An extension of the Fitzpatrick theory

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September 2014, 13(5): 2039-2058. doi: 10.3934/cpaa.2014.13.2039

An extension of the Fitzpatrick theory

Augusto Visintin ^1,

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

Abstract
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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.

Keywords: Maximal monotone operators, minimization., variational formulation, representative functions.

Mathematics Subject Classification: 35K60, 47H05, 49J40, 58.

Citation: Augusto Visintin. An extension of the Fitzpatrick theory. Communications on Pure and 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, J. Math. Anal. Appl., 58 (1977), 110.
[2]	J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley and Sons, New York, 1984.
[3]	G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), 1161-117.
[4]	C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, Wiley and Sons, Chichester, 1984.
[5]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.
[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, Trans. Amer. Math. Soc., 361 (2009), 5947-5965. doi: 10.1090/S0002-9947-09-04698-4.
[7]	H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.
[8]	H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
[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, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971-974, and ibid. 1197-1198.
[10]	F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Sympos. Pure Math., XVIII Part II. A.M.S., Providence, 1976.
[11]	F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294.
[12]	M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws, J. Convex Anal., 15 (2008), 87-104.
[13]	R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements, Set-Valued Analysis, 10 (2002), 297-316. doi: 10.1023/A:1020639314056.
[14]	R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), 2379-2383. doi: 10.1090/S0002-9939-03-07053-9.
[15]	I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles, Dunod Gauthier-Villars, Paris, 1974.
[16]	K. Fan, A generalization of Tychonoff's theorem, Math. Ann., 142 (1961), 305-310.
[17]	K. Fan, A minimax inequality and applications. Inequalities, III, In: Proc. Third Sympos., Univ. California, Los Angeles 1969, pp. 103-113. Academic Press, New York, 1972.
[18]	W. Fenchel, Convex Cones, Sets, and Functions, Princeton Univ., 1953.
[19]	S. Fitzpatrick, Representing monotone operators by convex functions, in Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59-65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988.
[20]	N. Ghoussoub, Selfdual Partial Differential Systems and their Variational Principles, Springer, 2008.
[21]	N. Ghoussoub, A variational theory for monotone vector fields, J. Fixed Point Theory Appl., 4 (2008), 107-135. doi: 10.1007/s11784-008-0083-4.
[22]	N. Ghoussoub and L. Tzou, A variational principle for gradient flows, Math. Ann., 330, (2004) 519-549. doi: 10.1007/s00208-004-0558-6.
[23]	J.-B. Hiriart-Urruty and C. Lemarechal, Fundamentals of Convex Analysis, Springer, Berlin, 2001. doi: 10.1007/978-3-642-56468-0.
[24]	Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Kluwer, Dordrecht, 1997. doi: 10.1016/0362-546X(85)90097-5.
[25]	B. Knaster, C. Kuratowski and S. Mazurkievicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math., 14 (1929), 132-138.
[26]	E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Anal., 9 (1985), 1381-1399.
[27]	J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.
[28]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
[29]	J.-E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal., 2 (2001), 243-247.
[30]	J.-E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal., 13 (2005), 21-46. doi: 10.1007/s11228-004-4170-4.
[31]	J.-E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product, Proc. Amer. Math. Soc., 136 (2008), 873-878. doi: 10.1090/S0002-9939-07-09176-9.
[32]	G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346.
[33]	B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035-A1038.
[34]	J.-P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules, C. R. Math. Acad. Sci. Paris, Ser. I, 338 (2004), 853-858. doi: 10.1016/j.crma.2004.03.017.
[35]	J.-P. Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., 58 (2004), 855-871. doi: 10.1016/j.na.2004.05.018.
[36]	T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., in press, 2014. doi: 10.1137/130909391.
[37]	R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1969.
[38]	T. Roubíček, Direct method for parabolic problems, Adv. Math. Sci. Appl., 10 (2000), 57-65.
[39]	T. Roubíček, Nonlinear Partial Differential Equations with Applications, second edition, Birkhäuser, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.
[40]	B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator, Proc. Amer. Math. Soc., 131 (2003), 3851-3859. doi: 10.1090/S0002-9939-03-07083-7.
[41]	A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators, Adv. Math. Sci. Appl., 18 (2008), 633-650.
[42]	A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47 (2013), 273-317. doi: 10.1007/s00526-012-0519-y.
[43]	A. Visintin, Scale-transformations and homogenization of maximal monotone relations, with applications, Asymptotic Analysis, 82 (2013), 233-270.
[44]	A. Visintin, Weak structural stability of pseudo-monotone equations, in press.
[45]	E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. II/B: Nonlinear Monotone Operators, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
[46]	E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. IV: Applications to Mathematical Physics, Springer, New York, 1988. doi: 10.1007/978-1-4612-4566-7.

show all references

References:

[1]	G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl., 58 (1977), 110.
[2]	J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley and Sons, New York, 1984.
[3]	G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), 1161-117.
[4]	C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, Wiley and Sons, Chichester, 1984.
[5]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.
[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, Trans. Amer. Math. Soc., 361 (2009), 5947-5965. doi: 10.1090/S0002-9947-09-04698-4.
[7]	H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.
[8]	H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
[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, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971-974, and ibid. 1197-1198.
[10]	F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Sympos. Pure Math., XVIII Part II. A.M.S., Providence, 1976.
[11]	F. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294.
[12]	M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws, J. Convex Anal., 15 (2008), 87-104.
[13]	R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements, Set-Valued Analysis, 10 (2002), 297-316. doi: 10.1023/A:1020639314056.
[14]	R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), 2379-2383. doi: 10.1090/S0002-9939-03-07053-9.
[15]	I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles, Dunod Gauthier-Villars, Paris, 1974.
[16]	K. Fan, A generalization of Tychonoff's theorem, Math. Ann., 142 (1961), 305-310.
[17]	K. Fan, A minimax inequality and applications. Inequalities, III, In: Proc. Third Sympos., Univ. California, Los Angeles 1969, pp. 103-113. Academic Press, New York, 1972.
[18]	W. Fenchel, Convex Cones, Sets, and Functions, Princeton Univ., 1953.
[19]	S. Fitzpatrick, Representing monotone operators by convex functions, in Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59-65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988.
[20]	N. Ghoussoub, Selfdual Partial Differential Systems and their Variational Principles, Springer, 2008.
[21]	N. Ghoussoub, A variational theory for monotone vector fields, J. Fixed Point Theory Appl., 4 (2008), 107-135. doi: 10.1007/s11784-008-0083-4.
[22]	N. Ghoussoub and L. Tzou, A variational principle for gradient flows, Math. Ann., 330, (2004) 519-549. doi: 10.1007/s00208-004-0558-6.
[23]	J.-B. Hiriart-Urruty and C. Lemarechal, Fundamentals of Convex Analysis, Springer, Berlin, 2001. doi: 10.1007/978-3-642-56468-0.
[24]	Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Kluwer, Dordrecht, 1997. doi: 10.1016/0362-546X(85)90097-5.
[25]	B. Knaster, C. Kuratowski and S. Mazurkievicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math., 14 (1929), 132-138.
[26]	E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Anal., 9 (1985), 1381-1399.
[27]	J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.
[28]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
[29]	J.-E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal., 2 (2001), 243-247.
[30]	J.-E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal., 13 (2005), 21-46. doi: 10.1007/s11228-004-4170-4.
[31]	J.-E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product, Proc. Amer. Math. Soc., 136 (2008), 873-878. doi: 10.1090/S0002-9939-07-09176-9.
[32]	G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346.
[33]	B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035-A1038.
[34]	J.-P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules, C. R. Math. Acad. Sci. Paris, Ser. I, 338 (2004), 853-858. doi: 10.1016/j.crma.2004.03.017.
[35]	J.-P. Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., 58 (2004), 855-871. doi: 10.1016/j.na.2004.05.018.
[36]	T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., in press, 2014. doi: 10.1137/130909391.
[37]	R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1969.
[38]	T. Roubíček, Direct method for parabolic problems, Adv. Math. Sci. Appl., 10 (2000), 57-65.
[39]	T. Roubíček, Nonlinear Partial Differential Equations with Applications, second edition, Birkhäuser, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.
[40]	B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator, Proc. Amer. Math. Soc., 131 (2003), 3851-3859. doi: 10.1090/S0002-9939-03-07083-7.
[41]	A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators, Adv. Math. Sci. Appl., 18 (2008), 633-650.
[42]	A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47 (2013), 273-317. doi: 10.1007/s00526-012-0519-y.
[43]	A. Visintin, Scale-transformations and homogenization of maximal monotone relations, with applications, Asymptotic Analysis, 82 (2013), 233-270.
[44]	A. Visintin, Weak structural stability of pseudo-monotone equations, in press.
[45]	E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. II/B: Nonlinear Monotone Operators, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
[46]	E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. IV: Applications to Mathematical Physics, Springer, New York, 1988. doi: 10.1007/978-1-4612-4566-7.

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