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## Delta-convex mappings between Banach spaces and applications

Autorzy
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
autor
• 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).
• [3] A. D. Aleksandrov, Surfaces represented by the differences of convex functions, Dokl. Akad. Nauk SSSR (N. S.) 72 (1950), 613-616 (in Russian).
• [4] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57 (1976), 147-190.
• [5] M. G. Arsove, Functions representable as the difference of subharmonic functions, Trans. Amer. Math. Soc. 75 (1953), 327-365.
• [6] J. M. Borwein, Generic differentiability of order-bounded convex operators, J. Austral. Math. Soc. Ser. B 28 (1986), 22-29.
• [7] N. Bourbaki, Éleménts de Mathématique, Variétés différentielles et analytiques, Paris 1967, 1971.
• [8] H. Busemann and W. Feller, Krümmungseigenschaften konvexer Flächen, Acta Math. 66 (1936), 1-47.
• [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).
• [12] V. F. Demjanov and A. M. Rubinov, On quasi-differentiable mappings, Math. Operations-forsch. u. Stat. ser. Optimization 14 (1983), 3-21.
• [13] V. F. Demjanov and L. V. Vasiljev, Nondifferentiable Optimization, Springer-Verlag, New York 1985.
• [14] J. Diestel and J. J. Uhl, Jr, The Radon-Nikodym theorem for Banach spaces valued measures. Rocky Mountain J. Math. 6 (1976), 1-46.
• [15] N. Dunford and J. T. Schwartz, Linear Operators, 1. General theory, New York 1958.
• [16] M. Fabián and D. Preiss, On the Clarke's generalized Jacobian, Proceedings or the 14-th Winter school on abstract analysis, to appear in Supplemento ai Rendiconti del Circolo Mathematico di Palermo.
• [17] S. Fučik, Solvability of Nonlinear Equations and Boundary Value Problems, Prague 1980.
• [18] P. Hartman, On functions representable as a difference of convex functions, Pacific J. Math. 9 (1959), 707-713.
• [19] J. -B. Hiriart-Urruty, Generalized differentiability, duality and optimization for problems dealing with differences of convex functions, preprint.
• [20] Hoàng Tuy, Global minimization of a difference of two convex functions, Lecture Notes in Econom. and Math. Systems 226, Springer-Verlag 1984, 98-118.
• [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.
• [23] E. M. Landis, On functions representable as the difference of two convex functions, Dokl. Akad. Nauk SSSR (N. S.) 80 (1951), 9-11.
• [24] J. Lukeš, J. Malý and L. Zajíček, Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Mathematics 1189, Springer-Verlag, 1986.
• [25] F. Mignot, Contrôle dans les inéquations variatonelles elliptiques, J. Functional Analysis 22 (1976), 130-185.
• [26] A. Nijenhuis, Strong derivatives and inverse mappings, Amer. Math. Monthly 81 (1974), 969-980.
• [27] A. V. Pogorelov, Surfaces of hounded extrinsic curvature, 1956 (in Russian).
• [28] B. H. Pourciau, Analysis and optimization of Lipschitz continuous mappings. J. Optim. Theory Appl. 22 (1977), 311-351.
• [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.
• [30] D. Preiss and L. Zajíček, Stronger estimates of smallness of sets of Fréchet nondifferentiability of convex functions, Proceedings of the 11-th Winter school on Abstract Analysis, in Supplemento ai Rend. Circ. Mat. Palermo (2) no. 3 (1984), 219-223.
• [31] A. W. Roberts and D. E. Varberg, Convex functions, New York and London 1973.
• [32] A. Shapiro, On functions representable as a difference of two convex functions in inequality constrained optimization, Research report University of South Africa (1983).
• [33] M. M. Vajnberg, Variational methods for the study of nonlinear operators (in Russian), Moscow 1956, English translation: Holden-Day, San Francisci 1964.
• [34] L. Veselý, On the multiplicity points of monotone operators on separable Banach spaces. Comment. Math. Univ. Carolinae 27 (1986), 551-570.
• [35] L. Veselý, A short proof of a theorem on compositions of d.c. mappings, to appear in Proc. Amer. Math. Soc.
• [36] L. Veselý, On the multiplicity points of monotone operators on separable Banach spaces, II, Comment. Math. Univ. Carolinae 28 (1987), 295-299.
• [37] L. Zajíček, On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29 (1979), 340-348.
• [38] L. Zajíček, On differentiation of metric projections in finite dimensional Banach spaces, Czechoslovak Math. J. 33 (1983), 325-336.
• [39] V. A. Zalgaller, On the representation of a function of two variables as the difference of convex functions, Vestn. Leningrad. Univ. Ser. Mat. Mekh. 18 (1963), 44-45 (in Russian).
Języki publikacji
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Identyfikator YADDA
bwmeta1.element.zamlynska-f664bd7a-845c-42c8-b543-2b96d75206ba
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ISBN
83-01-09395-1
ISSN
0012-3862
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