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 Open Access
Existence and uniqueness of positive solutions for a fractional differential equation with integral boundary conditions
 Yan Qiao^{1} and
 Zongfu Zhou^{1}Email author
https://doi.org/10.1186/s136620160772z
© Qiao and Zhou 2016
Received: 8 November 2015
Accepted: 26 January 2016
Published: 13 April 2016
Abstract
We study a class of boundary value problems for a fractional differential equation with integral boundary conditions. By means of \(u_{0}\)positive operator we obtain results on the existence and uniqueness of positive solutions for the boundary value problem.
Keywords
 \(u_{0}\)positive operator
 integral boundary condition
 positive solution
 fractional differential equation
1 Introduction
Nowadays, fractional differential equations become more and more important. They play an important role in engineering, science,economics, and so on. More and more people pay attention to the study of theory and applications of fractional differential equations [1–8]. Many efforts have also been made to develop the theory of fractional evolution equations: we refer the readers to [9–11]. A lot of papers are devoted to the positive solutions of boundary value problem for fractional differential equations, such as [12–14].
 (A_{1}):

\(p:(0,1)\rightarrow[0,+\infty)\) is a continuous function nonvanishing identically on any subinterval of \((0,1) \) with$$\int_{0}^{1}p(s)\,ds< +\infty. $$
 (A_{2}):

\(f:[0,1]\times\mathbb{R}\rightarrow[0,+\infty)\) is continuous, and \(q:(0,1)\rightarrow[0,+\infty)\) is continuous and Lebesgue integrable.
 (A_{3}):

\(l:(0,1)\rightarrow[0,+\infty)\) is continuous, and \(0\leq\int _{0}^{1}l(t)t^{p1}\,dt<1\).
2 Preliminaries and relevant lemmas
In order to obtain the main results of this work, we present some necessary definitions and several fundamental lemmas.
Definition 2.1
Definition 2.2
Definition 2.3
Lemma 2.1
Let E be a Banach space. Suppose that \(S:E\rightarrow E\) is a completely continuous linear operator and \(S(P)\subset P\). If there exist \(\psi\in E\setminus\{P\}\) and a constant \(c>0\) such that \(cS\psi\geq\psi\), then the spectral radius \(r(S)\neq0\), and S has a positive eigenfunction φ corresponding to its first eigenvalue \(\lambda_{1}=(r(S))^{1}\), that is, \(\varphi=\lambda_{1}S\varphi\).
Let \(E=C[0,1]\), which is a Banach space with norm \(\Vert x\Vert =\max_{t\in [0,1]}\vert x(t)\vert \).
Set \(P=\{x\in E x(t)\geq0, \forall t\in[0,1]\}\). In the rest of this paper, the partial ordering in \(C[0,1]\) is always given by P.
Lemma 2.2
([13])
Lemma 2.3
Proof
Let \(L=\int_{0}^{1}l(t)t^{p1}\,dt\) and \(Q(s)=\frac{\int _{s}^{1}l(u)(us)^{p1}\,du}{1L}\). By (A_{3}) we know that \(0\leq L<1\).
Remark 2.1
If \(G(t,s)\) is defined by (2.2), then \(G(t,s)\geq0\) for \(t,s\in(0,1)\).
It is easy to show that \(T:E\rightarrow E\) is a linear completely continuous operator and \(T(P)\subset P\). It is not difficult to see that the solution for (1.4) is, equivalently, a fixed point of A in E.
Lemma 2.4
The operator \(A:E\rightarrow E\) defined in (2.4) is completely continuous, and \(A(P)\subset P\).
Proof
Obviously, \(A:E\rightarrow E\), and \(A(P)\subset P\). The continuity of A in P is obvious. For any bounded set \(D\subset P\), \(A(D)\) is bounded, so that the functions in \(A(D)\) are uniformly bounded. It is easy to prove that A is equicontinuous. By the AscoliArzelà theorem A is completely continuous. □
Lemma 2.5
T is a \(u_{0}\)positive operator, and \(u_{0}(t)=t^{p1}\).
Proof
The inequalities imply that T is a \(u_{0}\)positive operator and \(u_{0}(t)=t^{p1}\). This proof is completed. □
Lemma 2.6
Let T be defined in (2.3). Then the spectral radius \(r(T)\neq0\), and T has a positive eigenfunction \(\varphi^{\ast}(t)\) corresponding to its first eigenvalue \(\lambda_{1}=(r(T))^{1}\).
Proof
Let \(\psi(t)=t^{p1}\) and \(c=\{\int _{0}^{1}[\frac{L}{\Gamma(p)(1L)}(1s)^{p1}+Q(s)]p(s)\psi(s)\,ds\} ^{1}>0\). Then from the proof of Lemma 2.5 we have \(cT\psi\geq\psi\). Thereby, from Lemma 2.1 we get that \(r(T)\neq0 \) and that T has a positive eigenfunction \(\varphi^{\ast}(t)\) corresponding to its first eigenvalue \(\lambda_{1}=(r(T))^{1}\), that is, \(\varphi^{\ast }(t)=\lambda_{1}T\varphi^{\ast}\). This completes the proof. □
From Lemma 2.5 and Definition 2.3 we get the following lemma.
Lemma 2.7
3 Main results
Theorem 3.1
Proof
By the completeness of E and the closeness of P there exists \(x^{\ast}\in P\) such that \(\lim_{n\rightarrow\infty}x_{n}=x^{\ast}\).
Passing to the limit in \(x_{n+1}=Ax_{n}\), we get \(x^{\ast}=Ax^{\ast}\), and it follows that \(x^{\ast}\) is a fixed point of A in P.
From \(\Vert xy\Vert \leq k^{n}\beta_{2}\lambda_{1}\Vert \varphi^{\ast}\Vert \) and \(\lim_{n\rightarrow\infty}k^{n}\beta_{2}\lambda_{1}\Vert \varphi^{\ast}\Vert =0\) we have \(\Vert xy\Vert \leq0\), and thus \(x=y\). Therefore, \(x^{\ast}\) is the unique fixed point of A in P or, equivalently, \(x^{\ast}\) is the unique positive solution of (1.4). The proof is completed. □
4 Conclusions
The method of a \(u_{0}\)positive operator is an important tool in boundary value problems for fractional differential equations. We established the existence of positive solutions for a fractional differential problem with integral boundary conditions by means of a \(u_{0}\)positive operator.
Declarations
Acknowledgements
This article was supported by the National Natural Science Foundation of China (Grant No. 11371027) and Anhui Provincial Natural Science Foundation (1608085MA12). The authors would like to thank the referees for their valuable suggestions and comments.
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
 Delbosco, D, Rodino, L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Appl. 204, 609625 (1996) MathSciNetMATHGoogle Scholar
 Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999) MATHGoogle Scholar
 Zhang, S: The existence of a positive solution for a nonlinear fractional differential equation. J. Math. Anal. Appl. 252, 804812 (2000) MathSciNetView ArticleMATHGoogle Scholar
 Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 9731033 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Liang, S, Zhang, J: Existence of multiple positive solutions for mpoint fractional boundary value problems on an infinite interval. Math. Comput. Model. 54, 13341346 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Ladde, GS, Lakshmikantham, V, Vatsala, AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, London (1985) MATHGoogle Scholar
 Zhang, S: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl. 278(1), 136148 (2003) MathSciNetView ArticleMATHGoogle Scholar
 Zhou, Y: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014) View ArticleMATHGoogle Scholar
 Zhou, Y, Jiao, F, Pecaric, J: On the Cauchy problem for fractional functional differential equations in Banach spaces. Topol. Methods Nonlinear Anal. 42, 119136 (2013) MathSciNetMATHGoogle Scholar
 Zhou, Y, Shen, XH, Zhang, L: Cauchy problem for fractional evolution equations with Caputo derivative. Eur. Phys. J. Spec. Top. 222, 17471764 (2013) Google Scholar
 Zhou, Y, Zhang, L, Shen, XH: Existence of mild solutions for fractional evolution equations. J. Integral Equ. Appl. 25, 557585 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, X, Liu, L, Wu, Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 12631274 (2012) MathSciNetView ArticleMATHGoogle 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. TMA 72, 710719 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Cui, Y: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 4854 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
 Krasnosel’skii, MA: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964) Google Scholar
 Guo, D, Sun, J: Nonlinear Integral Equations. Shandong Science and Technology Press, Jinan (1987) (in Chinese) Google Scholar
 Zhang, S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 59, 13001309 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Babakhani, A, DaftardarGejji, V: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 278, 434442 (2003) MathSciNetView ArticleMATHGoogle Scholar
 Agarwal, RP, Zhou, Y, He, YY: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59(3), 10951100 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Bai, C: Positive solutions for nonlinear fractional differential equations with coefficient that changes sign. Nonlinear Anal. TMA 64, 677685 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, X: Positive solutions for a class of singular fractional differential equation with infinitepoint boundary value conditions. Appl. Math. Lett. 39, 2227 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Wang, G: Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval. Appl. Math. Lett. 47, 17 (2015) MathSciNetView ArticleMATHGoogle Scholar