Open Access Research Article

Generalizations of the Lax-Milgram Theorem

Dimosthenis Drivaliaris1* and Nikos Yannakakis2

Author Affiliations

1 Department of Financial and Management Engineering, University of the Aegean, 31 Fostini Street, Chios 82100, Greece

2 Department of Mathematics, School of Applied Mathematics and Natural Sciences, National Technical University of Athens, Iroon Polytexneiou 9, Zografou 15780, Greece

For all author emails, please log on.

Boundary Value Problems 2007, 2007:087104 doi:10.1155/2007/87104


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/087104


Received:12 December 2006
Revisions received:8 March 2007
Accepted:19 April 2007
Published:21 May 2007

© 2007 Drivaliaris and Yannakakis

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular, we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter. We also give two applications to singular differential equations.

References

  1. An, LH, Du, PX, Duc, DM, Tuoc, PV: Lagrange multipliers for functions derivable along directions in a linear subspace. Proceedings of the American Mathematical Society. 133(2), 595–604 (2005). Publisher Full Text OpenURL

  2. Hayden, TL: The extension of bilinear functionals. Pacific Journal of Mathematics. 22, 99–108 (1967)

  3. Hayden, TL: Representation theorems in reflexive Banach spaces. Mathematische Zeitschrift. 104(5), 405–406 (1968). Publisher Full Text OpenURL

  4. Megginson, RE: An Introduction to Banach Space Theory, Graduate Texts in Mathematics,p. xx+596. Springer, New York, NY, USA (1998)

  5. Banach, S: Théorie des Opérations Linéaires, Monografje Matematyczne, Warsaw, Poland (1932).

  6. Dieudonné, J: La dualité dans les espaces vectoriels topologiques. Annales Scientifiques de l'École Normale Supérieure. Troisième Série. 59, 107–139 (1942)

  7. Brezis, H: Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Annales de l'Institut Fourier. Université de Grenoble. 18(1), 115–175 (1968). Publisher Full Text OpenURL

  8. Zeidler, E: Nonlinear Functional Analysis and Its Applications. II/B,p. xviii+467. Springer, New York, NY, USA (1990)