Abstract
By using the coincidence degree theory, we present an existence result for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions at resonance.
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1 Introduction
In recent years, due to the application in many fields, fractional calculus has attracted more and more attention from researchers, and many meaningful results have been obtained [1, 2, 9–14, 17, 18, 24–26]. Since some nature phenomena are naturally modeled by fractional differential equation boundary value problems, it is important to study the problems of nonlinear fractional differential equations boundary value problems. On the other hand, the study of a coupled system involving fractional differential equations boundary value problems is also important as such systems occur in various problems of applied nature, for instance, see [2, 22, 31].
Recently, Su [31] discussed a two-point boundary value problem for a coupled system of fractional differential equations
where \(1<\alpha,\beta<2,\mu,\nu>0,\alpha-\nu\geq1,\beta-\mu \geq1, f,g:[0,1]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) are given functions and \(D_{0+}^{\alpha}\) is the standard Riemann–Liouville derivative of order α. Bashir Ahmad et al. [2] considered a three-point boundary value problem for a coupled system of nonlinear fractional differential equations given by
where \(1<\alpha,\beta<2,p,q,\gamma>0,0<\eta<1,\alpha-q\geq1,\beta -p\geq1, \gamma\eta^{\alpha-1}<1,\gamma\eta^{\beta-1}<1, f,g:[0,1]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) are given functions and \(D_{0+}^{\alpha}\) is the standard Riemann–Liouville derivative of order α. For more fractional-order boundary value problems and boundary value problems, we refer the reader to [3, 4, 6, 8, 15, 16, 19–21, 23, 27, 29, 30, 32, 33, 36].
It should be noted that all of the above papers mentioned deal with non-resonance case. However, there are few papers that consider the coupled system of nonlinear fractional differential equations with boundary conditions at resonance. In [5], Bai investigated the nonlinear nonlocal problem
where \(1<\alpha\leq2,0<\beta\eta^{\alpha-1}<1\). If \(\beta\eta ^{\alpha-1}=1\), resonance occurs, this case was considered in [34]. In [34] the authors considered the existence of solutions of the fractional order ordinary differential equation
with boundary value conditions \(I_{0+}^{2-\alpha}u(0)=0, u(1)=\sigma u(\eta)\), where \(1< \alpha\leq2\) is a real number, \(D_{0+}^{\alpha}\) and \(I_{0+}^{\alpha}\) are the standard Riemann–Liouville derivative and integral, respectively, and \(\sigma\eta^{\alpha-1}=1\). Under such conditions, the kernel of the linear operator \(L=D_{0+}^{\alpha}\) is of one dimension. The case that the kernel of the linear operator \(L=D_{0+}^{\alpha}\) is of two dimensions was considered in [7].
In [22], Jiang studied the solvability for a coupled system of fractional differential equations at resonance. In [35], the authors investigated a three-point boundary value problem for a coupled system of nonlinear fractional differential equations given by
where \(1< \alpha,\beta\leq 2,0<\eta_{1},\eta_{2}<1,\sigma_{1},\sigma_{2}>0,\sigma_{1}\eta_{1}^{\alpha -1}=\sigma_{2}\eta_{2}^{\beta-1}=1\), D is the standard Riemann–Liouville fractional derivative. System (1.1) happens to be at resonance in the sense that the associated linear homogeneous coupled system
has \((u(t),\upsilon(t))=(c_{1}t^{\alpha-1},c_{2}t^{\beta-1}), c_{1},c_{2}\in\mathbb{R}\) as a nontrivial solution.
Enlightened by the above contributions, in this paper we investigate the multi-point boundary value problem at resonance for a coupled system of nonlinear fractional differential equations given by
where \(2< \alpha,\beta\leq3,0<\xi,\eta\leq1, 0<\eta_{i},\xi _{j}<1\ (1\leq i\leq m_{1},1\leq j\leq m_{2}),m_{1}\geq2,m_{2}\geq2\), and \(f,g:[0,1]\times\mathbb{R}^{3}\rightarrow\mathbb{R}\) satisfy the Carathéodory conditions. \(D_{0+}^{\alpha}\) and \(I_{0+}^{\alpha}\) are the standard Riemann–Liouville fractional derivative and fractional integral, respectively, and
We assume, in addition, that
where Γ is the gamma function. Due to conditions (1.4) and (1.5), the coupled system (1.2)–(1.3) happens to be at resonance in the sense that the associated linear homogeneous coupled system
with boundary value conditions (1.3) has \((u(t),\upsilon(t))=(c_{11}t^{\alpha-1}+c_{12}t^{\alpha -2},c_{21}t^{\beta-1}+ c_{22}t^{\beta-2}), c_{ij}\in\mathbb{R}\) as nontrivial solutions.
Since the associated linear homogeneous coupled system about (1.2)–(1.3) has nontrivial solutions \((u(t),\upsilon (t))=(c_{11}t^{\alpha-1}+c_{12}t^{\alpha-2},c_{21}t^{\beta -1}+c_{22}t^{\beta-2}), c_{ij}\in\mathbb{R}\), it is more complex than [35].
The rest of this paper is organized as follows. We present some notations and lemmas in Sect. 2 and establish a theorem of existence of a solution for the coupled system (1.2)–(1.3) in Sect. 3.
2 Background materials and methods
In this section, we present some necessary basic knowledge about fractional calculus theory and a fixed point theorem.
The definitions and properties of fractional integral and derivative can be found in many literature works [24].
Definition 2.1
([24])
The Riemann–Liouville fractional integral \(I_{0+}^{\alpha}f\) and derivative \(D_{0+}^{\alpha}y\) of order \(\alpha\ (\alpha>0)\) are defined by
and
respectively, where \(n=[\alpha]+1\).
The properties of fractional calculus we will use are listed below.
Assume that \(u\in C(0,1) \cap L^{1}(0,1)\) and \(D_{0+}^{\alpha}u\in C(0,1) \cap L^{1}(0,1)\) with \(\alpha> 0\). Then
where \(C_{i}\in\mathbb{R},i=1,2,\ldots,N\), \(N=[\alpha+1]-1\). If \(\alpha>0,\beta>0\), then for a continuous function f, the equality \((I_{0+}^{\alpha}I_{0+}^{\beta}f)(x)=(I_{0+}^{\alpha+\beta}f)(x)\) is satisfied. Let \(\alpha>0,m\in\mathbb{N}\), and \(D=d/dx\). If the fractional derivatives \((D_{0+}^{\alpha}y)(x)\) and \((D_{0+}^{\alpha+m}y)(x)\) exist, then \((D^{m}D_{0+}^{\alpha}y)(x)=(D_{0+}^{\alpha+m}y)(x)\). If \(\alpha>0\), then for a continuous function f, \((D_{0+}^{\alpha}I_{0+}^{\alpha}f)(x)=f(x)\) is satisfied. If \(\alpha>\beta>0\), then for a continuous function f, it has \((D_{0+}^{\beta}I_{0+}^{\alpha}f)(x)=(I_{0+}^{\alpha-\beta}f)(x)\).
Now, we present some notations and a fixed point theorem.
Let \(Y,Z\) be real Banach spaces, \(L:\operatorname{dom}(L)\subset Y\rightarrow Z\) be a Fredholm map of index zero, and \(P:Y\rightarrow Y,Q:Z\rightarrow Z\) be continuous projectors such that
We can conclude that \(L|_{\operatorname{dom}(L)\cap\operatorname{Ker}(P)}:\operatorname {dom}(L)\cap\operatorname{Ker}(P)\rightarrow \operatorname{Im}(L)\) is invertible. Denote the inverse of the map by \(K_{P}\). If Ω is an open bounded subset of Y such that \(\operatorname{dom}(L)\cap \Omega\neq \emptyset\) and \(QN(\overline{\Omega})\) is bounded and \(K_{P}(I-Q)N:\overline{\Omega}\rightarrow Y\) is compact, then the map \(N:Y\rightarrow Z\) will be called L-compact on Ω̅.
The theorem we used is Theorem 2.4 of [28].
Theorem 2.1
Suppose that L is a Fredholm operator of index zero and N is L-compact on Ω̅, and the following conditions are satisfied:
-
(i)
\(Lx \neq\lambda Nx\) for every \((x,\lambda) \in [(\operatorname{dom}(L)\backslash\operatorname{Ker}(L))\cap\partial \Omega]\times(0,1)\);
-
(ii)
\(Nx\notin \operatorname{Im}(L)\) for every \(x\in\operatorname {Ker}(L)\cap\partial\Omega\);
-
(iii)
\(\operatorname{deg} (JQN|_{\operatorname {Ker}(L)},\Omega\cap\operatorname{Ker}(L),0 )\neq0\), where \(Q:Z\rightarrow Z\) is a projection as above with \(\operatorname{Im}(L)=\operatorname{Ker}(Q)\) and \(J:\operatorname{Im}(Q)\rightarrow\operatorname {Ker}(L)\) is any isomorphism.
Then the equation \(Lx=Nx\) has at least one solution in \(\operatorname {dom}(L)\cap\overline{\Omega}\).
In the following, the Banach space \(C[0,1]\) with the norm \(\|x\|_{\infty}=\max_{t\in[0,1]}|x(t)|\) and \(L^{1}[0,1]\) with the norm \(\|x\|_{1}=\int_{0}^{1}|x(t)|\,\mathrm{d}t\) will be used. Given \(\mu>0\) and \(N=[\mu]+1\), one can define a linear space
where \(x\in C[0,1]\) and \(c_{i}\in \mathbb{R},i=1,\ldots,N-1\). It can prove that with the norm \(\|u\|_{C^{\mu}}=\|D_{0+}^{\mu}u\|_{\infty}+\cdots+\|D_{0+}^{\mu-(N-1)} u\|_{\infty}+ \|u\|_{\infty}\), \(C^{\mu}[0,1]\) is a Banach space [34].
Lemma 2.1
([34])
\(F\subset C^{\mu}[0,1]\) is a sequentially compact set if and only if F is uniformly bounded and equicontinuous. Here uniformly bounded means that there exists \(M>0\) such that, for every \(u\in F\), it has
and equicontinuous means that for \(\forall\varepsilon>0\), \(\exists\delta>0\) such that, for any \(t_{1},t_{2}\in[0,1]\),
Let \(Z_{1}=L^{1}[0,1]\) with the norm \(\|\cdot\|_{1}\). Fractional functional spaces \(Y_{1}=C^{\alpha-1}[0,1]\) and \(Y_{2}=C^{\beta-1}[0,1]\) defined by (2.1) are equipped with the norms \(\|u\|_{Y_{1}}=\|D_{0+}^{\alpha-1} u\|_{\infty}+\|D_{0+}^{\alpha-2} u\| _{\infty}+\|u\|_{\infty}\) and \(\|\upsilon\|_{Y_{2}}=\|D_{0+}^{\beta-1}\upsilon\|_{\infty}+\| D_{0+}^{\beta-2}\upsilon\|_{\infty}+\|\upsilon\|_{\infty}\), respectively. Then \(Y=Y_{1}\times Y_{2}\) is a Banach space with the norm defined by \(\|(u,\upsilon)\|_{Y}=\max\{\|u\|_{Y_{1}},\|\upsilon\|_{Y_{2}}\}\) and \(Z=Z_{1}\times Z_{1}\) is a Banach space with the norm defined by \(\|(x,y)\|_{Z}=\max\{\|x\|_{1},\|y\|_{1}\}\).
Definition 2.2
A pair of functions \((u,\upsilon)\in Y\) is called a solution of the coupled system of multi-point boundary value problem (1.2)–(1.3) if \(D_{0+}^{\alpha-1}u\) and \(D_{0+}^{\beta -1}\upsilon\) are all absolutely continuous on \((0,1)\), \((u,\upsilon)\) satisfies (1.2) almost everywhere on \((0,1)\) and satisfies boundary conditions (1.3).
Definition 2.3
The map \(f:[0,1]\times\mathbb{R}^{n}\rightarrow \mathbb{R}\) satisfies the Carathéodory conditions with respect to \(L^{1}[0,1]\) if the following conditions hold:
-
(i)
for each \(z\in\mathbb{R}^{n}\), the mapping \(t\rightarrow f(t,z)\) is Lebesgue measurable;
-
(ii)
for almost each \(t\in[0,1]\), the mapping \(z\rightarrow f(t,z)\) is continuous on \(\mathbb{R}^{n}\);
-
(iii)
for each \(r>0\), there exists \(\rho_{r}\in L^{1}([0,1],\mathbb{R})\) such that, for a.e. \(t\in[0,1]\) and every \(|z|\leq r\), we have \(|f(t,z)|\leq\rho_{r}(t)\).
Define \(L_{1}: \operatorname{dom}(L_{1})\cap Y_{1}\rightarrow Z_{1}\) by
with
Define \(L_{2}: \operatorname{dom}(L_{2})\cap Y_{2}\rightarrow Z_{1}\) by
with
Define L to be the linear operator from \(\operatorname{dom}(L)\cap Y\) to Z with
and
Define \(N:Y\rightarrow Z\) as
where \(N_{1}:Y_{2}\rightarrow Z_{1}\) is defined by
and \(N_{2}:Y_{1}\rightarrow Z_{1}\) is defined by
Then the coupled system of boundary value problem (1.2)–(1.3) can be written by
Lemma 2.2
The mapping \(L:\operatorname{dom}(L)\subset Y\rightarrow Z\) is a Fredholm operator of index zero.
Proof
It is clear that \(\operatorname{Ker}(L)=\{(c_{11}t^{\alpha-1}+c_{12}t^{\alpha -2},c_{21}t^{\beta-1}+c_{22}t^{\beta-2})|c_{ij}\in \mathbb{R},i,j=1,2\}\cong\mathbb{R}^{4}\).
Let \((x,y)\in \operatorname{Im}(L)\), then there exists \((u,\upsilon)\in\operatorname {dom}(L)\) such that \((x,y)=L(u,\upsilon)\), that is, \(u\in Y_{1},x=D_{0+}^{\alpha}u\), and \(\upsilon\in Y_{2},y=D_{0+}^{\beta}\upsilon\). Thus we have
By the boundary conditions (1.3), we obtain \(c_{3}=0\), \(c_{1},c_{2}\) can be any constants, and x satisfies
By the properties of fractional calculus, \(D_{0+}^{\alpha -1}I_{0+}^{\alpha}=I_{0+}^{1}\), thus the above two equalities can be reduced to
Similarly, we can derive that y satisfies
On the other hand, suppose that \(x,y\in Z_{1}\) satisfy (2.9)–(2.10) and (2.11)–(2.12), respectively. Let \(u(t)=I_{0+}^{\alpha}x(t),\upsilon(t)=I_{0+}^{\beta}y(t)\), then a basic calculation shows that \(u\in\operatorname{dom}(L_{1})\), \(D_{0+}^{\alpha}u(t) =x(t)\), and \(\upsilon\in\operatorname {dom}(L_{2}),D_{0+}^{\beta}\upsilon(t) =y(t)\). That is, \((x,y)\in \operatorname{Im}(L)\). From the above argument, we can derive that
Consider continuous linear mappings \(Q_{1}^{\alpha}:Z_{1}\rightarrow Z_{1}\) and \(Q_{2}^{\alpha}:Z_{1}\rightarrow Z_{1}\) defined by
Continuous linear mappings \(Q^{\beta}_{1}:Z_{1}\rightarrow Z_{1}\) and \(Q_{2}^{\beta}:Z_{1}\rightarrow Z_{1}\) are defined by
Using the above definitions, we construct the following auxiliary maps \(R^{\alpha}_{1},R^{\alpha}_{2},R^{\beta}_{1}\), and \(R^{\beta}_{2}:Z_{1}\rightarrow Z_{1}\) by
Since conditions (1.6)–(1.7) hold, the mappings \(Q^{\alpha}:Z_{1}\rightarrow Z_{1}\) and \(Q^{\beta}:Z_{1}\rightarrow Z_{1}\) defined by
are well defined. Thus, we can define the continuous linear mapping \(Q:Z\rightarrow Z\) by
Recall (1.6) and note that
Similarly, we can derive that \(R^{\alpha}_{1}(R^{\alpha}_{2}gt^{\alpha -2})=0\) and
For \((x,y)\in Z\), it follows from the above relations that
that is, the map Q is idempotent. In fact, Q is a continuous linear projector and \((x,y)\in \operatorname{Im}(L)\) is equivalent to \(Q(x,y)=(0,0)\). In fact, \(\operatorname{Im}(L)=\operatorname{Ker}(Q)\).
Take \((x,y)\in Z\) in the form \((x,y)=((x,y)-Q(x,y))+Q(x,y)\) so that \((x,y)-Q(x,y)\in \operatorname{Im}(L)=\operatorname{Ker}(Q)\) and \(Q(x,y)\in \operatorname{Im}(Q)\). Thus, \(Z=\operatorname{Im}(L)+\operatorname{Im}(Q)\). For every \((x,y)\in \operatorname{Im}(Q)\) has the form \((x,y)=(c_{11}t^{\alpha-1}+c_{12}t^{\alpha-2},c_{21}t^{\beta -1}+c_{22}t^{\beta-2}),c_{ij}\in\mathbb{R}\ (i,j=1,2)\). If \((x,y)=(c_{11}t^{\alpha-1}+c_{12}t^{\alpha-2},c_{21}t^{\beta -1}+c_{22}t^{\beta-2})\) satisfies (2.9)–(2.10) and (2.11)–(2.12) respectively, then we have \(c_{ij}=0\ (i,j=1,2)\). Hence, \(\operatorname{Im}(L)\cap \operatorname{Im}(Q)=(0,0)\) and, in fact, \(Z=\operatorname{Im}(L)\oplus \operatorname{Im}(Q)\).
Now, \(\operatorname{Ind}L= \operatorname{dim} \operatorname{Ker}(L)- \operatorname{codim} \operatorname{Im}(L)=0\), and so L is a Fredholm operator of index zero. □
Let the operators \(P_{1}:Y_{1}\rightarrow Y_{1}\), \(P_{2}:Y_{2}\rightarrow Y_{2}\), and \(P:Y\rightarrow Y\) be defined by
and
respectively. Note that \(P_{1},P_{2}\), and P are continuous linear projectors and
It is clear that \(Y=\operatorname{Ker}(L)\oplus\operatorname {Ker}(P)\) and, for every \((u,\upsilon)\in Y\),
Define \(K_{P}:\operatorname{Im}(L)\rightarrow\operatorname{dom}(L)\cap\operatorname {Ker}(P)\) by
For \((x,y)\in \operatorname{Im}(L)\), we have
and for \((u,\upsilon)\in\operatorname{dom}(L)\cap\operatorname {Ker}(P)\), we have \(u\in\operatorname{dom}(L_{1}),D_{0+}^{\alpha-1}u(0)=D_{0+}^{\alpha -2}u(0)=0\) and \(\upsilon\in \operatorname{dom}(L_{2})\), \(D_{0+}^{\beta-1}\upsilon(0)=D_{0+}^{\beta -2}\upsilon(0)=0\), so the coefficients \(c_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\) in the expressions \(u=I_{0+}^{\alpha}D_{0+}^{\alpha}u+c_{1}t^{\alpha-1}+c_{2}t^{\alpha -2}+c_{3}t^{\alpha-3}\) and \(\upsilon=I_{0+}^{\beta}D_{0+}^{\beta}\upsilon+c_{4}t^{\beta -1}+c_{5}t^{\beta-2}+c_{6}t^{\beta-3}\) are all equal to zero. Thus, we obtain
This shows that \(K_{P}=(L|_{\operatorname{dom}(L)\cap \operatorname{Ker}(P)})^{-1}\). Again, since for every \((x,y)\in \operatorname{Im}(L)\),
With arguments similar to those in [34], we obtain the following lemma.
Lemma 2.3
\(K_{P}(I-Q)N:Y\rightarrow Y\) is completely continuous.
3 Results and discussion
In this section, we shall prove existence results for the coupled system of fractional boundary value problem (1.2)–(1.3).
For convenience, let us set the following notations:
Assume that the following conditions are satisfied:
- (\(H_{1}\)):
-
There exists a constant \(A>0\) such that, for \((u,\upsilon)\in\operatorname{dom}(L)\backslash\operatorname {Ker}(L)\), if \(|D_{0+}^{\alpha-1}u(t)|+|D_{0+}^{\alpha-2}u(t)|>A\) for all \(t\in [0,1]\), then
$$Q^{\beta}_{1}N_{2}u(t)\neq0 \quad\mbox{or}\quad Q^{\beta}_{2}N_{2}u(t)\neq0, $$and if \(|D_{0+}^{\beta-1}\upsilon(t)|+|D_{0+}^{\beta-2}\upsilon (t)|>A\) for all \(t\in[0,1]\), then
$$Q^{\alpha}_{1}N_{1}\upsilon(t)\neq0 \quad\mbox{or} \quad Q^{\alpha}_{2}N_{1}\upsilon (t)\neq0. $$ - (\(H_{2}\)):
-
There exist functions \(a_{1},b_{1},d_{1},e_{1},r_{1}\in L^{1}[0,1]\) and a constant \(\theta_{1}\in[0,1)\) such that, for all \((x,y,z)\in\mathbb{R}^{3}\) and a.e. \(t\in [0,1]\), one of the following inequalities is satisfied:
$$\begin{aligned} & \bigl\vert f(t,x,y,z) \bigr\vert \leq a_{1}(t) \vert x \vert+b_{1}(t) \vert y \vert+d_{1}(t) \vert z \vert+e_{1}(t) \vert z \vert^{\theta_{1}} +r_{1}(t), \end{aligned}$$(3.2)$$\begin{aligned} & \bigl\vert f(t,x,y,z) \bigr\vert \leq a_{1}(t) \vert x \vert+b_{1}(t) \vert y \vert+d_{1}(t) \vert z \vert+e_{1}(t) \vert y \vert^{\theta_{1}} +r_{1}(t), \end{aligned}$$(3.3)$$\begin{aligned} & \bigl\vert f(t,x,y,z) \bigr\vert \leq a_{1}(t) \vert x \vert+b_{1}(t) \vert y \vert+d_{1}(t) \vert z \vert+e_{1}(t) \vert x \vert^{\theta_{1}} +r_{1}(t). \end{aligned}$$(3.4)There exist functions \(a_{2},b_{2},d_{2},e_{2},r_{2}\in L^{1}[0,1]\) and a constant \(\theta_{2}\in[0,1)\) such that, for all \((x,y,z)\in\mathbb{R}^{3}\) and a.e. \(t\in [0,1]\), one of the following inequalities is satisfied:
$$\begin{aligned} & \bigl\vert g(t,x,y,z) \bigr\vert \leq a_{2}(t) \vert x \vert+b_{2}(t) \vert y \vert+d_{2}(t) \vert z \vert+e_{2}(t) \vert z \vert^{\theta_{2}} +r_{2}(t), \end{aligned}$$(3.5)$$\begin{aligned} & \bigl\vert g(t,x,y,z) \bigr\vert \leq a_{2}(t) \vert x \vert+b_{2}(t) \vert y \vert+d_{2}(t) \vert z \vert+e_{2}(t) \vert y \vert^{\theta_{2}} +r_{2}(t), \end{aligned}$$(3.6)$$\begin{aligned} & \bigl\vert g(t,x,y,z) \bigr\vert \leq a_{2}(t) \vert x \vert+b_{2}(t) \vert y \vert+d_{2}(t) \vert z \vert+e_{2}(t) \vert x \vert^{\theta_{2}} +r_{2}(t). \end{aligned}$$(3.7) - (\(H_{3}\)):
-
There exists a constant \(B>0\) such that, for every \(c_{1},c_{2},c_{3},c_{4}\in\mathbb{R}\) satisfying \(\sum_{i=1}^{4}c_{i}^{2}>B\), at least one of the following expressions holds:
$$\begin{aligned} &R^{\alpha}_{1}N_{1} \bigl(c_{3}t^{\beta-1}+c_{4}t^{\beta-2} \bigr)\neq0,\qquad R^{\alpha}_{2}N_{1} \bigl(c_{3}t^{\beta-1}+c_{4}t^{\beta-2} \bigr)\neq0, \\ &R^{\beta}_{1}N_{2} \bigl(c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2} \bigr)\neq0,\qquad R^{\beta}_{2}N_{2} \bigl(c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2} \bigr)\neq0. \end{aligned}$$And for \(c_{1},c_{2},c_{3},c_{4}\in\mathbb{R}\) satisfying \(\sum_{i=1}^{4}c_{i}^{2}>B\),
$$\begin{aligned} R\doteq{}& \bigl[c_{1}R^{\alpha}_{1}N_{1} \bigl(c_{3}t^{\beta-1}+c_{4}t^{\beta-2} \bigr)+c_{2}R^{\alpha}_{2}N_{1} \bigl(c_{3}t^{\beta-1}+c_{4}t^{\beta-2} \bigr) \bigr] \\ &{}\times \bigl[c_{3}R^{\beta}_{1}N_{2} \bigl(c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2} \bigr)+c_{4}R^{\beta}_{2}N_{2} \bigl(c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2} \bigr) \bigr] \leq0. \end{aligned}$$(3.8)
Remark 3.1
\(R^{\alpha}_{i}N_{1}(at^{\beta-1}+bt^{\beta-2})\) and \(R^{\beta}_{i}N_{2}(at^{\alpha-1}+bt^{\alpha-2})\) from \((H_{3})\) stand for the images of \(\upsilon(t)=at^{\beta-1}+bt^{\beta-2},u(t)=at^{\alpha -1}+bt^{\alpha-2}\) under the maps \(R^{\alpha}_{i}N_{1}\) and \(R^{\beta}_{i}N_{2}\), respectively.
Theorem 3.1
If \((H_{1})\)–\((H_{3})\) hold, then the coupled system of fractional multi-point boundary value problem (1.2)–(1.3) has at least one solution provided that
Proof
Our proof can be divided into four steps.
Step 1: Set
Then, for \((u,\upsilon)\in\Omega_{1},L(u,\upsilon)=\lambda N(u,\upsilon)\), thus \(\lambda\neq0,N(u,\upsilon)\in \operatorname{Im}(L)=\operatorname{Ker}(Q)\), hence \(QN(u,\upsilon)=(Q^{\alpha}N_{1}\upsilon,Q^{\beta}N_{2}u)=(0,0)\) by the definition of Q. Thus we have \(Q_{1}^{\alpha}N_{1}\upsilon(t)=Q^{\alpha}_{2}N_{1}\upsilon(t)=0\) and \(Q_{1}^{\beta}N_{2}u(t)=Q^{\beta}_{2}N_{2}u(t)=0\) for all \(t\in[0,1]\). It follows from \((H_{1})\) that there exist \(t_{0},t_{1}\in[0,1]\) such that \(|D_{0+}^{\alpha-1}u(t_{0})|+|D_{0+}^{\alpha-2}u(t_{0})|\leq A\) and \(|D_{0+}^{\beta-1}\upsilon(t_{1})|+|D_{0+}^{\beta-2}\upsilon (t_{1})|\leq A\). Now
so that
Similar to the above argument, we can also obtain
Now by (3.11)–(3.14) and (2.31), we have
Note that \((I-P)(u,\upsilon)\in \operatorname{Im}(K_{P})=\operatorname{dom}(L)\cap \operatorname{Ker}(P)\) for \((u,\upsilon)\in\Omega_{1}\). Then
Using (3.15) and (3.16), we obtain
Without loss of generality, we assume that (3.2) and (3.5) in \((H_{1})\) hold, then from (3.17), the proof can be divided into four cases.
Case 1. \(\|(u,\upsilon)\|_{Y}\leq\epsilon_{1}\|N_{1}\upsilon\|_{1}+\rho_{1}A\). From (3.2), we have
Thus, from \(\|\upsilon\|_{\infty},\|D_{0+}^{\beta-2}\upsilon\|_{\infty}, \|D_{0+}^{\beta-1}\upsilon\|_{\infty}\leq\|(u,\upsilon)\|_{Y}\), and (3.17), we obtain
Again from (3.18), (3.19), one has
Since \(\theta_{1}\in[0,1)\), from the above last inequality, there exists \(M_{1}>0\) such that \(\|D_{0+}^{\beta-1}\upsilon\|_{\infty}\leq M_{1}\), thus from (3.20), there exists \(M_{2}>0\) such that \(\|D_{0+}^{\beta-2}\upsilon\|_{\infty}\leq M_{2}\). Again from (3.19), there exists \(M_{3}>0\) such that \(\|\upsilon\|_{\infty}\leq M_{3}\). Thus, from (3.18), there exists \(M_{4}>0\) such that \(\|(u,\upsilon)\|_{Y}\leq M_{4}\). Therefore \(\Omega_{1}\) is bounded.
Case 2. \(\|(u,\upsilon)\|_{Y}\leq\epsilon_{2}\|N_{2}u\|_{1}+\rho_{2}A\). The proof is similar to that of case 1. Here, we omit it.
Case 3. \(\|(u,\upsilon)\|_{Y}\leq\rho_{1}\|N_{1}\upsilon\|_{1}+\mu_{2}\|N_{2}u\| _{1}+\rho_{1}A\). From (3.2) and (3.5), we have
Thus, from \(\|u\|_{\infty},\|D_{0+}^{\alpha-2}u\|_{\infty},\|D_{0+}^{\alpha-1}u\| _{\infty},\|\upsilon\|_{\infty}, \|D_{0+}^{\beta-2}\upsilon\|_{\infty},\|D_{0+}^{\beta-1}\upsilon\| _{\infty}\leq\|(u,\upsilon)\|_{Y}\), and (3.22), we obtain
Again, from (3.22), (3.23), we have
and
If \(\rho_{1}\|e_{1}\|_{1}\|D_{0+}^{\beta-1}\upsilon\|_{\infty}^{\theta_{1}} \geq\mu_{2}\|e_{2}\|_{1}\|D_{0+}^{\alpha-1}u\|_{\infty}^{\theta_{2}}\), then from (3.28) we have
Since \(\theta_{1}\in[0,1)\), from the above last inequality, there exists \(M_{1}>0\) such that \(\|D_{0+}^{\beta-1}\upsilon\|_{\infty}\leq M_{1}\), thus from (3.22)–(3.28), there exists \(M_{2}>0\) such that \(\|u\|_{\infty}\), \(\|\upsilon\|_{\infty}\), \(\|D_{0+}^{\alpha-2}u\| _{\infty}\), \(\|D_{0+}^{\beta-2}\upsilon\|_{\infty}\), and \(\|D_{0+}^{\alpha-1}u\|_{\infty}\) are all less than \(M_{2}\), hence \(\|(u,\upsilon)\|_{Y}\leq3(M_{1}+M_{2})\). Therefore \(\Omega_{1}\) is bounded.
If \(\rho_{1}\|e_{1}\|_{1}\|D_{0+}^{\beta-1}\upsilon\|_{\infty}^{\theta_{1}} \leq\mu_{2}\|e_{2}\|_{1}\|D_{0+}^{\alpha-1}u\|_{\infty}^{\theta_{2}}\), then from (3.27), similar to the above argument, we can also prove that \(\Omega_{1}\) is bounded.
Case 4. \(\|(u,\upsilon)\|_{Y}\leq\rho_{2}\|N_{2}u\|_{1}+\mu_{1}\|N_{1}\upsilon\| _{1}+\rho_{2}A\). The proof is similar to that of case 3. Here, we omit it.
From the above argument, we have proved that \(\Omega_{1}\) is bounded.
Step 2: Let
For \((u,\upsilon)\in\Omega_{2},(u,\upsilon)\in\operatorname {Ker}(L)=\{(u,\upsilon)\in \operatorname{dom}(L)|u=c_{11}t^{\alpha-1}+c_{12}t^{\alpha -2},\upsilon=c_{21}t^{\beta-1}+c_{22}t^{\beta-2}, c_{ij}\in\mathbb {R},i,j=1,2,t\in [0,1]\}\), and \(QN(c_{11}t^{\alpha-1}+c_{12}t^{\alpha -2},c_{21}t^{\beta-1}+c_{22}t^{\beta-2})=(0,0)\), thus
By \((H_{3})\), \(c_{11}^{2}+c_{12}^{2}+c_{21}^{2}+c_{22}^{2}\leq B\), that is, \(\Omega_{2}\) is bounded.
Step 3: We define the isomorphism \(J:\operatorname{Im}(Q)\rightarrow \operatorname{Ker}(L)\) by
Let
For every \((c_{11}t^{\alpha-1}+c_{12}t^{\alpha-2},c_{21}t^{\beta -1}+c_{22}t^{\beta-2})\in\Omega_{3}\) with \(c_{11}^{2}+c_{12}^{2}+c_{21}^{2}+c_{22}^{2}>0\),
If \(\lambda=1\), then \(c_{ij}=0,i,j=1,2\). If \(\lambda=0\), then by Step 2, \(c_{11}^{2}+c_{12}^{2}+c_{21}^{2}+c_{22}^{2}< B\). If \(0<\lambda<1\) and \(c_{11}^{2}+c_{12}^{2}+c_{21}^{2}+c_{22}^{2}>B\), then by \((H_{3})\),
which, in either case, is a contradiction, that is, \(\Omega_{3}\) is bounded.
Step 4: Now we prove that the conditions of Theorem 2.1 are all satisfied. Set Ω to be a bounded open set of Y such that \(\bigcup_{i=1}^{3}\overline{\Omega}_{i}\subset\Omega\). By Lemma 2.3, the operator \(K_{P}(I-Q)N:\overline{\Omega}\rightarrow Y\) is compact, thus N is L-compact on Ω̅. Then, by the above argument, we have
-
(i)
\(L(u,\upsilon)) \neq\lambda N(u,\upsilon)\) for every \(((u,\upsilon),\lambda) \in [(\operatorname{dom}(L)\backslash\operatorname{Ker}(L))\cap\partial \Omega]\times(0,1)\);
-
(ii)
\(N(u,\upsilon)\notin \operatorname{Im}(L)\) for every \((u,\upsilon)\in \operatorname{Ker}(L)\cap \partial\Omega\).
Let \(H((u,\upsilon),\lambda)= \lambda I (u,\upsilon)+(1-\lambda)JQN(u,\upsilon)\), where I is the identical operator. According to the above argument, we know
thus, by the homotopy property of degree
Thus (iii) of Theorem 2.1 is satisfied. Then, by Theorem 2.1, \(L(u,\upsilon)=N(u,\upsilon)\) has at least one solution in \(\operatorname{dom}(L)\cap\overline{\Omega}\), so that the coupled system (1.2)–(1.3) has at least one solution in Y. The proof is finished. □
4 Conclusions
The linear operator \(L=0\) with boundary conditions at resonance with the kernel of four dimensions was considered and an existence result for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions at resonance was obtained by using the coincidence degree theory.
References
Agarwal, R.P., Lakshmikantham, V., Nieto, J.J.: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. TMA 72, 2859–2862 (2010)
Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)
Bai, Z., Chen, Y., Lian, H., Sun, S.: On the existence of blow up solutions for a class of fractional differential equations. Fract. Calc. Appl. Anal. 17(4), 1175–1187 (2014)
Bai, Z., Dong, X., Yin, C.: Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl. 2016, 63 (2016). https://doi.org/10.1186/s13661-016-0573-z
Bai, Z., Lü, H.: Positive solutions of boundary value problems of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)
Bai, Z., Zhang, S., Su, S., Yin, C.: Monotone iterative method for fractional differential equations. Electron. J. Differ. Equ. 2016, 06 (2016)
Bai, Z., Zhang, Y.: The existence of solutions for a fractional multi-point boundary value problem. Comput. Math. Appl. 60, 2364–2372 (2010)
Bai, Z., Zhang, Y.: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 5(218), 1719–1725 (2011)
Baleanu, D., Mustafa, O.G.: On the asymptotic integration of a class of sublinear fractional differential equations. J. Math. Phys. 50, 123520 (2009). https://doi.org/10.1063/1.3271111
Baleanu, D., Mustafa, O.G.: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59, 1835–1841 (2010)
Baleanu, D., Mustafa, O.G., Agarwal, R.P.: On the solution set for a class of sequential fractional differential equations. J. Phys. A, Math. Theor. 43, 385209 (2010). https://doi.org/10.1088/1751-8113/43/38/385209
Bonilla, B., Rivero, M., Rodriguez-Germa, L., Trujillo, J.J.: Fractional differential equations as alternative models to nonlinear differential equations. Appl. Math. Comput. 187, 79–88 (2007)
Cabada, A., Hamdi, Z.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 1(228), 251–257 (2014)
Chang, Y.K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009)
Chen, T., Liu, W., Hu, Z.: A boundary value problem for fractional differential equation with p-Laplacian operator at resonance. Nonlinear Anal. TMA 6(75), 3210–3217 (2012)
Chen, T., Liu, W., Liu, J.: Solvability of periodic boundary value problem for fractional p-Laplacian equation. Appl. Math. Comput. 1(244), 422–431 (2014)
Chen, Y., An, H.: Numerical solutions of coupled Burgers equations with time and space fractional derivatives. Appl. Math. Comput. 200, 87–95 (2008)
Chen, Y., Tang, X.: Solvability of sequential fractional order multi-point boundary value problems at resonance. Appl. Math. Comput. 14(218), 7638–7648 (2012)
Cui, Y.: Existence of solutions for coupled integral boundary value problem at resonance. Publ. Math. (Debr.) 89(1–2), 73–88 (2016)
Gafiychuk, V., Datsko, B., Meleshko, V.: Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220, 215–225 (2008)
Jiang, W.: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. TMA 5(74), 1987–1994 (2011)
Jiang, W.: Solvability for a coupled system of fractional differential equations at resonance. Nonlinear Anal., Real World Appl. 5(13), 2285–2292 (2012)
Jiang, W., Huang, X., Wang, B.: Boundary value problems of fractional differential equations at resonance. Phys. Proc. 25, 965–972 (2012)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Lakshmikantham, V., Leela, S.: Nagumo-type uniqueness result for fractional differential equations. Nonlinear Anal. TMA 71, 2886–2889 (2009)
Lakshmikantham, V., Vatsala, A.S.: Theory of fractional differential inequalities and applications. Commun. Appl. Anal. 11, 395–402 (2007)
Liu, B., Zhao, Z.: A note on multi-point boundary value problems. Nonlinear Anal. TMA 67, 2680–2689 (2007)
Mawhin, J.: Topological degree and boundary value problems for nonlinear differential equations. In: Fitzpatrick, P.M., Martelli, M., Mawhin, J., Nussbaum, R. (eds.) Topological Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, vol. 1537, pp. 74–142. Springer, Berlin (1991)
Miller, K.S.: Fractional differential equations. J. Fract. Calc. 3, 49–57 (1993)
Song, Q., Bai, Z.: Positive solutions of fractional differential equations involving the Riemann–Stieltjes integral boundary condition. Adv. Differ. Equ. 2018(1), 183 (2018). https://doi.org/10.1186/s13662-018-1633-8
Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)
Sun, Y., Zhao, M.: Positive solutions for a class of fractional differential equations with integral boundary conditions. Appl. Math. Lett. 34, 17–21 (2014)
Xu, N., Liu, W.: Iterative solutions for a coupled system of fractional differential-integral equations with two-point boundary conditions. Appl. Math. Comput. 244, 903–911 (2014)
Zhang, Y., Bai, Z.: Existence of solutions for nonlinear fractional three-point boundary value problems at resonance. J. Appl. Math. Comput. 36, 417–440 (2011)
Zhang, Y., Bai, Z., Feng, T.: Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Comput. Math. Appl. 4(61), 1032–1047 (2011)
Zou, Y., Liu, L., Cui, Y.: The existence of solutions for four-point boundary value problems of fractional differential equations at resonance. Abstr. Appl. Anal. 2014, Article ID 314083 (2014)
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The authors express their thanks to the editors for their valuable suggestions.
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This research was supported by the Fundamental Research Funds for the Central Universities under grants FRF-TP-15-100A1 and FRF-BR-17-014A.
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Zhang, Y. Existence results for a coupled system of nonlinear fractional multi-point boundary value problems at resonance. J Inequal Appl 2018, 198 (2018). https://doi.org/10.1186/s13660-018-1792-x
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DOI: https://doi.org/10.1186/s13660-018-1792-x