# American Institute of Mathematical Sciences

• Previous Article
Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit
• CPAA Home
• This Issue
• Next Article
The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction
September  2014, 13(5): 2039-2058. doi: 10.3934/cpaa.2014.13.2039

## An extension of the Fitzpatrick theory

 1 Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento) - Italia

Received  January 2014 Revised  March 2014 Published  June 2014

In the seminal work [MR 1009594], Fitzpatrick proved that for any maximal monotone operator $\alpha: V\to {\mathcal P}(V')$ ($V$ being a real Banach space) there exists a lower semicontinuous, convex representative function $f_\alpha: V \times V'\to R\cup \{+\infty\}$ such that \begin{eqnarray} f_\alpha(v,v') \ge \langle v',v\rangle \quad\;\forall (v,v'), \qquad\quad f_\alpha(v,v') = \langle v',v\rangle \;\;\Leftrightarrow\;\;\; v'\in \alpha(v). \end{eqnarray}
Here we assume that $\alpha_v$ is a maximal monotone operator for any $v\in V$, and extend the Fitzpatrick theory to provide a new variational formulation for either stationary or evolutionary (nonmonotone) inclusions of the form $\alpha_v(v) \ni v'$. For any $v'\in V'$, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.
Citation: Augusto Visintin. An extension of the Fitzpatrick theory. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2039-2058. doi: 10.3934/cpaa.2014.13.2039
##### References:
 [1] G. Allen, Variational inequalities, complementarity problems, and duality theorems,, \emph{J. Math. Anal. Appl.}, 58 (1977). Google Scholar [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley and Sons, (1984). Google Scholar [3] G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations,, \emph{Differential Integral Equations}, 6 (1993), 1161. Google Scholar [4] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems,, Wiley and Sons, (1984). Google Scholar [5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (1976). Google Scholar [6] H. H. Bauschke and X. Wang, The kernel average for two convex functions and its applications to the extension and representation of monotone operators,, \emph{Trans. Amer. Math. Soc.}, 361 (2009), 5947. doi: 10.1090/S0002-9947-09-04698-4. Google Scholar [7] H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité,, \emph{Ann. Inst. Fourier (Grenoble)}, 18 (1968), 115. Google Scholar [8] H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,, North-Holland, (1973). Google Scholar [9] H. Brezis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps et II. Le cas dépendant du temps,, \emph{C. R. Acad. Sci. Paris S\'er. A-B}, 282 (1976), 971. Google Scholar [10] F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces,, Proc. Sympos. Pure Math., XVIII (1976). Google Scholar [11] F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces,, \emph{J. Functional Analysis}, 11 (1972), 251. Google Scholar [12] M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, \emph{J. Convex Anal., 15 (2008), 87. Google Scholar [13] R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements,, \emph{Set-Valued Analysis}, 10 (2002), 297. doi: 10.1023/A:1020639314056. Google Scholar [14] R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product,, \emph{Proc. Amer. Math. Soc., 131 (2003), 2379. doi: 10.1090/S0002-9939-03-07053-9. Google Scholar [15] I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles,, Dunod Gauthier-Villars, (1974). Google Scholar [16] K. Fan, A generalization of Tychonoff's theorem,, \emph{Math. Ann.}, 142 (1961), 305. Google Scholar [17] K. Fan, A minimax inequality and applications. Inequalities, III,, In: \emph{Proc. Third Sympos.}, (1969), 103. Google Scholar [18] W. Fenchel, Convex Cones, Sets, and Functions,, Princeton Univ., (1953). Google Scholar [19] S. Fitzpatrick, Representing monotone operators by convex functions,, in \emph{Workshop/Miniconference on Functional Analysis and Optimization (Canberra, (1988), 59. Google Scholar [20] N. Ghoussoub, Selfdual Partial Differential Systems and their Variational Principles,, Springer, (2008). Google Scholar [21] N. Ghoussoub, A variational theory for monotone vector fields,, \emph{J. Fixed Point Theory Appl.}, 4 (2008), 107. doi: 10.1007/s11784-008-0083-4. Google Scholar [22] N. Ghoussoub and L. Tzou, A variational principle for gradient flows,, \emph{Math. Ann.}, 330 (2004), 519. doi: 10.1007/s00208-004-0558-6. Google Scholar [23] J.-B. Hiriart-Urruty and C. Lemarechal, Fundamentals of Convex Analysis,, Springer, (2001). doi: 10.1007/978-3-642-56468-0. Google Scholar [24] Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I,, Kluwer, (1997). doi: 10.1016/0362-546X(85)90097-5. Google Scholar [25] B. Knaster, C. Kuratowski and S. Mazurkievicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe,, \emph{Fund. Math.}, 14 (1929), 132. Google Scholar [26] E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions,, \emph{Nonlinear Anal.}, 9 (1985), 1381. Google Scholar [27] J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder,, \emph{Bull. Soc. Math. France}, 93 (1965), 97. Google Scholar [28] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,, Dunod, (1969). Google Scholar [29] J.-E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators,, \emph{J. Nonlinear Convex Anal.}, 2 (2001), 243. Google Scholar [30] J.-E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions,, \emph{Set-Valued Anal.}, 13 (2005), 21. doi: 10.1007/s11228-004-4170-4. Google Scholar [31] J.-E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product,, \emph{Proc. Amer. Math. Soc.}, 136 (2008), 873. doi: 10.1090/S0002-9939-07-09176-9. Google Scholar [32] G. J. Minty, Monotone (nonlinear) operators in Hilbert space,, \emph{Duke Math. J.}, 29 (1962), 341. Google Scholar [33] B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs,, \emph{C. R. Acad. Sci. Paris Sér. A-B}, 282 (1976). Google Scholar [34] J.-P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules,, \emph{C. R. Math. Acad. Sci. Paris, 338 (2004), 853. doi: 10.1016/j.crma.2004.03.017. Google Scholar [35] J.-P. Penot, The relevance of convex analysis for the study of monotonicity,, \emph{Nonlinear Anal.}, 58 (2004), 855. doi: 10.1016/j.na.2004.05.018. Google Scholar [36] T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence,, \emph{SIAM J. Control Optim.}, (2014). doi: 10.1137/130909391. Google Scholar [37] R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1969). Google Scholar [38] T. Roubíček, Direct method for parabolic problems,, \emph{Adv. Math. Sci. Appl.}, 10 (2000), 57. Google Scholar [39] T. Roubíček, Nonlinear Partial Differential Equations with Applications,, second edition, (2013). doi: 10.1007/978-3-0348-0513-1. Google Scholar [40] B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 3851. doi: 10.1090/S0002-9939-03-07083-7. Google Scholar [41] A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators,, \emph{Adv. Math. Sci. Appl.}, 18 (2008), 633. Google Scholar [42] A. Visintin, Variational formulation and structural stability of monotone equations,, \emph{Calc. Var. Partial Differential Equations}, 47 (2013), 273. doi: 10.1007/s00526-012-0519-y. Google Scholar [43] A. Visintin, Scale-transformations and homogenization of maximal monotone relations, with applications,, \emph{Asymptotic Analysis}, 82 (2013), 233. Google Scholar [44] A. Visintin, Weak structural stability of pseudo-monotone equations,, in press., (). Google Scholar [45] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. II/B: Nonlinear Monotone Operators,, Springer, (1990). doi: 10.1007/978-1-4612-0985-0. Google Scholar [46] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. IV: Applications to Mathematical Physics,, Springer, (1988). doi: 10.1007/978-1-4612-4566-7. Google Scholar

show all references

##### References:
 [1] G. Allen, Variational inequalities, complementarity problems, and duality theorems,, \emph{J. Math. Anal. Appl.}, 58 (1977). Google Scholar [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley and Sons, (1984). Google Scholar [3] G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations,, \emph{Differential Integral Equations}, 6 (1993), 1161. Google Scholar [4] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems,, Wiley and Sons, (1984). Google Scholar [5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (1976). Google Scholar [6] H. H. Bauschke and X. Wang, The kernel average for two convex functions and its applications to the extension and representation of monotone operators,, \emph{Trans. Amer. Math. Soc.}, 361 (2009), 5947. doi: 10.1090/S0002-9947-09-04698-4. Google Scholar [7] H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité,, \emph{Ann. Inst. Fourier (Grenoble)}, 18 (1968), 115. Google Scholar [8] H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,, North-Holland, (1973). Google Scholar [9] H. Brezis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps et II. Le cas dépendant du temps,, \emph{C. R. Acad. Sci. Paris S\'er. A-B}, 282 (1976), 971. Google Scholar [10] F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces,, Proc. Sympos. Pure Math., XVIII (1976). Google Scholar [11] F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces,, \emph{J. Functional Analysis}, 11 (1972), 251. Google Scholar [12] M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, \emph{J. Convex Anal., 15 (2008), 87. Google Scholar [13] R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements,, \emph{Set-Valued Analysis}, 10 (2002), 297. doi: 10.1023/A:1020639314056. Google Scholar [14] R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product,, \emph{Proc. Amer. Math. Soc., 131 (2003), 2379. doi: 10.1090/S0002-9939-03-07053-9. Google Scholar [15] I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles,, Dunod Gauthier-Villars, (1974). Google Scholar [16] K. Fan, A generalization of Tychonoff's theorem,, \emph{Math. Ann.}, 142 (1961), 305. Google Scholar [17] K. Fan, A minimax inequality and applications. Inequalities, III,, In: \emph{Proc. Third Sympos.}, (1969), 103. Google Scholar [18] W. Fenchel, Convex Cones, Sets, and Functions,, Princeton Univ., (1953). Google Scholar [19] S. Fitzpatrick, Representing monotone operators by convex functions,, in \emph{Workshop/Miniconference on Functional Analysis and Optimization (Canberra, (1988), 59. Google Scholar [20] N. Ghoussoub, Selfdual Partial Differential Systems and their Variational Principles,, Springer, (2008). Google Scholar [21] N. Ghoussoub, A variational theory for monotone vector fields,, \emph{J. Fixed Point Theory Appl.}, 4 (2008), 107. doi: 10.1007/s11784-008-0083-4. Google Scholar [22] N. Ghoussoub and L. Tzou, A variational principle for gradient flows,, \emph{Math. Ann.}, 330 (2004), 519. doi: 10.1007/s00208-004-0558-6. Google Scholar [23] J.-B. Hiriart-Urruty and C. Lemarechal, Fundamentals of Convex Analysis,, Springer, (2001). doi: 10.1007/978-3-642-56468-0. Google Scholar [24] Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I,, Kluwer, (1997). doi: 10.1016/0362-546X(85)90097-5. Google Scholar [25] B. Knaster, C. Kuratowski and S. Mazurkievicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe,, \emph{Fund. Math.}, 14 (1929), 132. Google Scholar [26] E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions,, \emph{Nonlinear Anal.}, 9 (1985), 1381. Google Scholar [27] J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder,, \emph{Bull. Soc. Math. France}, 93 (1965), 97. Google Scholar [28] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,, Dunod, (1969). Google Scholar [29] J.-E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators,, \emph{J. Nonlinear Convex Anal.}, 2 (2001), 243. Google Scholar [30] J.-E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions,, \emph{Set-Valued Anal.}, 13 (2005), 21. doi: 10.1007/s11228-004-4170-4. Google Scholar [31] J.-E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product,, \emph{Proc. Amer. Math. Soc.}, 136 (2008), 873. doi: 10.1090/S0002-9939-07-09176-9. Google Scholar [32] G. J. Minty, Monotone (nonlinear) operators in Hilbert space,, \emph{Duke Math. J.}, 29 (1962), 341. Google Scholar [33] B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs,, \emph{C. R. Acad. Sci. Paris Sér. A-B}, 282 (1976). Google Scholar [34] J.-P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules,, \emph{C. R. Math. Acad. Sci. Paris, 338 (2004), 853. doi: 10.1016/j.crma.2004.03.017. Google Scholar [35] J.-P. Penot, The relevance of convex analysis for the study of monotonicity,, \emph{Nonlinear Anal.}, 58 (2004), 855. doi: 10.1016/j.na.2004.05.018. Google Scholar [36] T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence,, \emph{SIAM J. Control Optim.}, (2014). doi: 10.1137/130909391. Google Scholar [37] R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1969). Google Scholar [38] T. Roubíček, Direct method for parabolic problems,, \emph{Adv. Math. Sci. Appl.}, 10 (2000), 57. Google Scholar [39] T. Roubíček, Nonlinear Partial Differential Equations with Applications,, second edition, (2013). doi: 10.1007/978-3-0348-0513-1. Google Scholar [40] B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 3851. doi: 10.1090/S0002-9939-03-07083-7. Google Scholar [41] A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators,, \emph{Adv. Math. Sci. Appl.}, 18 (2008), 633. Google Scholar [42] A. Visintin, Variational formulation and structural stability of monotone equations,, \emph{Calc. Var. Partial Differential Equations}, 47 (2013), 273. doi: 10.1007/s00526-012-0519-y. Google Scholar [43] A. Visintin, Scale-transformations and homogenization of maximal monotone relations, with applications,, \emph{Asymptotic Analysis}, 82 (2013), 233. Google Scholar [44] A. Visintin, Weak structural stability of pseudo-monotone equations,, in press., (). Google Scholar [45] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. II/B: Nonlinear Monotone Operators,, Springer, (1990). doi: 10.1007/978-1-4612-0985-0. Google Scholar [46] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. IV: Applications to Mathematical Physics,, Springer, (1988). doi: 10.1007/978-1-4612-4566-7. Google Scholar
 [1] Augusto VisintiN. On the variational representation of monotone operators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046 [2] Dalila Azzam-Laouir, Warda Belhoula, Charles Castaing, M. D. P. Monteiro Marques. Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evolution Equations & Control Theory, 2020, 9 (1) : 219-254. doi: 10.3934/eect.2020004 [3] JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042 [4] Dag Lukkassen, Annette Meidell, Peter Wall. Multiscale homogenization of monotone operators. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 711-727. doi: 10.3934/dcds.2008.22.711 [5] Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic & Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010 [6] Luca Lussardi, Stefano Marini, Marco Veneroni. Stochastic homogenization of maximal monotone relations and applications. Networks & Heterogeneous Media, 2018, 13 (1) : 27-45. doi: 10.3934/nhm.2018002 [7] Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks & Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181 [8] Woocheol Choi. Maximal functions of multipliers on compact manifolds without boundary. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1885-1902. doi: 10.3934/cpaa.2015.14.1885 [9] Xiao Ding, Deren Han. A modification of the forward-backward splitting method for maximal monotone mappings. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 295-307. doi: 10.3934/naco.2013.3.295 [10] Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070 [11] Gunther Dirr, Hiroshi Ito, Anders Rantzer, Björn S. Rüffer. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2497-2526. doi: 10.3934/dcdsb.2015.20.2497 [12] Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383 [13] Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 [14] Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35 [15] Chjan C. Lim, Junping Shi. The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 717-740. doi: 10.3934/dcdsb.2009.11.717 [16] Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751 [17] Sihong Su. A new construction of rotation symmetric bent functions with maximal algebraic degree. Advances in Mathematics of Communications, 2019, 13 (2) : 253-265. doi: 10.3934/amc.2019017 [18] A. C. Eberhard, J-P. Crouzeix. Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory. Journal of Industrial & Management Optimization, 2007, 3 (2) : 233-255. doi: 10.3934/jimo.2007.3.233 [19] Radu Ioan Boţ, Christopher Hendrich. Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators. Inverse Problems & Imaging, 2016, 10 (3) : 617-640. doi: 10.3934/ipi.2016014 [20] Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

2018 Impact Factor: 0.925