## Abstract and Applied Analysis

### Positive and Nondecreasing Solutions to an m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation

#### Abstract

We are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractional m-point boundary value problem: ${D}_{{0}^{+}}^{\alpha }u(t)+f(t,u(t))=0,\mathrm{ }0, where ${D}_{{0}^{+}}^{\alpha }$ denotes the standard Riemann-Liouville fractional derivative, $f:[0,1]×[0,\infty )\to [0,\infty )$ is a continuous function, ${a}_{i}\ge 0$ for $i=1,2,\dots ,m-2$, and $0<{\xi }_{1}<{\xi }_{2}<\cdots <{\xi }_{m-2}<1$. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also presented to illustrate the main results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 826580, 15 pages.

Dates
First available in Project Euclid: 1 April 2013

https://projecteuclid.org/euclid.aaa/1364845185

Digital Object Identifier
doi:10.1155/2012/826580

Mathematical Reviews number (MathSciNet)
MR2872307

Zentralblatt MATH identifier
1234.34006

#### Citation

Cabrera, I. J.; Harjani, J.; Sadarangani, K. B. Positive and Nondecreasing Solutions to an m -Point Boundary Value Problem for Nonlinear Fractional Differential Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 826580, 15 pages. doi:10.1155/2012/826580. https://projecteuclid.org/euclid.aaa/1364845185

#### References

• L. M. B. C. Campos, “On the solution of some simple fractional differential equations,” International Journal of Mathematics and Mathematical Sciences, vol. 13, no. 3, pp. 481–496, 1990.
• K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
• L. Yi and D. Shusen, “A class of analytic functions defined by fractional derivation,” Journal of Mathematical Analysis and Applications, vol. 186, no. 2, pp. 504–513, 1994.
• D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.
• M. El-Shahed, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, Article ID 10368, 8 pages, 2007.
• S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis, vol. 71, no. 11, pp. 5545–5550, 2009.
• J. Caballero, J. Harjani, and K. Sadarangani, “Existence and uniqueness of positive solution for a boundary value problem of fractional order,” Abstract and Applied Analysis, Article ID 165641, 12 pages, 2011.
• W. Zhong, “Positive solutions for multipoint boundary value problem of fractional differential equations,” Abstract and Applied Analysis, Article ID 601492, 15 pages, 2010.
• S. Liang and J. Zhang, “Existence and čommentComment on ref. [9?]: Please update the information of this reference, if possible.uniqueness of positive solutions to m-point boundary value problem for nonlinear fractional differential equation,” Journal of Applied Mathematics and Computing. In press.
• A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis, vol. 72, no. 5, pp. 2238–2242, 2010.
• J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 3403–3410, 2009.
• J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
• D. O'Regan and A. Petruşel, “Fixed point theorems for generalized contractions in ordered metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1241–1252, 2008.
• A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001.
• S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
• 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.
• A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
• J. Caballero Mena, J. Harjani, and K. Sadarangani, “Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems,” Boundary Value Problems, Article ID 421310, 10 pages, 2009.
• J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 3403–3410, 2009.
• J. Caballero, J. Harjani, and K. Sadarangani, “Uniqueness of positive solutions for a class of fourth-order boundary value problems,” Abstract and Applied Analysis, Article ID 543035, 13 pages, 2011.