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Solvability of boundary value problems of nonlinear fractional differential equations
Advances in Difference Equations volume 2015, Article number: 36 (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.
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.
Bai and Lü [28] studied the following twopoint boundary value problem of fractional differential equations:
where \(1<\alpha\leq2\) is a real number and \(D^{\alpha}_{0^{+}}\) is the standard RiemannLiouville fractional derivative. They obtained the existence of positive solutions by means of the Krasnosel’skii fixed point theorem and the LeggettWilliams fixed point theorem.
Zhang [29] considered the existence and multiplicity of positive solutions for the nonlinear fractional boundary value problem
where \(1<\alpha\leq2\) is a real number, \(f: [0,1]\times[0,+\infty)\to[0,+\infty)\) is continuous and \({}^{\mathrm{C}} D^{\alpha}_{0^{+}}\) is the standard Caputo fractional derivative. The author obtained the existence and multiplicity results of positive solutions by means of the Krasnosel’skii fixed point theorem and the LeggettWilliams fixed point theorem.
Liang and Zhang [30] investigated the following nonlinear fractional boundary value problem:
where \(3<\alpha\leq4\) is a real number, \(f\in C([0,1]\times[0,+\infty),(0,+\infty))\) and \(D^{\alpha}_{0^{+}}\) is the standard RiemannLiouville fractional derivative. By means of the lower and upper solution method and fixed point theorems, some results on the existence of positive solutions are obtained for the above fractional boundary value problems.
Yu and Jiang [31] discussed the following twopoint boundary value problem of fractional differential equations:
where \(2<\alpha\leq3\) is a real number and \(D^{\alpha}_{0^{+}}\) is the standard RiemannLiouville fractional derivative. By the properties of the Green’s function, they gave some results of multiple positive solutions for singular and nonsingular boundary value problems by means of the LeraySchauder nonlinear alternative, a fixed point theorem on cones, and a mixed monotone method.
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.
Motivated by all the works above, in this paper we discuss the boundary value problem
where \(2<\alpha\leq3\) is a real number and \(D^{\alpha}_{0^{+}}\) is the RiemannLiouville fractional differentiation. Using the lower and upper solution method and the LeggettWilliams fixed point theorem, we give some new existence criteria for the boundary value problem (1.1) and (1.2). Finally, we present an example to demonstrate our results.
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.
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])
The RiemannLiouville fractional derivative of order \(\alpha>0\) of a continuous function \(f:(0,+\infty)\to\mathbb{R}\) is given by
where \(n=[\alpha]+1\), \([\alpha]\) denotes the integer part of number α, provided that the right side is pointwise defined on \((0,+\infty)\).
Definition 2.2
([32])
The RiemannLiouville fractional integral of order \(\alpha>0\) of a function \(f:(0,+\infty)\to \mathbb{R}\) is given by
provided that the right side is pointwise defined on \((0,+\infty)\).
From the definition of the RiemannLiouville derivative, we can obtain the following statement.
Lemma 2.1
Let \(\alpha>0\). If we assume \(u\in C(0,1)\cap L^{1} (0,1)\), then the fractional differential equation
has \(u(t)=c_{1}t^{\alpha1}+c_{2}t^{\alpha2}+\cdots+c_{n}t^{\alphan}\), \(c_{i}\in\mathbb{R}\), \(i=1,2,\ldots,n\), as the unique solution, where n is the smallest integer greater than or equal to α.
Lemma 2.2
Assume that \(u\in C(0,1)\cap L^{1} (0,1)\) with a fractional derivative of order \(\alpha>0\) that belongs to \(C(0,1)\cap L^{1} (0,1)\). Then
where n is the smallest integer greater than or equal to α.
In the following, we present the Green’s function of the fractional differential equation boundary value problem.
Lemma 2.3
([31])
Let \(h\in{C[0,1]}\) and \(2<\alpha\leq3\). The unique solution of problem
is
where
Here G is called the Green’s function of the boundary value problem (2.1) and (2.2).
The following properties of the Green’s function play important roles in this paper.
Lemma 2.4
([31])
The function G defined by (2.3) satisfies the following conditions:

(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
A function β is called a lower solution of the fractional boundary value problem (1.1) and (1.2), if \(\beta\in C[0,1]\) and \(\beta(t)\) satisfies
where \(f\in C([0,1]\times[0,+\infty),(0,+\infty))\).
Definition 2.4
A function γ is called a upper solution of the fractional boundary value problem (1.1) and (1.2), if \(\gamma\in C[0,1]\) and \(\gamma(t)\) satisfies
where \(f\in C([0,1]\times[0,+\infty),(0,+\infty))\).
The following definition is about the nonnegative continuous concave functional.
Definition 2.5
The map θ is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that \(\theta:P \to[0,\infty)\) is continuous and
for all \(x, y\in P\) and \(0\leq t\leq 1\).
The following lemma is fundamental in the proofs of our main results.
Lemma 2.5
([33])
Let P be a cone in a real Banach space E, \(P_{c}=\{x\in P \x\\leq c\}\), θ a nonnegative continuous concave functional on P such that \(\theta(x)\leq\x\\), for all \(x\in\overline{P}_{c}\), and \(P(\theta,b,d)=\{x\in P b\leq\theta(x), \x\\leq d\}\). Suppose \(A: \overline{P}_{c} \to\overline{P}_{c}\) is completely continuous and there exist constants \(0< a<b<d\leq c\) such that

(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 \(\Ax\>d\).
Then A has at least three fixed points \(x_{1}\), \(x_{2}\), \(x_{3}\) with
Remark 2.2
If we have \(d=c\), then condition (C1) of Lemma 2.5 implies condition (C3) of Lemma 2.5.
For convenience, we set \(q(t)=t^{\alpha1}(1t)\), \(k(s)=s(1s)^{\alpha1}\), then
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
Since \(u\in C[0,1]\), there exists \(M'>0\) so that \(u(t)\leq M'\) for \(t\in[0,1]\). Taking
In view of Lemma 2.3, we have
By direct computation, we have
Thus we finish the proof of Lemma 3.1. □
Theorem 3.1
The fractional boundary value problem (1.1) and (1.2) has a positive solution u if the following conditions are satisfied:
 (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
At first, we will prove that the functions \(\beta(t)=k_{1}g(t)\), \(\gamma(t)=k_{2}g(t)\) are lower and upper solutions of (1.1) and (1.2), respectively, where \(0< k_{1}\leq\min \{\frac{1}{a_{2}}, (a_{1})^{\frac{\mu}{1\mu}} \}\), \(k_{2}\geq\max \{\frac{1}{a_{1}}, (a_{2})^{\frac{\mu}{1\mu}} \}\),
and
In view of Lemma 2.3 and Remark 2.1, we know that \(g(t)\) is a positive solution of the following problem:
From the conclusion of Lemma 3.1, we know that
Thus, by virtue of the assumptions of Theorem 3.1, one shows that
Therefore, we have
It implies that
Obviously, \(\beta(t)=k_{1}g(t)\), \(\gamma(t)=k_{2}g(t)\) satisfy the boundary conditions (1.2). So, \(\beta(t)=k_{1}g(t)\), \(\gamma(t)=k_{2}g(t)\) are lower and upper solutions of (1.1) and (1.2), respectively.
Next, we will prove the fractional boundary value problem
has a solution, where
Thus, we consider that the operator \(T: C[0,1]\to C[0,1]\) is defined as follows:
where \(G(t,s)\) is defined as (2.3). It is clear that T is continuous in \(C[0,1]\). Since the function \(f(t,u)\) in nondecreasing in u, this shows that, for any \(u\in C([0,1],[0,+\infty))\),
The operator \(T: C[0,1]\to C[0,1]\) is continuous in view of continuity of \(G(t,s)\) and \(g(t,u(t))\). By means of the ArzelaAscoli theorem, T is a compact operator. Therefore, from the LeraySchauder fixed point theorem, the operator T has a fixed point, i.e., the fractional boundary value problem (3.1) has a solution.
Finally, we will prove that the fractional boundary value problem (1.1) and (1.2) has a positive solution.
Suppose that \({u^{*}}(t)\) is a solution of the fractional boundary value problem (3.1). Since the function \(f(t,u)\) is nondecreasing in u, we know that
Thus,
where \(z(t)=\gamma(t){u^{*}}(t)\). By Remark 2.1, \(z(t)\geq0\), i.e., \({u^{*}}(t)\leq\gamma(t)\) for \(t\in[0,1]\). Similarly, \(\beta(t)\leq{u^{*}}(t)\) for \(t\in[0,1]\). Therefore, \({u^{*}}(t)\) is a positive solution of the fractional boundary value problem (1.1) and (1.2). We have finished the proof of Theorem 3.1. □
In the following, we present a simple example to explain our results.
Example 3.1
Consider the boundary value problem
and
Proof
Since \(k^{\mu}\leq1\) for \(0<\mu<1\) and \(0\leq k\leq1\). It is easy to verify that
Thus, by Theorem 3.1, we know that the boundary value problem (3.2) has a positive solution u. □
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.
Let the Banach space \(E=C[0,1]\) be endowed with the norm \(\u\=\max_{0\leq t\leq1}u(t)\). Define the cone \(P\subset E\) by
Let the nonnegative continuous concave functional θ on the cone P be defined by
Suppose that u is a solution of the boundary value problem (1.1) and (1.2). Then
We define an operator \(A: P\to E\) as follows:
By Lemma 2.4, we have
Thus, \(A(P)\subset P\).
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. □
For convenience, we denote
Theorem 4.1
Suppose \(f(t,u)\) is continuous on \([0,1]\times[0,+\infty)\) and there exist constants \(0< a<b<c\) such that the following assumptions hold:

(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 \((t,u)\in[0,1]\times[0,c]\).
Then the boundary value problem (1.1) and (1.2) has at least three positive solutions \(u_{1}\), \(u_{2}\), \(u_{3}\) with
Proof
We show that all the conditions of Lemma 2.5 are satisfied.
If \(u\in\overline{P}_{c}\), then \(\u\\leq c\). Assumption (B3) implies \(f(t,u(t))\leq Mc\) for \(0\leq t\leq1\). Consequently,
Hence, \(A:\overline{P}_{c}\to\overline{P}_{c}\). In the same way, if \(u\in \overline{P}_{a}\), then assumption (C2) of Lemma 2.5 is satisfied.
To check condition (C1) of Lemma 2.5, we choose \(u(t)=(b+c)/2\), \(0\leq t\leq1\). It is easy to see that \(u(t)=(b+c)/2\in P(\theta,b,c)\), \(\theta(u)=\theta((b+c)/2)>b\), consequently, \(\{u\in P(\theta,b,c)\theta(u)>b\}\neq\emptyset\). Hence, if \(u\in P(\theta,b,c)\), then \(b\leq u(t)\leq c\) for \(1/4\leq t\leq3/4\). Thus,
i.e., \(\theta(Au)>b\) for all \(u\in P(\theta,b,c)\). This shows that condition (C1) of Lemma 2.5 is satisfied.
By Lemma 2.5 and Remark 2.1, the boundary value problem (1.1) and (1.2) has at least three positive solutions \(u_{1}\), \(u_{2}\), and \(u_{3}\), satisfying
The proof is complete. □
Corollary 4.1
Suppose \(f(t,u)\) is continuous on \([0,1]\times[0,+\infty)\) and there exist constants \(0< a_{1}\leq\sigma b_{1}<(\alpha1)\sigma b_{1}<b_{1}<c_{1}\) such that the following assumptions hold:

(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 \((t,u)\in[0,1]\times[0,c_{1}]\).
Then the boundary value problem (1.1) and (1.2) has at least three positive solutions \(u_{1}\), \(u_{2}\), \(u_{3}\), satisfying
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
If condition (B3) in Theorem 4.1 is replaced by
 (B3′):

\(\limsup_{u\to\infty} \frac{f(t,u)}{u}< M\).
Then the conclusion of Theorem 4.1 also holds.
Proof
We only need to show there exists a number \(c'\) with \(c'>c\) and \(A: \overline{P}_{c'}\to\overline{P}_{c'}\).
From (B3′), we know there exist R and \(\varepsilon< M\), such that
Let \(L=\max_{u\in[0,R]}f(t,u)\), \(t\in[0,1]\).
In view of (4.1), it is easy to see that
Now let \(c'\) be such that
Then for arbitrary \(u\in\overline{P}_{c'}\), \(t\in[0,1]\), from (4.2) and (4.3), we obtain
Thus, \(A: \overline{P}_{c'}\to\overline{P}_{c'}\). □
Theorem 4.3
Suppose that there exist constants \(0<{a'_{1}}<\sigma{b'_{1}}<(\alpha1)\sigma {b'_{1}}<{b'_{1}}<{c'_{1}}<{a'_{2}}<\sigma{b'_{2}}<(\alpha1)\sigma {b'_{2}}<{b'_{2}}<{c'_{2}}<\cdots<{a'_{n}}\), \(n\in\mathbb{N}\), for \(i=1,2,\ldots,n\), such that

(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 \((t,u)\in [1/4,3/4]\times[\sigma{b'_{i}},{c'_{i}}]\).
Then the boundary value problem (1.1) and (1.2) has at least \(2n1\) positive solutions.
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.
When \(n=2\), it is clear that Corollary 4.1 holds (with \(c'={a'_{2}}\)). Then we can obtain at least three positive solutions \(u_{1}\), \(u_{2}\), and \(u_{3}\), satisfying
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
Consider the boundary value problem
where
We have
Choosing \(a=\frac{1}{12}\), \(b=1\), \(c=14\), we have
From Theorem 4.1, the boundary value problem (4.4) and (4.5) has at least three positive solutions \(u_{1}\), \(u_{2}\), \(u_{3}\) satisfying
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].
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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.
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Chen, W., Zhao, Y. Solvability of boundary value problems of nonlinear fractional differential equations. Adv Differ Equ 2015, 36 (2015). https://doi.org/10.1186/s1366201503732
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DOI: https://doi.org/10.1186/s1366201503732
MSC
 34A08
 34B18
Keywords
 fractional differential equation
 boundary value problem
 positive solution
 fractional Green’s function
 fixed point theorem
 lower and upper solution method