The concept of adjusted sublevel set for a quasiconvex function was introduced by Aussel and Hadjisavvas who proved the local existence of a norm-to-weak*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$&lt;^&gt;*$$\end{document} upper semicontinuous base-valued submap of the normal operator associated with the adjusted sublevel set. When the space is finite-dimensional, a globally defined upper semicontinuous base-valued submap is obtained by taking the intersection of the unit sphere, which is compact, with the normal operator, which is closed. Unfortunately, this technique does not work in the infinite-dimensional case. We propose a partition of unity technique to overcome this problem in Banach spaces. An application is given to a quasiconvex quasioptimization problem through the use of a new existence result for generalized quasivariational inequalities which is based on the Schauder fixed point theorem.

### A Continuity Result for the Adjusted Normal Cone Operator

#### Abstract

The concept of adjusted sublevel set for a quasiconvex function was introduced by Aussel and Hadjisavvas who proved the local existence of a norm-to-weak*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<^>*$$\end{document} upper semicontinuous base-valued submap of the normal operator associated with the adjusted sublevel set. When the space is finite-dimensional, a globally defined upper semicontinuous base-valued submap is obtained by taking the intersection of the unit sphere, which is compact, with the normal operator, which is closed. Unfortunately, this technique does not work in the infinite-dimensional case. We propose a partition of unity technique to overcome this problem in Banach spaces. An application is given to a quasiconvex quasioptimization problem through the use of a new existence result for generalized quasivariational inequalities which is based on the Schauder fixed point theorem.
##### Scheda breve Scheda completa Scheda completa (DC)
2024
File in questo prodotto:
Non ci sono file associati a questo prodotto.
##### Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/226919
• ND
• 1
• 0