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

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

Published electronically:
September 15, 2011

MathSciNet review:
2846321

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Brezis and Mawhin proved the existence of at least one $T$ periodic solution for differential equations of the form \begin{equation}\notag (\phi (u^{\prime }))^{\prime }-g(t,u)=h(t)\tag *{(0.1)} \end{equation} when $\phi :(-a,a)\rightarrow \mathbb {R},$ $0<a<\infty$, is an increasing homeomorphism with $\phi (0)=0$, $g$ is a Carathéodory function $T$ periodic with respect to $t$, $2\pi$ periodic with respect to $u$, of mean value zero with respect to $u$, and $h\in L_{loc}^{1}(\mathbb {R})$ is $T$ 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 $u_{0}$ in a closed convex subset $\mathcal {K}$ of the space of $T$ periodic Lipschitz functions. However, $u_{0}$ may not be an interior point of $\mathcal {K}$, 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 $T$ periodic solution to (0.1) when $g(t,u)=u$ to show that $u_{0}$ is indeed a $T$ periodic solution of (0.1). Here we make full use of the variational structure of the problem to obtain Brezis and Mawhin’s result.

- Haïm Brezis and Jean Mawhin,
*Periodic solutions of the forced relativistic pendulum*, Differential Integral Equations**23**(2010), no. 9-10, 801–810. MR**2675583** - Jean Mawhin and Michel Willem,
*Critical point theory and Hamiltonian systems*, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. MR**982267** - 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** - 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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
34C25,
49J40,
58Exx

Retrieve articles in all journals with MSC (2010): 34C25, 49J40, 58Exx

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

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