On Extensions of Kenderov's Single-Valuedness Result for Monotone Maps and Quasimonotone Maps

D. Aussel (1) , Y. García (2)

1 PROMES - Procédés, Matériaux et Energie Solaire
2 DynamE - Dynamiques européennes

One of the most famous single-valuedness results for set-valued maps is due to Kenderov [Fund. Math., LXXXVIII (1975), pp. 61--69] and states that a monotone set-valued operator is single-valued at any point where it is lower semicontinuous. This has been extended in Christensen and Kenderov [Math. Scand., 54 (1984), pp. 70--78] to monotone maps satisfying a so-called $*$-property. Our aim in this work is twofold: first, to prove that the $*$-property assumption can be weakened, and second, to emphasize that these classical single-valuedness results for monotone operators can be obtained, in very simple way, as direct consequences of counterpart results proved for quasi-monotone operators in terms of single-directionality.

Keywords: single-valuedness; quasi-monotone map; monotone map; Lipschitz-like property; single-directional property

Domains: Engineering Sciences [physics]

D. Aussel, Y. García. On Extensions of Kenderov's Single-Valuedness Result for Monotone Maps and Quasimonotone Maps. SIAM Journal on Optimization, 2014, 24 (2), pp.702-7013. ⟨10.1137/120880215⟩. ⟨hal-01176923⟩