Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions

Ming-Xing Wang; Alberto Cabada; Juan J. Nieto

Annales Polonici Mathematici (1993)

  • Volume: 58, Issue: 3, page 221-235
  • ISSN: 0066-2216

Abstract

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The purpose of this paper is to study the periodic boundary value problem -u''(t) = f(t,u(t),u'(t)), u(0) = u(2π), u'(0) = u'(2π) when f satisfies the Carathéodory conditions. We show that a generalized upper and lower solution method is still valid, and develop a monotone iterative technique for finding minimal and maximal solutions.

How to cite

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Ming-Xing Wang, Alberto Cabada, and Juan J. Nieto. "Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions." Annales Polonici Mathematici 58.3 (1993): 221-235. <http://eudml.org/doc/262275>.

@article{Ming1993,
abstract = {The purpose of this paper is to study the periodic boundary value problem -u''(t) = f(t,u(t),u'(t)), u(0) = u(2π), u'(0) = u'(2π) when f satisfies the Carathéodory conditions. We show that a generalized upper and lower solution method is still valid, and develop a monotone iterative technique for finding minimal and maximal solutions.},
author = {Ming-Xing Wang, Alberto Cabada, Juan J. Nieto},
journal = {Annales Polonici Mathematici},
keywords = {upper and lower solutions; monotone iterative technique; Carathéodory function; periodic solutions; boundary value problem; generalized upper and lower solution method; minimal and maximal solutions},
language = {eng},
number = {3},
pages = {221-235},
title = {Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions},
url = {http://eudml.org/doc/262275},
volume = {58},
year = {1993},
}

TY - JOUR
AU - Ming-Xing Wang
AU - Alberto Cabada
AU - Juan J. Nieto
TI - Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions
JO - Annales Polonici Mathematici
PY - 1993
VL - 58
IS - 3
SP - 221
EP - 235
AB - The purpose of this paper is to study the periodic boundary value problem -u''(t) = f(t,u(t),u'(t)), u(0) = u(2π), u'(0) = u'(2π) when f satisfies the Carathéodory conditions. We show that a generalized upper and lower solution method is still valid, and develop a monotone iterative technique for finding minimal and maximal solutions.
LA - eng
KW - upper and lower solutions; monotone iterative technique; Carathéodory function; periodic solutions; boundary value problem; generalized upper and lower solution method; minimal and maximal solutions
UR - http://eudml.org/doc/262275
ER -

References

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  1. [1] A. Adje, Sur et sous-solutions généralisées et problèmes aux limites du second ordre, Bull. Soc. Math. Belgique Sér. B 42 (1990), 347-368. Zbl0724.34018
  2. [2] J. Bebernes, A simple alternative problem for finding periodic solutions of second order ordinary differential systems, Proc. Amer. Math. Soc. 42 (1974), 121-127. Zbl0286.34055
  3. [3] J. Bebernes and R. Fraker, A priori bounds for boundary sets, ibid. 29 (1971), 313-318. Zbl0215.44003
  4. [4] J. Bebernes and W. Kelley, Some boundary value problems for generalized differential equations, SIAM J. Appl. Math. 25 (1973), 16-23. Zbl0237.34101
  5. [5] J. Bebernes and M. Martelli, On the structure of the solution set for periodic boundary value problems, Nonlinear Anal. 4 (1980), 821-830. Zbl0453.34019
  6. [6] J. Bebernes and K. Schmitt, Periodic boundary value problems for systems of second order differential equations, J. Differential Equations 13 (1973), 32-47. Zbl0253.34020
  7. [7] A. Cabada and J. J. Nieto, A generalization of the monotone iterative technique for nonlinear second-order periodic boundary value problems, J. Math. Anal. Appl. 151 (1990), 181-189. Zbl0719.34039
  8. [8] A. Cabada and J. J. Nieto, Extremal solutions of second-order nonlinear periodic boundary value problems, Appl. Math. Comput. 40 (1990), 135-145. Zbl0723.65056
  9. [9] G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, New York, 1985. Zbl0658.35003
  10. [10] J. J. Nieto, Nonlinear second-order periodic boundary value problems with Carathé- odory functions, Applicable Anal. 34 (1989), 111-128. 
  11. [11] D. R. Smart, Fixed Points Theorems, Cambridge University Press, Cambridge, 1974. Zbl0297.47042
  12. [12] M. X. Wang, Monotone method for nonlinear periodic boundary value problems, J. Beijing Inst. Technol. 9 (1989), 74-81 (in Chinese). Zbl0717.34021

Citations in EuDML Documents

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  1. Wenjie Gao, Junyu Wang, On a nonlinear second order periodic boundaryvalue problem with Carathéodory functions
  2. Daniel Wagner, Stefan Wopperer, Bol-loops of order 3 · 2 n
  3. Daqing Jiang, Junyu Wang, A generalized periodic boundary value problem for the one-dimensional p-Laplacian
  4. Staněk, Svatoslav, On solvability of nonlinear boundary value problems for the equation ( x ' + g ( t , x , x ' ) ) ' = f ( t , x , x ' ) with one-sided growth restrictions on f
  5. Lingbin Kong, Daqing Jiang, Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

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