On Characterizations of Relatively P– and P0– Properties in Nonsmooth Functions

Authors

Mohamed A. Tawhid, Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, V2C 0C8, Canada\\ Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Moharam Bey 21511, Alexandria, EgyptFollow

Author Country (or Countries)

Egypt

Abstract

For H-differentiable function f from a closed rectangle Q in Rn into Rn, a result of Song, Gowda and Ravindran [On Characterizations of P- and P0-Properties in Nonsmooth Functions. Mathematics of Operations Research. 25: 400-408 (2000)] asserts that f is a P(P0)− function on Q if the HQ-differential TQ(x) at each x ∈ Q consisting of P(P0)− matrices. In this paper, we introduce the concepts of relatively P(P0)− properties in order to extend these results to nonsmooth functions when the underlying functions are H-differentiable.We give characterizations of relatively P(P0)− of vector nonsmooth functions. Also, our results give characterizations of relatively P(P0)− when the underlying functions are C1-functions, semismooth-functions, and for locally Lipschitzian functions. Moreover, we show useful applications of our results by giving illustrations to generalized complementarity problems.

Recommended Citation

A. Tawhid, Mohamed (2015) "On Characterizations of Relatively P– and P0– Properties in Nonsmooth Functions," Applied Mathematics & Information Sciences: Vol. 09: Iss. 6, Article 10.
Available at: https://digitalcommons.aaru.edu.jo/amis/vol09/iss6/10