 Research Article
 Open Access
 Published:
Positive Solutions of mPoint Boundary Value Problems for Fractional Differential Equations
Advances in Difference Equations volume 2011, Article number: 571804 (2011)
Abstract
We discuss the existence of minimal and maximal positive solutions for fractional differential equations with multipoint boundary value conditions, and new results are given. An example is also given to illustrate the abstract results.
1. Introduction
Recently, [1] discussed the existence of positive solutions for the following boundary value problem of fractional order differential equation
where is the standard RiemannLiouville fractional derivative of order , , , , , and satisfies Carathéodorytype conditions. Moreover, [2] considered the following nonlinear point boundary value problem of fractional type:
where takes values in a reflexive Banach space ,
with and denotes the th Pseudoderivative of , denotes the Pseudo fractional differential operator of order , is a continuous realvalued function on , and is a vectorvalued Pettisintegrable function.
In this paper, we consider the existence of minimal and maximal positive solutions for the following multiplepoint boundary value problem:
where is the standard RiemannLiouville fractional derivative,
, is continuous, , , , , , and
New results on the problem will be obtained.
Recall the following wellknown definition and lemma (for more details on cone theory, see [3]).
Definition 1.1.
Let be a real Banach space. Then,

(a)
a nonempty convex closed set is called a cone if it satisfied the following two conditions:

(i)
implies ,

(ii)
implies , where denotes the zero element of .

(b)
a cone is said to be normal if there exists a constant such that implies .
Lemma 1.2.
Assume that with a fractional derivative of order that belongs to . Then,
where is the smallest integer greater than or equal to .
2. Main Results
Let and . Then, is the Banach space endowed with the norm and is normal cone.
We list the following assumptions to be used in this paper.
there exist two nonnegative realvalued functions , such that
for implies .
In the following, we will prove our main results.
Lemma 2.1.
Let . Then, the fractional differential equation
has a unique solution which is given by
where
in which
where
Proof.
Using Lemma 1.2, we have
It follows from the condition that .
Thus,
This, together with the relation , yields
From the boundary value condition , we deduce that
Thus,
The proof is complete.
Lemma 2.2.
If , then function in Lemma 2.1 satisfies the following conditions:
(i) , for ,
(ii) , for s,,
where
in which
Proof.
When , we have
Thus, for .
Furthermore, we conclude that
So, for . This, together with for , yields for .
Observing the express of , , and , we see that holds.
The proof is complete.
Remark 2.3.
From the express of and , we see that
Thus,
Now, we define an operator by
Theorem 2.4.
Let condition be satisfied. Suppose that . Then, problem (1.4) has at least one positive solution.
Proof.
Let , where
Step 1.
, for any
which implies that .
Step 2.
is continuous.
It is obvious from .
Step 3.
is equicontinuous.
From (2.11) and (2.18), for any , , , we conclude that
As , the righthand side of the above inequality tends to zero, so, is equicontinuous.
By the ArzeláAscoli theorem, we conclude that the operator is completely continuous. Thus, our conclusion follows from Schauder fixed point theorem, and the proof is complete.
Theorem 2.5.
Besides the hypotheses of Theorem 2.4, we suppose that holds. Then, BVP (1.4) has minimal positive solution in and maximal positive solution in ; Moreover, , as uniformly on , where
Proof.
By Theorem 2.4, we know that BVP (1.4) has at least one positive solution in .
Step 1.
BVP (1.4) has a positive solution in , which is minimal positive solution.
From (2.18) and (2.22), one can see that
This, together with , yields that
From and the proof of Theorem 2.4, it may be concluded that and .
Let
Thus,
By the complete community of , we know that is relatively compact. So, there exists a and a subsequence
such that converges to uniformly on . Since is normal and is nondecreasing, it is easily seen that the entire sequence converges to uniformly on . being closed convex set in and imply that .
From
and , we see that
By (2.30), (2.22), and Lebesgue's dominated convergence theorem, we get
Let be any positive solution of BVP (1.4) in . It is obvious that .
Thus,
Taking limits as in (2.32), we get .
Step 2.
BVP (1.4) has a positive solution in , which is maximal positive solution.
Let
It is obvious that
Thus, and .
By (2.18), (2.23), and , we have
This, together with , yields that
Using a proof similar to that of Step 1, we can show that
Let be any positive solution of BVP (1.4) in .
Obviously,
This, together with , implies
Taking limits as in (2.39), we obtain .
The proof is complete.
On the other hand, we note that in these years, going with the significant developments of various differential equations in abstract spaces (cf., e.g., [3–17] and references therein), fractional differential equations in Banach spaces have also been investigated by many authors (cf. e.g., [1, 2, 18–26] and references therein). In our coming papers, we will present more results on fractional differential equations in Banach spaces.
3. An Example
Example 3.1.
Consider the following boundary value problem
where , , , , ,
, . By computation, we deduce that
From Remark 2.3, we get
Therefore,
On the one hand, it is obvious that . Thus, is satisfied.
For , we see that , which implies that holds.
Hence, by Theorem 2.5, BVP (3.1) has minimal and maximal positive solutions in .
Furthermore, we can conclude that
References
 1.
Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Computers & Mathematics with Applications 2010,59(3):13631375.
 2.
Salem HAH:On the fractional order point boundary value problem in reflexive Banach spaces and weak topologies. Journal of Computational and Applied Mathematics 2009,224(2):565572. 10.1016/j.cam.2008.05.033
 3.
Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.
 4.
Barbu V: Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2010:x+272.
 5.
Batty CJK, Liang J, Xiao TJ: On the spectral and growth bound of semigroups associated with hyperbolic equations. Advances in Mathematics 2005,191(1):110. 10.1016/j.aim.2004.01.005
 6.
Chicone C, Latushkin Y: Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs. Volume 70. American Mathematical Society, Providence, RI, USA; 1999:x+361.
 7.
Engel KJ, Nagel R: OneParameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics. Volume 194. Springer, New York, NY, USA; 2000:xxii+586.
 8.
Diagana T: Pseudo Almost Periodic Functions in Banach Spaces. Nova Science, New York, NY, USA; 2007:xiv+132.
 9.
Diagana T, Mophou GM, N'Guérékata GM: On the existence of mild solutions to some semilinear fractional integrodifferential equations. Electronic Journal of Qualitative Theory of Differential Equations 2010,2010(58):117.
 10.
Liang J, Nagel R, Xiao TJ: Approximation theorems for the propagators of higher order abstract Cauchy problems. Transactions of the American Mathematical Society 2008,360(4):17231739.
 11.
Liang J, Zhang J, Xiao TJ: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. Journal of Mathematical Analysis and Applications 2008,340(2):14931499. 10.1016/j.jmaa.2007.09.065
 12.
Lorenzi A, Mola G: Identification of unknown terms in convolution integrodifferential equations in a Banach space. Journal of Inverse and IllPosed Problems 2010,18(3):321355. 10.1515/JIIP.2010.013
 13.
Xiao TJ, Liang J: The Cauchy Problem for HigherOrder Abstract Differential Equations, Lecture Notes in Mathematics. Volume 1701. Springer, Berlin, Germany; 1998:xii+301.
 14.
Xiao TJ, Liang J: Second order parabolic equations in Banach spaces with dynamic boundary conditions. Transactions of the American Mathematical Society 2004,356(12):47874809. 10.1090/S0002994704037043
 15.
Xiao TJ, Liang J: Complete second order differential equations in Banach spaces with dynamic boundary conditions. Journal of Differential Equations 2004,200(1):105136. 10.1016/j.jde.2004.01.011
 16.
Xiao TJ, Liang J: Second order differential operators with FellerWentzell type boundary conditions. Journal of Functional Analysis 2008,254(6):14671486. 10.1016/j.jfa.2007.12.012
 17.
Xiao TJ, Liang J, Zhang J: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 2008,76(3):518524. 10.1007/s002330079011y
 18.
Agarwal RP, Belmekki M, Benchohra M: A survey on semilinear differential equations and inclusions involving RiemannLiouville fractional derivative. Advances in Difference Equations 2009, 2009:47.
 19.
Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Analysis: Theory, Methods & Applications 2010,72(6):28592862. 10.1016/j.na.2009.11.029
 20.
Henderson J, Ouahab A: Fractional functional differential inclusions with finite delay. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):20912105. 10.1016/j.na.2008.02.111
 21.
Henderson J, Ouahab A: Impulsive differential inclusions with fractional order. Computers & Mathematics with Applications 2010,59(3):11911226.
 22.
Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):33373343. 10.1016/j.na.2007.09.025
 23.
Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):26772682. 10.1016/j.na.2007.08.042
 24.
Li F: Mild solutions for fractional differential equations with nonlocal conditions. Advances in Difference Equations 2010, 2010:9.
 25.
Lv ZW, Liang J, Xiao TJ: Solutions to fractional differential equations with nonlocal initial condition in Banach spaces. Advances in Difference Equations 2010, 2010:10.
 26.
Mophou GM: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2010,72(34):16041615. 10.1016/j.na.2009.08.046
Author information
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Received
Accepted
Published
DOI
Keywords
 Banach Space
 Fractional Order
 Fractional Derivative
 Fractional Type
 Fractional Differential Equation