Abstract
In this paper, we investigate the existence and uniqueness of solutions to the coupled system of nonlinear fractional differential equations
where \(D^{\nu}_{0^{+}}\) is the standard Riemann-Liouville fractional derivative of order ν, \(t\in(0,1)\), \(\nu_{1}, \nu_{2} \in(n-1,n]\) for \(n>3\) and \(n \in\mathbf{N} \), and \(\lambda_{1}, \lambda_{2} > 0\), with the multi-point boundary value conditions: \(y^{(i)}_{1}(0)=0=y^{(i)}_{2}(0)\), \(0 \leq i \leq n-2\); \(D^{\beta}_{0^{+}}y_{1}(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta }_{0^{+}}y_{1}(\xi_{i})\); \(D^{\beta}_{0^{+}}y_{2}(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta }_{0^{+}}y_{2}(\xi_{i})\), where \(n-2 < \beta< n-1\), \(0 < \xi_{1} < \xi_{2} < \cdots< \xi_{m-2} <1\), \(b_{i} \geq0\) (\(i=1,2,\ldots,m-2\)) with \(\rho_{1}: = \sum^{m-2}_{i=1} b_{i}\xi^{\nu_{1}-\beta-1}_{i}<1\), and \(\rho_{2}: =\sum^{m-2}_{i=1} b_{i}\xi^{\nu_{2}-\beta-1}_{i}<1\). Our analysis relies on the Banach contraction principle and Krasnoselskii’s fixed point theorem.
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1 Introduction
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The first definition of fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus, was mentioned already in 1695 by Leibniz and L’Hospital. With the help of fractional calculus, we can describe natural phenomena and mathematical models more accurately. The fractional differential equations play an important role in various fields of engineering, physics, economics and biological sciences, etc. (see [1–4] for example). In consequence, the subject of fractional differential equations is gaining much importance and attention. For more details on basic theory of fractional differential equations, one can see the monographs of Diethelm [1], Kilbas et al. [2], Miller and Ross [3], Podlubny [4] and Tarasov [5], and the papers [6–13] and the references therein.
As is known to all, the initial and boundary value problems for nonlinear fractional differential equations arise from the study of models of control, porous media, electrochemistry, viscoelasticity, electromagnetics, etc. Recently, the existence and uniqueness of solutions of initial and boundary value problems for nonlinear fractional equations have been extensively studied (see [14–20]), and some are coupled systems of nonlinear fractional differential equations (see [8, 14, 17, 21, 22]).
In [15], Mophou studied the mild solutions to impulsive fractional differential equations
where \(0< \alpha<1\), the operator \(A: D(A) \subset X \rightarrow X\) is a generator of \(C_{0}\)-semigroup \((T(t))_{t\geq0}\) on a Banach space X, \(D_{t}^{\alpha}\) is the Caputo fractional derivative, \(f: I\times X\rightarrow X\) is a given continuous function, \(I_{k}:X\rightarrow X\), \(0=t_{0}< t_{1}<\cdots <t_{m}<t_{m+1}=T\), \(\Delta x|_{t=t_{k}}=x(t^{+}_{k})-x(t^{-}_{k})\). Some existence and uniqueness results for the equations were established by means of Krasnoselskii’s fixed point theorem.
By using the same fixed point theorem, Goodrich [23] considered the existence of a positive solution to the following system of differential equations of fractional order:
where \(D^{\nu}_{0^{+}}\) is the standard Riemann-Liouville fractional derivative of order ν, \(t\in (0,1)\), \(\nu_{1}, \nu_{2} \in(n-1,n]\) for \(n>3\) and \(n \in\mathbf{N} \), and \(\lambda_{1}, \lambda_{2} > 0\), with the following boundary value conditions:
under the assumptions that \(a_{1}\), \(a_{2}\), f, g are nonnegative and continuous.
Very recently, Sun et al. [14] considered the coupled system of multi-term nonlinear fractional differential equations
where \(t\in(0,1]\), \(\alpha>\beta_{1}>\beta_{2}>\cdots>\beta_{N}>0\), \(\sigma>\rho_{1}>\rho _{2}\cdots>\rho_{N}>0\), \(n_{1}=[\alpha]+1\), \(n_{2}=[\sigma]+1\), \(\beta_{q}, \rho_{q}<1\) and \(q\in \{1,2,\ldots,N\}\). By using the Schauder fixed point theorem and the Banach contraction principle, some results of existence and uniqueness of solutions for the coupled system are obtained.
However, to our knowledge, there are few works that deal with multi-point boundary value problems for a coupled system of nonlinear fractional differential equations. The purpose of this article is to investigate the solutions for the coupled system of nonlinear fractional differential equations (1) with the multi-point boundary conditions:
where \(n-2 < \beta< n-1\), \(0 < \xi_{1} < \xi_{2} < \cdots< \xi_{m-2} <1\), \(b_{i} \geq0\) (\(i=1,2,\ldots,m-2\)) with \(\rho_{1}: =\sum^{m-2}_{i=1} b_{i}\xi^{\nu_{1}-\beta-1}_{i}<1\), and \(\rho_{2}: =\sum^{m-2}_{i=1} b_{i}\xi^{\nu_{2}-\beta-1}_{i}<1\).
Motivated by the above-mentioned works and recent works on coupled systems of fractional differential equations, we consider the existence and uniqueness of solutions of coupled system (1)-(4) by means of the Banach contraction principle and Krasnoselskii’s fixed point theorem. In our paper, we do not suppose that \(a_{1}\), \(a_{2}\), f, g are nonnegative.
With this context in mind, the outline of this paper is as follows. In Section 2 we recall certain results from the theory of continuous fractional calculus. In Section 3 we provide some conditions under which problem (1)-(4) will have a unique solution or at least one solution.
2 Preliminaries
For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems.
Definition 2.1
(see [4])
Let \(\nu>0\) with \(\nu\in\mathbf{R}\). Suppose that \(y:[a,+\infty)\rightarrow\mathbf{R}\). Then the νth Riemann-Liouville fractional integral is defined to be
whenever the right-hand side is defined. Similarly, with \(\nu>0\) and \(\nu\in\mathbf{R}\), we define the νth Riemann-Liouville fractional derivative to be
where \(n\in\mathbf{N}\) is the unique positive integer satisfying \(n-1\leq\nu< n\) and \(t>a\).
Lemma 2.2
(see [23])
Let \(\alpha\in\mathbf{R}\). Then \(D^{n}D^{\alpha}_{a^{+}}y(t)=D^{n+\alpha}_{a^{+}}y(t)\), for each \(n\in N_{0}\), where \(y(t)\) is assumed to be sufficiently regular so that both sides of the equality are well defined. Moreover, if \(\beta\in (-\infty,0]\) and \(\gamma\in[0,+\infty)\), then \(D^{\gamma}_{a^{+}}D^{\beta }_{a^{+}}y(t)=D^{\gamma+\beta}_{a^{+}}y(t)\).
Lemma 2.3
(see [23])
The general solution to \(D^{\nu }_{a^{+}}y(t)=0\), where \(n-1<\nu\leq n\) and \(\nu> 0\), is the function \(y(t)=c_{1}t^{\nu-1}+c_{2}t^{\nu-2}+\cdots+c_{n}t^{\nu- n}\), where \(c_{i}\in\mathbf{R}\) for each i.
Lemma 2.4
Let \(h\in C^{n}([0,1])\) be given. Then the unique solution to problem \(-D^{\nu}_{0^{+}}y(t)=h(t)\) together with the boundary conditions \(y^{(i)}(0)=0\) and \(D^{\beta}_{0^{+}}y(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta}_{0^{+}}y(\xi_{i})\), where \(n-2 < \beta< n-1\) and \(0\leq i \leq n-2\), is
where
is the Green function for this problem, where \(\rho=\sum^{m-2}_{i=1} b_{i}\xi^{\nu-\beta-1}_{i}<1\) and \(\xi_{0}=0\), \(\xi _{m-1}=1\).
Proof
We know that the general solution to our problem is
we immediately observe that the boundary value condition \(y^{(i)}(0)=0\), \(0 \leq i \leq n-2\), implies that \(c_{2}=c_{3}=\cdots=c_{n}=0\). On the other hand, \(D^{\beta}_{0^{+}}y(1)= \sum^{m-2}_{i=1}b_{i}D^{\beta}_{0^{+}}y(\xi_{i})\) implies that
That is to say,
Therefore, the unique solution is
The proof is complete. □
3 Main results
This section deals with the existence and uniqueness of solutions to problem (1)-(4).
Let E represent the Banach space of \(C[0,1]\) when equipped with the usual supremum norm \(\| \cdot\| \). Then put \(X:=E\times E\), where X is equipped with the norm
for \((y_{1},y_{2})\in X\). Observe that X is also a Banach space (see [24]). In addition, define two operators \(T_{1},T_{2}:X\rightarrow E\) by
and
where \(G_{1}(t,s)\) is the Green function of Lemma 2.4 with ν replaced by \(\nu_{1}\) and ρ replaced by \(\rho_{1}\), and likewise, \(G_{2}(t,s)\) is the Green function of Lemma 2.4 with ν replaced by \(\nu_{2}\) and ρ replaced by \(\rho_{2}\). Now, we define an operator \(S:X\rightarrow X\) by
We claim that whenever \((y_{1},y_{2})\in X\) is a fixed point of the operator defined in (7), it follows that \(y_{1}(t)\) and \(y_{2}(t)\) solve problem (1)-(4). We shall look for fixed points of the operator S, seeing as these fixed points coincide with solutions of problem (1)-(4).
To establish the main results, we need the following assumptions:
(H1) \(f,g:[0,1]\times\mathbf{R}\times\mathbf{ R}\rightarrow \mathbf{R}\) and \(a_{1},a_{2}:[0,1] \rightarrow\mathbf{R}\);
(H2) f, g, \(a_{1}\), \(a_{2}\) are continuous;
(H3) there exist positive functions \(L_{1}(t)\) and \(L_{2}(t)\) such that
for all \(t\in[0,1]\) and \((y_{1},y_{2}),(u_{1},u_{2}) \in X\).
Further, we set
(H4) The parameters \(\lambda_{1}\), \(\lambda_{2}\) satisfy \(\lambda _{1},\lambda_{2} < \Lambda\), where
(H5) The parameters \(\lambda_{1}\), \(\lambda_{2}\) satisfy \(\lambda _{1},\lambda_{2} < \Lambda\), where
(H6) There exists \(\mu\in L^{1}([0,1],\mathbf{R}^{+})\) such that
We are ready to state the existence and uniqueness result.
Theorem 3.1
Suppose that conditions (H1)-(H4) are satisfied. Then the boundary value problem (1)-(4) has a unique solution.
Proof
Let us set
and choose
Now, we show that \(S(\Omega_{r}) \subset\Omega_{r}\), where \(\Omega_{r} =\{ (y_{1},y_{2}) \in X :\|(y_{1},y_{2})\|\leq r \}\), and S is a contraction. In fact, for all \((y_{1},y_{2}) \in\Omega_{r}\), we obtain
that is to say, \(\| T_{1}(y_{1},y_{2}) \|\leq\frac {r}{2}\).
Then, for \((y_{1},y_{2}),(u_{1},u_{2}) \in X\) and for each \(t\in[0,1]\), we obtain
That is to say, \(\| T_{1}(y_{1},y_{2})- T_{1}(u_{1},u_{2})\|\leq\frac{1}{4}\| (y_{1}-u_{1},y_{2}-u_{2}) \|=\frac{1}{4} \| (y_{1},y_{2})-(u_{1},u_{2})\|\).
Hence, we find that \(T_{1}:\Omega_{r} \rightarrow B_{\frac{r}{2}}\) and \(T_{1}\) is a contraction, where \(B_{\frac{r}{2}}=\{ y\in B:\| y \|\leq\frac{r}{2} \}\). Similarly, we have \(T_{2}:\Omega_{r} \rightarrow B_{\frac{r}{2}}\) and \(T_{2}\) is a contraction. Consequently, for any \((y_{1},y_{2})\in\Omega_{r}\),
i.e., \(S(\Omega_{r}) \subset\Omega_{r}\). And, for \((y_{1},y_{2}),(u_{1},u_{2}) \in\Omega_{r}\) and for each \(t\in[0,1]\),
So, \(S:\Omega_{r} \rightarrow\Omega_{r} \) and S is a contraction. Thus, the conclusion of the theorem follows from the contraction mapping principle. □
Our next result is based on the following well-known fixed point theorem due to Krasnoselskii.
Lemma 3.2
(Krasnoselskii [25])
Let K be a closed convex and nonempty subset of a Banach space E. Let T, S be the operators such that:
-
(i)
\(Tx+Sy\in K\) whenever \(x,y\in K\);
-
(ii)
T is compact and continuous;
-
(iii)
S is a contraction mapping.
Then there exists \(z\in K\) such that \(z=Tz+Sz\).
Now we are ready to state and prove the following existence result.
Theorem 3.3
Suppose that conditions (H1)-(H3), (H5), (H6) are satisfied. Then there exists at least one solution of the boundary value problem (1)-(4).
Proof
Let us fix
and consider \(\Omega_{r} =\{ (y_{1},y_{2}) \in X :\| (y_{1},y_{2}) \|\leq r \}\). We define the operators \(Q_{1}\) and \(Q_{2}\) on \(\Omega_{r}\) as
where
For all \((y_{1},y_{2}),(u_{1},u_{2}) \in\Omega_{r} \), we find that
Thus, \(Q_{1}(y_{1},y_{2})+Q_{2}(u_{1},u_{2})\in\Omega_{\frac{r}{2}}\) for all \((y_{1},y_{2}),(u_{1},u_{2}) \in\Omega_{r}\). From assumption (H3), we have
where \((y_{1},y_{2}),(u_{1},u_{2}) \in X\), \(t \in[0,1] \). So, \(Q_{2}\) is a contraction mapping. We next consider the operator \(Q_{1}\). Evidently, the continuity of f implies that the operator \(Q_{1}\) is continuous. Also, \(Q_{1}\) is uniformly bounded on \(\Omega_{r}\) as
Now, we show that \(Q_{1}(y_{1},y_{2})(t)\) is equicontinuous. In fact, since \(a_{1}\), f are bounded on the compact set \([0,1]\) and \([0,1]\times\Omega_{r}\), respectively, we can define
and we have, for any \(t_{1},t_{2}\in[0,1]\),
which is independent of \((y_{1},y_{2})\). Therefore, \(Q_{1}\) is equicontinuous on \(\Omega_{r}\). Hence, by the Arzela-Ascoli theorem, \(Q_{1}\) is compact on \(\Omega_{r}\). Similarly, we set
where
And we can obtain a similar conclusion to the operator \(T_{1}\). Now, we let
Therefore, we have
In view of the proof above, we get the following.
(I) For all \((y_{1},y_{2}),(u_{1},u_{2}) \in\Omega_{r} \), we find that
Thus, \(A(y_{1},y_{2})+B(u_{1},u_{2})\in\Omega_{r}\) for all \((y_{1},y_{2}),(u_{1},u_{2}) \in\Omega_{r}\).
(II) For \((y_{1},y_{2}),(u_{1},u_{2}) \in X\), \(t \in[0,1]\),
So, B is a contraction mapping.
(III) The continuity of \(Q_{1}\) and \(P_{1}\) implies that the operator A is continuous. Also, A is uniformly bounded on \(\Omega _{r}\) as
Moreover, for any \(t_{1},t_{2}\in[0,1]\),
which is independent of \((y_{1},y_{2})\). Therefore, A is equicontinuous on \(\Omega_{r}\). Hence, by the Arzela-Ascoli theorem, A is compact on \(\Omega_{r}\).
Thus all the assumptions of Lemma 3.2 are satisfied, so \(S(y_{1},y_{2})=A(y_{1},y_{2})+B(y_{1},y_{2})\) has at least one fixed point. Hence, we obtain that (1)-(4) has at least one solution. □
4 Conclusions
There are few works that deal with multi-point boundary value problems for a coupled system of nonlinear fractional differential equations. In this article, we study multi-point boundary value problems for a coupled system of nonlinear fractional differential equations (1)-(4). By using Green’s function, the Banach contraction principle and Krasnoselskii’s fixed point theorem, we establish some new existence, uniqueness theorems of solutions for multi-point boundary value problems for a coupled system of nonlinear fractional differential equations (1)-(4).
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Acknowledgements
The research was supported by the Youth Science Foundation of China (11201272) and the Science Foundation of Shanxi Province (2010021002-1).
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Zhai, C., Hao, M. Multi-point boundary value problems for a coupled system of nonlinear fractional differential equations. Adv Differ Equ 2015, 147 (2015). https://doi.org/10.1186/s13662-015-0487-6
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DOI: https://doi.org/10.1186/s13662-015-0487-6