Journal of Convex Analysis 20 (2013), No. 1, 143--155
Copyright Heldermann Verlag 2013
Monotone Operators and "Bigger Conjugate" Functions
Heinz H. Bauschke
Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada
heinz.bauschke@ubc.ca
Jonathan M. Borwein
CARMA, University of Newcastle, Newcastle, NSW 2308, Australia
jonathan.borwein@newcastle.edu.au
Xianfu Wang
Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada
shawn.wang@ubc.ca
Liangjin Yao
Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada
ljinyao@interchange.ubc.ca
We study a question posed by Stephen Simons in his 2008 monograph involving "bigger conjugate" (BC) functions and the partial infimal convolution. As Simons demonstrated in his monograph, these function have been crucial to the understanding and advancement of the state-of-the-art of harder problems in monotone operator theory, especially the sum problem.
In this paper, we provide some tools for further analysis of BC-functions which allow us to answer Simons' problem in the negative. We are also able to refute a similar but much harder conjecture which would have generalized a classical result of Brézis, Crandall and Pazy. Our work also reinforces the importance of understanding unbounded skew linear relations to construct monotone operators with unexpected properties.
Keywords: Adjoint, BC-function, Fenchel conjugate, Fitzpatrick function, linear relation, maximally monotone operator, monotone operator, multifunction, normal cone operator, partial infimal convolution.
MSC: 47A06, 47H05; 47B65, 47N10, 90C25
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