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August  2017, 10(4): 909-918. doi: 10.3934/dcdss.2017046

On the variational representation of monotone operators

Dipartimento di Matematica, dell'Università degli Studi di Trento, via Sommarive 14,38050 Povo di Trento, Italy

Received  May 2016 Revised  May 2016 Published  April 2017

Let
$V$
be a Banach space,
$z'\in V'$
, and
$\alpha: V\to {\mathcal P}(V')$
be a maximal monotone operator. A large number of phenomena can be modelled by inclusions of the form
$\alpha(u) \ni z'$
, or by the associated flow
$D_tu + \alpha(u) \ni z'$
. Fitzpatrick proved that there exists a lower semicontinuous, convex representative function
$f_\alpha: V \!\times\! V'\to \mathbb{R}\cup \{+\infty\}$
such that
$f_\alpha(v,v') \ge \langle v',v\rangle\quad\;\forall (v,v'), \qquad\quadf_\alpha(v,v') = \langle v',v\rangle\;\;\Leftrightarrow\;\;\; v'\in \alpha(v).$
This provides a variational formulation for the above inclusions. Here we use this approach to prove two results of existence of a solution, without using the classical theory of maximal monotone operators. This is based on a minimax theorem, and on the duality theory of convex optimization.
Citation: Augusto VisintiN. On the variational representation of monotone operators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046
References:
[1]

G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), 1161-1170. Google Scholar

[2]

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

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. doi: 10.1007/978-94-010-1537-0. Google Scholar

[4]

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. Google Scholar

[5]

H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.Google Scholar

[6]

H. Brezis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. Ⅰ. Le cas indépendant du temps and Ⅱ. Le cas dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971–974, and ibid. 1197–1198.Google Scholar

[7]

F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Sympos. Pure Math. , Volume ⅩⅧ, Part Ⅱ, A. M. S. , Providence 1976. doi: 10.1090/S0002-9904-1967-11820-2. Google Scholar

[8]

M. BuligaG. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws, J. Convex Anal., 15 (2008), 87-104. Google Scholar

[9]

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. Google Scholar

[10]

R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), 2379-2383. Google Scholar

[11]

I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles, Dunod GauthierVillars, Paris, 1974.Google Scholar

[12]

K. Fan, A minimax inequality and applications, Inequalities Ⅲ, Proc. Third Sympos. , Univ. California, Los Angeles 1969, pp. 103-113, Academic Press, New York 1972.Google Scholar

[13]

W. Fenchel, Convex Cones, Sets, and Functions, Princeton Univ. , 1953.Google Scholar

[14]

S. Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization, Canberra, 1988, Proc. Centre Math. Anal. Austral. Nat. Univ., 20 (1988), 59-65. Google Scholar

[15]

N. Ghoussoub, Selfdual Partial Differential Systems and their Variational Principles, Springer, 2008.Google Scholar

[16]

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. Google Scholar

[17]

N. Ghoussoub and L. Tzou, A variational principle for gradient flows, Math. Ann., 330 (2004), 519-549. doi: 10.1007/s00208-004-0558-6. Google Scholar

[18]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.Google Scholar

[19]

J. -E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal., 2 (2001), 243-247. Google Scholar

[20]

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. Google Scholar

[21]

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. Google Scholar

[22]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2. Google Scholar

[23]

B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, (1976), 282. Google Scholar

[24]

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. Google Scholar

[25]

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. Google Scholar

[26]

H. Rios, Étude de la question d'existence pour certains problèmes d'évolution par minimisa-tion d'une fonctionnelle convexe, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A83-A86. Google Scholar

[27]

T. RocheR. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., 52 (2014), 1071-1107. doi: 10.1137/130909391. Google Scholar

[28] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1969.
[29]

T. Roubíček, Direct method for parabolic problems, Adv. Math. Sci. Appl., 10 (2000), 57-65. Google Scholar

[30]

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. Google Scholar

[31]

A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators, Adv. Math. Sci. Appl., 18 (2008), 633-650. Google Scholar

[32]

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. Google Scholar

[33]

A. Visintin, An extension of the Fitzpatrick theory, Commun. Pure Appl. Anal., 13 (2014), 2039-2058. doi: 10.3934/cpaa.2014.13.2039. Google Scholar

[34]

A. Visintin, On Fitzpatrick's theory and stability of flows, Rend. Lincei Mat. Appl., 27 (2016), 151-180. doi: 10.4171/RLM/729. Google Scholar

[35]

A. Visintin, Structural compactness and stability of pseudo-monotone flows, forthcoming.Google Scholar

show all references

References:
[1]

G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), 1161-1170. Google Scholar

[2]

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

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. doi: 10.1007/978-94-010-1537-0. Google Scholar

[4]

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. Google Scholar

[5]

H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.Google Scholar

[6]

H. Brezis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. Ⅰ. Le cas indépendant du temps and Ⅱ. Le cas dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971–974, and ibid. 1197–1198.Google Scholar

[7]

F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Sympos. Pure Math. , Volume ⅩⅧ, Part Ⅱ, A. M. S. , Providence 1976. doi: 10.1090/S0002-9904-1967-11820-2. Google Scholar

[8]

M. BuligaG. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws, J. Convex Anal., 15 (2008), 87-104. Google Scholar

[9]

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. Google Scholar

[10]

R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), 2379-2383. Google Scholar

[11]

I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnelles, Dunod GauthierVillars, Paris, 1974.Google Scholar

[12]

K. Fan, A minimax inequality and applications, Inequalities Ⅲ, Proc. Third Sympos. , Univ. California, Los Angeles 1969, pp. 103-113, Academic Press, New York 1972.Google Scholar

[13]

W. Fenchel, Convex Cones, Sets, and Functions, Princeton Univ. , 1953.Google Scholar

[14]

S. Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization, Canberra, 1988, Proc. Centre Math. Anal. Austral. Nat. Univ., 20 (1988), 59-65. Google Scholar

[15]

N. Ghoussoub, Selfdual Partial Differential Systems and their Variational Principles, Springer, 2008.Google Scholar

[16]

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. Google Scholar

[17]

N. Ghoussoub and L. Tzou, A variational principle for gradient flows, Math. Ann., 330 (2004), 519-549. doi: 10.1007/s00208-004-0558-6. Google Scholar

[18]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.Google Scholar

[19]

J. -E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal., 2 (2001), 243-247. Google Scholar

[20]

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. Google Scholar

[21]

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. Google Scholar

[22]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2. Google Scholar

[23]

B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, (1976), 282. Google Scholar

[24]

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. Google Scholar

[25]

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. Google Scholar

[26]

H. Rios, Étude de la question d'existence pour certains problèmes d'évolution par minimisa-tion d'une fonctionnelle convexe, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A83-A86. Google Scholar

[27]

T. RocheR. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., 52 (2014), 1071-1107. doi: 10.1137/130909391. Google Scholar

[28] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1969.
[29]

T. Roubíček, Direct method for parabolic problems, Adv. Math. Sci. Appl., 10 (2000), 57-65. Google Scholar

[30]

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. Google Scholar

[31]

A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators, Adv. Math. Sci. Appl., 18 (2008), 633-650. Google Scholar

[32]

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. Google Scholar

[33]

A. Visintin, An extension of the Fitzpatrick theory, Commun. Pure Appl. Anal., 13 (2014), 2039-2058. doi: 10.3934/cpaa.2014.13.2039. Google Scholar

[34]

A. Visintin, On Fitzpatrick's theory and stability of flows, Rend. Lincei Mat. Appl., 27 (2016), 151-180. doi: 10.4171/RLM/729. Google Scholar

[35]

A. Visintin, Structural compactness and stability of pseudo-monotone flows, forthcoming.Google Scholar

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