Journal of Convex Analysis 13 (2006), No. 3, 721--738
Copyright Heldermann Verlag 2006
Extended Sums and Extended Compositions of Monotone Operators
Yboon García
Université des Antilles et de la Guyane, Campus de Fouillole, 97159 Pointe-à-Pitre, Guadeloupe, France
yboon.garcia@univ-ag.fr
Marc Lassonde
Université des Antilles et de la Guyane, Campus de Fouillole, 97159 Pointe-à-Pitre, Guadeloupe, France
marc.lassonde@univ-ag.fr
Julian P. Revalski
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street Block 8, 1113 Sofia, Bulgaria
revalski@math.bas.bg
We study extended notions for sums of monotone operators and for precompositions of monotone operators with continuous linear mappings. First, we establish some new properties related to the notion of extended sum recently proposed by Revalski and Théra, among them the monotonicity of this sum provided the operators involved are maximal monotone. Then, we show that the equivalence which exists between usual sums and compositions remains valid also for the extended operations. This allows us to obtain some known and new results for extended compositions via the corresponding properties of extended sums.
Keywords: Monotone operators, extended sums of monotone operators, compositions, enlargements, subdifferentials.
MSC: 47H05, 52A41, 26B25
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