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Proceedings of the American Mathematical Society

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On a result of Brezis and Mawhin
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by R. Manásevich and J. R. Ward PDF
Proc. Amer. Math. Soc. 140 (2012), 531-539 Request permission

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.
References
  • 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, DOI 10.1007/978-1-4757-2061-7
  • 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
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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
  • 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
  • © Copyright 2011 American Mathematical Society
  • 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
  • MathSciNet review: 2846321