 Research
 Open access
 Published:
Positive solutions for periodic boundary value problem of fractional differential equation in Banach spaces
Advances in Difference Equations volume 2018, Article number: 322 (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.
1 Introduction
Fractional derivatives and integrals are generalizations of traditional integerorder 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 fractionalorder models are more accurate than integerorder models, there is a higher degree of freedom in the fractionalorder 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 [22], by means of the method of upper and lower solutions and the associated monotone iterative, the author proved the existence and uniqueness of the solution for the initial value problem
where \(0< T<+\infty\), \(0<\alpha\leq1\) is a real number, \(D^{\alpha}\) is the Riemann–Liouville fractional derivative. In 2010, Wei, Dong and Che [23], using the method of upper and lower solutions and its associated monotone iterative technique, proved the existence and uniqueness of the solution to the periodic boundary value problem for a class of fractional differential equations in real space \(\mathbb{R}\).
Motivated by the aforementioned work, in this paper, we consider the existence and uniqueness of positive solutions for the following periodic boundary value problem (PBVP) of a nonlinear fractional differential equation in Banach space E:
where \(0<\alpha\leq1\) is a real number, \(D^{\alpha}\) is the Riemann–Liouville fractional derivative, and \(f:[0,\omega]\times E\rightarrow E\) is a continuous function.
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])
The Riemann–Liouville fractional integral of order \(\delta>0\) of a function \(u(t)\) is defined by
provided that the righthand side is defined pointwise, where \(\Gamma(\cdot)\) is the gamma function.
Definition 2.2
([24])
The Riemann–Liouville fractional derivative of order \(\delta>0\) of a function \(u(t)\) is defined by
where n is the smallest integer greater than or equal to δ, provided that the righthand side is defined pointwise. In particular, if \(\delta=n\), then \(D_{a}^{n}u(t)=u^{(n)}(t)\).
The MittagLeffler function plays a similar role in fractional calculus to the exponential function in the theory of integerorder differential equation. Thus, the MittagLeffler function in two parameters is defined as [25]
Note that the series converges uniformly in \(\mathbb{R}\).
Lemma 2.3
([26])
Let \(0<\alpha\leq 1\), \(\beta, \gamma>0\), \(M\in\mathbb {R}\) and \(a\in\mathbb {R}\). Then:

(i)
\(E_{\alpha,2\alpha}(Mt^{\alpha })=M^{1}t^{\alpha} (E_{\alpha,\alpha}(Mt^{\alpha})1/\Gamma(\alpha))\);

(ii)
\(I_{a}^{\gamma}(ta)^{\beta1}E_{\alpha,\beta }(M(ta)^{\alpha}) =(ta)^{\beta+\gamma1}E_{\alpha,\beta+\gamma}(M(ta)^{\alpha})\) 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.
Obviously, the periodic boundary value problem (1.1) is equivalent to the following:
where \(M>0\) is real number.
To prove our main results, for any \(h\in C_{1\alpha}(I,E)\), we consider the periodic boundary value problem (PBVP) of the linear equation in E,
Lemma 2.4
For any \(h\in C_{1\alpha}(I,E)\), the linear periodic boundary value problem (2.2) has a unique solution \(u\in C_{1\alpha}(I,E)\) given by
where the Green’s function is given by
Moreover, the operator \(P:C_{1\alpha}(I,E)\rightarrow C_{1\alpha }(I,E)\) is a linear bounded operator.
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
Let \(0<\alpha\leq1\) and \(M>0\), then the Green’s function (2.4) satisfies
Proof
Employing the results of Lemma 2.3, we have
This concludes the proof. □
Lemma 2.7
For any \(h\in C_{1\alpha}(I,E)\), the norm of the solution operator P satisfies
Proof
For any \(h\in C_{1\alpha}(I,E)\), due to the definition of the operator P and Lemma 2.6, we get
which means that \(\Ph\_{C_{1\alpha}}\leq\frac{1}{M} \h\_{C_{1\alpha}}\). Hence \(\P\_{C_{1\alpha}}\leq\frac{1}{M}\), namely (2.5) holds. □
3 Main results
Theorem 3.1
Let E be an ordered Banach space, whose positive cone K is normal, let \(f: I\times E\rightarrow E\) be a continuous mapping which is ωperiodic in t, and for any \(t\in I\), and \(f(t,\theta)\geq\theta\). Suppose that the following conditions are satisfied:

(H1)
There exists a constant \(M>0\), such that \(\theta\leq x_{1}\leq x_{2}\), we have
$$f(t,x_{2})f(t,x_{1})\geqM(x_{2}x_{1}), \quad t\in I. $$ 
(H2)
There exists a constant \(0< L< M\), such that \(\theta\leq x_{1}\leq x_{2}\), we have
$$f(t,x_{2})f(t,x_{1})\leqL(x_{2}x_{1}), \quad t\in I. $$
Then the periodic boundary value problem (1.1) has a unique positive solution.
Proof
Let the positive cone K be a normal with normal constant N in E; evidently, the closed convex of cone \(K_{C}\) in \(C_{1\alpha}(I,E)\) is deduced by cone K, namely
then \(K_{C}\) is also normal with the same normal constant N. Hence, \(C_{1\alpha}(I,E)\) is an order Banach space with the semiorder reduced by the normal cone \(K_{C}\). In the following, E comes with partial order ≤.
Denote \(h_{0}(t)=f(t,\theta)\), then \(h_{0}\geq\theta\) and \(h_{0}\in C_{1\alpha}(I,E)\), we consider the existence of solution for the linear periodic boundary value problem
By Lemma 2.4, for \(h_{0}\in C_{1\alpha}(I,E)\), we find that the linear periodic boundary value problem (3.1) has a unique solution \(\omega_{0}\in C_{1\alpha}(I,E)\) with \(\omega_{0}\geq\theta\).
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 \(\P\\leq\frac{1}{M}\).
Set \(F(u)=f(t,u)+Mu\), then \(F:C_{1\alpha}(I,E)\rightarrow C_{1\alpha}(I,E)\) is a continuous mapping, and \(F(\theta)=h_{0}\geq \theta\). We defined an order interval \([\theta,\omega_{0}]\) in \(C_{1\alpha}(I,E)\), by condition (H1), we see that F is restricted: it is an increasing operator on \([\theta,\omega_{0}]\). Setting \(\upsilon_{0}=\theta\), we make the iterative scheme
Since \(\omega_{0}\) is a solution of problem (3.1), we add \(M\omega_{0}L\omega_{0}\) on both sides of Eq. (3.1), thus, we see that \(\omega_{0}\) is also the corresponding solution of (2.2), when \(h=h_{0}+M\omega_{0}L\omega_{0}\), namely
In condition (H2), setting \(x_{1}=\theta\), \(x_{2}=\omega_{0}(t)\), we have
adding both sides of this inequality by \(M\omega_{0}\), we can obtain
Acting on (3.4) by P, combining this with the positivity of P and (3.3), we have
Since \(P\circ F\) is an increasing operator on \([\theta,\omega_{0}]\), repeated acting to this inequality by \(P\circ F\) means that
so we have
Using the recursive approach, it follows that
by this and the normality of the cone \(K_{C}\), we conclude that
On the other hand, since \(0< ML< M\), choose \(\epsilon>0\), such that \(ML+\epsilon< M\). By (2.5), there exists \(N_{0}\in\mathbb {N}\), such that, for \(n\geq N_{0}\),
Hence, for \(n\geq N_{0}\), from (3.6) it follows that
By (3.5) and the above inequality, combining this with the principle of nested intervals, we can obtain the existence of a unique solution, \(u^{\ast}\in\bigcap_{n=0}^{\infty}[\upsilon_{n},\omega_{n}]\), such that
consequently, letting \(n\rightarrow\infty\) in (3.2), we see that \(u^{\ast}=P\circ F(u^{\ast})\). By the definition of P, it is easy to see that \(u^{\ast}\) is the corresponding solution of the linear periodic boundary value problem (2.2), when \(h(t)=f(t,u^{\ast}(t))+Mu^{\ast}(t)\), and therefore, it is a positive solution of the periodic boundary value problem (1.1).
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. □
References
Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with threepoint boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)
Belmekki, M., Nieto, J.J., RodríguezLópez, R.: Existence of periodic solutions for a nonlinear fractional differential equations. Bound. Value Probl. 2009, 324561 (2009)
Mouffak, B., Alberto, C., Djamila, S.: An existence result for nonlinear fractional differential equations on Banach spaces. Bound. Value Probl. 2009, 628916 (2009)
Bonilla, B., Rivero, M., RodríguezGermá, L., Trujillo, J.J.: Fractional differential equations as alternative models to nonlinear differential equations. Appl. Math. Comput. 187, 79–88 (2007)
Chen, P., Zhang, X., Li, Y.: Study on fractional nonautonomous evolution equations with delay. Comput. Math. Appl. 73, 794–803 (2017)
Chen, P., Zhang, X., Li, Y.: Approximation technique for fractional evolution equations with nonlocal integral conditions. Mediterr. J. Math. 14, 1–16 (2017)
Chen, P., Zhang, X., Li, Y.: A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Commun. Pure Appl. Anal. 17, 1975–1992 (2018)
Hao, X.: Positive solution for singular fractional differential equations involving derivatives. Adv. Differ. Equ. 2016, 139 (2016)
Liu, L.L., Zhang, X., Liu, L., Wu, Y.: Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions. Adv. Differ. Equ. 2016, 154 (2016)
Guo, L., Liu, L., Wu, Y.: Existence of positive solutions for singular higherorder fractional differential equations with infinitepoint boundary conditions. Bound. Value Probl. 2016, 114 (2016)
Hao, X., Wang, H., Liu, L., Cui, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and pLaplacian operator. Bound. Value Probl. 2017, 182 (2017)
Zhu, B., Liu, L., Wu, Y.: Local and global existence of mild solutions for a class of nonlinear fractional reaction–diffusion equations with delay. Appl. Math. Lett. 61, 73–79 (2016)
Aydi, H., Marasi, H.R., Piri, H., Talebia, A.: A solution to the new Caputo–Fabrizio fractional KDV equation via stability. J. Math. Anal. 8(4), 147–155 (2017)
Afshari, H., Marasi, H., Aydi, H.: Existence and uniqueness of positive solutions for boundary value problems of fractional differential equations. Filomat 31(9), 2675–2682 (2017)
Marasia, H., Piria, H., Aydi, H.: Existence and multiplicity of solutions for nonlinear fractional differential equations. J. Nonlinear Sci. Appl. 9, 4639–4646 (2016)
Hao, X., Wang, H.: Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions. Open Math. 16, 581–596 (2018)
Yan, F., Zuo, M., Hao, X.: Positive solution for a fractional singular boundary value problem with pLaplacian operator. Bound. Value Probl. 2018, 51 (2018)
Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 1–104 (2004)
Vinagre, B.M., Podlubny, I., Hernández, A., Feliu, V.: Some approximations of fractional order operators used in control theory and applications. Fract. Calc. Appl. Anal. 3, 231–248 (2000)
Oldham, K.B.: Fractional differential equations in electrochemistry. Adv. Eng. Softw. 41, 9–12 (2010)
Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Physica A 278, 107–125 (2000)
Zhang, S.: Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives. Nonlinear Anal. 71, 2087–2093 (2009)
Wei, Z., Dong, W., Che, J.: Periodic boundary value problems for fractional differential equations involving a Riemann–Liouville fractional derivative. Nonlinear Anal. 73, 3232–3238 (2010)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Cabada, A., Kisela, T.: Existence of positive periodic solutions of some nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 50, 51–67 (2017)
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 NWNULKQN146.
Author information
Authors and Affiliations
Contributions
Each of the authors contributed to each part of this study equally and approved the final version of this manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Kong, Y., Chen, P. Positive solutions for periodic boundary value problem of fractional differential equation in Banach spaces. Adv Differ Equ 2018, 322 (2018). https://doi.org/10.1186/s1366201817883
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366201817883