Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems
© J. Caballero Mena et al. 2009
Received: 24 April 2009
Accepted: 14 June 2009
Published: 19 July 2009
We establish the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equation. Our analysis relies on a fixed point theorem in partially ordered sets.
Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function (see, e.g., [1, 2]) and to problems of analyticity in the complex domain . Moreover, Delbosco and Rodino  considered the existence of a solution for the nonlinear fractional differential equation , where and , is a given continuous function in . They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Recently, Zhang  considered the existence of positive solution for equation , where and is a given continuous function by using the sub- and super-solution methods.
In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution to boundary-value problem of the nonlinear fractional differential equation
Note that this problem was considered in  where the authors proved the existence of one positive solution for (1.1) by using Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone and assuming certain hypotheses on the function . In  the uniqueness of the solution is not treated.
In this paper we will prove the existence and uniqueness of a positive and nondecreasing solution for the problem (1.1) by using a fixed point theorem in partially ordered sets.
2. Preliminaries and Previous Results
For the convenience of the reader, we present here some notations and lemmas that will be used in the proofs of our main results.
The following lemmas appear in .
The following lemmas appear in .
Note that for and (see ).
Now, we present some results about the fixed point theorems which we will use later. These results appear in .
then we have the following theorem .
Note that this space can be equipped with a partial order given by
In  it is proved that with the classic metric given by
3. Main Result
Then one's problem (1.1) has an unique nonnegative solution.
In what follows we check that hypotheses in Theorems 2.8 and 2.9 are satisfied.
Finally, take into account that for the zero function, , by Theorem 2.8 our problem (1.1) has at least one nonnegative solution. Moreover, this solution is unique since satisfies condition (2.10) (see comments at the beginning of this section) and Theorem 2.9.
In [6, lemma 3.2] it is proved that is completely continuous and Schauder fixed point theorem gives us the existence of a solution to our problem (1.1).
In the sequel we present an example which illustrates Theorem 3.1.
Theorem 3.1 give us that our fractional differential (3.10) has an unique nonnegative solution.
This example give us uniqueness of the solution for the fractional differential equation appearing in  in the particular case and
This completes the proof.
Remark 3.5 gives us the following theorem which is a better result than that [6, Theorem 3.3] because the solution of our problem (1.1) is positive in and strictly increasing.
Under assumptions of Theorem 3.1, our problem (1.1) has a unique nonnegative and strictly increasing solution.
This research was partially supported by "Ministerio de Educación y Ciencia" Project MTM 2007/65706.
- Campos LMBC: On the solution of some simple fractional differential equations. International Journal of Mathematics and Mathematical Sciences 1990, 13(3):481–496. 10.1155/S0161171290000709MATHMathSciNetView ArticleGoogle Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.Google Scholar
- Ling Y, Ding S: A class of analytic functions defined by fractional derivation. Journal of Mathematical Analysis and Applications 1994, 186(2):504–513. 10.1006/jmaa.1994.1313MATHMathSciNetView ArticleGoogle Scholar
- Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 1996, 204(2):609–625. 10.1006/jmaa.1996.0456MATHMathSciNetView ArticleGoogle Scholar
- Zhang S: The existence of a positive solution for a nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2000, 252(2):804–812. 10.1006/jmaa.2000.7123MATHMathSciNetView ArticleGoogle Scholar
- Qiu T, Bai Z: Existence of positive solutions for singular fractional differential equations. Electronic Journal of Differential Equations 2008, 2008(146):1–9.MathSciNetGoogle Scholar
- Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
- Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(7–8):3403–3410. 10.1016/j.na.2009.01.240MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proceedings of the American Mathematical Society 2007, 135(8):2505–2517. 10.1090/S0002-9939-07-08729-1MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22(3):223–239. 10.1007/s11083-005-9018-5MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Mathematica Sinica 2007, 23(12):2205–2212. 10.1007/s10114-005-0769-0MATHMathSciNetView ArticleGoogle Scholar
- O'Regan D, Petruşel A: Fixed point theorems for generalized contractions in ordered metric spaces. Journal of Mathematical Analysis and Applications 2008, 341(2):1241–1252. 10.1016/j.jmaa.2007.11.026MATHMathSciNetView ArticleGoogle Scholar
- Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems—I. Applicable Analysis 2001, 78(1–2):153–192. 10.1080/00036810108840931MATHMathSciNetView ArticleGoogle Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
- Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
- Belmekki M, Nieto JJ, Rodríguez-López R: Existence of periodic solution for a nonlinear fractional differential equation. Boundary Value Problems. In pressGoogle Scholar
- Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009, 49(3–4):605–609. 10.1016/j.mcm.2008.03.014MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.