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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

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C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems,, Wiley and Sons, (1984).   Google Scholar

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V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (1976).   Google Scholar

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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

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H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité,, \emph{Ann. Inst. Fourier (Grenoble)}, 18 (1968), 115.   Google Scholar

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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

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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

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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

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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

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J.-B. Hiriart-Urruty and C. Lemarechal, Fundamentals of Convex Analysis,, Springer, (2001).  doi: 10.1007/978-3-642-56468-0.  Google Scholar

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Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I,, Kluwer, (1997).  doi: 10.1016/0362-546X(85)90097-5.  Google Scholar

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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

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