Abstract

We consider a family of enlargements of maximal monotone operators in a reflexive Banach space. Each enlargement, depending on a parameter ε≥0\varepsilon\ge 0, is a continuous point-to-set mapping E(ε,x)E(\varepsilon,x) whose graph contains the graph of the given operator TT. The enlargements are also continuous in ε\varepsilon, and they coincide with TT for ε=0\varepsilon=0. The family contains both a maximal and a minimal enlargement, denoted as TeT^e and TseT^{se} respectively. We address the following questions: a) which are the operators which are not enlarged by TeT^e, i.e., such that T(⋅)=Te(ε,⋅)T(\cdot)=T^e(\varepsilon,\cdot) for some ε>0\varepsilon>0? b) same as (a) but for TseT^{se} instead of TeT^e. c) Which operators are fully enlargeable by TeT^e, in the sense that for all xx and all ε>0\varepsilon>0 there exists δ>0\delta>0 such that all points whose distance to T(x)T(x) is less than δ\delta belong to Te(ε,x)T^e(\varepsilon,x)? We prove that the operators not enlarged by TeT^e are precisely the point-to-point affine operators with skew symmetric linear part; those not enlarged by TseT^{se} are the point-to-point and affine operators, and the operators fully enlarged by TeT^e are those operators TT whose Fitzpatrick function is continuous in its second argument at pairs belonging to the graph of TT.

Author information

Contact details are reproduced from the original publication and may be historical.

Regina Sandra Burachik

Programa de Engenharia de Sistemas e Computação, Coordenação de Programas de Pós-Graduação em Engenharia (COPPE), Universidade Federal de Rio de Janeiro (UFRJ), C.P. 68511, Rio de Janeiro, RJ, CEP 21945-970, Brazil

regi@cos.ufrj.br

Alfredo Noel Iusem

Inst. Matemática Pura e Aplicada, Estrada Doña Castorina 110, Rio de Janeiro, CEP 22460-320, Brazil

iusp@impa.br

Keywords

  • Maximal monotone operators
  • enlargements

Mathematics Subject Classification

46N10, 47H05

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Suggested citation

R. S. Burachik, A. N. Iusem. “On Non-Enlargeable and Fully Enlargeable Monotone Operators.” Journal of Convex Analysis 13 (2006), No. 3/4, 603–622. https://doi.org/10.68381/jca13047

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press.