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Existence results for a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions
Boundary Value Problems volume 2017, Article number: 88 (2017)
Abstract
By means of coincidence degree theory, we present the existence of solutions of a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions. This paper enriches and extends some existing literature. Finally, an example is given to support our results.
1 Introduction
In this paper, we study the existence of solutions for higher-order nonlinear fractional differential equations with p-Laplacian operator:
where the p-Laplacian operator is defined as \(\phi_{p}(s)= \vert s \vert ^{p-2}s\), \(p>1\), \(\phi_{q}(s)=\phi _{p}^{-1}(s)\), \(\frac{1}{p}+\frac{1}{q}=1\), \(0< \beta_{1}, \beta_{2} <1\), \(n-1<\alpha_{1},\alpha_{2}<n\), \(0<\xi_{1}<\xi_{2}<\cdots<\xi_{i}<\cdots<1\), \(0<\eta_{1}<\eta_{2}<\cdots<\eta_{i}<\cdots<1\), \(\sum_{i=1}^{\infty}a_{i}= \sum_{i=1}^{\infty}b_{i}=1\), \(\sum_{i=1}^{\infty} \vert a_{i} \vert <\infty\), \(\sum_{i=1}^{\infty} \vert b_{i} \vert <\infty\), \(D_{0+}^{\alpha_{1}}\), \(D_{0+}^{\beta_{1}}\), \(D_{0+}^{\alpha_{2}}\), \(D_{0+}^{\beta_{2}}\) denote the Caputo fractional derivatives and \(f,g:[0,1]\times\mathbb{R}^{n }\rightarrow \mathbb{R}\) are continuous.
The theory of fractional differential equations is a branch of differential equation theory, which occurs more frequently in different research areas and engineering, such as fluid mechanics, control system, viscoelasticity, chemistry, electromagnetic, etc. (see [1–5]). In the last few decades, many authors devoted their attention to the study of resonant boundary value problems for nonlinear fractional differential equations, see [6–19]. Meanwhile, some important results relative to the existence of solutions for a coupled system of fractional differential equations with p-Laplacian operator at resonance have been obtained, see [11–16].
In [15], Hu et al. considered the two-point boundary value problem for nonlinear fractional differential equations with p-Laplacian operator at resonance:
where \(\phi_{p}(s)= \vert s \vert ^{p-2}s\), \(p>1\) is the p-Laplacian operator, \(0<\alpha,\beta<1\), \(1<\alpha+\beta<2\), \(D_{0^{+}}^{\alpha}\), \(D_{0^{+}}^{\beta}\) \(D_{0^{+}}^{\gamma}\) \(D_{0^{+}}^{\delta}\) denote the Caputo fractional derivatives and \(f,g:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}\) are continuous.
In [16], Cheng et al. considered the two-point boundary value problem for nonlinear fractional p-Laplacian differential equations with \(\operatorname{Ker} L=n\geq2\):
where \(\phi_{p}(s)= \vert s \vert ^{p-2}s\), \(p>1\) is the p-Laplacian operator, \(0<\gamma<1\), \(n-1<\alpha,\beta<n\), \(D_{0^{+}}^{\alpha}\), \(D_{0^{+}}^{\beta}\), \(D_{0^{+}}^{\gamma}\) denote the Caputo fractional derivatives and \(f,g:[0,1]\times\mathbb{R} \rightarrow \mathbb{R}\) are continuous.
In recent years, the subject of infinite-point boundary value problems of fractional differential equations which can extend many previous results have attracted more attention. Most of the results are mainly at nonresonance. For the resonance case, the existing results of fractional differential equations with infinite-point boundary value problems are few. We refer the reader to [20–23] and the references cited therein.
From the above work, we see that recent study on a coupled system of fractional p-Laplacian differential equations is mainly at two-point boundary value problem. The theory for fractional p-Laplacian differential equations with multi-point and even infinite-point at resonance has yet been sufficiently developed. To the best of our knowledge, this is the first paper to study higher order fractional differential equations with p-Laplacian and infinite-point boundary value conditions at resonance. Motivated by the works above, we consider the existence of solutions for BVP (1.1).
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) by the coincidence degree theory due to Mawhin [24]. 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
provided that the right-hand side is pointwise defined on \((0,\infty)\).
Definition 2.2
[1]
The Caputo fractional derivative of order \(\alpha>0\) of a function \(f\in \mathit{AC}^{n-1}[0,1]\) is given by
where \(n-1<\alpha\leq n\), provided that the right-hand side is pointwise defined on \((0,\infty)\).
Lemma 2.1
[1]
Let \(n-1<\alpha\leq n\), \(u\in \mathit{AC}^{n-1}[0,1]\), then
where \(c_{i}\in\mathbb{R}\), \(i=0,1,\dots, n-1\).
Lemma 2.2
[1]
If \(\beta>0\), \(\alpha+\beta>0\), then the equation
is satisfied for an integrable function f.
Lemma 2.3
[23]
For any \(u,v \geq0\), then
Firstly, we briefly recall some definitions on the coincidence degree theory. For more details, see [14].
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
It follows that
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 Ω̅ 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 Ω̅. Suppose that the following conditions are satisfied:
-
(1)
\(Lx \neq\lambda Nx\) for each \((x,\lambda) \in [(\operatorname{dom}L\backslash\operatorname{Ker} L) \cap \partial\Omega ]\times (0,1)\);
-
(2)
\(Nx \notin\operatorname{Im} L\) for each \(x \in\operatorname{Ker} L \cap\partial\Omega\);
-
(3)
\(\deg(JQN|_{\operatorname{Ker} L}, \Omega\cap\operatorname{Ker} L, 0)\neq0\), where \(Q:Z\to Z\) is a continuous projection as above with \(\operatorname{Im}L= \operatorname{Ker} Q\) and \(J:\operatorname{Im}Q\to \operatorname{Ker} L\) 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 begin to prove the existence of solutions to problem (1.1). Consider the functions \(\phi_{1}(z) = \sum_{i=1}^{\infty} a_{i}\xi_{i}^{z}\), \(\phi_{2}(z) = \sum_{i=1}^{\infty} b_{i}\eta _{i}^{z}\), \(z\in[0,\infty)\). According to \(\sum_{i=1}^{\infty} \vert a_{i} \vert <\infty\), \(\sum_{i=1}^{\infty} \vert b_{i} \vert <\infty\), one has the series are (uniformly) convergent and thus \(\phi _{1}\), \(\phi_{2}\) are continuous on \([0,\infty)\).
The following assumption will be used in our main results:
- \((\mathrm{H}_{0})\) :
-
There exist \(z_{0}\), \(\tilde{z}_{0}\) with \(z_{0}\geq\alpha_{1}\), \(\tilde{z}_{0}\geq \alpha_{2}\) such that \(\phi_{1}(z_{0})\cdot\phi_{2}(\tilde{z}_{0})\neq0\).
The following lemma is fundamental in the proofs of our main results.
Lemma 3.1
Problem (1.1) is equivalent to the following equation:
Proof
By Lemma 2.1, \(D_{0+}^{\beta_{1}} \phi_{p}( D_{0+}^{\alpha_{1}} u(t))=f (t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0+}^{\alpha_{2}-(n-1)}v(t) )\) has the following solution:
Substituting \(t=0\) into the above formula, by \(D_{0^{+}}^{\alpha _{1}}u(0)=0\), we obtain \(c=0\). Then we have
Applying the operator \(\phi_{q}\) to the both sides of (3.2) respectively, we have
By a similar argument, we have
is equivalent to
Therefore, BVP (1.1) is rewritten by (3.1)
It is easy to verify that equation (1.1) has a solution \((u,v) \) if and only if \((u,v)\) solves equation (3.1). □
Let \(E=C[0,1]\) with the norm \(\Vert x \Vert _{\infty}=\max_{0 \le t \le1} \vert x(t) \vert \). Now, we set \(X_{1} =\{u(t): u(t), D_{0+}^{\alpha_{1}-i}u(t)\in E, i=1,2,\ldots,n-1\}\) with the norm
and \(X_{2} =\{v(t): v(t), D_{0+}^{\alpha_{2}-i}v(t)\in E, i=1,2,\ldots,n-1\} \) with the norm
Let \(Y=X_{1}\times X_{2}\) with the norm \(\Vert (u,v) \Vert _{Y}=\max\{ \Vert u \Vert _{X_{1}}, \Vert v \Vert _{X_{2}} \} \) and \(Z=E\times E\) with the norm \(\Vert (x,y) \Vert _{Z}=\max \{ \Vert x \Vert _{\infty}, \Vert y \Vert _{\infty}\}\).
Clearly, X and Y are Banach spaces.
Define the linear operator \(L_{1}:\operatorname{dom}L_{1}\rightarrow E\) by setting
and
Define the linear operator \(L_{2}\) from \(\operatorname{dom}L_{2} \rightarrow E\) by setting
and
Define the operator \(L: \operatorname{dom}L \rightarrow Z\) with
and
Let \(N:Y\to Z\) be the Nemytskii operator
where \(N_{1}:X\to E\) is defined by
and \(N_{2}:X\to E\) is defined by
Then BVP (3.1) can be written as \(L(u,v)=N(u,v)\).
Lemma 3.2
L is defined as above, then
Proof
For \((u,v)\in\operatorname{Ker}L\), then \(L_{1}u=L_{2}v=0\). By Lemma 2.1, the equation \(D_{0 +}^{\alpha_{1}}u(t)=0\) has solution
In view of \(u^{(i)}(0)=0\), \(i=1,2,\ldots,n-1\), we get \(c_{i}=0\), \(i=1,2,\ldots,n-1\). Then \(u(t)=c_{0}\). Similarly, for \(v\in\operatorname{Ker}L_{2}\), we have \(v(t)=d_{0}\in\mathbb{R}\). Thus, we obtain (3.3).
Next we prove that (3.4) holds. Let \((x,y)\in\operatorname{Im}L\), so there exists \((u,v)\in\operatorname{dom}L\) such that \(x(t)=D_{0 +}^{\alpha _{1}}u(t)\), \(y(t)=D_{0 +}^{\alpha_{2}}v(t)\). By Lemma 2.1, we have
In view of \(u^{(i)}(0)=v^{(i)}(0)=0\), \(i=1,2,\ldots,n-1\), we get \(c_{i}=d_{i}=0\), \(i=1,2,\ldots, n-1\). Hence,
According to \(u(0)=\sum_{i=1}^{\infty}a_{i}u(\xi_{i})\) and \(v(0)=\sum_{i=1}^{\infty}b_{i}v(\eta_{i})\), we have
that is,
On the other hand, suppose that \((x,y)\) satisfies the above equations. Let \(u(t)=I_{0 +}^{\alpha_{1}}x(t)\) and \(v(t)=I_{0 +}^{\alpha_{2}}y(t)\), we can prove \((u ,v )\in\operatorname{dom} L\) and \(L (u ,v )=(x,y)\). Then (3.4) holds. □
Lemma 3.3
The mapping \(L:\operatorname{dom} L \subset Y\rightarrow Z\) is a Fredholm operator of index zero.
Proof
The linear continuous projector operator \(P(u,v)=(P_{1}u,P_{2}v)\) can be defined as
Obviously, \(P_{1}^{2}=P_{1}\) and \(P_{2}^{2}=P_{2}\).
It is clear that
It follows from \((u,v)=(u,v)-P(u,v)+P(u,v)\) that \(Y=\operatorname{Ker}P+ \operatorname{Ker}L\). For \((u,u)\in\operatorname{Ker} L\cap\operatorname{Ker}P\), then \(u=c_{0}\), \(v=d_{0}\), \(c_{0},d_{0}\in\mathbb{R}\). Furthermore, by the definition of KerP, we have \(c_{0}= d_{0}=0\). Thus, we get
By \((\mathrm{H}_{0})\), the linear operator \(Q(x,y)=(Q_{1}x,Q_{2}y)\) can be defined as
where \(\theta_{1}=z_{0}-\alpha_{1}\), \(\theta_{2}=\tilde{z}_{0}-\alpha_{2}\).
Obviously, \(Q(x,y)= (Q_{1}x(t),Q_{2}y(t) )\cong\mathbb{R}^{2}\).
For \(x(t)\in E\), we have
Similarly, \(Q_{2}^{2}=Q_{2}\), that is to say, the operator Q is idempotent. It follows from \((x,y)=(x,y)-Q(x,y)+Q(x,y)\) that \(Z=\operatorname{Im}L+ \operatorname{Im}Q\). Moreover, by \(\operatorname{Ker}Q=\operatorname{Im}L \) and \(Q_{2}^{2}=Q_{2}\), we get \(\operatorname{Im}L\cap\operatorname{Im}Q=\{(0,0)\}\). Hence,
Now, \(\operatorname{Ind}L = \operatorname{\operatorname{dim} \operatorname{Ker}} L - \operatorname{\operatorname{codim} \operatorname{Im}}L = 0\), so L is a Fredholm mapping of index zero. □
For every \((u,v)\in Y\),
Furthermore, the operator \(K_{P}:\operatorname{Im}L\to \operatorname {dom}L \cap\operatorname{Ker} P\) can be defined
For \((x,y)\in\operatorname{Im} L \), we have
On the other hand, for \((u,v)\in\operatorname{dom} L \cap\operatorname{Ker} P \), according to Lemma 2.1, we have
By the definitions of domL and KerP, one has \(u^{(i)}(0)=v^{(i)}(0)\), \(i=0,1,\ldots,n-1\), which implies that \(c_{i}=d_{i}\), \(i=0,1,\ldots,n-1\). Thus, we obtain
Combining (3.6) and (3.7), we get \(K_{P}=(L_{\operatorname{dom} L \cap\operatorname{Ker} P })^{-1}\).
For \((x,y)\in\operatorname{Im} L\), we have
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 (3.8) we have
By similar arguments as in [11] or [12], we have the following lemma. We omit the proof of it.
Lemma 3.4
\(K_{P}(I-Q)N:Y\rightarrow Y \) is completely continuous.
For simplicity of notation, we set
Theorem 3.1
Assume that \((\mathrm{H}_{0})\) and the following conditions hold.
-
(H1)
There exist nonnegative functions \(\psi(t),\tilde{\psi}(t),\varphi _{i}(t),\tilde{\varphi}_{i}(t) \in E\), \(i=1,2,\ldots,n-1\), such that for \(t\in[0,1]\), \((u_{1},u_{2},\ldots,u_{n}),(v_{1},v_{2},\ldots,v_{n}) \in\mathbb {R}^{n}\), one has
$$\begin{aligned}& \bigl\vert f(t,u_{1},u_{2},\ldots,u_{n}) \bigr\vert \leq \psi(t)+ \varphi_{1}(t) \vert u_{1} \vert ^{p-1} +\cdots+\varphi _{n-1}(t) \vert u_{n} \vert ^{p-1}, \\& \bigl\vert g(t,v_{1},v_{2},\ldots,v_{n}) \bigr\vert \leq\tilde{\psi }(t)+ \tilde{\varphi}_{1}(t) \vert v_{1} \vert ^{p-1} +\cdots +\tilde{\varphi}_{n-1}(t) \vert v_{n} \vert ^{p-1}. \end{aligned}$$ -
(H2)
There exists \(A>0\) such that if \(\vert u \vert >A \) or \(\vert v \vert >A\), \(\forall t\in[0,1] \), one has
$$\begin{aligned} &u\cdot \Biggl[ \sum_{i=1}^{\infty}a_{i}\phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr]\big| _{t=\xi_{i}} \Biggr]>0, \\ &v\cdot \Biggl[ \sum_{i=1}^{\infty}b_{i}\phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr]\big|_{t=\eta_{i}} \Biggr]>0, \end{aligned}$$or
$$\begin{aligned} &u\cdot \Biggl[ \sum_{i=1}^{\infty}a_{i}\phi_{q} \bigl[I_{0+}^{\beta_{1}}f \bigl(t,v(t),D_{0+}^{\alpha_{2}-1}v(t), \ldots,D_{0^{+}}^{\alpha_{2}-(n-1)}v(t) \bigr) \bigr]\big|_{t=\xi_{i}} \Biggr]< 0, \\ &v\cdot \Biggl[ \sum_{i=1}^{\infty}b_{i}\phi_{q} \bigl[I_{0+}^{\beta_{2}}g \bigl(t,u(t),D_{0+}^{\alpha_{1}-1}u(t), \ldots,D_{0^{+}}^{\alpha_{1}-(n-1)}u(t) \bigr) \bigr]\big|_{t=\eta_{i}} \Biggr]< 0. \end{aligned}$$
Then BVP (3.1) has at least a solution in X provided that
where \(c= (\sum_{i=1}^{n-1} \Vert \varphi_{i}(t) \Vert _{\infty})^{q-1}\) and \(\tilde{c}= (\sum_{i=1}^{n-1} \Vert \tilde{\varphi}_{i}(t) \Vert _{\infty})^{q-1}\).
Proof
According to the definitions of \(N_{1}\) and \(N_{2}\), we have the following inequalities.
For \(1< p\leq2\), one has
and
By the similar proof of (3.12) and (3.13), one has
Let
First, we give a proof that for \(1< p\leq2\), \(\Omega_{1}\) is bounded.
Let \(L(u,v)=\lambda N(u,v)\in\operatorname{Im}L=\operatorname{Ker}Q\), that is, \(L_{1}u=\lambda N_{1}v\in\operatorname{Ker}Q_{1}\) and \(L_{2}v=\lambda N_{2}u\in \operatorname{Ker}Q_{2}\). By the definition of \(\operatorname{Ker}Q_{1}\) and \(\operatorname{Ker}Q_{2}\), we have
According to (H2), there exist \(t_{0},t_{1}\in(0,1)\) such that \(\vert u(t_{0}) \vert \leq A\) and \(\vert v(t_{1}) \vert \leq A\). Again, \(L_{1}u=\lambda N_{1}v\), \(u\in\operatorname{dom}L_{1}\setminus\operatorname{Ker}L_{1}\), that is, \(D_{0^{+}}^{\alpha_{1}}u=\lambda N_{1}v\), we have
Substituting \(t=t_{0} \) into the above equation, we get
So, we obtain
Together with \(\vert u (t_{0}) \vert \leq A\) and (3.12), we have
Similarly, by (3.13), we obtain
For \((u,v)\in\Omega_{1}\), by (3.5) and (3.9), we have
The following proof is divided into four cases.
Case 1. \(\Vert (u,v) \Vert _{Y} \leq \vert {{u} }(0) \vert + \Vert N_{1}v \Vert _{\infty}\).
According to (3.10), we can derive
Thus, \(\Omega_{1}\) is bounded.
Case 2. \(\Vert (u,v) \Vert _{Y} \leq \vert {{u} }(0) \vert + \Vert N_{2}u \Vert _{\infty}\).
By (3.10), we can derive
Then \(\Omega_{1}\) is bounded.
Case 3. \(\Vert (u,v) \Vert _{Y} \leq \vert {{v}}(0) \vert + \Vert N_{1}v \Vert _{\infty}\).
According to (3.12) and (3.17), we have
By (3.10), we have
Then \(\Omega_{1}\) is bounded.
Case 4. \(\Vert (u,v) \Vert _{Y} \leq \vert {{v}}(0) \vert + \Vert N_{2}u \Vert _{\infty}\).
According to (3.13) and (3.17), we have
By (3.10), we get
Then \(\Omega_{1}\) is bounded.
Therefore, we have proved that \(\Omega_{1}\) is bounded for \(1< p\leq2\). By similar arguments as the above proof, according to (3.11), (3.14) and (3.15), we can prove that \(\Omega_{1}\) is also bounded for \(p>2\). We omit the proof of it.
Let
Let \((u,v)\in\operatorname{Ker}L\), so we have \(u=c_{0} \), \(v=d_{0}\). In view of \(N(u,v)=(N_{1}v,N_{2}u)\in\operatorname{Im}L=\operatorname{Ker}Q\), we have \(Q_{1}(N_{1}v)=0\), \(Q_{2}(N_{2}u)=0\), that is,
By (H2), there exist constants \(t_{0},t_{1}\in[0,1]\) such that
Therefore, \(\Omega_{2}\) is bounded.
Let
For \((u,v)\in\operatorname{Ker}L\), so we have \(u=c_{0}\) and \(v=d_{0}\). By the definition of the set \(\Omega_{3}\), we have
If \(\lambda=0\), similar to the proof of the boundedness of \(\Omega_{2}\), we have \(\vert c_{0} \vert \leq A\) and \(\vert d_{0} \vert \leq A\). If \(\lambda=1\), we have \(c_{0}=d_{0}=0\). If \(\lambda\in(0,1)\), we also have \(\vert c_{0} \vert \leq A\) and \(\vert d_{0} \vert \leq A\). Otherwise, if \(\vert c_{0} \vert >A\) or \(\vert d_{0} \vert >A\), in view of the first part of (H2), we obtain
which contradict (3.18). Thus, \(\Omega_{3}\) is bounded.
If the second part of (H2) holds, then we can prove that the set
is bounded.
Finally, let Ω to be a bounded open set of Y such that \(\bigcup_{i=1}^{3}{\overline{\Omega}_{i}}\subset\Omega\). By Lemma 3.4, N is L-compact on Ω. Then, by the above arguments, we get
-
(1)
\(L(u,v)\neq\lambda N(u,v)\), for every \((u,v)\in[(\operatorname{dom}L\setminus{Ker}L)\cap\partial\Omega]\times(0,1)\);
-
(2)
\(N(u,v)\notin\operatorname{Im}L\) for every \((u,v)\in\operatorname{Ker}L\cap \partial\Omega\);
-
(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 degree, we obtain that
$$\begin{aligned} \deg (JQ N|_{\operatorname{Ker} L},\Omega\cap\operatorname{Ker} L,0 ) &= \deg \bigl(H( \cdot,0), \Omega\cap\operatorname{Ker} L,0 \bigr) \\ &= \deg \bigl(H(\cdot,1),\Omega\cap\operatorname{Ker} L,0 \bigr) \\ &= \deg (I,\Omega\cap\operatorname{Ker} L,0 ) \\ &=1\neq0. \end{aligned}$$
Applying Theorem 2.1, we conclude that \(Lu=Nu\) has at least one solution in \(\operatorname{dom}L\cap\overline{\Omega}\). □
4 Example
Let us consider the following fractional differential equations with p-Laplacian operator at resonance:
where
Corresponding to BVP (1.1), we have that \(\alpha_{1}=2.6\), \(\beta _{1}=0.6\), \(\alpha_{2}=2.8\), \(\beta_{2}=0.7\), \(n=3\), \(p=3\), \(q=1.5\), \(a= ({\Gamma(\alpha_{1}+1)})^{-1}= ({\Gamma(3.6)})^{-1}\approx0.269\), \(b= ({\Gamma(\beta_{1}+1)})^{1-q}=({\Gamma(1.6)})^{-0.5}\approx1.058\), \(\tilde{a}= ({\Gamma(\alpha_{2}+1)})^{-1}= ({\Gamma(3.8)})^{-1}\approx 0.213\), \(\tilde{b}= ({\Gamma(\beta_{2}+1)})^{1-q}=({\Gamma (1.7)})^{-0.5}\approx1.049\), \(a_{i}=\frac{1}{2^{i}}\), \(\xi_{i}=\frac{1}{2i}\), \(b_{i}=\frac{2}{3^{i}}\), \(\eta _{i}=\frac{1}{3i}\), \(i=1,2,\ldots\) . Then we have \(\sum_{i=1}^{\infty}a_{i}=\sum_{i=1}^{\infty} \vert a_{i} \vert =\sum_{i=1}^{\infty}b_{i}=\sum_{i=1}^{\infty} \vert b_{i} \vert =1\). Taking \(z_{0}=\tilde{z}_{0}=3\), we have
which implies that \((\mathrm{H}_{0})\) holds.
By a simple proof, we have
Choose \(\psi(t)=\frac{1}{5} \), \(\varphi_{1}(t)= \frac{1}{10}\), \(\varphi _{2}=\varphi_{3}=0\), \(\tilde{\psi}(t)=\frac{1}{8} \), \(\tilde{\varphi}_{1}(t)= \frac{1}{20}\), \(\tilde{\varphi}_{2}=\tilde{\varphi}_{3}=0\), then we have (H1) of Theorem 3.1 is satisfied.
By a simple computation, we have \(c= (\sum_{i=1}^{n-1} \Vert \varphi_{i}(t) \Vert _{\infty})^{q-1}=(\varphi_{1})^{q-1}=\sqrt{0.1}\approx0.316\), \(\tilde {c}= (\sum_{i=1}^{n-1} \Vert \tilde{\varphi}_{i}(t) \Vert _{\infty})^{q-1}= (\tilde{\varphi}_{1})^{q-1}=\sqrt{0.05}\approx0.224 \), \(\tilde{a}\tilde{b}\tilde{c}+bc\approx0.287\), \(abc+\tilde{b}\tilde {c}\approx0.298\), \(abc+bc\approx0.301\), \(\tilde{a}\tilde{b}\tilde {c}+\tilde{b}\tilde{c}\approx0.240\). So, (3.11) holds.
In addition, by choosing \(A=1 \), we have if \(u> 1\), or \(v> 1\), then f, g are positive functions. So, the first inequality of (H2) is satisfied.
Thus, all the conditions of Theorem 3.1 are satisfied; consequently, its conclusion implies that problem (4.1) has a solution on \([0, 1]\).
5 Conclusion
In this paper, we have obtained the existence of solutions for a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions at resonance. We base our analysis on the known coincidence degree theory. The issue on the existence of solutions of infinite-point boundary value problems is interesting. As applications, an example is presented to illustrate the main results. In the future, we will consider the positive solutions for the fractional infinite-point boundary value problems at resonance.
References
Kilbas, AA, Srivastava, HH, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Oldham, KB, Spanier, J: The Fractional Calculus. Academic Press, New York (1974)
Gaul, L, Klein, P, Kempfle, S: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81-88 (1991)
Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Ahmad, B, Agarwal, PR, Alsaedi, A: Fractional differential equations and inclusions with semiperiodic and three-point boundary conditions. Bound. Value Probl. 2016, 28 (2016)
Liu, R, Kou, C, Xie, X: Existence results for a coupled system of nonlinear fractional boundary value problems at resonance. Math. Probl. Eng. 2013, 1-9 (2013)
Ahmad, B, Ntouyas, SK, Agarwal, RP, Alsaedi, A: Existence results for sequential fractional integro-differential equations with nonlocal multi-point and strip conditions. Bound. Value Probl. 2016, 205 (2016)
Agarwal, RP, Ntouyas, SK, Ahmad, B, Alzahrani, AK: Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments. Adv. Differ. Equ. 2016, 92 (2016)
Jiang, W: Solvability of fractional differential equations with p-Laplacian at resonance. Appl. Math. Comput. 260, 48-56 (2015)
Kosmatov, N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135 (2010)
Hu, L, Zhang, S: On existence results for nonlinear fractional differential equations involving the p-Laplacian at resonance. Mediterr. J. Math. 13, 955-966 (2016)
Hu, L, Zhang, S, Shi, A: Existence result for nonlinear fractional differential equation with p-Laplacian operator at resonance. J. Appl. Math. Comput. 48, 519-532 (2015)
Shen, T, Liu, W, Chen, T, Shen, X: Solvability of fractional multi-point boundary-value problems with p-Laplacian operator at resonance. Electron. J. Differ. Equ. 2014, 58 (2014)
Chen, T, Liu, W, Hu, Z: A boundary value problem for fractional differential equation with p-Laplacian operator at resonance. Nonlinear Anal. 75, 3210-3217 (2012)
Hu, Z, Liu, W, Liu, J: Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance. Adv. Differ. Equ. 2013, 312 (2013)
Cheng, L, Liu, W, Ye, Q: Boundary value problem for a coupled system of fractional differential equations with p-Laplacian operator at resonance. Electron. J. Differ. Equ. 2014, 60 (2014)
Kosmatov, N, Jiang, W: Resonant functional problems of fractional order. Chaos Solitons Fractals 91, 573-579 (2016)
Aljoudi, S, Ahmad, B, Nieto, JJ, Alsaedi, A: A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions. Chaos Solitons Fractals 91, 39-46 (2016)
Alsaedi, A, Aljoudi, S, Ahmad, B: Existence of solutions for Riemann-Liouville type coupled systems of fractional integro-differential equations and boundary conditions. Electron. J. Differ. Equ. 2016, 211 (2016)
Zhang, X: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett. 39, 22-27 (2015)
Gao, H, Han, X: Existence of positive solutions for fractional differential equation with nonlocal boundary condition. Int. J. Differ. Equ. 2011, 328394 (2011)
Zhong, Q, Zhang, X: Positive solution for higher-order singular infinite-point fractional differential equation with p-Laplacian. Adv. Differ. Equ. 2016, 11 (2016)
Ge, F, Zhou, H, Kou, C: Existence of solutions for coupled fractional differential equation with infinitely many points boundary conditions at resonance on an unbounded domain. Differ. Equ. Dyn. Syst. 24, 1-17 (2016)
Mawhin, J: Topological degree and boundary value problems for nonlinear differential equations in topological methods for ordinary differential equations. Lect. Notes Math. 1537, 74-142 (1993)
Acknowledgements
We are grateful to the referees for their careful reading of the manuscript. Their comments and suggestions were of great importance in helping to improve the quality of the paper. This work is supported by the National Natural Science Foundation of China (Grant No. 11371364).
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Hu, L., Zhang, S. Existence results for a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions. Bound Value Probl 2017, 88 (2017). https://doi.org/10.1186/s13661-017-0819-4
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DOI: https://doi.org/10.1186/s13661-017-0819-4
MSC
- 26A33
- 34B15
Keywords
- fractional differential equation
- infinite-point boundary value conditions
- p-Laplacian
- resonance