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Generalizations of the Lax-Milgram Theorem

Boundary Value Problems volume 2007, Article number: 087104 (2007) Cite this article

Abstract

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.

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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 2005,133(2):595-604. 10.1090/S0002-9939-04-07711-1

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Hayden TL: Representation theorems in reflexive Banach spaces. Mathematische Zeitschrift 1968,104(5):405-406. 10.1007/BF01110432

    Article  MATH  MathSciNet  Google Scholar 

  4. Megginson RE: An Introduction to Banach Space Theory, Graduate Texts in Mathematics. Volume 183. Springer, New York, NY, USA; 1998:xx+596.

    Book  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  7. Brezis H: Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Annales de l'Institut Fourier. Université de Grenoble 1968,18(1):115-175. 10.5802/aif.280

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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Authors and Affiliations

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

    Dimosthenis Drivaliaris

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

    Nikos Yannakakis

Authors
  1. Dimosthenis Drivaliaris
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  2. Nikos Yannakakis
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Correspondence to Dimosthenis Drivaliaris.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Drivaliaris, D., Yannakakis, N. Generalizations of the Lax-Milgram Theorem. Bound Value Probl 2007, 087104 (2007). https://doi.org/10.1155/2007/87104

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  • DOI: https://doi.org/10.1155/2007/87104

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