigr\vert , \\& 0=\bigl\vert g\bigl(\eta,u_{1}(\eta),u_{1}'( \eta),\ldots,u_{1}^{(N-1)}(\eta)\bigr)- g\bigl( \eta,u_{2}(\eta),u_{2}'(\eta), \ldots,u_{2}^{(N-1)}(\eta)\bigr)\bigr\vert \\& \hphantom{0}\geq{{k}_{N-1}}\bigl\vert {{u}^{(N-1)}}(\eta) \bigr\vert -{{k}_{0}}\bigl\vert u(\eta )\bigr\vert -{{k}_{1}}\bigl\vert u'(\eta)\bigr\vert - \cdots-{{k}_{N-2}}\bigl\vert {{u}^{(N-2)}}(\eta)\bigr\vert . \end{aligned}$$

It follows from the two inequalities above that

$$\begin{aligned} \begin{aligned} \bigl\vert v^{(N-1)}(\xi)\bigr\vert &\leq \frac{l_{0}}{l_{N-1}}\bigl\vert v(\xi)\bigr\vert +\frac{l_{1}}{l_{N-1}}\bigl\vert v'(\xi) \bigr\vert +\cdots+ \frac{l_{N-2}}{l_{N-1}}\bigl\vert v^{(N-2)}(\xi)\bigr\vert \\ &\leq\frac{l_{0}}{l_{N-1}}\|v\|_{\infty}+ \sum_{i=1}^{N-2} \frac{l_{i}}{l_{N-1}}\bigl\Vert v^{(i)}\bigr\Vert _{\infty}\\ &\leq(p_{1}+q_{1})\|v\|_{X} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \bigl\vert {{u}^{(N-1)}}(\eta)\bigr\vert &\le \frac{{{k}_{0}}}{{{k}_{N-1}}}\|u \|_{\infty}+\sum_{i=1}^{N-2}{ \frac{{{k}_{i}}}{{{k}_{N-1}}}} \bigl\Vert {{u}^{(i)}}\bigr\Vert _{\infty} \\ &\leq(p_{2}+q_{2})\|u\|_{X}. \end{aligned}$$

(3.10)

By (3.9), we obtain

$$\begin{aligned} {{u}^{(N-1)}}(t) =&I_{0+}^{\alpha -(N-1)} \bigl[ f \bigl(t,{{v}_{1}}(t),{{v}'_{{{1}}}}(t),\ldots ,v_{1}^{(N-1)}(t) \bigr) \\ &{}-f \bigl(t,{{v}_{2}}(t),{{v}'_{{{2}}}}(t), \ldots ,v_{2}^{(N-1)}(t) \bigr) \bigr]-{{u}^{(N-1)}}(0). \end{aligned}$$

Substituting $t=\eta$ into the above equation, we get

$$\begin{aligned} {{u}^{(N-1)}}(\eta) =& I_{0+}^{\alpha-(N-1)} \bigl[ f \bigl(t,{{v}_{1}}(t),{v}'_{1}(t), \ldots,v_{1}^{(N-1)}(t)\bigr) \\ &{}-f \bigl(t,{{v}_{2}}(t),{{v}'_{{{2}}}}(t), \ldots ,v_{2}^{(N-1)}(t) \bigr) \bigr]_{t=\eta}-{{u}^{(N-1)}}(0). \end{aligned}$$

By ($\mathrm{H}'_{1}$), (3.10) and the definition of $\|v\|_{X}$, we have

$$\begin{aligned} \bigl\vert {{u}^{(N-1)}}(0) \bigr\vert &\le\bigl\vert {{u}^{(N-1)}}(\eta) \bigr\vert + \frac{1}{\Gamma(\alpha-N+1)} \int _{0}^{\eta}{{{(\eta -s)}^{\alpha -N}}}\, ds\cdot \sum_{i=0}^{N-1}{{{a}_{i}}}\bigl\Vert v^{(i)}\bigr\Vert _{\infty} \\ &\le\bigl\vert {{u}^{(N-1)}}(\eta) \bigr\vert + \|v\|_{X} \cdot \frac{1}{\Gamma(\alpha-N+1)} \int_{0}^{\eta}{{{(\eta -s)}^{\alpha -N}}}\, ds\cdot\sum_{i=0}^{N-1}{{{a}_{i}}} \\ &\leq(p_{2}+q_{2})\|u\|_{X}+\|v\|_{X} \cdot a\sum_{i=0}^{N-1}{{{a}_{i}}} \\ &= (p_{2}+q_{2})\|u\|_{X}+ r_{1}\|v \|_{X}. \end{aligned}$$

(3.11)

Similarly,

$$\begin{aligned} \bigl\vert {{v}^{(N-1)}}(0) \bigr\vert &\le \frac{{{l}_{0}}}{{{l}_{1}}}\|v \|_{\infty}+ \sum_{i=2}^{N-1}{ \frac{{{l}_{i}}}{{{l}_{1}}}}\bigl\Vert {{v}^{(i-1)}}\bigr\Vert _{\infty}+ \|u\|_{X}\cdot b\sum_{i=0}^{N-1}{{{b}_{i}}} \\ &= (p_{1}+q_{1})\|v\|_{X}+ r_{2}\|u \|_{X} . \end{aligned}$$

(3.12)

According to (3.3), (3.11), and (3.12), we have

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{Y} =&\bigl\Vert P(u,v)+(I-P) (u,v) \bigr\Vert _{Y} \leq\bigl\Vert P(u,v)\bigr\Vert _{Y}+ \bigl\Vert (I-P) (u,v)\bigr\Vert _{Y} \\ =&\max \bigl\{ \bigl\vert {u}^{(N-1)}(0)\bigr\vert ; \bigl\vert {v}^{(N-1)}(0)\bigr\vert \bigr\} + \max \bigl\{ a\Vert L_{1}u \Vert _{\infty}; b\|L_{2}v\|_{\infty}\bigr\} \\ \leq&\max \bigl\{ (p_{2}+q_{2})\|u\|_{X}+ r_{1}\|v\|_{X}+a\|L_{1}u\|_{\infty}, \\ &(p_{2}+q_{2})\|u\|_{X}+ r_{1}\|v \|_{X}+b\|L_{2}v\|_{\infty}, \\ &(p_{1}+q_{1})\|v\|_{X}+ r_{2}\|u \|_{X}+a\|L_{1}u\|_{\infty}, \\ &(p_{1}+q_{1})\|v\|_{X}+ r_{2}\|u \|_{X}+b\|L_{2}v\|_{\infty} \bigr\} . \end{aligned}$$

Our proof can be divided into four cases.

Case 1. $\|(u,v)\|_{Y}\leq (p_{2}+q_{2})\|u\|_{X}+ r_{1}\|v\|_{X}+a\|L_{1}u\|_{\infty }$.

By ($\mathrm{H}'_{1}$) and the definition of $\|(u,v)\|_{Y}$, we have

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{Y} &\leq(p_{2}+q_{2}) \|u\|_{X}+ r_{1}\|v\|_{X}+a\|L_{1}u \|_{\infty} \\ &\leq(p_{2}+q_{2})\|u\|_{X}+ r_{1}\|v \|_{X}+a \Biggl(a_{0}\|v\|_{\infty}+\sum _{i=1}^{N-1}a_{i}\bigl\Vert v^{(i)}\bigr\Vert _{\infty}\Biggr) \\ &\leq(p_{2}+q_{2})\|u\|_{X}+ 2r_{1}\|v \|_{X} \\ &\le(p_{2}+q_{2}+ 2r_{1})\bigl\Vert (u,v)\bigr\Vert _{Y} . \end{aligned}$$

(3.13)

By the assumption (3.8), the coefficient on the right side of (3.13) is less than 1. So we have $\|u\|_{X}=\|v\|_{X}=0$, i.e., $u_{1}=u_{2}$, $v_{1}=v_{2}$.

Case 2. $\|(u,v)\|_{Y}\leq(p_{1}+q_{1})\|v\|_{X}+ r_{2}\|u\|_{X}+b\|L_{2}v\|_{\infty}$. The proof is similar to Case 1. So we omit it.

Case 3. $\|(u,v)\|_{\infty}\leq(p_{2}+q_{2})\|u\|_{X}+ r_{1}\|v\|_{X}+b\|L_{2}v\|_{\infty}$.

By ($\mathrm{H}'_{1}$) and the definition of $\|(u,v)\|_{Y}$, we have

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{Y}& \leq(p_{2}+q_{2})\|u\|_{X}+ r_{1}\|v \|_{X}+b\|L_{2}v\|_{\infty} \\ &\leq(p_{2}+q_{2})\|u\|_{X}+ r_{1}\|v \|_{X}+b \Biggl({{b}_{0}} {{\Vert u \Vert }_{\infty}}+{{\sum_{i=1}^{N-1}{{{b}_{i}} \bigl\Vert {{u}^{(i)}} \bigr\Vert _{\infty}} }} \Biggr) \\ &\leq(p_{2}+q_{2}+r_{2})\|u\|_{X}+ r_{1}\|v\|_{X} \\ &\leq(p_{2}+q_{2}+ r_{1}+r_{2})\bigl\Vert (u,v)\bigr\Vert _{Y}. \end{aligned}$$

(3.14)

By our assumption (3.8), the coefficients on the right side of (3.14) are all less than 1. So we have $\|u\|=\|v\|=0$, so that $u_{1}=u_{2}$, $v_{1}=v_{2}$.

Case 4. $\|(u,v)\|_{Y}\leq(p_{1}+q_{1})\|v\|_{X}+ r_{2}\|u\|_{X}+a\|L_{1}u\|_{\infty}$. The proof is similar to Case 3. Here, we omit it.

By the above argument, we have derived that BVP (1.1) and (1.2) has exactly one solution. The proof is finished. □

4 Example

Let us consider the following coupled system of fractional differential equations at resonance:

$$ \textstyle\begin{cases} D_{0^{+}}^{3.5}u(t)=f(t,v,v',v'',v'''), \quad 0< t< 1, \\ D_{0^{+}}^{3.6}v(t)=g(t,u,u',u'',u'''),\quad 0< t< 1, \\ u(0)=u'(0)= u''(0)=0,\qquad u'''(0)=u'''(1), \\ v(0)=v'(0)=v''(0)=0,\qquad v'''(0)=v'''(1), \end{cases} $$

(4.1)

where

$$\begin{aligned}& f(t,x_{1},x_{2},x_{3},x_{4})= \frac{t}{5}+\arctan x_{1}+\frac{1}{9}e^{-|x_{2}|}+ \cos x_{3}+\frac{1}{8}x_{4}, \\& g(t,y_{1},y_{2},y_{3},y_{4})=t^{2}+ \frac{1}{4}\sin y_{1}+ \sin^{2}(y_{2}y_{3})+ \frac{1}{2}(y_{4})^{\frac{1}{3}}. \end{aligned}$$

Corresponding to BVP (1.1) and (1.2), we have $\alpha=3.5$ and $\beta=3.6$. Take $a_{1}=\frac{59}{45}+\frac{\pi}{2}$, $a_{2}=\frac{9}{4}$, $b_{i}=c_{i}=0$, $i=1,2,3$, $b_{4}=\frac{1}{8}$, $c_{4}=\frac {1}{2}$, $\theta_{i}= \lambda_{i}=1$, $i=1,2,3$, $\theta_{4}=1$, $\lambda_{4}=\frac{1}{3}$, and $A=16$. Then we can calculate that (H₁)-(H₃) hold.

By Theorem 3.1, we see that BVP (4.1) has at least one solution.