Journal of Applied Mathematics

Upper and Lower Solution Method for Fractional Boundary Value Problems on the Half-Line

Dandan Yang and Chuanzhi Bai

Full-text: Open access

Abstract

We establish the existence of unbounded solutions for nonlinear fractional boundary value problems on the half-line. By the upper and lower solution method technique, sufficient conditions for the existence of solutions for the fractional boundary value problems are established. An example is presented to illustrate our main result.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 349025, 11 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808289

Digital Object Identifier
doi:10.1155/2013/349025

Mathematical Reviews number (MathSciNet)
MR3142568

Zentralblatt MATH identifier
06950628

Citation

Yang, Dandan; Bai, Chuanzhi. Upper and Lower Solution Method for Fractional Boundary Value Problems on the Half-Line. J. Appl. Math. 2013 (2013), Article ID 349025, 11 pages. doi:10.1155/2013/349025. https://projecteuclid.org/euclid.jam/1394808289


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References

  • R. P. Agrwal and D. O'Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publisher, Dodrecht, The Netherlands, 2001.
  • R. P. Agrwal and D. O'Regan, “Infinite interval problems modeling phenomena which arise in the theory of plasma and electrical potential theory,” Studies in Applied Mathematics, vol. 111, no. 3, pp. 339–358, 2003.
  • J. V. Baxley, “Existence and uniqueness for nonlinear boundary value problems on infinite intervals,” Journal of Mathematical Analysis and Applications, vol. 147, no. 1, pp. 122–133, 1990.
  • C. Bai and J. Fang, “On positive solutions of boundary value problems for second-order functional differential equations on inifinite intervals,” Journal of Mathematical Analysis and Applications, vol. 282, no. 2, pp. 711–731, 2003.
  • C. Bai and C. Li, “Unbounded upper and lower solution method for third-order boundary-value problems on the half-line,” Electronic Journal of Differential Equations, vol. 2009, no. 119, pp. 1–12, 2009.
  • H. Lian and F. Geng, “Multiple unbounded solutions for a boundary value problem on infinite intervals,” Boundary Value Problems, vol. 2011, no. 51, pp. 1–8, 2011.
  • S. Liang and J. Zhang, “The existence of countably many positive solutions for some nonlinear three-point boundary problems on the half-line,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 9, pp. 3127–3139, 2009.
  • Y. Liu, “Existence and unboundedness of positive solutions for singular boundary value problems on half-line,” Applied Mathematics and Computation, vol. 144, no. 2-3, pp. 543–556, 2003.
  • B. Liu, L. Liu, and Y. Wu, “Unbounded solutions for three-point boundary value problems with nonlinear boundary conditions on [0, +$\infty $),” Nonlinear Analysis, Theory, Methods and Applications, vol. 73, no. 9, pp. 2923–2932, 2010.
  • J. Li, B. Liu, and L. Liu, “Solutions for a boundary value problem at resonance on [0,$\infty $),” Mathematical and Computer Modelling, vol. 58, no. 11-12, pp. 1769–1776, 2013.
  • H. Lian and W. Ge, “Solvability for second-order three-point boundary value problems on a half-line,” Applied Mathematics Letters, vol. 19, no. 10, pp. 1000–1006, 2006.
  • R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 6, pp. 2859–2862, 2009.
  • B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers and Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009.
  • C. Bai, “Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 384, no. 2, pp. 211–231, 2011.
  • C. Bai, “Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance,” Journal of Applied Mathematics and Computing, vol. 39, no. 1-2, pp. 421–443, 2012.
  • K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.
  • I. Podlubny, Fractional Differential Equations, in: Mathematics in Sciences and Engineering, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
  • V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media,, Springer, New York, NY, USA, 2011.
  • Y. Chen and X. Tang, “Positive solutions of fractional differential equations at resonance on the half-line,” Boundary Value Problems, vol. 2012, no. 64, pp. 1–13, 2012.
  • C. Kou, H. Zhou, and Y. Yan, “Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 17, pp. 5975–5986, 2011.
  • X. Su and S. Zhang, “Unbounded solutions to a boundary value problem of fractional order on the half-line,” Computers and Mathematics with Applications, vol. 61, no. 4, pp. 1079–1087, 2011.
  • A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 2, pp. 580–586, 2009.
  • D. Baleanu, O. G. Mustafa, and D. O'Regan, “On a fractional differential equation with infinitely many solutions,” Advances in Difference Equations, vol. 2012, no. 145, pp. 1–6, 2012.
  • X. Liu and M. Jia, “Multiple solutions of nonlocal boundary value problems for fractional differential equations on the half-line,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2011, no. 56, pp. 1–14, 2011.
  • S. Liang and J. Zhang, “Existence of three positive solutions of m-point boundary value problems for some nonlinear fractional differential equations on an infinite interval,” Computers and Mathematics with Applications, vol. 61, no. 11, pp. 3343–3354, 2011.
  • Y. Lin, B. Ahmad, and R. P. Agarwal, “Existence of solutions for a coupled system of nonlinear fractional differential equations with fractional boundary conditions on the half-line,” Advances in Difference Equations, vol. 2013, no. 46, pp. 1–19, 2013.
  • X. Su, “Solutions to boundary value problem of fractional order on unbounded domains in a Banach space,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 8, pp. 2844–2852, 2011.
  • G. Wang, B. Ahmad, and L. Zhang, “A coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain,” Abstract and Applied Analysis, vol. 2012, Article ID 248709, 11 pages, 2012.
  • L. Zhang, G. Wang, B. Ahmad, and R. P. Agarwal, “Nonlinear fractional integro-differential equations on unbounded domains in a Banach space,” Journal of Computational and Applied Mathematics, vol. 249, pp. 51–56, 2013.
  • X. Zhao and W. Ge, “Unbounded solutions for a fractional boundary value problems on the infinite interval,” Acta Applicandae Mathematicae, vol. 109, no. 2, pp. 495–505, 2010.
  • 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. Meehan and D. O'Regan, “Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals,” Nonlinear Analysis, Theory, Methods and Applications, vol. 35, no. 3, pp. 355–387, 1999.
  • B. Yan, D. O'Regan, and R. P. Agarwal, “Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity,” Journal of Computational and Applied Mathematics, vol. 197, no. 2, pp. 365–386, 2006.