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

top
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

top

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

top
  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

top
  1. Wenjie Gao, Junyu Wang, On a nonlinear second order periodic boundaryvalue problem with Carathéodory functions
  2. Hugo Carrasco, Feliz Minhós, Sufficient conditions for the solvability of some third order functional boundary value problems on the half-line
  3. Daniel Wagner, Stefan Wopperer, Bol-loops of order 3 · 2 n
  4. Daqing Jiang, Junyu Wang, A generalized periodic boundary value problem for the one-dimensional p-Laplacian
  5. 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
  6. Lingbin Kong, Daqing Jiang, Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.