Abstract
This paper deals with a periodic boundary value problem for a second order functional differential equation. We obtain the existence of extreme solutions under new concept of upper and lower solutions. Also, a mistake in a recent paper (Ding et al. in J. Math. Anal. Appl. 298:341–351, 2004) is corrected.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ladde, G.S., Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman Advanced Publishing Program. Pitman, London (1985)
Nieto, J.J., Rodriguez-Lopez, R.: Existence and approximation of solutions for nonlinear function differential equations with periodic boundary value conditions. J. Comput. Appl. Math. 40, 433–442 (2000)
Nieto, J.J., Rodriguez-Lopez, R.: Remark on periodic boundary value problems for function differential equations. J. Comput. Appl. Math. 158, 339–353 (2003)
He, Z., He, X.: Periodic boundary value problems for first order impulsive integro-differential equations of mixed type. J. Math. Anal. Appl. 296, 8–20 (2004)
Ding, W., Mi, J.R., Han, M.: Periodic boundary value problems for the first order impulsive functional differential equations. Appl. Math. Comput. 165, 443–456 (2005)
Ding, W., Han, M., Yan, J.: Periodic boundary value problems for the second order functional equations. J. Math. Anal. Appl. 298, 341–351 (2004)
Henderson, J.: Boundary Value Problems for Functional Equations. World Scientific, Singapore (1995)
Haddock, J.R., Nkashama, M.N.: Periodic boundary value problems and monotone iterative methods for functional differential equations. Nonlinear Anal. 22(3), 267–276 (1994)
Jankowski, T.: Existence of solutions of boundary value problems for differential equations with delayed arguments. J. Comput. Appl. Math. 156, 239–252 (2003)
Jankowski, T.: Advanced differential equations with nonlinear boundary conditions. J. Math. Anal. Appl. 304, 490–503 (2005)
Jankowski, T.: Solvability of three point boundary value problems for second order ordinary differential equations with deviating arguments. J. Math. Anal. Appl. 312, 620–636 (2005)
Nieto, J.J.: Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Anal. TMA 51(2), 1223–1232 (2002)
Nieto, J.J., Cabada, A.: A generalized upper and lower solution method for nonlinear second order ordinary differential equations. J. Appl. Math. Stoch. Anal. 5, 157–165 (1992)
Nieto, J.J., Rodriguez-Lopez, R.: Monotone method for first-order functional differential equations. Comput. Math. Appl. 52, 471–484 (2006)
Wang, W., Yang, X., Shen, J.: Boundary value problems involving upper and lower solutions in reverse order. J. Comput. Appl. Math. 230, 1–7 (2009)
Zuo, W., Jiang, D., O’Regan, D., Agarwal, R.P.: A monotone method for fourth order periodic boundary value problems and periodic solutions of functional differential equations. Methods Appl. Anal 12, 19–28 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Wang, W., Shen, J. & Nieto, J.J. Periodic boundary value problems for second order functional differential equations. J. Appl. Math. Comput. 36, 173–186 (2011). https://doi.org/10.1007/s12190-010-0395-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-010-0395-6
Keywords
- Periodic boundary value problems
- Upper and lower solutions
- Comparison theorem
- Functional differential equations