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Delta-convex mappings between Banach spaces and applications

Seria
Rozprawy Matematyczne tom/nr w serii: 289 wydano: 1989
Zawartość
Warianty tytułu
Abstrakty
EN
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.
EN

CONTENTS
0. Introduction and notations...................................................5
1. Basic properties of delta-convex mappings.........................8
2. Delta-convex curves..........................................................15
3. Differentiability of delta-convex mappings.........................17
   A. First derivative...............................................................17
   B. Second derivative of mappings $F: R^n → Y$...............23
4. Superpositions and inverse mappings..............................26
5. Inverse mappings in finite-dimensional case.....................31
6. 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..................41
7. Some open problems........................................................44
References...........................................................................47
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 289
Liczba stron
48
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCLXXXIX
Daty
wydano
1989
Twórcy
  • aculty of Mathematics and Physics, Charles University, Sokolovski 83, 186 00 Praha 8, Czechoslovakia
autor
  • Faculty of Mathematics and Physics, Charles University, Sokolovski 83, 186 00 Praha 8, Czechoslovakia
Bibliografia
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  • [2] A. D. Aleksandrov, On surfaces represented as the difference of convex functions, Izv. Akad. Nauk. Kaz. SSR 60, Ser. Math. Mekh. 3 (1949), 3-20 (in Russian).
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  • [9] H. Cartan, Calcul différentiel, Forms différentielles, Paris 1967.
  • [10] F. H. Clarke, On the inverse function theorem, Pacific J. Math. 67 (1976), 97-102.
  • [11] V. F. Demjanov and A. M. Rubinov, On quasi-differentiable functionals, Dokl. Akad. Nauk SSSR 250 (1980), 21-25 (in Russian).
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  • [17] S. Fučik, Solvability of Nonlinear Equations and Boundary Value Problems, Prague 1980.
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  • [21] Ch. O. Kiselman, Fonctions delta-convexes, delta-sousharmoniques et delta-plurisoushar-moniques, Lecture Notes in Mathematics 578, Springer-Verlag 1977, 93-107.
  • [22] K. Kuratowski, Topology, Vol. I (transl.), Academic Press, New York 1966.
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  • [29] D. Preiss and L. Zajíček, Fréchet differentiation of convex functions in a Banach space with a separable dual, Proc. Amer. Math. Soc. 91 (1984), 202-204.
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  • [31] A. W. Roberts and D. E. Varberg, Convex functions, New York and London 1973.
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Języki publikacji
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Uwagi
Identyfikator YADDA
bwmeta1.element.zamlynska-f664bd7a-845c-42c8-b543-2b96d75206ba
Identyfikatory
ISBN
83-01-09395-1
ISSN
0012-3862
Kolekcja
DML-PL
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