Existence and multiplicity of positive solutions for singular fractional differential equations with integral boundary value conditions
 Ying He^{1}Email author
Received: 13 May 2015
Accepted: 14 December 2015
Published: 29 January 2016
Abstract
Keywords
MSC
1 Introduction
Differential equations with fractional derivative have been proved to be strong tools in the modeling of many physical phenomena. In consequence the subject of fractional differential equations is gaining much importance and attention [1–3]. Some recent investigations have shown that many physical systems can be represented more accurately using fractional derivative formulations. For details, see [4–10].
But up to now, there are few papers that have considered the multiplicity of positive solutions with two integral boundary conditions and a nonlinear term f possessing a singularity at \(u=0\). Motivated by the results mentioned above, the aim of this paper is to establish the multiplicity of positive solutions for singular fractional differential equations with two integral boundary value conditions (1.1).
In this paper, in analogy with boundary value problems for differential equations of integer order, we first of all derive the corresponding Green’s function known as the fractional Green’s function. Here we give some properties that relate the expressions of \(G(t,s)\) and \(G(1,s)\). It is well known that cones play an important role in applying the Green’s function in research areas. Consequently problem (1.1) is reduced to an equivalent Fredholm integral equation. Finally, by using the LeraySchauder nonlinear alternative and a fixedpoint theorem in cones, the existence and multiplicity of positive solutions are obtained.
2 Background materials and Green’s function
For the reader’s convenience, we present some necessary definitions from fractional calculus, both theory and lemmas. These definitions can be found in the recent literature such as [14].
Definition 2.1
[14]
Definition 2.2
[14]
Lemma 2.1
[14]
Lemma 2.2
[14]
In the following we present the Green’s function of a fractional differential equation with integral boundary conditions.
Lemma 2.3
Proof
Lemma 2.4
 (1)
\(G(1,s)=0\), for \(s\in[0,1]\) if and only if \(\eta=0\);
 (2)
\(G(1,s)>0\), for \(s\in(0,1)\) and \(\eta\in(0,2)\);
 (3)
\(tG(1,s)\leq G(t,s)\leq M_{0} G(1,s)\), for \(3<\alpha<4\), \(s\in(0,1)\) and \(\eta\in(0,2)\) where \(M_{0}=\frac{\alpha(\eta+2)}{2\eta(\alpha1)}\);
 (4)
\(G(t,s)>0\), for \(t,s\in(0,1)\) and \(\eta\in(0,2)\).
Proof
Observing the expression of \(G(1,s)\), it is clear that (1) and (2) hold.
Theorem 2.1
(LeraySchauder alternative)
 (i)
A has a fixed point in U̅, or
 (ii)
there is a \(u\in\partial U\) and \(\lambda\in(0,1) \) with \(u=\lambda AU+(1\lambda)p\).
Theorem 2.2
 (i)
\(\Tu\\geq\u\\), \(u\in P\cap\partial\Omega_{1}\), and \(\Tu\\leq\ u\\), \(u\in P\cap\partial\Omega_{2}\); or
 (ii)
\(\Tu\\leq\u\\), \(u\in P\cap\partial\Omega_{1}\), and \(\Tu\\geq \u\\), \(u\in P\cap\partial\Omega_{2}\).
3 Main results
Theorem 3.1
 (H_{1}):

\(f:[0,1]\times(0,\infty)\rightarrow[0,\infty)\) is continuous andwith \(g(u)>0\) is nonincreasing and \(h(u)/g(u)\) is nondecreasing in \(u\in (0,\infty)\);$$0\leq f(t,u)=g(u)+h(u),\quad\textit{for } (t,u)\in[0,1]\times(0,\infty), $$
 (H_{2}):

there exists a constant \(K_{0}>0\) such that \(g(ab)\leq K_{0}g(a)g(b)\) for all \(a,b\geq0\);
 (H_{3}):

\(\int_{0}^{1}g(s)\,ds<\infty\);
 (H_{4}):

there exists a positive number r such that$$\biggl\{ 1+\frac{h(r)}{g(r)}\biggr\} M_{0}K_{0}g\biggl( \frac{r}{M_{0}}\biggr) \int_{0}^{1}G(1,s)g(s)\,ds< r; $$
 (H_{5}):

there exists a positive number \(R>r\) with$$\biggl(1\frac{2}{\alpha}\biggr)g(R) \int_{0}^{1}G(1,s) \biggl\{ 1+\frac{h(\frac {R}{M_{0}}s)}{g(\frac{R}{M_{0}}s)} \biggr\} \,ds\geq R. $$
Proof
Finally, we show that T is equicontinuous.
It follows from Theorem 2.2, (3.3), and (3.4) that T has a fixed point \(\widetilde{u}\in K\cap(\overline{\Omega_{2}}\setminus\Omega _{1})\). Clearly this fixed point is a positive solution of (3.1) satisfying \(r\leq\\widetilde{u}\\leq R \). □
Theorem 3.2
 (H_{6}):

for each \(L>0\), there exists a function \(\varphi_{L}\in C[0,1]\), \(\varphi_{L}>0\), for \(t\in(0,1)\), such that \(f(t,u)>\varphi_{L}(t)\), for \((t,u)\in(0,1)\times(0,L]\). Then (3.1) has a solution u with \(0<\u\<r\).
Proof
Theorem 3.3
Suppose that (H_{1})(H_{6}) are satisfied. Then problem (3.1) has two positive solutions u, ũ with \(0<\u\<r\leq\\widetilde{u}\\leq R\).
Proof
From the proof of Theorem 3.1, we see that (3.1) has a positive solution \(\widetilde{u(t)}\) with \(r\leq\\widetilde{u}\\leq R\), and by Theorem 3.2, we see that (3.1) has another positive solution \(u(t)\) with \(0<\ u\<r\). Thus (3.1) has at least two positive solutions. □
Example 3.1
 (i)
if \(b<1\), then (3.10) has at least one nonnegative solution for each \(\omega>0\);
 (ii)
if \(b>1\), then (3.10) has at least one nonnegative solution for each \(0<\omega<\omega_{1}\), where \(\omega_{1}\) is some positive constant;
 (iii)
if \(b>1\), then (3.10) has at least two nonnegative solutions for each \(0<\omega<\omega_{1}\).
Proof
Declarations
Acknowledgements
The author sincerely thanks the editor and reviewers for their valuable suggestions and useful comments to improve the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Wang, Y, Liu, L, Wu, Y: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal., Theory Methods Appl. 74, 35993605 (2011) MATHMathSciNetView ArticleGoogle Scholar
 Bai, Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal., Theory Methods Appl. 72, 916924 (2010) MATHView ArticleGoogle Scholar
 Xu, J, Wei, Z, Dong, W: Uniqueness of positive solutions for a class of fractional boundary value problems. Appl. Math. Lett. 25, 590593 (2012) MATHMathSciNetView ArticleGoogle Scholar
 Jiang, D, Yuan, C: The positive properties of the Green function for Dirichlettype boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal., Theory Methods Appl. 72, 710719 (2010) MATHMathSciNetView ArticleGoogle Scholar
 Li, S, Zhang, X, Wu, Y, Caccetta, L: Extremal solutions for pLaplacian differential systems via iterative computation. Appl. Math. Lett. 26, 11511158 (2013) MATHMathSciNetView ArticleGoogle Scholar
 Zhao, X, Chai, C, Ge, W: Existence and nonexistence results for a class of fractional boundary value problems. J. Appl. Math. Comput. 41, 1731 (2013) MATHMathSciNetView ArticleGoogle Scholar
 Zhang, X, Liu, L, Wu, Y, Lu, Y: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 46804691 (2013) MathSciNetView ArticleGoogle Scholar
 AlRefai, M, Hajji, M: Monotone iterative sequences for nonlinear boundary value problems of factional order. Nonlinear Anal. 74, 35313559 (2011) MATHMathSciNetView ArticleGoogle Scholar
 Yao, Q: Successively iterative method of nonlinear Neumann boundary value problems. Appl. Math. Comput. 217, 23012306 (2010) MATHMathSciNetView ArticleGoogle Scholar
 Li, CF, Luo, XN, Zhou, Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 13631375 (2010) MATHMathSciNetView ArticleGoogle Scholar
 Cabada, A, Wang, G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403411 (2012) MATHMathSciNetView ArticleGoogle Scholar
 Zhang, X, Liu, L, Wiwatanpataphee, B, Wu, Y: The eigenvalue for a class of singular pLaplacian fractional differential equations involving the RiemannStieltjes integral boundary condition. Appl. Math. Comput. 235, 412422 (2014) MathSciNetView ArticleGoogle Scholar
 Zhou, WX, Zhang, JG, Li, JM: Existence of multiple positive solutions for singular boundary value problems of nonlinear fractional differential equations. Adv. Differ. Equ. 2014, 97 (2014) MathSciNetView ArticleGoogle Scholar
 Bai, Z, Qiu, T: Existence of positive solutions for singular fractional differential equation. Appl. Math. Comput. 215, 27612767 (2009) MATHMathSciNetView ArticleGoogle Scholar