# Delta-convex mappings between Banach spaces and applications

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

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topL. 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

top- Miroslav Zelený, An example of a ${\mathcal{C}}^{1,1}$ function, which is not a d.c. function
- D. Pavlica, Morse-Sard theorem for delta-convex curves
- Jakub Duda, Curves with finite turn
- David Pavlica, A d.c. ${C}^{1}$ function need not be difference of convex ${C}^{1}$ functions
- Luděk Zajíček, Fréchet differentiability, strict differentiability and subdifferentiability
- Jakub Duda, On inverses of $\delta $-convex mappings
- Libor Veselý, Luděk Zajíček, On vector functions of bounded convexity
- , Report of Meeting
- Jan Rataj, Luděk Zajíček, Properties of distance functions on convex surfaces and applications
- Jakub Duda, Luděk Zajíček, Curves in Banach spaces which allow a ${C}^{1,\mathrm{BV}}$ parametrization or a parametrization with finite convexity

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