Positive Solutions of m-Point Boundary Value Problems for Fractional Differential Equations
© Zhi-Wei Lv. 2011
Received: 28 November 2010
Accepted: 19 January 2011
Published: 9 February 2011
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.
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 ).
2. Main Results
We list the following assumptions to be used in this paper.
In the following, we will prove our main results.
The proof is complete.
The proof is complete.
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
- 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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