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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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 $ T$ periodic solution for differential equations of the form

$\displaystyle \notag (\phi(u^{\prime}))^{\prime}-g(t,u)=h(t)$ (0.1)

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.


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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
PII: S 0002-9939(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