On Rockafellar ’ s conjecture on the maximality of the sum of two maximal monotone operators in general Banach spaces

By using two recently published results concerning monotone operators we prove the maximality of the sum of two maximal monotone operators in general Banach spaces under a weak regularity condition. This provides an affirmative answer to Rockafellar’s celebrated conjecture stated in 1970.

[1]  H. Attouch,et al.  SOMME PONCTUELLE D’OPÉRATEURS MAXIMAUX MONOTONES , 2009 .

[2]  B. Svaiter,et al.  Bronsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces , 2008, 0802.1895.

[3]  S. Simons.  From Hahn-Banach to monotonicity , 2008 .

[4]  Jonathan M. Borwein,et al.  Maximality of sums of two maximal monotone operators in general Banach space , 2007 .

[5]  Ernö Robert Csetnek,et al.  A new condition for maximal monotonicity via representative functions , 2007 .

[6]  Sorin-Mihai Grad,et al.  Weaker Constraint Qualifications in Maximal Monotonicity , 2007 .

[7]  Sorin-Mihai Grad,et al.  Maximal Monotonicity for the Precomposition with a Linear Operator , 2006, SIAM J. Optim..

[8]  J. Borwein,et al.  A Variation on the Banach-Dieudonne Theorem with Application to Maximal Monotone Operators , 2007 .

[9]  J. Borwein,et al.  A General Sum Theorem for Maximality of Monotone Operators , 2007 .

[10]  M. D. Voisei.  A Maximality Theorem for the Sum of Maximal Monotone Operators in Non-Re exive Banach Spaces , 2006 .

[11]  M. D. Voisei.  Calculus rules for maximal monotone operators in general Banach spaces , 2006 .

[12]  J. Borwein.  Maximal Monotonicity via Convex Analysis , 2006 .

[13]  C. Zalinescu,et al.  Some problems about the representation of monotone operators by convex functions , 2005 .

[14]  Jean-Paul Penot,et al.  The relevance of convex analysis for the study of monotonicity , 2004 .

[15]  Stephen Simons,et al.  Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem , 2005 .

[16]  Regina Sandra Burachik,et al.  MAXIMAL MONOTONICITY, CONJUGATION AND THE DUALITY PRODUCT , 2003, 0802.1654.

[17]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[18]  Teemu Pennanen,et al.  Dualization of Generalized Equations of Maximal Monotone Type , 1999, SIAM J. Optim..

[19]  Andrei Verona,et al.  Regular Maximal Monotone Operators , 1998 .

[20]  Liang Ju Chu.  On the sum of monotone operators , 1996 .

[21]  R. R. Phelps,et al.  Some properties of maximal monotone operators on nonreflexive Banach spaces , 1995 .

[22]  M. E. Verona,et al.  Remarks on subgradients and ɛ-subgradients , 1993 .

[23]  S. Fitzpatrick.  Representing monotone operators by convex functions , 1988 .

[24]  R. Rockafellar,et al.  On the maximal monotonicity of subdifferential mappings. , 1970 .

[25]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .