Delta-convex mappings between Banach spaces and applications

L. Veselý, L.; Zajíček, L.

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Tytuł książki

Delta-convex mappings between Banach spaces and applications

Autorzy

L. L. Veselý L. Zajíček

Seria

Rozprawy Matematyczne tom/nr w serii: 289 wydano: 1989

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Abstrakty

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.

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

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Seria

Rozprawy Matematyczne tom/nr w serii: 289

Liczba stron

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCLXXXIX

Daty

wydano

1989

Twórcy

autor

L. L. Veselý

aculty of Mathematics and Physics, Charles University, Sokolovski 83, 186 00 Praha 8, Czechoslovakia

autor

L. Zajíček

Faculty of Mathematics and Physics, Charles University, Sokolovski 83, 186 00 Praha 8, Czechoslovakia

Bibliografia

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[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).

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83-01-09395-1

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0012-3862

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