### Abstract

The extension to infinite dimensional domains of Clarke's generalized Jacobian is the focus of this paper. First, a generalization of a Fabian-Preiss theorem to the infinite dimensional setting is obtained. As a consequence, a new formula relating the Clarke's generalized Jacobians corresponding to finite dimensional spaces K, L with K ⊆ L is established. Furthermore, in the infinite dimensional case, basic properties pertaining the generalized Jacobian are developed and then an identification of this set-valued map is produced. Applications of these results in the form of chain rules including sum and product rules, and a computational formula for continuous selections are derived.

Original language | English |
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Pages (from-to) | 55-75 |

Number of pages | 21 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 344 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 1 2008 |

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### Keywords

- Chain rule
- Characterization of the generalized Jacobian
- Formula for a continuous selection
- Generalized Jacobian
- Sum rule

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Infinite dimensional generalized Jacobian : Properties and calculus rules.** / Páles, Z.; Zeidan, Vera.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 344, no. 1, pp. 55-75. https://doi.org/10.1016/j.jmaa.2008.02.044

}

TY - JOUR

T1 - Infinite dimensional generalized Jacobian

T2 - Properties and calculus rules

AU - Páles, Z.

AU - Zeidan, Vera

PY - 2008/8/1

Y1 - 2008/8/1

N2 - The extension to infinite dimensional domains of Clarke's generalized Jacobian is the focus of this paper. First, a generalization of a Fabian-Preiss theorem to the infinite dimensional setting is obtained. As a consequence, a new formula relating the Clarke's generalized Jacobians corresponding to finite dimensional spaces K, L with K ⊆ L is established. Furthermore, in the infinite dimensional case, basic properties pertaining the generalized Jacobian are developed and then an identification of this set-valued map is produced. Applications of these results in the form of chain rules including sum and product rules, and a computational formula for continuous selections are derived.

AB - The extension to infinite dimensional domains of Clarke's generalized Jacobian is the focus of this paper. First, a generalization of a Fabian-Preiss theorem to the infinite dimensional setting is obtained. As a consequence, a new formula relating the Clarke's generalized Jacobians corresponding to finite dimensional spaces K, L with K ⊆ L is established. Furthermore, in the infinite dimensional case, basic properties pertaining the generalized Jacobian are developed and then an identification of this set-valued map is produced. Applications of these results in the form of chain rules including sum and product rules, and a computational formula for continuous selections are derived.

KW - Chain rule

KW - Characterization of the generalized Jacobian

KW - Formula for a continuous selection

KW - Generalized Jacobian

KW - Sum rule

UR - http://www.scopus.com/inward/record.url?scp=43049102695&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43049102695&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2008.02.044

DO - 10.1016/j.jmaa.2008.02.044

M3 - Article

AN - SCOPUS:43049102695

VL - 344

SP - 55

EP - 75

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -