# Positive Solutions of m-Point Boundary Value Problems for Fractional Differential Equations

- Zhi-Wei Lv
^{1, 2}Email author

**2011**:571804

https://doi.org/10.1155/2011/571804

© Zhi-Wei Lv. 2011

**Received: **28 November 2010

**Accepted: **19 January 2011

**Published: **9 February 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

with and denotes the th Pseudo-derivative of , denotes the Pseudo fractional differential operator of order , is a continuous real-valued function on , and is a vector-valued Pettis-integrable function.

New results on the problem will be obtained.

Recall the following well-known definition and lemma (for more details on cone theory, see [3]).

Definition 1.1.

Lemma 1.2.

## 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 real-valued functions , such that

In the following, we will prove our main results.

Lemma 2.1.

Proof.

It follows from the condition that .

The proof is complete.

Lemma 2.2.

If , then function in Lemma 2.1 satisfies the following conditions:

Proof.

So, for . This, together with for , yields for .

Observing the express of , , and , we see that holds.

The proof is complete.

Remark 2.3.

Theorem 2.4.

Let condition be satisfied. Suppose that . Then, problem (1.4) has at least one positive solution.

Proof.

Step 1.

Step 2.

Step 3.

As , the right-hand side of the above inequality tends to zero, so, is equicontinuous.

By the Arzelá-Ascoli 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.

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 and the proof of Theorem 2.4, it may be concluded that and .

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 .

Let be any positive solution of BVP (1.4) in . It is obvious that .

Taking limits as in (2.32), we get .

Step 2.

BVP (1.4) has a positive solution in , which is maximal positive solution.

Let be any positive solution of BVP (1.4) in .

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

## Authors’ Affiliations

## References

- 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):1363-1375.MathSciNetView ArticleMATHGoogle Scholar - 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):565-572. 10.1016/j.cam.2008.05.033MathSciNetView ArticleMATHGoogle Scholar - 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.MATHGoogle Scholar - Barbu V:
*Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics*. Springer, New York, NY, USA; 2010:x+272.View ArticleMATHGoogle Scholar - Batty CJK, Liang J, Xiao T-J:
**On the spectral and growth bound of semigroups associated with hyperbolic equations.***Advances in Mathematics*2005,**191**(1):1-10. 10.1016/j.aim.2004.01.005MathSciNetView ArticleMATHGoogle Scholar - 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.View ArticleMATHGoogle Scholar - Engel K-J, Nagel R:
*One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics*.*Volume 194*. Springer, New York, NY, USA; 2000:xxii+586.MATHGoogle Scholar - Diagana T:
*Pseudo Almost Periodic Functions in Banach Spaces*. Nova Science, New York, NY, USA; 2007:xiv+132.MATHGoogle Scholar - Diagana T, Mophou GM, N'Guérékata GM:
**On the existence of mild solutions to some semilinear fractional integro-differential equations.***Electronic Journal of Qualitative Theory of Differential Equations*2010,**2010**(58):1-17.MathSciNetMATHGoogle Scholar - Liang J, Nagel R, Xiao T-J:
**Approximation theorems for the propagators of higher order abstract Cauchy problems.***Transactions of the American Mathematical Society*2008,**360**(4):1723-1739.MathSciNetView ArticleMATHGoogle Scholar - Liang J, Zhang J, Xiao T-J:
**Composition of pseudo almost automorphic and asymptotically almost automorphic functions.***Journal of Mathematical Analysis and Applications*2008,**340**(2):1493-1499. 10.1016/j.jmaa.2007.09.065MathSciNetView ArticleMATHGoogle Scholar - Lorenzi A, Mola G:
**Identification of unknown terms in convolution integro-differential equations in a Banach space.***Journal of Inverse and Ill-Posed Problems*2010,**18**(3):321-355. 10.1515/JIIP.2010.013MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
*The Cauchy Problem for Higher-Order Abstract Differential Equations, Lecture Notes in Mathematics*.*Volume 1701*. Springer, Berlin, Germany; 1998:xii+301.View ArticleGoogle Scholar - Xiao T-J, Liang J:
**Second order parabolic equations in Banach spaces with dynamic boundary conditions.***Transactions of the American Mathematical Society*2004,**356**(12):4787-4809. 10.1090/S0002-9947-04-03704-3MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
**Complete second order differential equations in Banach spaces with dynamic boundary conditions.***Journal of Differential Equations*2004,**200**(1):105-136. 10.1016/j.jde.2004.01.011MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
**Second order differential operators with Feller-Wentzell type boundary conditions.***Journal of Functional Analysis*2008,**254**(6):1467-1486. 10.1016/j.jfa.2007.12.012MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J, Zhang J:
**Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces.***Semigroup Forum*2008,**76**(3):518-524. 10.1007/s00233-007-9011-yMathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Belmekki M, Benchohra M:
**A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative.***Advances in Difference Equations*2009,**2009:**-47.Google Scholar - 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):2859-2862. 10.1016/j.na.2009.11.029MathSciNetView ArticleMATHGoogle Scholar - Henderson J, Ouahab A:
**Fractional functional differential inclusions with finite delay.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(5):2091-2105. 10.1016/j.na.2008.02.111MathSciNetView ArticleMATHGoogle Scholar - Henderson J, Ouahab A:
**Impulsive differential inclusions with fractional order.***Computers & Mathematics with Applications*2010,**59**(3):1191-1226.MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V:
**Theory of fractional functional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(10):3337-3343. 10.1016/j.na.2007.09.025MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V, Vatsala AS:
**Basic theory of fractional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(8):2677-2682. 10.1016/j.na.2007.08.042MathSciNetView ArticleMATHGoogle Scholar - Li F:
**Mild solutions for fractional differential equations with nonlocal conditions.***Advances in Difference Equations*2010,**2010:**-9.Google Scholar - Lv Z-W, Liang J, Xiao T-J:
**Solutions to fractional differential equations with nonlocal initial condition in Banach spaces.***Advances in Difference Equations*2010,**2010:**-10.Google Scholar - Mophou GM:
**Existence and uniqueness of mild solutions to impulsive fractional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(3-4):1604-1615. 10.1016/j.na.2009.08.046MathSciNetView ArticleMATHGoogle Scholar

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