 Research
 Open Access
Solvability of boundary value problems of nonlinear fractional differential equations
 Weiqi Chen^{1, 2} and
 Yige Zhao^{3}Email author
https://doi.org/10.1186/s1366201503732
© Chen and Zhao; licensee Springer. 2015
 Received: 8 October 2014
 Accepted: 13 January 2015
 Published: 31 January 2015
Abstract
In this paper, we study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem \(D^{\alpha}_{0^{+}}u(t)+f(t,u(t))=0 \), \(0< t<1\), \(u(0)=u(1)=u'(0)=0\), where \(2<\alpha\leq3\) is a real number, \(D^{\alpha}_{0^{+}}\) is the RiemannLiouville fractional derivative. By the properties of the Green’s function, the lower and upper solution method and the LeggettWilliams fixed point theorem, some new existence criteria are established. As applications, examples are presented to illustrate the main results.
Keywords
 fractional differential equation
 boundary value problem
 positive solution
 fractional Green’s function
 fixed point theorem
 lower and upper solution method
MSC
 34A08
 34B18
1 Introduction
Fractional differential equations have been of great interest. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in selfsimilar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science; see [1–5]. Recently, there have appeared some papers dealing with the existence of solutions of fractional differential equations by the use of techniques of nonlinear analysis (fixed point theorems, LeraySchauder theory, Adomian decomposition method, etc.); see [6–9]. Especially, boundary value problems for fractional differential equations have attracted considerable attention; see [10–31]. As is well known, the aim of finding solutions to boundary value problems is of main importance in various fields of applied mathematics. Recently, there seems to be a new interest in the study of the boundary value problems for fractional differential equations.
From the above works, we can see that, although the fractional boundary value problems have been investigated by some authors, the lower and upper solution method and the fixed point theorem due to LeggettWilliams are seldom considered. In addition, in the latter work the multiplicity of the solutions was not employed. Furthermore, the solution technique of upper and lower solutions was not studied, and it was also assumed that \(2<\alpha\leq3\). This paper will fill up the gap.
The plan of the paper is as follows. In Section 2, we shall give some definitions and lemmas to prove our main results. In Section 3, we establish the existence of a single positive solution for the boundary value problem (1.1) and (1.2) by the lower and upper solution method. In Section 4, we establish the existence of multiple positive solutions for the boundary value problem (1.1) and (1.2) by the LeggettWilliams fixed point theorem. Examples are presented to illustrate the main results in Section 3 and Section 4, respectively. In Section 5, we give the conclusion of the paper.
2 Preliminaries
For the convenience of the reader, we give some background materials from fractional calculus theory to facilitate the analysis of problem (1.1) and (1.2). These materials can be found in the recent literature; see [31–33].
Definition 2.1
([32])
Definition 2.2
([32])
From the definition of the RiemannLiouville derivative, we can obtain the following statement.
Lemma 2.1
Lemma 2.2
In the following, we present the Green’s function of the fractional differential equation boundary value problem.
Lemma 2.3
([31])
The following properties of the Green’s function play important roles in this paper.
Lemma 2.4
([31])
 (1)
\(G(t,s)=G(1s,1t)\), for \(t, s\in(0,1)\);
 (2)
\(t^{\alpha1}(1t)s(1s)^{\alpha1}\leq{\Gamma(\alpha)}G(t,s)\leq (\alpha1)s(1s)^{\alpha1}\), for \(t, s\in(0,1)\);
 (3)
\(G(t,s)>0\), for \(t, s\in(0,1)\);
 (4)
\(t^{\alpha1}(1t)s(1s)^{\alpha1}\leq{\Gamma(\alpha)}G(t,s)\leq (\alpha1)(1t)t^{\alpha1}\), for \(t, s\in(0,1)\).
Remark 2.1
Obviously, by Lemma 2.4, we have \(u(t)\geq0\) if \(h(t)\geq0\) on \(t\in[0,1]\), where \(u(t)\) and \(h(t)\) are defined as (2.1).
Now we introduce the following two definitions concerned with the upper and lower solutions of the fractional boundary value problem (1.1) and (1.2).
Definition 2.3
Definition 2.4
The following definition is about the nonnegative continuous concave functional.
Definition 2.5
The following lemma is fundamental in the proofs of our main results.
Lemma 2.5
([33])
 (C1)
\(\{x\in P(\theta,b,d)\theta(x)>b\}\neq\emptyset\) and \(\theta(Ax)>b\) for \(x\in P(\theta,b,d)\);
 (C2)
\(\Ax\< a\) for \(x\leq a\);
 (C3)
\(\theta(Ax)>b\) for \(x\in P(\theta,b,c)\) with \(\Ax\>d\).
Remark 2.2
If we have \(d=c\), then condition (C1) of Lemma 2.5 implies condition (C3) of Lemma 2.5.
3 Single positive solution
In this section, we establish the existence of single positive solution for the boundary value problem (1.1) and (1.2) by the lower and upper solution method. In this section, we set \(f\in C([0,1]\times[0,+\infty),(0,+\infty))\). As an application, an example is given to illustrate the main results.
Lemma 3.1
If u is a positive solution of (1.1) and (1.2), then there exist two constants r and R such that \(r\rho(t)\leq u(t)\leq R\rho(t)\), where \(\rho(t)=\int_{0}^{1}G(t,s)\,ds\).
Proof
Theorem 3.1
 (H_{ f }):

\(f(t,u)\in C([0,1]\times[0,+\infty), \mathbb{R}^{+})\) is nondecreasing relative to u, \(f(t,\rho(t))\not\equiv0\) for \(t\in(0,1)\) and there exists a positive constant \(\mu<1\) such that$$k^{\mu}f(t,u)\leq f(t,ku), \quad \forall 0\leq k\leq1. $$
Proof
Finally, we will prove that the fractional boundary value problem (1.1) and (1.2) has a positive solution.
In the following, we present a simple example to explain our results.
Example 3.1
4 Multiple positive solutions
In this section, we establish the existence of multiple positive solutions for the boundary value problem (1.1) and (1.2) by the LeggetWilliams fixed point theorem. In this section, we set \(f\in C([0,1]\times[0,+\infty),[0,+\infty))\). As an application, an example is given to illustrate the main results.
Then we have the following lemma.
Lemma 4.1
\(A:P\to P\) is completely continuous.
Proof
The operator \(A:P\to P\) is continuous in view of the continuity of \(G(t,s)\) and \(f(t,u(t))\). By means of the ArzelaAscoli theorem, \(A:P\to P\) is completely continuous. □
Theorem 4.1
 (B1)
\(f(t,u)< Ma\), for \((t,u)\in[0,1]\times[0,a]\);
 (B2)
\(f(t,u)\geq Nb\), for \((t,u)\in[1/4,3/4]\times[b,c]\);
 (B3)
\(f(t,u)\leq Mc\), for \((t,u)\in[0,1]\times[0,c]\).
Proof
We show that all the conditions of Lemma 2.5 are satisfied.
The proof is complete. □
Corollary 4.1
 (B4)
\(f(t,u)< Ma_{1}\), for \((t,u)\in[0,1]\times[0,a_{1}]\);
 (B5)
\(f(t,u)\geq\widetilde{N}b_{1}\), for \((t,u)\in[1/4,3/4]\times [\sigma b_{1},c_{1}]\);
 (B6)
\(f(t,u)\leq Mc_{1}\), for \((t,u)\in[0,1]\times[0,c_{1}]\).
Proof
If we choose \(a=a_{1}\), \(b=\sigma b_{1}\), and \(c=c_{1}\), then from Theorem 4.1, the conclusion holds. □
Theorem 4.2
Proof
We only need to show there exists a number \(c'\) with \(c'>c\) and \(A: \overline{P}_{c'}\to\overline{P}_{c'}\).
Let \(L=\max_{u\in[0,R]}f(t,u)\), \(t\in[0,1]\).
Thus, \(A: \overline{P}_{c'}\to\overline{P}_{c'}\). □
Theorem 4.3
 (B7)
\(f(t,u)< M{a'_{i}}\), for \((t,u)\in[0,1]\times[0,{a'_{i}}]\);
 (B8)
\(f(t,u)\geq\widetilde{N}{b'_{i}}\), for \((t,u)\in [1/4,3/4]\times[\sigma{b'_{i}},{c'_{i}}]\).
Proof
When \(n=1\), it is immediate from condition (B7) that \(A: \overline{P}_{{a'_{1}}}\to\overline{P}_{{a'_{1}}}\), which means that A has at least one point \(u_{1}\in\overline{P}_{{a'_{1}}}\) by the Schauder fixed point theorem.
In this way, we finish the proof by induction. The proof is complete. □
In the following, we present a simple example to illustrate our results.
Example 4.1
5 Conclusions
In this paper, we have studied the existence of positive solutions for a boundary value problem of nonlinear fractional differential equations involving the RiemannLiouville fractional derivative. The existence of a single positive solution for the given problem has been obtained by using the properties of the Green’s function and the lower and upper solution method, while the existence of multiple positive solutions is based on the LeggettWilliams fixed point theorem. The main results are well illustrated with the help of examples. Our results improve the work presented in [31].
Declarations
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the National Natural Science Foundation of China (G61374065, G61374002), and the Research Fund for the Taishan Scholar Project of Shandong Province of China.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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