Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Cover of the book
Tytuł książki

Orlicz type category theorems for functional and differential equations

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 206 wydano: 1983
Zawartość
Warianty tytułu
Abstrakty
EN

CONTENTS
Introduction.....................................................................................................................................5
  1. General results.........................................................................................................................7
  1.1. Residual sets.........................................................................................................................7
  1.2. Generic properties of abstract functional equations..............................................................8
II. Differential equations................................................................................................................13
  2.1. Continuous differential equations without existence............................................................13
  2.2 Existence, uniqueness and continuous dependence............................................................18
  2.3. Successive approximations..................................................................................................21
  2.4. Remarks..............................................................................................................................26
III. Non-expansive mappings in Banach spaces............................................................................28
  3.1. Generic properties..............................................................................................................28
  3.2. The density result...............................................................................................................30
  3.3. Supplementary remarks......................................................................................................32
IV. Asymptotic equilibria for accretive operators...........................................................................33
  4.1. Notation and preliminaries...................................................................................................33
  4.2. Lemmas...............................................................................................................................35
  4.3. Category theorem................................................................................................................37
  4.4 Application to the fixed-point theory......................................................................................38
  4.5 Further results......................................................................................................................41
V. Hyperbolic equations................................................................................................................43
  5.1. Notation...............................................................................................................................43
  5.2. Generic property of existence, uniqueness and continuous dependence...........................43
  5.3. Generic property of the convergence of successive approximations...................................45
  5.4. A density theorem................................................................................................................47
  5.5. Remarks..............................................................................................................................48
VI. Generic asymptotic stability.....................................................................................................50
  6.1. Stability and asymptotic stability of stationary equations.....................................................50
  6.2. Stability and asymptotic stability of non-stationary equations..............................................54
  6.3. Stability by the Lyapunov function method..........................................................................55
  6.4. Generic stability...................................................................................................................57
VII. Functional integral equations.................................................................................................58
  7.1. Notation and auxiliary lemmas.............................................................................................58
  7.2. Category theorems..............................................................................................................61
  7.3. Some generalizations..........................................................................................................63
VIII. Functional differential equations............................................................................................64
  8.1. Notation and preliminaries...................................................................................................64
  8.2. Existence of unlimited solutions...........................................................................................65
  8.3. Continuous dependence.....................................................................................................67
  8.4. Existence and uniqueness as a generic property................................................................72
  8.5. Convergence of successive approximations as a generic property.....................................75
References..................................................................................................................................79
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 206
Liczba stron
81
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCVI
Daty
wydano
1983
Twórcy
autor
Bibliografia
  • [1] A. Alexiewicz and W. Orlicz, Some remarks on the existence and uniqueness of solutions of hyperbolic equation $z_{xy} = f(x, y, z, z_x, z_y)$, Studia Math. 25 (1956), pp. 201-215.
  • [2] B. Baillon, Un théorèm de type ergodique pour les contractions non linèaires dans un espace de Hilbert, C. R. Acad. Sc. Paris, 280 (1975), Série A. pp. 1511-1514.
  • [3] V. Barbu, Nonlinear Semigroups and Differential Equations, Ed. Academiei, Bucaresti 1976.
  • [4] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA 54 (1965), pp. 1041-1044.
  • [5] A. Cellina, On the nonexistence of solutions of differential equations in nonreflexive spaces, Bull. Amer. Math. Soc. 78 (1972), pp. 1069-1072.
  • [6] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Company, Boston 1965.
  • [7] T. Costello, Generic properties of differential equations, SIAM J. Math. Anal. 4 (1973), pp. 245-249.
  • [8] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, Mc Graw-Hill, New York 1955.
  • [9] M. M. Day, On the basic problem in normed spaces, Proc. Math. Soc. 13 (1962), pp. 655-658.
  • [10] F. S. De Blasi and J. Myjak, Sur la convergence des approximations successives pour les contractions non linèaires dans un espace de Banach, C. R. Acad. Sc. Paris, 283 (1976), Série A, pp. 185-188.
  • [11] F. S. De Blasi and J. Myjak, Generic properties of hyperbolic partial differential equations, J. London Math. Soc. 15 (2) (1977), pp. 113-118.
  • [12] F. S. De Blasi and J. Myjak, Two density properties of ordinary differential equations, Atti Acc. Naz. Lincei, Rend. Cl. Sci. Mat. Nat. Ser. VIII, Vol. LXI 1976, pp. 387-391.
  • [13] F. S. De Blasi and J. Myjak, On generic asymptotic stability of differential equations in Banach spaces, Proc. Amer. Math. Soc. 65 (1977), pp. 45-51.
  • [14] F. S. De Blasi and J. Myjak, Generic properties of differential equations in Banach spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), pp. 395-400.
  • [15] F. S. De Blasi and J. Myjak, The generic property of existence of solutions for a class of multivalued differential equations in Hilbert spaces, Funkcialaj Ekvacioj, 21 (1978), pp. 271-278.
  • [16] F. S. De Blasi and J. Myjak, Some generic properties of functional differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), pp. 437-451.
  • [17] F. S. De Blasi and J. Myjak, Generic properties of contraction semigroups and fixed points of nonexpansive operators, Proc. Amer. Math. Soc. 77 (1979), pp. 341-347.
  • [18] F. S. De Blasi and J. Myjak, Generic flows generated by continuous vector fields in Banach spaces, Advances in Math, (to appear).
  • [19] F. S. De Blasi and J. Myjak, and M. Kwapisz, Generic properties of functional equations, J. Nonlinear Analysis, Theory Methods and Appl. 2 (1977), pp. 1-11.
  • [20] K. Deimling, Zeros of accretive operators, manuscripta math. 13 (1974), pp. 365-374.
  • [21] J. Dieudonné, Deux example singuliers d'équations différentielles, Acta Sci. Math. B. (Szeged) 12 (1950), pp. 38-40.
  • [22] J. Dugundji, Topology, Allyn and Baycon, Boston 1966.
  • [23] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), pp. 353-367.
  • [24] J. Gatica and W. Kirk, Fixed point theorems for Lipschitzian pseudocontractive mappings, Proc. Amer. Math. Soc. 36 (1972), pp. 111-115.
  • [25] D. Gödhe, Zum Prinzip der Kontraktiven Abbildung, Math. Nach. 30 (1966), pp. 251-258.
  • [26] P. Hartman and A. Winter, On hyperbolic partial differential equations, Amer. J. Math. 74 (1952), pp. 834-864.
  • [27] R. C. Haworth and R. A. McCoy, Baire spaces, Dissertationes Math. 141, Warszawa 1977.
  • [28] T. Kato, Nonlinear semigroups and evolutions equations, J. Math. Soc. Japan 19 (1967), pp. 508-520.
  • [29] W. A. Kirk, A fixed point theory for mappings which do not increace distance, Amer. Math. Monthly 72 (1965), pp. 1004-1006.
  • [30] W. A. Kirk, Fixed point theorems for nonlinear nonexpansive and generalized contraction mappings, Pacific J. Math. 38 (1971), pp. 89-94.
  • [31] M. Kisielewicz, The Orlicz type theorem for differential integral equations with a lagging argument, Ann. Polon. Math. 31 (1975), pp. 67-71.
  • [32] M. Kisielewicz, Description of a class of differential equations with set-valued solutions, Atti, Acc. Naz. Lincei, Rend. Cl. Sci. Math. Nat. 58 (1975), pp. 338-341.
  • [33] M. Kwapisz, On the existence and uniqueness of solutions of a certain integral-functional equations, Ann. Polon. Math. 31 (1975), pp. 23-41.
  • [34] M. Kwapisz and J. Turo, Some integral functional equations, Funkcialaj Ekvacioj 18 (1975), pp. 107-162.
  • [35] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Academic Press, New York 1969.
  • [36] A. Lasota and J. Yorke, The generic property of existence of solutions of differential equations in Banach space, J. Differential Equations, 13 (1973), pp. 1-12.
  • [37] R. H. Martin, Jr., Differential equations on closed subsets of a Banach space, Trans, Amer. Math. Soc. 179 (1973), pp. 399-414.
  • [38] R. H. Martin, Jr., Nonlinear operators and differential equations in Banach spaces.
  • [39] M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differential Gleichungen, Proc. Phys. Math. Soc. Japan 24 (1942), pp. 551-559.
  • [40] W. Orlicz, Zur Theorie des Differentialgleichung y' = f(t,x). Bull. Acad. Polon. Sci. Sér. A (1932), pp. 221-228.
  • [41] W. Orlicz, and S. Szufla, Generic properties of infinite systems of integral equations in Banach spaces, Ann. Polon. Math, (to appear).
  • [42] A. Pelczar, Some functional-differential equations, Dissertationes Math. 100, Warszawa 1973.
  • [43] G. Pianigiani, A density result for differential equations in Banach spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. (to appear).
  • [44] J. Piórek, On integral equations in a Banach spaces, Ann. Polon. Math. 36 (1979), pp. 111-121.
  • [45] J. Piórek, On the asymptotically solvable linear boundary value problem, Zeszyty Naukowe UJ, Prace matematyczne 19 (1977), pp. 137-143.
  • [46] S. Reich, On fixed point theorems obtained from existence theorems for differential equations, J. Math. Anal. Appl. 54 (1976), pp. 26-36.
  • [47] G. Sansone and R. Conti, Non-linear differential equations, Pergamon Press, Oxford 1964.
  • [48] F. A. Valetine, A Lipschitz condition preserving extension for a vector function, Ainer. J. Math. 67 (1945), pp. 83-93.
  • [49] G. Vidossich, How to get zeros of nonlinear operators using the theory of ordinary differential equations, Atas Semana Analise Functional, S. Paulo 1973.
  • [50] G. Vidossich, Existence, uniqueness and approximations of fixed points as a generic property, Bol. Soc. Brasil. Math. 5 (1974), pp. 17-29.
  • [51] G. Vidossich, Most of the successive approximations do converge. J. Math. Anal. Appl. 45 (1974), pp. 127-131.
  • [52] G. Vidossich, Applications of topology to analysis. On the topological properties of the set of fixed points of nonlinear operators, Confer. Sem. Bari 126 (1971), pp. 1-62.
  • [53] J. Yorke, A continuous differential equations in Hilbert space without existence, Funkcial, Ekvac. 13 (1970), pp. 19-21.
  • [54] М. М. Вайнберг, Вариационный метод и метод монотонных операторов, Наука, Москва 1972.
  • [55] А. Н. Годунов, Контпример к теореме Пеано в бесконечномерном гильбертовом пространстве, Вестник МГУ, серия матем. и техн. 5 (1972), pp. 31-34.
  • [56] А. Н. Годунов, О теореме Пеано в банаховых пространствах , функциональный анализ и его приложения 9 (1975), pp. 59-60.
  • [57] G. J. Butler, The generecity of the convergence of successive approximations in Banach spaces, Funk. Ekvacioj 22 (1979), pp. 231-240.
  • [58] G. J. Butler, Almost all 1-set contractions have a fixed points, Proc. Amer. Math. Soc. 74 (1979), pp. 353-357.
  • [59] F. S. De Blasi, Some generic properties in fixed point theory, J. Math. Anal. Appl. 71 (1979), pp. 161-166.
  • [60] F. S. De Blasi and J. Myjak, Orlicz type category results for differential equations in Banach spaces, Comm. Math, (to appear).
  • [61] T. Dominguez Benavides, Aplicaciones de las categorias de Baire a la existencia de solucion de una ecuacion diferencial en un espacio de Banach, Revista Real Acad. Ciencias Exactas, Fis. Nat. Madrid (to appear).
  • [62] S. Giuntini, Most of the generalized Euler's polygonals do converge, Zeszyty Naukowe UJ, Prace Matematyczne (to appear).
  • [63] M. Kisielewicz, The generic property of existence of solutions of differential equations in Banach space, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 29 (1981), pp. 231-233.
  • [64] M. V. Marchi, On non existence of fixed points for nonlinear operators in infinite dimensional Banach space, Raporti technici, Instituto Mat. Appl. Università di Padova, RT 13 (1980).
  • [65] J. Myjak and J. Schinas, Remark on generic properties of differential equations, (in preparation).
  • [66] D. D. Myškis, Some problems in the theory of differential equations with deviating argument, Uspehi Mat. Nauk 32 (1977), pp. 173-202.
  • [67] W. Orlicz and S. Szufla, On the convergence of successive approximations for nonlinear equations in function spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 27 (1979), pp. 153-156.
  • [68] S. Szufla, The generic property of existence of fixed points of nonlinear mappings in junction spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), pp. 711-714.
  • [69] S. Szufla, The generic property of existence of solutions of integral equations in Banach spaces. Math. Nachr. 93 (1979), pp. 305-312.
Języki publikacji
EN
Uwagi
Identyfikator YADDA
bwmeta1.element.zamlynska-50083acd-55dd-4765-a303-be4b214965f2
Identyfikatory
ISBN
82-01-02522-0
ISSN
0012-3862
Kolekcja
DML-PL
Zawartość książki

rozwiń roczniki

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.