Stability of generalized monotone maps with respect to their characterizations

Abstract

We show that the known types of generalized monotone maps are not stable with respect to their characterizations (i.e. the characterizations are not maintained if an arbitrary map of this type is disturbed by an element with sufficiently small norm) and introduce s-quasimonotone maps, which are stable with respect to their characterization. For gradient maps, s-quasimonotonicity is related to s-quasiconvexity (introduced by Phu in Optimization, 38, 1996) of the underlying function. A necessary and sufficient condition for a univariate polynomial to be s-quasimonotone is given. Furthermore, some stability properties of s-quasiconvex functions are presented.

Keywords:

• Generalized convexity
• Generalized monotonicity
• s-Quasiconvex function
• s-Quasimonotone map
• Stability

Keywords:

• Mathematics Subject Classifications 2000:
• Primary: 26A51
• 26B25
• Secondary: 52A41
• 90C26

Acknowledgments

The author gratefully acknowledges the many helpful suggestions of Professor Dr. Sc. Hoang Xuan Phu and Dr. Nguyen Ngoc Hai during the preparation of the article. The author also thanks the anonymous referees for the helpful suggestions. This article was partially done within the framework of the Associateship Scheme and Mathematics Research Fellowship of the the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The financial support from the Swedish International Development Cooperation Agency is acknowledged.