Delta-convex mappings between Banach spaces and applications

L. L. Veselý; L. Zajíček

  • Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1989

Abstract

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We investigate delta-convex mappings between normed linear spaces. They provide a generalization of functions which are representable as a difference of two convex functions (labelled as 5-convex or d.c. functions) and are considered in many articles. We show that delta-convex mappings have many good differentiability properties of convex functions and the class of them is very stable. For example, the class of locally delta-convex mappings is closed under superpositions and (in some situations) under inverses. Some operators which occur naturally in the theory of integral and differential equations are shown to be delta-convex. As an application of our general results, we show that some "solving operators" of such equations are delta-convex and consequently have good differentiability properties. An implicit function theorem for quasi-differentiable functions is an another application.CONTENTS0. Introduction and notations...................................................51. Basic properties of delta-convex mappings.........................82. Delta-convex curves..........................................................153. Differentiability of delta-convex mappings.........................17   A. First derivative...............................................................17   B. Second derivative of mappings F : R n Y ...............234. Superpositions and inverse mappings..............................265. Inverse mappings in finite-dimensional case.....................316. Examples and applications................................................34   A. Three counterexamples.................................................34   B. Nemyckii and Hammerstein operators............................36   C. Weak solution of a differential equation.........................38   D. Quasidifferentiable functions and mappings..................417. Some open problems........................................................44References...........................................................................47

How to cite

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L. L. Veselý, and L. Zajíček. Delta-convex mappings between Banach spaces and applications. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1989. <http://eudml.org/doc/268419>.

@book{L1989,
abstract = {We investigate delta-convex mappings between normed linear spaces. They provide a generalization of functions which are representable as a difference of two convex functions (labelled as 5-convex or d.c. functions) and are considered in many articles. We show that delta-convex mappings have many good differentiability properties of convex functions and the class of them is very stable. For example, the class of locally delta-convex mappings is closed under superpositions and (in some situations) under inverses. Some operators which occur naturally in the theory of integral and differential equations are shown to be delta-convex. As an application of our general results, we show that some "solving operators" of such equations are delta-convex and consequently have good differentiability properties. An implicit function theorem for quasi-differentiable functions is an another application.CONTENTS0. Introduction and notations...................................................51. Basic properties of delta-convex mappings.........................82. Delta-convex curves..........................................................153. Differentiability of delta-convex mappings.........................17   A. First derivative...............................................................17   B. Second derivative of mappings $F: R^n → Y$...............234. Superpositions and inverse mappings..............................265. Inverse mappings in finite-dimensional case.....................316. Examples and applications................................................34   A. Three counterexamples.................................................34   B. Nemyckii and Hammerstein operators............................36   C. Weak solution of a differential equation.........................38   D. Quasidifferentiable functions and mappings..................417. Some open problems........................................................44References...........................................................................47},
author = {L. L. Veselý, L. Zajíček},
keywords = {delta-convex functions; Asplund spaces; differentiability for locally delta-convex mappings; composition theorem; inverse function theorem; Nemyckii and Hammerstein operators},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Delta-convex mappings between Banach spaces and applications},
url = {http://eudml.org/doc/268419},
year = {1989},
}

TY - BOOK
AU - L. L. Veselý
AU - L. Zajíček
TI - Delta-convex mappings between Banach spaces and applications
PY - 1989
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - We investigate delta-convex mappings between normed linear spaces. They provide a generalization of functions which are representable as a difference of two convex functions (labelled as 5-convex or d.c. functions) and are considered in many articles. We show that delta-convex mappings have many good differentiability properties of convex functions and the class of them is very stable. For example, the class of locally delta-convex mappings is closed under superpositions and (in some situations) under inverses. Some operators which occur naturally in the theory of integral and differential equations are shown to be delta-convex. As an application of our general results, we show that some "solving operators" of such equations are delta-convex and consequently have good differentiability properties. An implicit function theorem for quasi-differentiable functions is an another application.CONTENTS0. Introduction and notations...................................................51. Basic properties of delta-convex mappings.........................82. Delta-convex curves..........................................................153. Differentiability of delta-convex mappings.........................17   A. First derivative...............................................................17   B. Second derivative of mappings $F: R^n → Y$...............234. Superpositions and inverse mappings..............................265. Inverse mappings in finite-dimensional case.....................316. Examples and applications................................................34   A. Three counterexamples.................................................34   B. Nemyckii and Hammerstein operators............................36   C. Weak solution of a differential equation.........................38   D. Quasidifferentiable functions and mappings..................417. Some open problems........................................................44References...........................................................................47
LA - eng
KW - delta-convex functions; Asplund spaces; differentiability for locally delta-convex mappings; composition theorem; inverse function theorem; Nemyckii and Hammerstein operators
UR - http://eudml.org/doc/268419
ER -

Citations in EuDML Documents

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  1. Miroslav Zelený, An example of a 𝒞 1 , 1 function, which is not a d.c. function
  2. D. Pavlica, Morse-Sard theorem for delta-convex curves
  3. Jakub Duda, Curves with finite turn
  4. David Pavlica, A d.c. C 1 function need not be difference of convex C 1 functions
  5. Luděk Zajíček, Fréchet differentiability, strict differentiability and subdifferentiability
  6. Jakub Duda, On inverses of δ -convex mappings
  7. Libor Veselý, Luděk Zajíček, On vector functions of bounded convexity
  8. , Report of Meeting
  9. Jan Rataj, Luděk Zajíček, Properties of distance functions on convex surfaces and applications
  10. Jakub Duda, Luděk Zajíček, Curves in Banach spaces which allow a C 1 , BV parametrization or a parametrization with finite convexity

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