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The Brézis-Browder Theorem Revisited and Properties of Fitzpatrick Functions of Order n

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Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 49))

Abstract

In this paper, we study maximal monotonicity of linear relations (set-valued operators with linear graphs) on reflexive Banach spaces. We provide a new and simpler proof of a result due to Brézis–Browder which states that a monotone linear relation with closed graph is maximal monotone if and only if its adjoint is monotone. We also study Fitzpatrick functions and give an explicit formula for Fitzpatrick functions of order n for monotone symmetric linear relations.

AMS 2010 Subject Classification: 47A06, 47H05

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Acknowledgements

The author thanks Dr. Heinz Bauschke and Dr. Xianfu Wang for valuable discussions. The author also thanks the two anonymous referees for their careful reading and their pertinent comments.

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Authors and Affiliations

  1. Department of Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C., V1V 1V7, Canada

    Liangjin Yao

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  1. Liangjin Yao
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Correspondence to Liangjin Yao .

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Editors and Affiliations

  1. Okanagan Campus, Department of Mathematics and Statistic, University of British Columbia, Kelowna, V1V 1V7, British Columbia, Canada

    Heinz H. Bauschke

  2. School of Mathematics & Statistics, Division of Information Technology, University of South Australia, Mawson Lakes Blvd., Mawson Lakes, 5095, South Australia, Australia

    Regina S. Burachik

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Place Jussieu 4, Paris, 75005, France

    Patrick L. Combettes

  4. Dept. Physics, Lab. Atomic & Solid State, Cornell University, Clark Hall, Ithaca, 14853-2501, New York, USA

    Veit Elser

  5. Institut für Numerische und Angewandte M, Universität Göttingen, Lotzestr. 16-18, Göttingen, 37073, Germany

    D. Russell Luke

  6. Faculty of Mathematics, Dept. Combinatorics & Optimization, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada

    Henry Wolkowicz

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Yao, L. (2011). The Brézis-Browder Theorem Revisited and Properties of Fitzpatrick Functions of Order n . In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_18

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