On a result of Brezis and Mawhin

Authors:
R. Manásevich and J. R. Ward

Journal:
Proc. Amer. Math. Soc. **140** (2012), 531-539

MSC (2010):
Primary 34C25, 49J40, 58Exx

Published electronically:
September 15, 2011

MathSciNet review:
2846321

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Abstract: Brezis and Mawhin proved the existence of at least one periodic solution for differential equations of the form

(0.1) |

when , is an increasing homeomorphism with , is a Carathéodory function periodic with respect to , periodic with respect to , of mean value zero with respect to , and is periodic and has mean value zero. Their proof was partly variational. First it was shown that the corresponding action integral had a minimum at some point in a closed convex subset of the space of periodic Lipschitz functions. However, may not be an interior point of , so it may not be a critical point of the action integral. The authors used an ingenious argument based on variational inequalities and uniqueness of a periodic solution to (0.1) when to show that is indeed a periodic solution of (0.1). Here we make full use of the variational structure of the problem to obtain Brezis and Mawhin's result.

**[1]**Haïm Brezis and Jean Mawhin,*Periodic solutions of the forced relativistic pendulum*, Differential Integral Equations**23**(2010), no. 9-10, 801–810. MR**2675583****[2]**Jean Mawhin and Michel Willem,*Critical point theory and Hamiltonian systems*, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. MR**982267****[3]**Paul H. Rabinowitz,*Some minimax theorems and applications to nonlinear partial differential equations*, Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, New York, 1978, pp. 161–177. MR**0501092****[4]**Paul H. Rabinowitz,*Minimax methods in critical point theory with applications to differential equations*, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**845785**

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Additional Information

**R. Manásevich**

Affiliation:
Centro de Modelamiento Matemático and Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile

Email:
manasevi@dim.uchile.cl

**J. R. Ward**

Affiliation:
Department of Mathematics, The University of Alabama at Birmingham, Birmingham, Alabama 35294

Email:
jrward@uab.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11311-X

Received by editor(s):
October 17, 2010

Published electronically:
September 15, 2011

Additional Notes:
The first author was partially supported by Fondap and Basal-CMM grants and Milenio grant P05-004F

Communicated by:
Yingfei Yi

Article copyright:
© Copyright 2011
American Mathematical Society