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Set-Valued and Variational Analysis

, Volume 21, Issue 4, pp 603–616 | Cite as

Maximality of the Sum of a Maximally Monotone Linear Relation and a Maximally Monotone Operator

  • Jonathan M. Borwein
  • Liangjin YaoEmail author
Article

Abstract

The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A + B provided that A, B are maximally monotone and A is a linear relation, as soon as Rockafellar’s constraint qualification holds: \({\operatorname{dom}}\,A\cap{\operatorname{int}}\,{\operatorname{dom}}\,B\neq\varnothing\). Moreover, A + B is of type (FPV).

Keywords

Constraint qualification Convex set Fitzpatrick function Linear relation Maximally monotone operator Monotone operator Monotone operator of type (FPV) Multifunction Normal cone operator Rockafellar’s sum theorem Set-valued operator 

Mathematics Subject Classifications (2010)

Primary 47A06 47H05; Secondary 47B65 47N10 90C25 

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.CARMAUniversity of NewcastleNewcastleAustralia

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