Periodic solutions for third order ordinary differential equations

Juan J. Nieto

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 3, page 495-499
  • ISSN: 0010-2628

Abstract

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In this paper, we introduce the concept of upper and lower solutions for third order periodic boundary value problems. We show that the monotone iterative technique is valid and obtain the extremal solutions as limits of monotone sequences. We first present a new maximum principle for ordinary differential inequalities of third order that is interesting by itself.

How to cite

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Nieto, Juan J.. "Periodic solutions for third order ordinary differential equations." Commentationes Mathematicae Universitatis Carolinae 32.3 (1991): 495-499. <http://eudml.org/doc/21795>.

@article{Nieto1991,
abstract = {In this paper, we introduce the concept of upper and lower solutions for third order periodic boundary value problems. We show that the monotone iterative technique is valid and obtain the extremal solutions as limits of monotone sequences. We first present a new maximum principle for ordinary differential inequalities of third order that is interesting by itself.},
author = {Nieto, Juan J.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {periodic solution; maximum principle; upper and lower solutions; monotone method; periodic solutions; periodic boundary value problem; third-order ordinary differential equation; uniqueness},
language = {eng},
number = {3},
pages = {495-499},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Periodic solutions for third order ordinary differential equations},
url = {http://eudml.org/doc/21795},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Nieto, Juan J.
TI - Periodic solutions for third order ordinary differential equations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 3
SP - 495
EP - 499
AB - In this paper, we introduce the concept of upper and lower solutions for third order periodic boundary value problems. We show that the monotone iterative technique is valid and obtain the extremal solutions as limits of monotone sequences. We first present a new maximum principle for ordinary differential inequalities of third order that is interesting by itself.
LA - eng
KW - periodic solution; maximum principle; upper and lower solutions; monotone method; periodic solutions; periodic boundary value problem; third-order ordinary differential equation; uniqueness
UR - http://eudml.org/doc/21795
ER -

References

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  1. Aftabizadeh A.R., Gupta C.P., Xu J.M., Existence and uniqueness theorems for three-point boundary value problems, SIAM J. Math. Anal. 20 (1989), 716-726. (1989) Zbl0704.34019MR0990873
  2. Aftabizadeh A.R., Gupta C.P., Xu J.M., Periodic boundary value problems for third order ordinary differential equations, Nonlinear Anal. 14 (1990), 1-10. (1990) Zbl0706.34018MR1028242
  3. Afuwape A.U., Omari P., Zanolin F., Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems, J. Math. Anal. Appl. 143 (1989), 35-56. (1989) Zbl0695.47044MR1019448
  4. Afuwape A.U., Zanolin F., An existence theorem for periodic solutions and applications to some third order nonlinear differential equations, preprint. 
  5. Agarwal R.P., Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986. Zbl0921.34021MR1021979
  6. Agarwal R.P., Existence-uniqueness and iterative methods for third-order boundary value problems, J. Comp. Appl. Math. 17 (1987), 271-289. (1987) Zbl0617.34008MR0883170
  7. Cabada A., Nieto J.J., A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems, J. Math. Anal. Appl. 151 (1990), 181-189. (1990) Zbl0719.34039MR1069454
  8. Ezeilo J.O.C., Nkashama M.N., Resonant and nonresonant oscillations for some third order nonlinear ordinary differential equations, Nonlinear Anal. 12 (1988), 1029-1046. (1988) Zbl0676.34021MR0962767
  9. Gregus M., Third Order Linear Differential Equations, D. Reidel, Dordrecht, 1987. Zbl0602.34005MR0882545
  10. Ladde G.S., Lakshmikantham V., Vatsala A.S., Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985. Zbl0658.35003MR0855240
  11. Lakshmikantham V., Nieto J.J., Sun Y., An existence result about periodic boundary value problems of second order differential equations, Appl. Anal., to appear. MR1121320
  12. Nieto J.J., Nonlinear second order periodic boundary value problems, J. Math. Anal. Appl. 130 (1988), 22-29. (1988) Zbl0678.34022MR0926825
  13. Rudolf B., Kubacek Z., Remarks on J. J. Nieto's paper: Nonlinear second order periodic boundary value problems, J. Math. Anal. Appl. 146 (1990), 203-206. (1990) Zbl0713.34015MR1041210
  14. Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. Zbl0871.35001MR0953967

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