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On generalized differential equations in Banach spaces

Seria
Rozprawy Matematyczne tom/nr w serii: 310 wydano: 1991
Zawartość
Warianty tytułu
Abstrakty
EN
CONTENTS
Introduction . . . . . . . . 5
I. Fundamental problems for generalized differential equations at nonsingular points
§1. Introduction . . . . . . . . 6
§2. Cauchy problem at nonsingular points for generalized differential equations of the first order . . . . . . . . 6
§3. Dependence of solution on parameters and initial conditions . . . . . . . . 8
II. Total solutions of generalized linear differential equations
§1. Introduction . . . . . . . . 11
§2. Form of solutions of generalized linear differential equations . . . . . . . . 11
§3. Stability of generalized linear differential equations . . . . . . . . 15
III. Fundamental problems for generalized differential equations at singular points
§1. Introduction . . . . . . . . 19
§2. Initial conditions at singular points and dependence of solutions upon initial conditions and parameters . . . . . . . . 19
§3. Form of solutions in a vicinity of a singular point . . . . . . . . 26
IV. Existence and form of solutions of generalized linear differential equations connected with geometrical properties of holomorphic mappings
§1. Introduction . . . . . . . . 29
§2. Holomorphic solutions of generalized differential equation connected with spiral-like mappings . . . . . . . . 31
§3. Existence and form of solutions of generalized differential equations which define close-to-starlike mappings . . . . . . . . 37
§4. Univalent subordination chains and solutions of a generalized equation of Löwner . . . . . . . . 39
V. The generalized form of the Frobenius theorem
§1. Introduction . . . . . . . . 44
§2. A necessary condition and a sufficient condition for existence and uniqueness . . . . . . . . 45
§3. The generalized Frobenius equation and its integrability conditions in Euclidean spaces . . . . . . . . 47
References . . . . . . . . 49
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 310
Liczba stron
50
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCX
Daty
wydano
1991
otrzymano
1989-12-28
poprawiono
1991-01-10
Twórcy
  • Institute of Mathematics, Technical University of Łódź, Al. Politechniki 11, 90-924 Łódź, Poland
Bibliografia
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  • [Ap2] L. N. Apostolova, The Poincaré lemma for flat RC-structures, in preparation.
  • [BS] J. Bochnak and J. Siciak, Analytic functions in topological vector spaces, Studia Math. 39 (1971), 77-112.
  • [Ca] H. Cartan, Calcul différentiel. Formes différentielles, Hermann, Paris 1967 (Russian transl.: Mir, Moscow 1971).
  • [DK] Yu. Daletskiĭ and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Nauka, Moscow 1970 (in Russian).
  • [DPS] H. Dębiński, T. Poreda and A. Szadkowska, On the stability of the generalized linear differential equations of the first order in Banach spaces, Demonstratio Math. 23 (4) (1990), 1-11.
  • [D] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York 1960 (Russian transl.: Mir, Moscow 1964).
  • [Di] S. Dineen, Complex Analysis in Locally Convex Spaces, North-Holland, Amsterdam 1981.
  • [G] K. R. Gurganus, Φ-like holomorphic functions in $ℂ^n$ and Banach spaces, Trans. Amer. Math. Soc. 205 (1975), 389-406.
  • [Ha] L. A. Harris, Schwarz's Lemma in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 64 (4) (1965), 1014-1017.
  • [HS] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, Berlin 1965.
  • [H] L. Hörmander, The Frobenius-Nirenberg theorem, Ark. Mat. 5 (1965), 425-432.
  • [KaP] J. Kalina and T. Poreda, The generalized form of the Frobenius theorem, submitted.
  • [Ka] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508-520.
  • [Ko] J. J. Kohn, Integration of complex vector fields, Bull. Amer. Math. Soc. 78 (1972), 1-11.
  • [KP] E. Kubicka and T. Poreda, On the parametric representation of starlike maps of the unit ball in $ℂ^n$ into $ℂ^n$, Demonstratio Math. 21 (2) (1988), 345-355.
  • [L] G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Academic Press, New York 1972.
  • [Lu] G. Lumer, Semi-inner product spaces, Trans. Amer. Math. Soc. 100 (1961), 29-33.
  • [LP] G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679-698.
  • [M] K. Maurin, Analysis, part I, PWN, Warszawa 1976.
  • [Mu] J. Mujica, Complex Analysis in Banach Spaces, North-Holland, Amsterdam 1986.
  • [N] L. Nachbin, Topology on Spaces of Holomorphic Mappings, Ergeb. Math. Grenzgeb. 47, Springer, Berlin 1969.
  • [Pf] J. A. Pfaltzgraff, Subordination chains and univalence of holomorphic mappings on $ℂ^n$, Math. Ann. 210 (1974), 55-68.
  • [PS] J. A. Pfaltzgraff and T. J. Suffridge, Close-to-starlike holomorphic functions of several variables, Pacific J. Math. 57 (1975), 271-279.
  • [Pm] Ch. Pommerenke, Über die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965), 159-173.
  • [Po1] T. Poreda, Generalized differential equations for maps of Banach spaces, Comment. Math. 30 (1) (1990), 141-146.
  • [Po2] T. Poreda, On the geometrical properties of starlike maps of Banach spaces, submitted.
  • [Po3] T. Poreda, On some topological properties of the class of normalized and starlike maps of the unit polydisc in $ℂ^n$, Acta. Univ. Lodz. Folia Math. 3 (1989), 87-93.
  • [Po4] T. Poreda, On the univalent subordination chains of holomorphic mappings in Banach spaces, Comment. Math. 28 (2) (1989), 295-304.
  • [Po5] T. Poreda, On the univalent holomorphic maps of the unit polydisc in $ℂ^n$ which have the parametric representation I - the geometrical properties, Ann. Univ. Mariae Curie-Skłodowska Sect. A 41 (1987), 105-113.
  • [Po6] T. Poreda, On the univalent holomorphic maps of the unit polydisc in $ℂ^n$ which have the parametric representation II - the necessary conditions and the sufficient conditions, ibid., 114-121.
  • [PoS] T. Poreda and A. Szadkowska, On the holomorphic solutions of certain differential equations of first order for the mappings of the unit ball in $ℂ^n$ into $ℂ^n$, Demonstratio Math. 22 (4) (1989), 983-996.
  • [Se] Z. Semadeni, Banach Spaces of Continuous Functions, PWN, Warszawa 1971.
  • [Su] T. J. Suffridge, Starlike and convex maps in Banach spaces, Pacific J. Math. 46 (1973), 575-589.
  • [Su1] T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, in: Complex Analysis, Kentucky 1976, Lecture Notes in Math. 599, Springer 1977, 146-159.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: Primary 35F99, 34G99.
Identyfikator YADDA
bwmeta1.element.zamlynska-122d45fb-f825-4f05-a3c3-af1874790d96
Identyfikatory
ISBN
83-85116-11-7
ISSN
0012-3862
Kolekcja
DML-PL
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