On some concepts of generalized differentials

Benedetto Piccoli; Ewa Girejko

doi:10.1007/s11228-006-0026-4

On some concepts of generalized differentials

Benedetto Piccoli, Ewa Girejko

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In this paper, we study some concepts of generalized differentials for set-valued maps and introduce some new ones. In particular we first focus on the concept of Generalized Differential Quotients, briefly GDQs. It is shown that minimal GDQs are unique for scalar single-valued functions, then GDQs are compared with contingent and Dini derivatives, finally some other results characterizing GDQs are given. A new definition of generalized differentiation theory is presented, namely weak GDQs that are a modification of GDQs. We clarify the relationships with other concepts of generalized differentiability: Clarke generalized Jacobians, path-integral generalized differentials and Warga derivate containers. Finally, some applications of GDQs end the paper.

Original language	English (US)
Pages (from-to)	163-183
Number of pages	21
Journal	Set-Valued Analysis
Volume	15
Issue number	2
DOIs	https://doi.org/10.1007/s11228-006-0026-4
State	Published - Jun 2007
Externally published	Yes

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

Keywords

CCA set-valued maps
Generalized differential quotients
Minimal generalized differential quotients

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10.1007/s11228-006-0026-4

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title = "On some concepts of generalized differentials",

abstract = "In this paper, we study some concepts of generalized differentials for set-valued maps and introduce some new ones. In particular we first focus on the concept of Generalized Differential Quotients, briefly GDQs. It is shown that minimal GDQs are unique for scalar single-valued functions, then GDQs are compared with contingent and Dini derivatives, finally some other results characterizing GDQs are given. A new definition of generalized differentiation theory is presented, namely weak GDQs that are a modification of GDQs. We clarify the relationships with other concepts of generalized differentiability: Clarke generalized Jacobians, path-integral generalized differentials and Warga derivate containers. Finally, some applications of GDQs end the paper.",

keywords = "CCA set-valued maps, Generalized differential quotients, Minimal generalized differential quotients",

author = "Benedetto Piccoli and Ewa Girejko",

note = "Funding Information: Acknowledgements This work was partially supported by a European Community Marie Curie Fellowship and in the framework of the CTS and partially by Bialystok Technical University under the grant W/IMF/1/04.",

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AU - Piccoli, Benedetto

AU - Girejko, Ewa

N1 - Funding Information: Acknowledgements This work was partially supported by a European Community Marie Curie Fellowship and in the framework of the CTS and partially by Bialystok Technical University under the grant W/IMF/1/04.

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N2 - In this paper, we study some concepts of generalized differentials for set-valued maps and introduce some new ones. In particular we first focus on the concept of Generalized Differential Quotients, briefly GDQs. It is shown that minimal GDQs are unique for scalar single-valued functions, then GDQs are compared with contingent and Dini derivatives, finally some other results characterizing GDQs are given. A new definition of generalized differentiation theory is presented, namely weak GDQs that are a modification of GDQs. We clarify the relationships with other concepts of generalized differentiability: Clarke generalized Jacobians, path-integral generalized differentials and Warga derivate containers. Finally, some applications of GDQs end the paper.

AB - In this paper, we study some concepts of generalized differentials for set-valued maps and introduce some new ones. In particular we first focus on the concept of Generalized Differential Quotients, briefly GDQs. It is shown that minimal GDQs are unique for scalar single-valued functions, then GDQs are compared with contingent and Dini derivatives, finally some other results characterizing GDQs are given. A new definition of generalized differentiation theory is presented, namely weak GDQs that are a modification of GDQs. We clarify the relationships with other concepts of generalized differentiability: Clarke generalized Jacobians, path-integral generalized differentials and Warga derivate containers. Finally, some applications of GDQs end the paper.

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On some concepts of generalized differentials

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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