Some remarks on the class of continuous (semi-)∈strictly quasiconvex functions

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Aguilar R.
Brioso X.
Calderón C.
Pando M.A.
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Springer Nature
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We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T  relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.
A major part of this work was accomplished during a research visit of the first author to the University of Antilles–Guyane (February 2005). The first author wishes to thank his hosts and the University of Antilles–Guyane for the hospitality. The research of the first author has been supported by the MEC grant (Project MTM2005-08572-C03-03, Spain), and that of the second author by the CONCYTEC (National Committee of Science and Technology) of Peru and by the CROUS (scholarship for graduate studies) of France. The authors wish to thank D. Aussel, N. Hadjisavvas and the two anonymous referees for careful reading.
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