- Research
- Open Access
Positive solutions for periodic boundary value problem of fractional differential equation in Banach spaces
- Yibo Kong^{1} and
- Pengyu Chen^{1}Email author
https://doi.org/10.1186/s13662-018-1788-3
© The Author(s) 2018
- Received: 29 December 2017
- Accepted: 7 September 2018
- Published: 13 September 2018
Abstract
This paper discusses the existence and uniqueness of positive solutions for a periodic boundary value problem of a fractional differential equation in an ordered Banach space E. The existence and uniqueness results of solutions for the associated linear periodic boundary value problem of the fractional differential equation are established, and the norm estimation of resolvent operator is accurately obtained. With the aid of this estimation, the existence and uniqueness results of positive solutions are obtained by using a monotone iterative technique.
Keywords
- Fractional differential equation
- Periodic boundary value problem
- Existence and uniqueness
MSC
- 26A33
- 34B15
- 34K30
1 Introduction
Fractional derivatives and integrals are generalizations of traditional integer-order differential and integral calculus. The history of fractional calculus reaches back to the end of 17th century, this idea has been a subject of interest not only among mathematicians but also among physicists and engineers; see [1–17] and the references therein for more comments and citations. Since fractional-order models are more accurate than integer-order models, there is a higher degree of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a memory term in the model. This memory term ensures the history and its impact to the present and future. Hence, fractional differential equations have been frequently used in economics, bioscience [18], system control theory [19], electrochemistry [20], diffusion process [21], signal and image processing, and so on. Recently, the monotone iterative technique in the presence of upper and lower solutions has appeared to be an important method for seeking solutions of nonlinear differential equations.
In the general case, the authors always established the upper and lower solution criteria under the assumption that for the studied problem there exist a couple of ordered lower and upper solutions, which is a strong assumption. The main purpose of this paper is to obtain the existence of positive solutions for the periodic boundary value problem of a nonlinear fractional differential equation directly from the characteristics of the nonlinear term \(f(t,u)\), without assuming the existence of the upper and lower solutions. In this paper, we first of all derive the corresponding fractional Green’s function. Then the corresponding linear periodic boundary value problem is reduced to an equivalent integral equation by using the Green’s function. Finally, we derive the sufficient conditions for nonlinear function f under which for the periodic boundary value problem (1.1) there exists a unique positive solution by using a monotone iterative technique.
2 Preliminaries
For the convenience of the reader, first we present the necessary definitions and some basic results.
Definition 2.1
([24])
Definition 2.2
([24])
Lemma 2.3
([26])
- (i)
\(E_{\alpha,2\alpha}(Mt^{\alpha })=M^{-1}t^{-\alpha} (E_{\alpha,\alpha}(Mt^{\alpha})-1/\Gamma(\alpha))\);
- (ii)
\(I_{a}^{\gamma}(t-a)^{\beta-1}E_{\alpha,\beta }(M(t-a)^{\alpha}) =(t-a)^{\beta+\gamma-1}E_{\alpha,\beta+\gamma}(M(t-a)^{\alpha})\) for \(t>a\);
- (iii)
\(E_{\alpha,\alpha}(Mt^{\alpha})\) is decreasing in t for \(M<0\) and increasing for \(M>0\) for all \(t>0\).
Let \(I=[0,\omega]\), we use \(C(I,E)\) to denote the Banach space of all continuous function on interval I with the norm \(\|u\|_{C}=\max_{t\in I}\|u(t)\|\). In our further consideration we utilize its generalization, namely, \(C_{1-\alpha}(I,E)=\{u\in C(I,E)|t^{1-\alpha}u(t)\in C(I,E),t\in I\}\) equipped with the norm \(\|u\|_{C_{1-\alpha}}=\|t^{1-\alpha}u(t)\|_{C}\). It is easy to verify that \(C_{1-\alpha}(I,E)\) is a Banach space.
Lemma 2.4
Proof
We can verify directly that the function \(u\in C_{1-\alpha}(I,E)\) defined by Eq. (2.3) is a solution of the linear periodic boundary value problem (2.2). Next, we prove that u is unique as a solution. Assume that \(u_{1},u_{2}\in C_{1-\alpha}(I,E)\) are two solutions of the linear periodic boundary value problem (2.2). From (2.3) one can easily see that \(u_{1}(t)=u_{2}(t)\) on I. Hence, the linear periodic boundary value problem (2.2) has a unique solution \(u(t)\) given by (2.3). Obviously, \(P:C_{1-\alpha}(I,E)\rightarrow C_{1-\alpha}(I,E)\) is a linear bounded operator. □
Remark 2.5
In Lemma 2.4, for all \(t\in (0,\omega]\), \(s\in[0,\omega)\), and for \(M>0\), we have \(G_{\alpha,M}(s,t)>0\). Hence, for any \(h\in C_{1-\alpha}^{+}(I,E)\), periodic resolvent operator \(P:C_{1-\alpha}(I,E)\rightarrow C_{1-\alpha}(I,E)\) is positive linear operator.
Lemma 2.6
Proof
Lemma 2.7
Proof
3 Main results
Theorem 3.1
- (H1)There exists a constant \(M>0\), such that \(\theta\leq x_{1}\leq x_{2}\), we have$$f(t,x_{2})-f(t,x_{1})\geq-M(x_{2}-x_{1}), \quad t\in I. $$
- (H2)There exists a constant \(0< L< M\), such that \(\theta\leq x_{1}\leq x_{2}\), we have$$f(t,x_{2})-f(t,x_{1})\leq-L(x_{2}-x_{1}), \quad t\in I. $$
Proof
We reconsider the linear periodic boundary value problem (2.2). By Lemma 2.4, for \(h\in C_{1-\alpha}(I,E)\), we see that the linear periodic boundary value problem (2.2) has a unique solution \(u=Ph\), and \(P:C_{1-\alpha}(I,E)\rightarrow C_{1-\alpha}(I,E)\) is a positive linear boundary operator with \(\|P\|\leq\frac{1}{M}\).
Next, we prove the uniqueness. Let \(u_{1}\), \(u_{2}\) be two arbitrary positive solutions of the PBVPs (1.1). Let P and F is the operator of the M corresponding in the above existence argumentation, then the operator F is order increasing on \([\theta,u_{i}]\) (\(i=1,2\)) of the order interval. In the iterative scheme of (3.2), the initial element \(\omega_{0}\) is replaced by \(u_{i}\), and we repeat the above argumentation process. Since \(P\circ F(u_{i})=u_{i}\), we have \(u_{i}=\omega_{n}\). By (3.7), letting \(n\rightarrow\infty\), we obtain \(\|u_{i}-\upsilon_{n}\|_{C_{1-\alpha}}\rightarrow0\), which means that \(u_{1}=u_{2}=\lim_{n\rightarrow\infty}\upsilon_{n}\). Therefore, PBVP (1.1) has a unique positive solution. □
Declarations
Acknowledgements
We would like to thank the referees very much for their valuable suggestions to improve this paper.
Funding
This work was partially supported by NNSF of China (11501455), NNSF of China (11661071), Key project of Gansu Provincial National Science Foundation (1606RJZA015) and Project of NWNU-LKQN-14-6.
Authors’ contributions
Each of the authors contributed to each part of this study equally and approved the final version of this manuscript.
Competing interests
The authors declare that they have no competing interests.
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
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