Abstract
The relationships among five classes of monotonicity, namely 3∗-, 3-cyclic, strictly, para-, and maximal monotonicity, are explored for linear operators and linear relations in Hilbert space. Where classes overlap, examples are given; otherwise their relationships are noted for linear operators in \({\mathbb{R}}^{2}\), \({\mathbb{R}}^{3}\), and general Hilbert spaces. Along the way, some results for linear relations are obtained.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Attouch, H., Damlamian, A.: On multivalued evolution equations in Hilbert spaces. Israel J. Math. 12(4), 373–390 (1972)
Auslender, A.A., Haddou, M.: An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities. Math. Program. 71, 77–100 (1995)
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., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Bauschke, H.H., Wang, W., Yao, L.: Monotone linear relations: maximality and Fitzpatrick functions. J. Convex Anal. 25, 673–686 (2009)
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.: Rectangularity and paramonotonicity of maximally monotone operators. Optimization. In press
Bello Cruz, J.Y., Iusem, A.N.: Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 46(2), 247–263 (2010)
Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13(3), 561–586 (2006)
Bressan, A., Staicu, V.: On nonconvex perturbations of maximal monotone differential inclusions. Set-Valued Var. Anal. 2, 415–437 (1994)
Brézis, H., Haraux, A.: Image d’une somme d’opérateurs monotones et applications. Israel J. Math. 23(2), 165–186 (1976)
Bruck, R.E., Jr.: An iterative solution of a variational inequality for certain monotone operators in Hilbert space. Bull. Amer. Math. Soc. 81(5), 890–892 (1975)
Burachik, R.S., Iusem, A.N.: An iterative solution of a variational inequality for certain monotone operators in a Hilbert space. SIAM J. Optim. 8, 197–216 (1998)
Burachik, R.S., Lopes, J.O., Svaiter, B.F.: An outer approximation method for the variational inequality problem. SIAM J. Control Optim. 43, 2071–2088 (2005)
Censor, Y., Iusem, A.N., Zenios, S.A.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. Program. 81(3), 373–400 (1998)
Chu, L.-J.: On the sum of monotone operators. Michigan Math. J. 43, 273–289 (1996)
Cross, R.: Monotone Linear Relations. M. Dekker, New York (1998)
Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)
Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669–713 (December 1997)
Ghoussoub, N.: Selfdual partial differential systems and their variational principles. Springer Monographs in Mathematics, vol. 14. Springer, New York (2010)
Hadjisavvas, N., Schaible, S.: On a generalization of paramonotone maps and its application to solving the stampacchia variational inequality. Optimization 55(5-6), 593–604 (October-December 2006)
Hartmann, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Math. 115(1), 271–310 (1966)
Iusem, A.N.: On some properties of paramonotone operators. J. Convex Anal. 5(2), 269–278 (1998)
Minty, G.J.: Monotone (nonlinear) operators in a Hilbert space. Duke Math. J. 29, 341–346 (1962)
Papageorgiou, N.S., Shahzad, N.: On maximal monotone differential inclusions in RN. Acta Math. Hungar. 78(3), 175–197 (1998)
Pennanen, T.: Dualization of monotone generalized equations, PhD thesis. University of Washington, Seattle, Washington (1999)
Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Grundlehren der Mathematischen Wissenschaften, vol. 317, 2nd edn. Springer, Berlin (2004)
Simons, S.: LC functions and maximal monotonicity. J. Nonlinear Convex Anal. 7, 123–138 (2006)
Yagi, A.: Generation theorem of semigroup for multivalued linear operators. Osaka J. Math. 28, 385–410 (1991)
Yamada, I., Ogura, N.: Hybrid steepest descent method for variational inequality problems over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications, II/B - Nonlinear Monotone Operators. Springer, New York (1990)
Acknowledgements
The author would like to thank the professional and editorial support of Philip Loewen.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to Jonathan Borwein on the occasion of his 60th birthday
Communicated by Heinz H. Bauschke.
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this paper
Cite this paper
Edwards, M.R. (2013). Five Classes of Monotone Linear Relations and Operators. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_17
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7621-4_17
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7620-7
Online ISBN: 978-1-4614-7621-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)