## Abstract and Applied Analysis

### Positive Solutions of a Singular Nonlocal Fractional Order Differential System via Schauder’s Fixed Point Theorem

#### Abstract

We establish the existence of positive solutions to a class of singular nonlocal fractional order differential system depending on two parameters. Our methods are based on Schauder’s fixed point theorem.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 457965, 8 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412279727

Digital Object Identifier
doi:10.1155/2014/457965

Mathematical Reviews number (MathSciNet)
MR3216052

Zentralblatt MATH identifier
07022417

#### Citation

Zhang, Xinguang; Mao, Cuiling; Wu, Yonghong; Su, Hua. Positive Solutions of a Singular Nonlocal Fractional Order Differential System via Schauder’s Fixed Point Theorem. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 457965, 8 pages. doi:10.1155/2014/457965. https://projecteuclid.org/euclid.aaa/1412279727

#### References

• A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elservier Science, Amsterdam, The Netherlands, 2006.
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
• J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering,, Springer, Dordrecht, The Netherlands, 2007.
• S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral and Derivatives, Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
• A. S. Perelson, “Modeling the interaction of the immune system with HIV,” in Mathematical and Statistical Approaches To AIDS Epidemiology, C. Castillo-Chavez, Ed., vol. 83 of Lecture Notes in Biomathematics, Springer, New York, NY, USA, 1989.
• C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1251–1268, 2011.
• X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012.
• B. Ahmad and J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009.
• C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers and Mathematics with Applications, vol. 61, no. 2, pp. 191–202, 2011.
• C. S. Goodrich, “Positive solutions to boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis, Theory, Methods and Applications, vol. 75, no. 1, pp. 417–432, 2012.
• A. A. M. Arafa, S. Z. Rida, and M. Khalil, “Fractional modeling dynamics of HIV and CD4$^{+}$ T-cells during primary infection,” Nonlinear Biomedical Physics, vol. 6, no. 1, article 1, 2012.
• X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a singular fractional differential system involving derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1400–1409, 2013.
• X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012.
• X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012.
• I. S. Jesus, J. A. Tenreiro MacHado, and J. Boaventure Cunha, “Fractional electrical impedances in botanical elements,” JVC/Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1389–1402, 2008.
• X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012.
• M. Rehman and R. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, pp. 1038–1044, 2010.
• X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4680–4691, 2013. \endinput