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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Bartz, S., Bauschke, H.H., Borwein, J.M., Reich, S., Wang, X.: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative. Nonlinear Anal. 66, 1198–1223 (2007)
Bauschke, H.H., Borwein, J.M.: Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators. Pacific J. Math. 189, 1–20 (1999)
Bauschke, H.H., Borwein, J.M., Wang, X.: Fitzpatrick functions and continuous linear monotone operators. SIAM J. Optim. 18, 789–809 (2007)
Bauschke, H.H., Lucet, Y., Wang, X.: Primal-dual symmetric antiderivatives for cyclically monotone operators. SIAM J. Control Optim. 46, 2031–2051 (2007)
Bauschke, H.H., Wang, X., Yao, L.: An answer to S. Simons’ question on the maximal monotonicity of the sum of a maximal monotone linear operator and a normal cone operator. Set-Valued Var. Anal. 17, 195–201 (2009)
Bauschke, H.H., Wang, X., Yao, L.: Monotone linear relations: maximality and Fitzpatrick functions. J. Convex Anal. 16, 673–686 (2009)
Bauschke, H.H., Wang, X., Yao, L.: Autoconjugate representers for linear monotone operators. Math. Program. Ser. B 123, 5–24 (2010)
Bauschke, H.H., Wang, X., Yao, L.: Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter. J. Math. Anal. Appl. 370, 224–241 (2010)
Bauschke, H.H., Wang, X., Yao, L.: On Borwein-Wiersma Decompositions of monotone linear relations. SIAM J. Optim. 20, 2636–2652 (2010)
Bauschke, H.H., Wang, X., Yao, L.: On the maximal monotonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function. To appear in Proceedings of the Haifa Workshop on Optimization Theory and Related Topics. Contemp. Math., Amer. Math. Soc., Providence, RI (2010). http://arxiv.org/abs/1001.0257v1
Borwein, J.M.: A note on ε-subgradients and maximal monotonicity. Pacific J. Math. 103, 307–314 (1982)
Borwein, J.M., Vanderwerff, J.D.: Convex Functions. Cambridge University Press (2010)
Boţ, R.I., Csetnek, E.R.: On extension results for n-cyclically monotone operators in reflexive Banach spaces. J. Math. Anal. Appl. 367, 693–698 (2010)
Brézis, H., Browder, F.E.: Linear maximal monotone operators and singular nonlinear integral equations of Hammerstein type. In: Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic, 31–42 (1978)
Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer (2008)
Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10, 297–316 (2002)
Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)
Cross, R.: Multivalued Linear Operators. Marcel Dekker (1998)
Fitzpatrick, S.: Representing monotone operators by convex functions. In: Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis 20, 59–65. Australian National University, Canberra, Australia (1988)
Haraux, A.: Nonlinear Evolution Equations – Global Behavior of Solutions. Springer, Berlin (1981)
Penot, J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58, 855–871 (2004)
Phelps, R.R.: Convex functions, Monotone Operators and Differentiability, 2nd edn. Springer (1993)
Phelps, R.R., Simons, S.: Unbounded linear monotone operators on nonreflexive Banach spaces. J. Convex Anal. 5, 303–328 (1998)
Rockafellar, R.T., Wets, R.J-B.: Variational Analysis. Springer (2004)
Simons, S.: Minimax and Monotonicity. Springer (1998)
Simons, S.: From Hahn-Banach to Monotonicity. Springer (2008)
Simons, S., Zălinescu, C.: A new proof for Rockafellar’s characterization of maximal monotone operators. Proc. Amer. Math. Soc. 132, 2969–2972 (2004)
Simons, S., Zălinescu, C.: Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal. 6, 1–22 (2005)
Svaiter, B.F.: Non-enlargeable operators and self-cancelling operators. J. Convex Anal. 17, 309–320 (2010)
Voisei, M.D.: The sum theorem for linear maximal monotone operators. Math. Sci. Res. J. 10, 83–85 (2006)
Voisei, M.D., Zălinescu, C.: Linear monotone subspaces of locally convex spaces. Set-Valued Var. Anal. 18, 29–55 (2010)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing (2002)
Zeidler, E.: Nonlinear Functional Analysis and its Application, Vol II/B Nonlinear Monotone Operators. Springer, Berlin (1990)
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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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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DOI: https://doi.org/10.1007/978-1-4419-9569-8_18
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