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Deterministic and stochastic differential equations in infinite-dimensional spaces

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Abstract

This article is a survey of deterministic and stochastic differential equations in infinite-dimensional spaces. We discuss the existence and uniqueness of solutions of such equations in general locally convex spaces. In particular, linear equations are considered. Some interesting connections between the solvability of deterministic and stochastic equations are studied.

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References

  1. Agase, S. B.: Existence and stability of ordinary differential equations in locally convex spaces,Nonlinear Anal. 5(7) (1981), 713–719.

    Google Scholar 

  2. Agmon, S. and Nirenberg, L.: Properties of solutions of ordinary differential equations in Banach spaces,Comm. Pure Appl. Math. 16 (1963), 121–239.

    Google Scholar 

  3. Ahmed, N. U.: Nonlinear stochastic differential inclusions on Banach space,Stoch. Anal. Appl. 12(1) (1994), 1–10.

    Google Scholar 

  4. Albeverio, S. and Cruzeiro, A. B.: Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids,Comm. Math. Phys. 129 (1990), 431–444.

    Google Scholar 

  5. Albeverio, S., Hoegh-Krohn, R., Fenstad, J. E., and Lindstrom, T.:Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986.

    Google Scholar 

  6. Albeverio, S. and Röckner, M.: Stochastic differential equations in infinite-dimensions: solutions via Dirichlet forms,Probab. Theor. Rel. Fields 89 (1991), 347–386.

    Google Scholar 

  7. Altman, M: Nonlinear equations of evolution in Banach spaces,Nonlinear Anal. 8(5) (1984), 491–499.

    Google Scholar 

  8. Ambrosetti, A.: Un teorema di esistenza per le equazioni differenziali negli spazi di Banach,Rend. Sem. Mat. Univ. Padova 39 (1967), 341–369.

    Google Scholar 

  9. Asano, K.: A note on the abstract Cauchy-Kowalewski theorem,Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), 102–105.

    Google Scholar 

  10. Astala, K.: On Peano's theorem in locally convex spaces,Studia Math. 73(3) (1982), 213–223.

    Google Scholar 

  11. Aubin, J. P. and Cellina, A.:Differential Inclusions, Springer, New York, 1984.

    Google Scholar 

  12. Averbuh, V. I. and Smolyanov, O. G.: Differentiation theory in linear topological spaces,Uspekhi Mat. Nauk 22(6) (1967), 201–260; English transl. inRussian Math. Surv. 22 (1967).

    Google Scholar 

  13. Averbuh, V. I. and Smolyanov, O. G.: Different definitions of derivative in linear topological spaces,Uspekhi Mat. Nauk 23(4) (1968), 67–116; English transl. inRussian Math. Surv. 23 (1968).

    Google Scholar 

  14. Banas, J. and Goebel, K.:Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.

    Google Scholar 

  15. Barbu, V.:Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

    Google Scholar 

  16. Barlow, M. T.: One-dimensional stochastic differential equation with no strong solution,J. London Math. Soc. (2) 26(2) (1982), 335–347.

    Google Scholar 

  17. Baouendi, M. S. and Goulaouic, C.: Remarks on the abstract form of nonlinear CauchyKovalevsky theorems,Comm. Partial Differential Equations 2 (1977), 1151–1162.

    Google Scholar 

  18. Beck, A.: Uniqueness of flow solutions of differential equations,Lecture Notes in Math. 318, Springer, New York (1973), 30–50.

    Google Scholar 

  19. Belopolskaya, Ya. I. and Dalecky, Yu. L.:Stochastic Equations and Differential Geometry, Kluwer Academic Publishers, Dordrecht, 1990.

    Google Scholar 

  20. Bessaga, Ch. and Pelchinskii, A.: On the embedding of nuclear spaces in the space of all differentiable functions on the line,Dokl. Akad. Nauk SSSR 134(4) (1960), 745–748; English transl. inSoviet Math. Dokl. 1 (1960).

    Google Scholar 

  21. Bessaga, Ch., Pelczynski, A., and Rolewicz, S.: On diametral approximative dimension and linear homogeneity of F-spaces,Bull. Acad. Polon. Sci. 9(9) (1961), 677–683.

    Google Scholar 

  22. Binding, P.: On infinite-dimensional differential equations,J. Differential Equations 24 (1977), 349–354.

    Google Scholar 

  23. Bogachev, V. I.: (1) Negligible sets in locally convex spaces,Mat. Zametki 36 (1984), English transl. inMath. USSR Notes 36 (1984), 519–526.

  24. Bogachev, V. I.: (2) Smooth measures, the Malliavin calculus and approximations in infinite-dimensional spaces,Acta Math. Univ. Carolinae, Math. Phys. 31(2) (1990), 9–23.

    Google Scholar 

  25. Bogachev, V. I.: (3) Infinite-Dimensional integration by parts and related problems, Preprint Univ. Bonn, Sonderforschungsbereich 256, n. 235 (1992).

  26. Bogachev, V. I.: (4) Deterministic and stochastic differential equations in infinite-dimensional spaces, Preprint 143 (1993), Centro Vito Volterra, Universita Roma II.

  27. Bogachev, V. I.: (5) Deterministic and stochastic differential equations in infinite-dimensional spaces, in: G. Choquetet al. (Eds),Séminaire d'Initiation a l'Analyse 1992/93.

  28. Bogachev, V. I. and Kobanenko, K. N.: Liens entre léquations différentielles stochastique et ordinaires en dimension infime,C. R. Acad. Sci. Paris (1995) (to appear).

  29. Bogachev, V. I. and Röckner, M.: Mehler formula and capacities for infinite-dimensional Ornstein-Uhlenbeck processes with general linear drift,Osaka J. Math. (1995) (to appear).

  30. Bogachev, V. I. and Shkarin, S. A.: On differentiable and Lipschitz mappings in Banach spaces,Mat. Zametki 44(5) (1988), 567–583; English transl. inMath. USSR Notes 44 (1988).

    Google Scholar 

  31. Bogachev, V. I. and Smolyanov, O. G.: Analytic properties of infinite-dimensional distributions,Uspekhi Mat. Nauk 45(3) (1990), 1–83; English transl. inRussian Math. Surveys 45(3) (1990), 1–104.

    Google Scholar 

  32. Bonet, J. and Perez Carreras, P.:Barrelled Locally Convex Spaces, Math. Stud., Vol. 131, North-Holland, Amsterdam, 1987.

    Google Scholar 

  33. Borisovich, Yu. G.: (1) Weak topology and periodic solutions of differential equations,Dokl. Akad. Nauk SSSR 136(6) (1961); English transl. inSoviet Math. Dokl. (1961).

  34. Borisovich, Yu. G.: Unicity of weak solutions of differential equations in Banach spaces and convergence of successive approximations,Funkt. Anal. i Teoret. Funkt., Kazanskii Univ. No. 1, (1963), 3–12 (in Russian).

    Google Scholar 

  35. Bownds, J. M. and Diaz, J. B.: Euler-Cauchy polygons and the local existence of solutions to abstract ordinary differential equations,Funkt. Ekvac. 15 (1972), 193–207.

    Google Scholar 

  36. Brezis, H. and Pazy, A.: Accretive sets and differential equations in Banach spaces,Israel J. Math. 9 (1970), 367–383.

    Google Scholar 

  37. Brezis, H.:Opérateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert, Math. Stud. Vol. 5, North-Holland, Amsterdam, 1973.

    Google Scholar 

  38. Browder, F.: Nonlinear equations of evolution,Ann. Math. 80 (1964), 485–523.

    Google Scholar 

  39. Caflisch, R. E.: A simplified version of the abstract Cauchy-Kowalewski theorem with weak singularities,Bull. Amer. Math. Soc. 23(2) (1990), 495–500.

    Google Scholar 

  40. Cellina, A.: On the existence of solutions of ordinary differential equations in Banach spaces,Funkt. Ekvac. 14(2) (1971), 129–136.

    Google Scholar 

  41. Cellina, A.: On the existence of solutions of differential equations in nonreflexive Banach spaces,Bull. Amer. Math. Soc. 78 (1972), 1069–1072.

    Google Scholar 

  42. Cellina, A.: On the local existence of solutions of ordinary differential equations,Bull. Acad. Polon. Sci. 20 (1972), 293–296.

    Google Scholar 

  43. Cellina, A.: (4) Differential equations in Banach spaces, Autumn Math. Course, Triest, Italy (1974).

  44. Chaljub-Simon, A., Lemmert, R., Schmidt, S., and Volkmann, P.:Gewöhnliche Differentialgleichungen mit quasimonoton wachsenden rechten Seiten in geordneten Banachräumen, General Inequalities 6, International Series of Numerical Mathematics, Vol. 103, Birkhäuser, Basel, 1992, pp. 307–320.

    Google Scholar 

  45. Chaljub-Simon, A. and Volkmann, P.: Un théorème d'existence pour les équations différentielles dans les espaces de Banach ordonnés,C. R. Acad. Sci. Paris,311, Ser. 1, (1990), 515–517.

    Google Scholar 

  46. Chaljub-Simon, A. and Volkmann, P.: Ordinary differential equations in Banach spaces with variable order cones,Recent Trends in Ordinary Differential Equations, WSSIA1 (1992), 63–70.

    Google Scholar 

  47. Chow, S.-N. and Schuur, J. D.: An existence theorem for ordinary differential equations in Banach spaces,Bull. Amer. Math. Soc. 77(1) (1971), 1018–1020.

    Google Scholar 

  48. Coddington, E. A. and Levinson, N.:Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

    Google Scholar 

  49. Corduneanu, C.: Equazioni differenziali negli spazi di Banach. Teoremi di esistenza e prolungabilita,Atti. Acad. Naz. Lincei Rend. Cl. Sci. Fis. Nat. 23 (1957), 226–230.

    Google Scholar 

  50. Cramer, E., Lakshmikantham, V., and Mitchell, A. R.: On the existence of weak solutions of differential equations in nonreflexive Banach spaces,Nonlinear Anal. 2 (1978), 169–177.

    Google Scholar 

  51. Crandall, M. G. and Liggett, T. M.: Generation of semigroups of nonlinear transformations on general Banach spaces,Amer. J. Math. 93 (1971), 265–298.

    Google Scholar 

  52. Crandall, M. G. and Pazy, A.: Nonlinear evolution equations in Banach spaces,Israel J. Math. 11 (1972), 57–94.

    Google Scholar 

  53. Cruzeiro, A. B.: Équations différentielles sur l'espace de Wiener et formules de Cameron-Martin nonlineaires,J. Funct. Anal. 54(2) (1983), 206–227.

    Google Scholar 

  54. Cruzeiro, A. B.: Unicité de solutions d'équations différentielles sur l'espace de Wiener,J. Funct. Anal. 58(3) (1984), 335–347.

    Google Scholar 

  55. Cruzeiro, A. B.: Estimations capacitaires sur l'espace de Wiener,Bull. Sci. Math., 2e sér.,110 (1986), 139–147.

    Google Scholar 

  56. Curtain, R. F. and Pritchard, A. J.:Infinite-Dimensional Linear System Theory, Lect. Notes in Control Inf. Sci. 8, Springer, New York, 1978.

    Google Scholar 

  57. Daleckii, Yu. L. and Krein, S. G.:Stability of Solutions of Differential Equations in Banach Spaces, Math. Monogr. 43, Amer. Math. Soc., Providence, 1974.

    Google Scholar 

  58. Daleckii, Yu. L. and Fomin, S. V.:Differential Equations and Measures in Infinite-Dimensional Spaces, Nauka, Moscow, 1983; English transl.: Kluwer Academic Publishers, Dordrecht, 1991.

    Google Scholar 

  59. Da Prato, G. and Zabczyk, J.:Stochastic Equations in Infinite-Dimensions, Cambridge Univ. Press, 1992.

  60. Dawson, D. A.: Stochastic evolution equations,Math. Biosciences 15 (1972), 287–316.

    Google Scholar 

  61. De Blasi, F. S.: Existence and stability of solutions for autonomous multivalued differential equations in Banach spaces,Atti Accad. Naz. Lincei Rend. (8),60(6) (1976), 767–774.

    Google Scholar 

  62. De Blasi, F. S.: On a property of the unit sphere in a Banach space,Bull. Math. Soc. Sci. Math. R. S. Roumanie 21 (1977), 259–262.

    Google Scholar 

  63. De Blasi, F. S. and Myjak, J.: Generic flows generated by continuous vector fields in Banach spaces,Adv. in Math. 50 (1983), 266–280.

    Google Scholar 

  64. De Blasi, F. S. and Myjak, J.: Generic properties of differential equations in a Banach space,Bull. Acad. Polon. Sci. Ser. Sci. Math. 26 (1978), 395–400.

    Google Scholar 

  65. De Blasi, F. S. and Myjak, J.: Some generic properties of functional differential equations in Banach spaces,J. Math. Anal. Appl. 67 (1978), 437–451.

    Google Scholar 

  66. De Blasi, F. S. and Myjak, J.: Two density properties of ordinary differential equations,Atti Acad. Naz. Lincei Rend. (8),61(5) (1976), 387–391.

    Google Scholar 

  67. De Blasi, F. S. and Myjak, J.: On generic asymptotic stability of ordinary differential equations in Banach spaces,Proc. Amer. Math. Soc. 65(1) (1977), 47–51.

    Google Scholar 

  68. De Blasi, F. S. and Myjak, J.: La convergence des approximations successives pour les équations différentielles dans les espaces de Banach est une proprieté générique,C. R. Acad. Sci. Paris, Ser. A-B,286(1) (1978), A29-A31.

    Google Scholar 

  69. De Blasi, F. S. and Myjak, J.: Quelques proprietés génériques des équations différentielles dans les espaces de Banach,C. R. Acad. Sci. Paris, Ser. A-B,287(7) (1978), A511-A513.

    Google Scholar 

  70. De Blasi, F. S. and Myjak, J.: Sur l'existence de selections continues,C. R. Acad. Sci. Paris 296(17) (1983), 737–739.

    Google Scholar 

  71. De Blasi, F. S. and Myjak, J.: Orlicz type category results for differential equations in Banach spaces,Comment. Math. Prace Mat. 23(2) (1983), 193–197.

    Google Scholar 

  72. De Blasi, F. S. and Myjak, J.: On the nonexistence of solutions for random differential equations in Banach spaces,Stoch. Anal. Appl. 2(1) (1984), 107–119.

    Google Scholar 

  73. De Blasi, F. S., Kwapisz, M., and Myjak, J.: Generic properties of functional equations,Nonlinear Anal. 2 (1977), 239–249.

    Google Scholar 

  74. De Blasi, F. S. and Pianigiani, G.: A Baire category approach to the existence of solutions of multivalued differential equations in Banach spaces,Funkt. Ekvac. 25(2) (1982), 153–162.

    Google Scholar 

  75. De Blasi, F. S. and Pianigiani, G.: Uniqueness for differential equations implies continuous dependence only in finite dimension,Bull. London Math. Soc. 18(4) (1986), 379–382.

    Google Scholar 

  76. Deimling, K.:Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.

    Google Scholar 

  77. Deimling, K.: On existence and uniqueness for Cauchy's problem in infinite-dimensional Banach spaces,Ann. Mat. Pura Appl. (IV),106 (1975), 1–12.

    Google Scholar 

  78. Deimling, K.:Multivalued Differential Equations, de Gruyter, Berlin, New York, 1992.

    Google Scholar 

  79. Deimling, K. and Lakshmikantham, V.: On existence of extremal solutions of differential equations in Banach spaces,Nonlinear Anal. 3(5) (1979), 563–568.

    Google Scholar 

  80. Deimling, K. and Lakshmikantham, V.: Existence and comparison theorems for differential equations in Banach spaces,Nonlinear Anal. 3(5) (1979), 569–575.

    Google Scholar 

  81. Deleanu, A.: Note on differential equations in locally convex spaces,Bull. Acad. Polon. Sci. 11(11) (1963), 667–680.

    Google Scholar 

  82. Dembart, B.: On the theory of semigroups on locally convex spaces,J. Funct. Anal. 16 (1974), 123–160.

    Google Scholar 

  83. De Pascale, E., Marino, G., and Pietramala, P.: The use of the E-metric space in the search of fixed points (to appear).

  84. Deville, R., Godefroy, G., and Zizler, V.:Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math. 64, Longman Scientific, New York, 1993.

    Google Scholar 

  85. Dieudonné, J.: Deux exemples singuliers d'équations différentielles,Acta Sci. Math. Szeged 12 (1950), 38–40.

    Google Scholar 

  86. Dominguez Benavides, T.: Generic existence of a solution for a differential equation in a scale of Banach spaces,Proc. Amer. Math. Soc. 86 (1982), 477–484.

    Google Scholar 

  87. Doss, H.: Liens entre équations différentielles stochastiques et ordinaires,Ann. Inst. H. Poincare B 13(2) (1977), 99–125.

    Google Scholar 

  88. Doss, H.: Liens entre équations différentielles stochastiques et ordinaires,C. R. Acad. Sci. Paris, Ser. A,283(13) (1976), 939–942.

    Google Scholar 

  89. Dragilev, M. M. and Kondakov, V. P.: On a class of nuclear spaces,Mat. Zametki 8(2) (1970), 169–179; English transl. inMath. USSR Notes 8 (1970), 569–574.

    Google Scholar 

  90. Driver, B.: A Cameron-Martin type quasi-invariance theorem for the Brownian motion on a compact manifold,J. Funct. Anal. 110 (1992), 272–376.

    Google Scholar 

  91. Du, S. W. and Lakshmikantham, V.: Monotone iterative technique for differential equations in a Banach space,J. Math. Anal. Appl. 87 (1982), 454–459.

    Google Scholar 

  92. Dubinsky, E.: Differential equations and differential calculus in Montel spaces,Trans. Amer. Math. 110(1) (1964), 1–21.

    Google Scholar 

  93. Dugundji, J.: An extension of Tietze's theorem,Pacific J. Math. 1 (1951), 353–367.

    Google Scholar 

  94. Dubois, J. and Morales, P.: On the Hukuhara-Kneser property for some Cauchy problems in locally convex topological vector spaces,Lect. Notes in Math. 964, Springer, New York (1982), 162–170.

    Google Scholar 

  95. Duchateau, P. and Treves, J. F.:An Abstract Cauchy-Kowalevska Theorem in Scales of Gevrey Classes, Symp. Math., Vol. 7, Academic Press, New York, 1971, pp. 135–163.

    Google Scholar 

  96. Edwards, R. E.:Functional Analysis, Holt, Rinehart and Winston, New York, 1965.

    Google Scholar 

  97. Elworthy, D.: Stochastic differential equations on manifolds,London Math. Soc. Lect. Notes, 1982.

  98. Evans, L. C.: Nonlinear evolution equations in an arbitrary Banach space,Israel J. Math. 26 (1977), 1–41.

    Google Scholar 

  99. Fattorini, H. O.: Ordinary differential equations in linear topological spaces, I,J. Differential Equations 5 (1969), 72–105.

    Google Scholar 

  100. Fattorini, H. O.: Sur quelques equations differentielles pour les distributions vectorielles,C. R. Acad. Sci. Paris, Ser. A-B 268 (1969), A707-A709.

    Google Scholar 

  101. Faulkner, G. D.: On the nonexistence of weak solutions to abstract differential equations in nonreflexive spaces,Nonlinear Anal. 2 (1978), 505–508.

    Google Scholar 

  102. Fernique, X.: Intégrabilité des vecteurs gaussiens,C. R. Acad. Sci. Paris, Ser. A 270 (1970), 1698.

    Google Scholar 

  103. Feyel, D. and De La Pradelle, A.: Processus browniens de dimension infinie, Laboratoire de Mathematiques Appliquees, Equipe d'Analyse et Probabilités, Université d'Ivry, Preprint No. 26 (1993).

  104. Flett, T. M.: Some applications of Zygmund's lemma to nonlinear differential equations in Banach and Hilbert spaces,Studia Math. 44 (1972), 335–344.

    Google Scholar 

  105. Garay, B. M.: Cross-sections of solution funnels in Banach spaces,Studia Math. 97(1) (1990), 13–26.

    Google Scholar 

  106. Garay, B. M.: Deleting homeomorphisms and the failure of Peano's existence theorem in infinite-dimensional Banach spaces,Funkt. Ekvac. 34 (1991), 85–93.

    Google Scholar 

  107. Garay, B. M. and Schaeffer, J. J.: More on uniqueness without continuous dependence in infinite-dimension,J. Differential Equations 64 (1986), 48–50.

    Google Scholar 

  108. Gelfand, I. M. and Shilov, G. E.:Generalized Functions, III. Some Problems of the Theory of Differential Equations, Nauka, Moscow, 1958.

    Google Scholar 

  109. Godunov, A. N. and Durkin, A. P.: On differential equations in linear topological spaces,Vestnik Moscow Univ., Ser. I, Mat. Mekh., No. 4 (1969), 39–47; English transl. inMoscow Univ. Math. Bull. 24 (1969).

    Google Scholar 

  110. Godunov, A. N.: A counter-example to Peano's theorem in infinite-dimensional Hilbert space,Vestnik Moscow Univ., Ser. I, Mat. Mekh., No. 5 (1972), 31–34; English transl. inMoscow Univ. Math. Bull. 27 (dy1974).

    Google Scholar 

  111. Godunov, A. N.: On linear differential equations in locally convex spaces,Vestnik Moscow Univ., No. 5 (1974), 31–39. English transl. inMoscow Univ. Math. Bull. 29 (1974).

    Google Scholar 

  112. Godunov, A. N.: Peano's theorem in infinite-dimensional Hilbert space fails even in a weakened formulation,Mat. Zametki 15(3) (1974), 476–477; English transl. inMath. USSR Notes 15 (1974).

    Google Scholar 

  113. Godunov, A. N.: On the Peano theorem in Banach spaces,Funkt. Anal. i ego Prilozhen. 9(1) (1975), 59–60; English transl. inFunctional Anal. Appl. 9 (1975).

    Google Scholar 

  114. Godunov, A. N.: An existence and uniqueness result for differential equations in a Hilbert space,Uspekhi Mat. Nauk 31(5) (1976), 235–236 (in Russian).

    Google Scholar 

  115. Godunov, A. N.: (6) Some properties of differential equations in locally convex spaces, Dissertation, Moscow State University, 1973 (in Russian).

  116. Goebel, K. and Rzymowski, W.: An existence theorem for the equationx′=f(x,t) in Banach spaces,Bull. Acad. Polon. Sci., Ser. Math. 18 (1970), 367–370.

    Google Scholar 

  117. Hadzic, O.: Existence of the solution of the equationx=G(x,s(x)) in locally convex spaces,Math. Balkanica 3 (1973), 118–123.

    Google Scholar 

  118. Hadzic, O.: (2) Fixed point theory in topological vector spaces, Inst. Math., Univ. Novi Sad, 1984.

  119. Hamilton, R. S.: The inverse function theorem of Nash and Moser,Bull. Amer. Math. Soc. 7(1) (1982), 65–222.

    Google Scholar 

  120. Herzog, G.: Über gewöhnliche Differentialgleichungen in Frécheträumen, Dissertation, Karlsruhe Univ., 1992.

  121. Hille, E.: Pathology of infinite systems of linear first order differential equations with constant coefficients,Ann. Mat. Pura Appl. 55 (1961), 133–148.

    Google Scholar 

  122. Hille, E. and Phillips, R. S.:Functional Analysis and Semigroups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, 1957.

    Google Scholar 

  123. Horst, E.: Differential equations in Banach spaces: five counterexamples,Arch. Math. 46 (1986), 440–444.

    Google Scholar 

  124. Hukuhara, M.: Théorèmes fondamentaux de la théorie des équations différentielles ordinaires dans l'espace vectoriel topologique,J. Fac. Sci. Univ. Tokyo, Sec. 1,8 (1959), 111–138.

    Google Scholar 

  125. Ikeda, N. and Watanabe, S.:Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1984.

    Google Scholar 

  126. Inaba, M.: On differential equations in locally convex spaces of some types,Kumamoto J. Sci. (A) 2 (1955), 119–124.

    Google Scholar 

  127. Inaba, M.: Differential equations in coordinated spaces,Kumamoto J. Sci. (A) 2 (1955), 233–243.

    Google Scholar 

  128. Itô, K.:Foundations of Stochastic Differential Equations in Infinite-Dimensional Spaces, CBMS-NSF, Reg. Conf. Series in Appl. Math. V. 47, SIAM, Philadelphia, 1984.

    Google Scholar 

  129. Kallianpur, G.: Infinite-dimensional stochastic differential equations and applications, in:Proc. Coxmexz'89, World Scientific, Singapore, 1990, pp. 227–238.

    Google Scholar 

  130. Kallianpur, G. and Pérez-Abreu, V.: Stochastic evolution equations driven by nuclear space valued martingales,Appl. Math. Optim. 17 (1988), 237–272.

    Google Scholar 

  131. Kamenskii, M. I.: On the Peano theorem in infinite-dimensional spaces,Mat. Zametki 11 (1972), 569–576; English transl. inMath. Notes 11 (1972), 347–351.

    Google Scholar 

  132. Kato, S.: Some remarks on nonlinear differential equations in Banach spaces,Hokkaido Math. J. 4 (1975), 205–226.

    Google Scholar 

  133. Kato, S.: On local and global existence theorem for a nonautonomous differential equation in a Banach space,Funkt. Ekvac. 19 (1976), 279–286.

    Google Scholar 

  134. Kato, S.: On the global existence of unique solutions of differential equations in a Banach space,Hokkaido Math. J. 7 (1978), 58–73.

    Google Scholar 

  135. Kato, S.: Some remarks on nonlinear ordinary differential equations in a Banach space,Nonlinear Anal. 5(1) (1981), 81–93.

    Google Scholar 

  136. Kato, T.: Nonlinear evolution equations in Banach spaces,Proc. Symp. Appl. Math. 17 (1965), 50–67.

    Google Scholar 

  137. Kato, T.: Nonlinear semigroups and evolution equations,J. Math. Soc. Japan 19 (1967), 508–520.

    Google Scholar 

  138. Kato, T.: Abstract differential equations and nonlinear mixed problems, Scuola Normale Superiore, Pisa, 1988.

    Google Scholar 

  139. Kenmochi, N. and Takahashi, T.: Nonautonomous differential equations in a Banach space,Nonlinear Anal. 4(6) (1980), 1109–1121.

    Google Scholar 

  140. Kholina, G. G.: One-parametric subgroups of differential locally convex groupuskuls,Mat. Zametki 33(6) (1983), 823–832; English transl. inMath. USSR Notes 33 (1983).

    Google Scholar 

  141. Khrennikov, A. Yu.: Stochastic integrals in locally convex spaces,Uspekhi Mat. Nauk 37(1) (1982), 161–162 (in Russian).

    Google Scholar 

  142. Khrennikov, A. Yu.: Itô's formula in a nuclear Fréchet space,Vestnik Moscow Univ., Ser. I, Mat. Mekh. No. 1 (1984), 14–20; English transl. inMoscow Univ. Math. Bull. 39 (1984).

    Google Scholar 

  143. Khrennikov, A. Yu.: A theorem of existence of a solution of a stochastic differential equation in a locally convex space,Theor. Probab. Appl. 30(1) (1985), 157–160.

    Google Scholar 

  144. Khrennikov, A. Yu.: Differential Equations in locally convex spaces and evolution pseudodifferential equations,Difer. Uravn. 22(2) (1986), 1596–1603; English transl. inDifferential Equations 22(2) (1986), 1116–1122.

    Google Scholar 

  145. Kibenko, A. V.: A contribution to the theory of ordinary differential equations in Banach spaces,Dokl. Akad. Nauk SSSR 136(5) (1961), 1019–1021; English transl. inSoviet Math. Dokl. (1961).

    Google Scholar 

  146. Kneser, H.: Über die Lösungen eines Systems gewöhnlichen Differentialgleichungen das der Lipschitzschen Bedingung nicht genügt,S.-B. Preuss Akad. Wiss. Math. Kl. (1923), 171–174.

  147. Knight, W. J.: Solutions of differential equations inB-spaces,Duke Math. J. 41 (1974) 437–442.

    Google Scholar 

  148. Komantsu, T.: Semi-groups of operators in locally convex spaces,J. Math. Soc. Japan 16 (1964), 230–262.

    Google Scholar 

  149. Komura, T.: Semi-groups of operators in locally convex spaces,J. Funct. Anal. 2(3) (1968), 258–296.

    Google Scholar 

  150. Komura, T. and Komura, Y.: Über die Einbettung der nuklearen Räumen in (s)A,Math. Ann. 162(2) (1966), 284–288.

    Google Scholar 

  151. Kotelenez, P.: A maximal inequality for stochastic convolution integrals on Hilbert spaces and space-time regularity of linear stochastic partial differential equations,Stochastics 21(4) (1987), 345–358.

    Google Scholar 

  152. Köthe, G.: (1) Die Stufenräume, eine einfache Klasse linear vollkommener Räume,Math. Z. 51 (1948), 317–345.

    Google Scholar 

  153. Köthe, G.: (2)Topological Vector Spaces, Springer, New York, 1969.

    Google Scholar 

  154. Krasnoselskii, M. A. and Krein, S. G.: On the theory of ordinary differential equations in Banach spaces,Trudi Seminar. Funk. Anal. Voronezh Univ. 2 (1956), 3–23 (in Russian).

    Google Scholar 

  155. Krasnoselskii, M. A. and Krein, S. G.: Nonlocal existence and uniqueness theorems for systems of ordinary differential equations,Dokl. Acad. Nauk SSSR 102 (1955), 13–16 (in Russian).

    Google Scholar 

  156. Krasnoselskii, M. A. and Zabreiko, P. P.:Geometrical Methods of Nonlinear Analysis, Nauka, Moscow, 1975 (Russian); English transl.: Springer, New York, 1974.

    Google Scholar 

  157. Krasnoselskii, M. A. and Pokrovskii, A. V.: Natural solutions of stochastic differential equations,Dokl. Akad. Nauk SSSR 240(2) (1978), 264–267; English transl. inSoviet Math. Dokl. (1978).

    Google Scholar 

  158. Krasnoselskii, M. A. and Pokrovskii, A. V.:Systems with Hysteresis, Nauka, Moscow, 1983 (in Russian).

    Google Scholar 

  159. Krylov, N. V.:Controled Diffusion Processes, Nauka, Moscow, 1977; English transl.: Springer, Berlin, 1980.

    Google Scholar 

  160. Kubiaczyk, I.: Kneser type theorems for ordinary differential equations in Banach spaces,J. Differential Equations 45 (1982), 139–146.

    Google Scholar 

  161. Lakshmikantham, V.: Stability and asymptotic behavior of solutions of differential equations in a Banach space, Lecture Notes, CIME, Italy, 1974.

    Google Scholar 

  162. Lakshmikantham, V. and Leela, S.:Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Oxford, 1981.

    Google Scholar 

  163. Lakshmikantham, V. and Leela, S.:Differential and Integral Inequalities, Vol. 1, Academic Press, New York, 1969.

    Google Scholar 

  164. Lasota, A. and Yorke, J.: The generic property of existence of solutions of differential equations in Banach spaces,J. Differential Equations 13 (1973), 1–12.

    Google Scholar 

  165. Lemmert, R.: On ordinary differential equations in locally convex spaces,Nonlinear Anal. 10(12) (1986), 1385–1390.

    Google Scholar 

  166. Lemmert, R.: Existenzsätze für gewöhnliche Differentialgleichungen in geordneten Banachräumen,Funkt. Ekvac. 32 (1989), 243–249.

    Google Scholar 

  167. Lemmert, R., Redheffer, R. M., and Volkmann, P.:Ein Existenzsatz für gewöhnliche Differentialgleichungen in geordneten Banachräumen, General Inequalities 5, International Series of Numerical Mathematics, Vol. 80, Birkhäuser, Basel, 1987, pp. 381–390.

    Google Scholar 

  168. Lemmert, R. and Weckbach, A.: Charakterisierungen zeilenendlicher Matrizen mit abzahlbarem Spektrum,Math. Z. 188 (1984), 119–124.

    Google Scholar 

  169. Lewy, H.: An example of smooth linear partial differential equation without solution,Ann. Math. 66(2) (1957), 155–158.

    Google Scholar 

  170. Lions, J. L.:Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969.

    Google Scholar 

  171. Liptser, R. Sh. and Shiryaev, A. N.:Statistics of Random Processes, Nauka, Moscow, 1974; English transl.: Springer, New York, 1977.

    Google Scholar 

  172. Li, T. Y.: Existence of solutions for ordinary differential equations in Banach spaces,J. Differential Equations 18 (1975), 29–40.

    Google Scholar 

  173. Lobanov, S. G.: On linear representations of nonlinear differential equations,Vestnik Moscow Univ., Ser. I, Mat. Mekh., No. 1 (1977), 43–49; English transl. inMoscow Univ. Math. Bull. 32 (1977).

    Google Scholar 

  174. Lobanov, S. G.: An example of a nonnormable Fréchet space in which any continuous linear operator has an exponent,Uspekhi Mat. Nauk 34(4) (1979), 201–202 (in Russian).

    Google Scholar 

  175. Lobanov, S. G.: On the uniqueness of solutions of evolution differential equations in locally convex spaces,Mat. Zametki 26(4) (1979), 523–533. English transl. inMath. Notes USSR 26 (1979).

    Google Scholar 

  176. Lobanov, S. G.: On the solvability of linear ordinary differential equations in locally convex spaces,Vestnik Moscow Univ., Ser. I, Mat. Mekh. No. 2 (1980), 3–7; English transl. inMoscow Univ. Math. Bull. 35 (1980).

    Google Scholar 

  177. Lobanov, S. G.: (5) Some properties of ordinary differential equations in locally convex spaces, Dissertation, Moscow State University, 1979 (in Russian).

  178. Lobanov, S. G.: Chain rule and its converse for mappings in locally convex spaces,Mat. Zametki 45(1) (1989), 43–56; English transl. inMath. USSR Notes 45 (1989).

    Google Scholar 

  179. Lobanov, S. G.: Picard's theorem for ordinary differential equations in locally convex spaces,Izv. Russian Acad. Sci. 56(6) (1992), 1217–1243; English transl. inRussian Acad. Sci. Izv. Math. 41(3) (1993), 465–487.

    Google Scholar 

  180. Lobanov, S. G.: On Peano's theorem in Fréchet spaces,Differential Equations 6 (1992), 1085 (in Russian).

    Google Scholar 

  181. Lobanov, S. G.: On the structure of the solutions sets of ordinary differential equations in locally convex spaces,Differential Equations 6 (1992), 1086 (in Russian).

    Google Scholar 

  182. Lobanov, S. G.: Peano's theorem fails for any infinite-dimensional Fréchet space,Mat. Sbornik 184(2) (1993), 83–86.

    Google Scholar 

  183. Lovelady, D. L. and Martin, R. H., Jr.: A global existence theorem for a nonautonomous differential equations in a Banach space,Proc. Amer. Math. Soc. 35 (1972), 445–449.

    Google Scholar 

  184. Malliavin, P.: (1)Géometrie différentielle stochastique, Univ. de Montreal, 1978.

  185. Malliavin, P.: Infinite-dimensional analysis,Bull. Sci. Math., Ser. 2,117 (1993), 63–90.

    Google Scholar 

  186. Martin, R. H.:Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1978.

    Google Scholar 

  187. Martin, R. H.: Approximation and existence of solutions to ordinary differential equations in Banach spaces,Funkt. Ekvac. 16 (1973), 195–211.

    Google Scholar 

  188. Martin, R. H.: A global existence theorem for autonomous differential equations in a Banach spaces,Proc. Amer. Math. Soc. 26 (1970), 307–314.

    Google Scholar 

  189. Martin, R. H.: The logarithmic derivatives and equations of evolution in a Banach space,J. Math. Soc. Japan 22 (1970), 411–429.

    Google Scholar 

  190. Martin, R. H.: Generating an evolution system in a class of uniformly convex Banach spaces,J. Funct. Anal. 11 (1972), 62–76.

    Google Scholar 

  191. Martin, R. H.: Strongly dissipative operators and nonlinear equations in a Fréchet space,Proc. Amer. Math. Soc. 32 (1972), 161–168.

    Google Scholar 

  192. Martin, R. H.: Lyapunov functions and autonomous differential equations in a Banach space,Math. Syst. Theory 7 (1973), 66–72.

    Google Scholar 

  193. Martin, R. H.: Differential equations on closed subsets of a Banach space,Trans. Amer. Math. Soc. 179 (1973), 399–414.

    Google Scholar 

  194. Martin, R. H.: Remarks on ordinary differential equations involving dissipative and compact operators,J. London Math. Soc. 10(2) (1975), 61–65.

    Google Scholar 

  195. Massera, J. L. and Schaeffer, J. J.:Linear Differential Equations and Functional Spaces, Academic Press, New York, 1966.

    Google Scholar 

  196. Metivier, M.: Stochastic partial differential equations in infinite-dimensional spaces, Scuola Normale Superiore, Pisa, 1988.

    Google Scholar 

  197. Michael, E.: Continuous selections,Ann. Math., Ser. 2 63(2) (1956), 361–382; 64(3) (1956), 562-580;65(2) (1956), 375–390.

    Google Scholar 

  198. Michal, A. D. and Elconin, V.: Completely integrable differential equations in abstract spaces,Acta Math. 68 (1937), 71–107.

    Google Scholar 

  199. Millionshchikov, V. M.: (1) A contribution to the theory of differential equations dx/dtf=(x,t) in locally convex spaces,Dokl. Akad. Nauk SSSR 131(3) (1960), 510–513; English transl. inSoviet Math. Dokl. 1 (1960).

    Google Scholar 

  200. Millionshchikov, V. M.: A contribution to the theory of differential equations in locally convex spaces,Mat. Sb. USSR 57 (1962), 385–406 (in Russian).

    Google Scholar 

  201. Millionshchikov, V. M.: Theory of differential equations with a small factor in the derivatives in linear topological spaces,Dokl. Akad. Nauk SSSR 146(5) (1962), 1021–1024; English transl. inSoviet Math. Dokl. 3 (1962).

    Google Scholar 

  202. Minty, G.: On the extension of Lipschitz, Lipschitz-Holder continuous, and monotone functions,Bull. Amer. Math. Soc. 76 (1970), 334–339.

    Google Scholar 

  203. Mityagin, B. S.: Approximate dimension and bases in nuclear spaces,Russian Math. Surveys 16(4) (1961), 59–127.

    Google Scholar 

  204. Moench, H.: Boundary value problem for nonlinear differential equations of second order in Banach spaces,Nonlinear Anal. 4 (1980), 985–999.

    Google Scholar 

  205. Morales, P.: Properties of the set of global solutions for the Cauchy problem in a locally convex space,Lecture Notes in Math. 1151, Springer, New York, 1985, pp. 276–284.

    Google Scholar 

  206. Moser, J.: A rapidly convergent iteration method and nonlinear partial differential equations I,Ann. Scu. Norm. Super. Pisa 20 (1966), 265–315.

    Google Scholar 

  207. Mueller, G. O.: Continuity moduli criteria for ODE's in a Banach space,J. Differential Equations 45 (1982), 317–331.

    Google Scholar 

  208. Nazarov, V. I.: Solvability of the Cauchy problem for differential equations in locally convex spaces,Differential Equations 27(6) (1991), 688–695.

    Google Scholar 

  209. Nazaryan, A. L.: On continuous dependence of solutions of differential equations in Banach spaces on the initional data,Mat. Zametki 51(5) (1992), 81–86; English transl. inMath. USSR Notes 51 (1992).

    Google Scholar 

  210. Nemitskii, V. V., Vainberg, M. M., and Gusarova, R. S.: Operator differential equations, in:Matem. Anal. 1964, Itogi Nauki, Moscow, 1966, pp. 165–235 (in Russian).

    Google Scholar 

  211. Netes, V. A.: The existence and uniqueness theorem for ordinary differential equations in locally convex spaces,Mat. Zametki 25(6) (1979), 891–898; English transl. inMath. USSR Notes 25 (1979).

    Google Scholar 

  212. Nirenberg, L.: An abstract form of the nonlinear Cauchy-Kowalewski theorem,J. Differential Geometry 6 (1972), 561–576.

    Google Scholar 

  213. Nirenberg, L.:Topics in Nonlinear Functional Analysis, Courant Inst. of Math. Sci., New York, 1974.

    Google Scholar 

  214. Nishida, T.: A note on Nirenberg's theorem as an abstract form of the nonlinear CauchyKowalewsky theorem in a scale of Banach spaces,J. Differential Geom. 12 (1977), 629–633.

    Google Scholar 

  215. Olech, C.: On the existence and uniqueness of solutions of an ordinary differential equation in the case of Banach space,Bull. Acad. Polon. 8 (1960), 667–673.

    Google Scholar 

  216. Orlicz, W.: Zur Theorie der Differentialgleichungy′= f(x,y),Bull. Acad. Polon. (1932), 221–228.

  217. Orlicz, W. and Szufla, S.: Generic properties of infinite systems of integral equations in Banach spaces,Ann. Polon. Math. 40 (1981), 67–79.

    Google Scholar 

  218. Ovsyannikov, L. V.: A singular operator in the scale of Banach spaces,Dokl. Acad. Nauk SSSR 163(4) (1965), 819–822; English transl. inSoviet Math. Dokl. 6 (1965).

    Google Scholar 

  219. Ovsyannikov, L. V.: A nonlinear Cauchy problem in the scale of Banach spaces,Soviet. Math, Dokl. 12 (1971), 1497–1502.

    Google Scholar 

  220. Papageorgiou, N. S.: Weak solutions of differential equations in Banach spaces,Bull. Austral. Math. Soc. 33 (1986), 407–418.

    Google Scholar 

  221. Papageorgiou, N. S.: Kneser's theorem for differential equations in Banach spaces,Bull. Austral. Math. Soc. 33 (1986), 419–434.

    Google Scholar 

  222. Pardoux, E.:Filtrage non linéaire et équations aux dérivées partielles stochastiques associée, Ecole d'été de Probabilités de Saint-Flour, 1989.

  223. Pasika, E. E.: An example of a differential equation of the first order without continuous dependence on the initial data,Ukranian Math. J. 35(6) (1983), 786–788 (in Russian).

    Google Scholar 

  224. Pasika, E. E.: On the conditions of the solvability of the Cauchy problem of the first order in Banach spaces,Funkt. Anal. i ego Prilozh. 18(2) (1984), 71–72; English transl. inFunct. Anal. Appl. 18 (1984).

    Google Scholar 

  225. Pavel, N. H.:Differential Equations, Flow Invariance and Applications, Pitman Research Notes in Math. 113, Longman, London, 1984.

    Google Scholar 

  226. Pavel, N. H.:Nonlinear Evolution Operators and Semigroups, Lecture Notes in Math. 1260, Springer, New York, 1987.

    Google Scholar 

  227. Pazy, A.: A Trotter-Lie formula for compactly generated semigroups of nonlinear operators,J. Funct. Anal. 23 (1976), 353–361.

    Google Scholar 

  228. Pazy, A.:Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.

    Google Scholar 

  229. Pianigiani, G.: A density result for differential equations in Banach spaces,Bull. Acad. Polon. Sci., Ser. Math. 26 (1978), 791–793.

    Google Scholar 

  230. Pianigiani, G.: Existence of solutions for ordinary differential equations in Banach spaces,Bull. Acad. Polon. Sci., Ser. Sci. Math., Astron., Phys. 23 (1975), 853–857.

    Google Scholar 

  231. Pietsch, A.:Nukleare lokalkonvexe Räume, Akademie-Verlag, Berlin, 1965.

    Google Scholar 

  232. Phillips, R.: Integration in convex linear topological spaces,Trans. Amer. Math. Soc. 47 (1940), 114–171.

    Google Scholar 

  233. Prostov, Yu. I.: For homeomorphisms of nonnormable Fréchet spaces the theorem on the differentiability of an inverse function is invalid,Mat. Zametki 47(1–7) (1990), 170–178; English transl. inMath. USSR Notes 47 (1990).

    Google Scholar 

  234. Przeslawski, K. and Yost, D.: Continuity properties of selections and Michael's theorem,Michigan Math. J. 36 (1989), 113–134.

    Google Scholar 

  235. Pugh, C. C.: Funnel sections,J. Differential Equations 19 (1975), 270–295.

    Google Scholar 

  236. Pulvirenti, G.: Equazioni differenziali in uno pennello delle soluzioni in ipotesi di Caratheodory,Ann. Mat. Pura Appl. 56 (1961), 281–300.

    Google Scholar 

  237. Pulvirenti, G.: Equazioni differenziali in forma implicita in uno spazio di Banach,Ann. Mat. Pura Appl. 56 (1961), 177–199.

    Google Scholar 

  238. Radyno, Ya. V.: (1)Linear Equations and Bornology, Minsk, 1982 (in Russian).

  239. Radyno, Ya. V.: Linear differential equations in locally convex spaces,Differential Equations 13(8) (1977), 972–978;13(9) (1977), 1124–1131.

    Google Scholar 

  240. Raikov, D. A.: Appendix I to the Russian translation of the book by A. Robertson and V. Robertson,Topological Vector Spaces.

  241. Ren, J.: Analyse quasi-sure des équations différentielles stochastiques,Bull. Sci. Math. 2e sér.,114 (1990), 187–213.

    Google Scholar 

  242. Ricciardi, P. and Tubaro, L.: Local existence for differential equations in Banach spaces,Boll. Un. Mat. Ital. 8 (1973), 306–316.

    Google Scholar 

  243. Robbin, J. W.: On the existence theorem for differential equations,Proc. Amer. Math. Soc. 19(2) (1968), 1005–1006.

    Google Scholar 

  244. Rozovskii, B.:Evolution Stochastic Systems, Nauka, Moscow, 1981 (in Russian).

    Google Scholar 

  245. Rzepecki, B.: On the method of Euler polygons for the differential equation in a locally convex space,Bull. Acad. Polon. Sci., Ser. Math. 23(4) (1975), 411–414.

    Google Scholar 

  246. Sadovskii, B. N.: Limit-compact and condensing operators,Uspekhi Mat. Nauk 27(1) (1972), 81–146; English transl. inRussian Math. Surv. 27 (1972), 85–156.

    Google Scholar 

  247. Saint Raymond, J.: Une équation différentielle sans solution, Séminaire d'Initiation à l'Analyse G. Choquet, M. Rogalski, J. Saint Raymond 20e annee, 1980/81, Communication No. 2, 1–7.

  248. Schaefer, H. H.:Topological Vector Spaces, Springer, Berlin, 1971.

    Google Scholar 

  249. Schaeffer, J. J.: Uniqueness without continuous dependence in infinite-dimensions,J. Differential Equations 56 (1985), 426–428.

    Google Scholar 

  250. Schmidt, S.: Existenzsätze für gewöhnliche Differentialgleichungen in Banachräumen,Funkt. Ekvac. 35 (1992), 199–222.

    Google Scholar 

  251. Shkarin, S. A.: Some results on solvability of ordinary differential equations in locally convex spaces,Mat. Sb. 181(9) (1990), 1183–1195; English transl. inMath. USSR Sb. 71(1) (1992), 29–40.

    Google Scholar 

  252. Shkarin, S. A.: On the continuity points of Gateaux differentiable functions,Sibirsk. Mat. J. 33(5) (1992), 176–185; English transl. inSiberian Math. J. 33 (1992).

    Google Scholar 

  253. Shkarin, S. A.: On a problem of O. G. Smolyanov connected with infinite-dimensional Peano's theorem,Differential Equations 6 (1992), 1092 (in Russian).

    Google Scholar 

  254. Shkarin, S. A.: (4) Peano's theorem fails in infinite-dimensional Fréchet spaces,Funkt. Anal. i ego Prilozhen. 27(3) (1993); English transl.in Funct. Anal. Appl. 27 (1993).

  255. Skorokhod, A. V.:Studies in the Theory of Random Processes, Addison-Wesley, Reading, Mass., 1965.

    Google Scholar 

  256. Smolyanov, O. G.: Linear representations of evolution differential equations,Dokl. Akad. Nauk SSSR 221(6) (1975), 1288–1291; English transl. inSoviet Math. Dokl. 16 (1975).

    Google Scholar 

  257. Smolyanov, O. G.: A method of proving the uniqueness theorems for evolution differential equations,Mat. Zametki 25(2) (1979), 259–269; English transl. inMath. USSR Notes 25 (1979).

    Google Scholar 

  258. Smolyanov, O. G.: (3)Analysis on Topological Linear Spaces and Its Applications, Moscow State University, 1979 (in Russian).

  259. Smolyanov, O. G.: Integral invariants connected with the evolution differential equations,Uspekhi Mat. Nauk 40(5) (1985), 209 (in Russian).

    Google Scholar 

  260. Sobczyk, A.: Projektion of the spacem on its subspace c0,Bull. Amer. Math. Soc. 55 (1944), 153–169.

    Google Scholar 

  261. Staicu, V.: Continuous selections of solution sets to evolution equations,Proc. Amer. Math. Soc. 113 (1991), 403–413.

    Google Scholar 

  262. Stankovic, B.: L'équation différentielle vectorielle,Acad. Serbe Sci., Publ. Inst. Math., New Ser. 6 (1966), 77–82.

    Google Scholar 

  263. Stankovic, B.: Differential equations in locally convex spaces,Mat. Vestnik 7(4) (1970), 551–558.

    Google Scholar 

  264. Stankovic, B.: Two theorems on differential equations in locally convex spaces,Bull. Sci., Sect. A Youg. 16(1–2) (1971), 1.

    Google Scholar 

  265. Stankovic, B.: Two theorems on differential equations in locally convex spaces,Acad. Serbe Sci. Art. Glas. 283(35) (1972), 33–42.

    Google Scholar 

  266. Stecenko, V. Ya.: K-regular cones,Dokl. Akad. Nauk SSSR 136(5) (1961), 1038–1040; English transl. inSoviet Math. Dokl. 2 (1961).

    Google Scholar 

  267. Sussmann, H.: On the gap between deterministic and stochastic ordinary differential equations,Ann. Probab. 6 (1978), 19–41.

    Google Scholar 

  268. Sussmann, H.: An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point,Bull. Amer. Math. Soc. 83 (1977), 296–298.

    Google Scholar 

  269. Szep, A.: Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces,Studia Sci. Math. Hungar. 6 (1971), 197–203.

    Google Scholar 

  270. Szufla, S.: Kneser's theorem for weak solutions of ordinary differential equations in reflexive Banach spaces,Bull. Acad. Polon. Sci., Ser. Math. 26 (1978), 407–413.

    Google Scholar 

  271. Szufla, S.: On the Kneser-Hukuhara property for integral equations in locally convex spaces,Bull. Austral. Math. Soc. 36 (1987), 353–360.

    Google Scholar 

  272. Szufla, S.: On the equationx′=f(t,x) in locally convex spaces,Math. Nachr. 118 (1984), 179–185.

    Google Scholar 

  273. Szufla, S.: On the existence of solutions of ordinary differential equations in Banach spaces,Boll. Un. Mat. Ital. 15-A (1978), 535–544.

    Google Scholar 

  274. Szufla, S.: Measure of noncompactness and ordinary differential equations in Banach spaces,Bull. Acad. Polon. Sci. Ser. Math. 19 (1971), 831–835.

    Google Scholar 

  275. Szufla, S.: Structure of the solutions set of ordinary differential equations in Banach space,Bull. Acad. Polon. Sci. Ser. Math. 21 (1973), 141–144.

    Google Scholar 

  276. Szufla, S.: Solutions sets for nonlinear equations,Bull. Acad. Polon. Sci. Ser. Math. 21 (1973), 971–976.

    Google Scholar 

  277. Szufla, S.: Some properties of the solutions set of ordinary differential equations,Bull. Acad. Polon. Sci. Ser. Math. 22 (1974), 675–678.

    Google Scholar 

  278. Szufla, S.: On the existence of solutions of Volterra integral equations in Banach space,Bull. Acad. Polon. Sci. Ser. Math. 22 (1974), 1209–1213.

    Google Scholar 

  279. Szufla, S.: Solutions sets of nonlinear integral equations,Funkt. Ekvac. 17 (1974), 67–71.

    Google Scholar 

  280. Szufla, S.: On the structure of solutions sets of differential and integral equations in Banach spaces,Ann. Polon. Math. 34(2) (1977), 165–177.

    Google Scholar 

  281. Szufla, S.: On Volterra integral equations in Banach spaces,Funkt. Ekvac. 20(3) (1977), 247–258.

    Google Scholar 

  282. Szufla, S.: On the equationx′=f(t, x) in Banach spaces,Bull. Acad. Polon. Sci. Ser. Math. 26(5) (1978), 401–406.

    Google Scholar 

  283. Szufla, S.: Some remarks on ordinary differential equations in Banach spaces,Bull. Acad. Polon. Sci., Ser. Sci. Math., Astron., Phys. 16 (1968), 795–800.

    Google Scholar 

  284. Tolstonogov, A. A.:Differential Inclusions in Banach Spaces, Nauka, Novosibirsk, 1986 (in Russian).

    Google Scholar 

  285. Treves, J. F.: An abstract nonlinear Cauchy-Kovalevska theorem,Trans. Amer. Math. Soc. 150 (1970), 77–92.

    Google Scholar 

  286. Tsar'kov, I. G.: Theorems on global existence of an implicit function and their applications,Dokl. Russian Akad. Nauk 324(4) (1992), 754–756; English transl. inSoviet Math. Dokl. 33 (1992).

    Google Scholar 

  287. Tychonoff, A.: Über einen Eixpunksatz,Math. Ann. 111 (1935), 762–776.

    Google Scholar 

  288. Uhl, R.: Beiträge zur Theorie der gewöhnlichen Differentialgleichungen in Banachräumen, Dissertation, Universität Karlsruhe (1993).

  289. Ushijima, T.: On the abstract Cauchy problems and semi-groups of linear operators in locally convex space,Sci. Pap. Coll. Gen. Educ. Univ. Tokyo 21(2) (1971), 93–122.

    Google Scholar 

  290. Ushijima, T.: On the generation and smoothness of semi-groups of linear operators,J. Fac. Sci. Univ. Tokyo, Sec. 1,19 (1972), 66–127.

    Google Scholar 

  291. Ustinov, S. M.: On differential invariants of evolution differential equations in locally convex spaces,Vestnik Moscow Univ., Ser. Math., Mech. No. 4 (1993), 78–81; English transl. inMoscow Univ. Math. Bull. 48 (1993).

    Google Scholar 

  292. Ustunel, A. S.: Stochastic analysis on nuclear spaces and its applications, in: L. Arnold and P. Kotelenez (Eds),Stochastic Space-Time Models and Limit Theorems, D. Reidel, Dordrecht, 1985, pp. 163–177.

    Google Scholar 

  293. Vainberg, M. M.:Variational Method and Method of Monotone Operators, Nauka, Moscow, 1972.

    Google Scholar 

  294. Veretennikov, A. Yu.: On strong solutions and explicit formulas for solutions of stochastic integral equations,Mat. Sb. 111(3) (1980), 434–452; English transl. inMath. USSR Sb.

    Google Scholar 

  295. Veretennikov, A. Yu.: On approximation of ordinary equations by stochastic equations,Mat. Zametki 33(6) (1983), 929–932; English transl. inMath. USSR Notes 33 (1983).

    Google Scholar 

  296. Veretennikov, A. Yu. and Kleptsina, M. L.: On the pathwise approach to stochastic differential equations, in: A. N. Shiryaev (Ed.),Statistics and Control of Random Processes, Nauka, Moscow, 1989, pp. 22–23 (in Russian).

    Google Scholar 

  297. Vidossich, G.: Existence, uniqueness and approximation of fixed points as a generic property,Bol. Soc. Brasil. Mat. 5 (1974), 17–29.

    Google Scholar 

  298. Vidossich, G.: Global convergence of successive approximations,J. Math. Anal. Appl. 45 (1974), 285–292.

    Google Scholar 

  299. Vil'ke, V. G.:Analytic and Qualitative Methods of the Mechanics of Systems with Infinite Number of Degrees of Freedom, Moscow State University, 1986 (in Russian).

  300. Volkmann, P.: Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen,Math. Z. 127 (1972), 157–164.

    Google Scholar 

  301. Volkmann, P.: Ein Existenzsatz für gewöhnliche Differentialgleichungen in Banachräumen,Proc. Amer. Math. Soc. 80 (1980), 297–300.

    Google Scholar 

  302. Volkmann, P.: Équations différentielles ordinaires dans les espaces des fonctions bornées,Czech. Math. J. 35(1985), 201–211.

    Google Scholar 

  303. Volkmann, P.: On the convergence of approximate solutions for an initial value problem in Banach spaces,Nonlinear Anal. 7 (1983), 217–222; Erratum:ibid. 8 (1985), 1009.

    Google Scholar 

  304. Volkmann, P.: Über die Existenz von Lösungen der Differentialgleichung u′=f(u) in einer abgeschlossenen Menge, wennf eineκ-Mengenkontraktion ist,Proc. Conf. Ordinary and Partial Differential Equations, Dundee 1976, Lecture Notes in Math. 564, Springer, New York, 1977.

    Google Scholar 

  305. Volkmann, P.: Ausdehnung eines Satzes von Max Müller auf unendliche Systeme von gewöhnlichen Differentialgleichungen,Funkt. Ekvac. 21 (1978), 81–96.

    Google Scholar 

  306. Walsh, J.: An introduction to stochastic partial differential equations, in:Ecole d'été de Probabilité de Saint Flour XIV, Lecture Notes in Math. 1180, Springer, New York, 1986, pp. 265–439.

    Google Scholar 

  307. Walter, W.: Ordinary differential inequalities in ordered Banach spaces,J. Differential Equations 9 (1971), 253–261.

    Google Scholar 

  308. Wazewski, T.: Sur l'existence et l'unicité des intégrales des équations différentielles ordinaires au cas de l'espace de Banach,Bull. Acad. Polon. Sci. 8 (1960), 301–305.

    Google Scholar 

  309. Wells, J. H. and Williams, L. R.:Embeddings and Extensions in Analysis, Ergebnisse Vol. 84, Springer, New York, 1975.

    Google Scholar 

  310. Wong, E. and Zakai, M.: On the relationship between ordinary and stochastic differential equations and applications to stochastic problems in control theory,Proc. Third IFAC Congress, Paper 3B (1966).

  311. Yamanaka, T.: Note on Kowalevskaja's system of partial differential equations,Comm. Math. Univ. St. Paul. 9 (1960), 7–10.

    Google Scholar 

  312. Yorke, J. A.: A continuous differential equation in Hilbert space without existence,Funkt. Ekvac. 13 (1970), 19–21.

    Google Scholar 

  313. Yosida, K.:Functional Analysis, 6th edn., Springer, New York, 1980.

    Google Scholar 

  314. Zizler, V.: Smooth extension of norms and complementability of subspaces,Arch. Math. 53(6) (1989), 585–589.

    Google Scholar 

  315. Bensoussan, A. and Temam, R.: Équations aux dérivées partielles stochastiques non linéaires, I,Israel J. Math. 11 (1972), 95–129.

    Google Scholar 

  316. Kuo, H.-H.:Gaussian Measures in Banach Spaces, Lect. Notes in Math. 463, Springer, New York, 1975.

    Google Scholar 

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  1. Moscow State University, 119899, Moscow, Russia

    V. I. Bogachev

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Bogachev, V.I. Deterministic and stochastic differential equations in infinite-dimensional spaces. Acta Appl Math 40, 25–93 (1995). https://doi.org/10.1007/BF01518365

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