×

Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations. (English) Zbl 1014.34049

The paper is devoted to develop a monotone iterative technique for some first- and second-order periodic boundary value problems for functional-differential equations (FDEs), assuming the existence of a couple of lower and upper solutions. For it, the authors prove some comparison results (maximum principles), following the ideas in [E. Liz and J. J. Nieto, J. Math. Anal. Appl. 200, No. 3, 680-686 (1996; Zbl 0855.34080)]. It is worth mentioning that in the case of first-order FDEs, the maximum principle (theorem 2.1) can be deduced from a more general result in [J. J. Nieto, Appl. Math. Lett. 15, No. 2, 173-179 (2002; Zbl 1014.34060)].
Reviewer: Eduardo Liz (Vigo)

MSC:

34K10 Boundary value problems for functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone iterative techniques for nonlinear differential equations, Pitman Advanced Publishing Program (1985), Pitman: Pitman London · Zbl 0658.35003
[2] Leela, S.; Oguztoreli, M. N., Periodic boundary value problems for differential equations with delay and monotone iterative methods, J. Math. Anal. Appl., 122, 301-307 (1987) · Zbl 0616.34062
[3] Haddock, J. R.; Nkashama, M. N., Periodic boundary value problems and monotone iterative methods for functional differential equations, Nonlinear Anal., 22, 267-276 (1994) · Zbl 0804.34062
[4] Liz, E.; Nieto, J. J., Periodic boundary value problems for a class of functional differential equations, J. Math. Anal. Appl., 200, 680-686 (1996) · Zbl 0855.34080
[5] Jiang, D.; Wang, J., On boundary value problems for singular second-order functional differential equations, J. Comput. Appl. Math., 116, 231-241 (2000) · Zbl 0952.34053
[6] Nieto, J. J.; Jiang, Y.; Jurang, Y., Comparison results and monotone iterative technique for impulsive delay differential equations, Acta Sci. Math. (Szeged), 65, 121-130 (1999) · Zbl 0936.34069
[7] Nieto, J. J.; Jiang, Y.; Jurang, Y., Monotone iterative method for functional- differential equations, Nonlinear Anal., 32, 741-747 (1998) · Zbl 0937.34053
[8] Nieto, J. J.; Rodrguez-Lopez, R., Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions, Comput. Math. Appl., 40, 433-442 (2000) · Zbl 0958.34055
[9] Wan, Z.; Chen, Y.; Chen, J., Remarks on the periodic boundary value problems for first differential equations, Comput. Math. Appl., 37, 49-55 (1999) · Zbl 0936.34013
[10] Weng, P.; Jiang, D., Existence of positive solutions for boundary value problem of second-order FDE, Comput. Math. Appl., 37, 1-9 (1999) · Zbl 0942.34058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.