Generalized Jacobian for functions with infinite dimensional range and domain

Z. Páles, Vera Zeidan

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon-Nikodým property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set.

Original languageEnglish
Pages (from-to)331-375
Number of pages45
JournalSet-Valued Analysis
Volume15
Issue number4
DOIs
Publication statusPublished - Dec 2007

Fingerprint

Generalized Jacobian
Radon
Derivatives
Range of data
Locally Lipschitz Function
Chain rule
Vectorization
Upper Semicontinuity
Mean value theorem
Extension Theorem
Infinite-dimensional Spaces
Dual space
Normed Space
Compactness
Restriction
Derivative
Theorem

Keywords

  • Chain rule
  • Characterization theorem
  • Continuous selection
  • Generalized Jacobian
  • Restriction rule
  • Sum rule

ASJC Scopus subject areas

  • Applied Mathematics
  • Analysis

Cite this

Generalized Jacobian for functions with infinite dimensional range and domain. / Páles, Z.; Zeidan, Vera.

In: Set-Valued Analysis, Vol. 15, No. 4, 12.2007, p. 331-375.

Research output: Contribution to journalArticle

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