- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 420514, 6 pages
Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem
King Abdulaziz University, Rabigh Campus, College of Sciences and Arts, Department of Mathematics, P.O. Box 344, Rabigh 21911, Saudi Arabia
Received 4 November 2012; Accepted 25 December 2012
Academic Editor: Chuanzhi Bai
Copyright © 2013 Habib Mâagli et al. 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.
We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .
Fractional differential equations arise in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetism. They also serve as an excellent tool for the description of hereditary properties of various materials and processes (see[1–3]). In consequence, the subject of fractional differential equations is gaining much importance and attention. Motivated by the surge in the development of this subject, we consider the following problem: where , , is a nonnegative continuous function on that may be singular at or and is the Riemann-Liouville fractional derivative. Then we study the existence and exact asymptotic behavior of positive solutions for this problem.
We recall that for a measurable function , the Riemann-Liouville fractional integral and the Riemann-Liouville derivative of order are, respectively, defined by provided that the right-hand sides are pointwise defined on . Here and means the integral part of the number and is the Euler Gamma function.
Several results are obtained for fractional differential equation with different boundary conditions (see [5–15] and the references therein), but none of them deals with the existence of a positive solution for problem (1).
Our aim in this paper is to establish the existence and uniqueness of a positive solution for problem (1) with a precise asymptotic behavior, where is the set of all functions such that is continuous on . Note that for , the solution for problem (1) blows up at .
To state our result, we need some notations. We will use to denote the set of Karamata functions defined on by for some , where and such that . It is clear that a function is in if and only if is a positive function in such that For two nonnegative functions and defined on a set , the notation , , means that there exists such that , for all . We denote also for .
Throughout this paper we assume that is nonnegative on and satisfies the following condition.
such that where , , satisfying In the sequel, we introduce the function defined on by Our main result is the following.
Theorem 1. Let and assume that satisfies . Then problem (1) has a unique positive solution satisfying for
This paper is organized as follows. Some preliminary lemmas are stated and proved in the next section, involving some already known results on Karamata functions. In Section 3, we give the proof of Theorem 1.
2. Technical Lemmas
Lemma 2. The following hold.(i)Let and , then one has (ii)Let , and . Then one has , , and .
Example 3. Let . Let , and let be a sufficiently large positive real number such that the function is defined and positive on , for some , where ( times. Then .
Lemma 4. Let and let be a function in defined on . One has the following.(i)If , then diverges and . (ii)If , then converges and .
Lemma 5. Let be defined on . Then one has If further converges, then one has
Proof. We distinguish two cases.
Case 1. We suppose that converges. Since the function is nonincreasing in , for some , it follows that, for each , we have It follows that . So we deduce (11).
Now put Using that , we obtain This implies that So (12) holds.
Case 2. We suppose that diverges. We have, for some , This implies that This proves (11) and completes the proof.
Lemma 7. Given and is such that the function is continuous and integrable on , then the boundary value problem has a unique solution given by where is the Green function for the boundary value problem (21).
Proof. Since is a solution of the equation , then . Consequently there exist two constants , such that . Using the fact that and , we obtain and . So In the following, we give some estimates on the function . So, we need the following lemma.
Lemma 8. For , and , one has
Proposition 9. On , one has
Proof. For , we have Since for , then, by applying Lemma 8 with and , we obtain which completes the proof.
In the sequel we put where and we aim to give some estimates on .
Proposition 10. Assume that with Then for ,
Proof. Using Proposition 9, we have
For , we use Lemma 4 and hypotheses (30) to deduce that
Hence, it follows from Lemma 2 and hypothesis (30) that, for , we have
Now, for , we use again Lemma 4 and hypothesis (30) to deduce that
Hence, it follows from Lemmas 2 and 5 that, for , we have
This together with (34) implies that (36) holds on .
3. Proof of Theorem 1
We begin this section by giving a preliminary result that will play a crucial role in the proof of Theorem 1.
Proposition 11. Assume that the function satisfies and put for . Then one has, for ,
Proof of Theorem 1. From Proposition 11, there exists such that for each
Put and let In order to use a fixed point theorem, we denote and we define the operator on by For this choice of , we can easily prove that for , we have and .
Now, we have Since the function is continuous on and the function is integrable on , we deduce that the operator is compact from to itself. It follows by the Schauder fixed point theorem that there exists such that . Put . Then and satisfies the equation Since the function is continuous and integrable on , then by Lemma 7 the function is a positive continuous solution of problem (1).
Finally, let us prove that is the unique positive continuous solution satisfying (8). To this aim, we assume that (1) has two positive solutions satisfying (8) and consider the nonempty set and put . Then and we have . It follows that and consequently Which implies by Lemma 7 that . By symmetry, we obtain also that . Hence and . Since , then and consequently .
Example 12. Let and be a positive continuous function on such that where and or and . Then, using Theorem 1, problem (1) has a unique positive continuous solution satisfying the following estimates.(i)If or and , then for , (ii)If , then for , (iii)If , then for , (iv)If , then for , (v)If , then for ,
The authors thank the anonymous referees for a careful reading of the paper and for their helpful suggestions.
- K. Diethelm and A. D. Freed, “On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity,” in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, F. Keil, W. Mackens, and H. Voss, Eds., pp. 217–224, Springer, Heidelberg, Germany, 1999.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
- W. Lin, “Global existence theory and chaos control of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 709–726, 2007.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- R. P. Agarwal, D. O'Regan, and S. Staněk, “Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 57–68, 2010.
- R. P. Agarwal, M. Benchohra, S. Hamani, and S. Pinelas, “Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half-line,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 18, no. 2, pp. 235–244, 2011.
- Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
- B. Ahmad, “Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 4, pp. 390–394, 2010.
- B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,” Boundary Value Problems, vol. 2011, article 36, 2011.
- J. Caballero, J. Harjani, and K. Sadarangani, “Positive solutions for a class of singular fractional boundary value problems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1325–1332, 2011.
- J. Deng and L. Ma, “Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 6, pp. 676–680, 2010.
- N. Kosmatov, “A singular boundary value problem for nonlinear differential equations of fractional order,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 125–135, 2009.
- Y. Liu, W. Zhang, and X. Liu, “A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann-Liouville derivative,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1986–1992, 2012.
- T. Qiu and Z. Bai, “Existence of positive solutions for singular fractional differential equations,” Electronic Journal of Differential Equations, vol. 149, 19 pages, 2008.
- Y. Zhao, S. Sun, Z. Han, and Q. Li, “Positive solutions to boundary value problems of nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 390543, 16 pages, 2011.
- V. Marić, Regular Variation and Differential Equations, vol. 1726 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.
- E. Seneta, Regularly Varying Functions, vol. 508 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1976.