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 ...............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
@book{bwmeta1.element.zamlynska-f664bd7a-845c-42c8-b543-2b96d75206ba,
author = {L. L. Vesel\'y and L. Zaj\'\i \v cek},
title = {Delta-convex mappings between Banach spaces and applications},
series = {GDML\_Books},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
address = {Warszawa},
year = {1989},
zbl = {0685.46027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.zamlynska-f664bd7a-845c-42c8-b543-2b96d75206ba}
}
L. L. Veselý; L. Zajíček. Delta-convex mappings between Banach spaces and applications. GDML_Books (1989), http://gdmltest.u-ga.fr/item/bwmeta1.element.zamlynska-f664bd7a-845c-42c8-b543-2b96d75206ba/