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

Abstract : 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.
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Journal articles

https://hal-univ-perp.archives-ouvertes.fr/hal-01176923
Contributor : Olivier Savoyat <>
Submitted on : Thursday, July 16, 2015 - 11:29:11 AM
Last modification on : Thursday, March 15, 2018 - 1:27:11 AM

### Citation

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

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