Abstract and Applied Analysis

Existence Results and the Monotone Iterative Technique for Nonlinear Fractional Differential Systems with Coupled Four-Point Boundary Value Problems

Yujun Cui and Yumei Zou

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Abstract

By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for nonlinear fractional differential systems with coupled four-point boundary value problems.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 242591, 6 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412606720

Digital Object Identifier
doi:10.1155/2014/242591

Mathematical Reviews number (MathSciNet)
MR3246322

Zentralblatt MATH identifier
07021985

Citation

Cui, Yujun; Zou, Yumei. Existence Results and the Monotone Iterative Technique for Nonlinear Fractional Differential Systems with Coupled Four-Point Boundary Value Problems. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 242591, 6 pages. doi:10.1155/2014/242591. https://projecteuclid.org/euclid.aaa/1412606720


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References

  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006.
  • V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, Mass, USA, 2009.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
  • 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.
  • H. Amann, “Parabolic evolution equations with nonlinear boundary conditions,” in Nonlinear Functional Analysis and Its Applications, Proceedings of Symposia in Pure Mathematics, pp. 17–27, Providence, RI, USA, 1986.
  • H. Amann, “Parabolic evolution equations and nonlinear boundary conditions,” Journal of Differential Equations, vol. 72, no. 2, pp. 201–269, 1988.
  • N. A. Asif and R. A. Khan, “Positive solutions to singular system with four-point coupled boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 386, no. 2, pp. 848–861, 2012.
  • Y. Cui and J. Sun, “On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system,” Electronic Journal of Qualitative Theory of Differential Equations, no. 41, 13 pages, 2012.
  • Y. Cui, L. Liu, and X. Zhang, “Uniqueness and existence of positive solutions for singular differential systems with coupled integral boundary value problems,” Abstract and Applied Analysis, vol. 2013, Article ID 340487, 9 pages, 2013.
  • Y. Cui and Y. Zou, “Monotone iterative method for differential systems with coupled integral boundary value problems,” Boundary Value Problems, vol. 2013, article 245, 2013.
  • K. Deng, “Blow-up rates for parabolic systems,” Zeitschrift für Angewandte Mathematik und Physik, vol. 47, pp. 132–143, 1996.
  • K. Deng, “Global existence and blow-up for a system of heat equations with non-linear boundary conditions,” Mathematical Methods in the Applied Sciences, vol. 18, no. 4, pp. 307–315, 1995.
  • L. Zhigui and X. Chunhong, “The blow-up rate for a system of heat equations with nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 5, pp. 767–778, 1998.
  • M. Pedersen and Z. Lin, “Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition,” Applied Mathematics Letters, vol. 14, no. 2, pp. 171–176, 2001.
  • C. Yuan, D. Jiang, D. O'Regan, and R. P. Agarwal, “Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 13, 17 pages, 2012.
  • Y. Zou, L. Liu, and Y. Cui, “The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance,” Abstract and Applied Analysis, vol. 2014, Article ID 314083, 8 pages, 2014.
  • M. Al-Refai and M. Ali Hajji, “Monotone iterative sequences for nonlinear boundary value problems of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 11, pp. 3531–3539, 2011.
  • T. Jankowski, “Boundary problems for fractional differential equations,” Applied Mathematics Letters, vol. 28, pp. 14–19, 2014.
  • L. Lin, X. Liu, and H. Fang, “Method of upper and lower solutions for fractional differential equations,” Electronic Journal of Differential Equations, vol. 2012, 2012.
  • J. D. Ramirez and A. S. Vatsala, “Monotone iterative technique for fractional differential equations with periodic boundary conditions,” Opuscula Mathematica, vol. 29, no. 3, pp. 289–304, 2009.
  • A. Shi and S. Zhang, “Upper and lower solutions method and a fractional differential equation boundary value problem,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 30, 13 pages, 2009.
  • G. Wang, “Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments,” Journal of Computational and Applied Mathematics, vol. 236, no. 9, pp. 2425–2430, 2012.
  • Z. Wei, Q. Li, and J. Che, “Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 260–272, 2010.
  • S. Zhang and X. Su, “The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1269–1274, 2011.
  • S. Zhang, “Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2087–2093, 2009. \endinput