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1995 | 62 | 1 | 13-21
Tytuł artykułu

Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

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EN
Abstrakty
EN
We investigate the structure of the set of solutions of the Cauchy problem x' = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in $C_{w}(I,E)$, the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions, so we also obtain Kneser-type theorems for these classes of solutions.
Twórcy
  • Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
  • Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
Bibliografia
  • [1] O. Arino, S. Gautier and J. P. Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkc. Ekvac. 27 (1984), 273-279.
  • [2] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math. 60, Marcel Dekker, New York, 1980.
  • [3] J. Banaś and J. Rivero, On measure of weak noncompactness, Ann. Mat. Pura Appl. 125 (1987), 213-224.
  • [4] M. Cichoń, Weak solutions of differential equations in Banach spaces, Discuss. Math. 15 (1994) (in press).
  • [5] E. Cramer, V. Lakshmikantham and A. R. Mitchell, On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal. 2 (1978), 169-177.
  • [6] S. J. Daher, On a fixed point principle of Sadovskii, Nonlinear Anal., 643-645.
  • [7] F. S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R. S. Roumanie 21 (1977), 259-262.
  • [8] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.
  • [9] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Side, Nauka, Moscow, 1985 (in Russian).
  • [10] W. J. Knight, Solutions of differential equations in B-spaces, Duke Math. J. 41 (1974), 437-442.
  • [11] I. Kubiaczyk, Kneser type theorems for ordinary differential equations in Banach spaces, J. Differential Equations 45 (1982), 139-146.
  • [12] I. Kubiaczyk and S. Szufla, Kneser's theorem for weak solutions of ordinary differential equations in Banach spaces, Publ. Inst. Math. (Beograd) 32 (1982), 99-103.
  • [13] A. R. Mitchell and Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, in: Nonlinear Equations in Abstract Spaces, V. Lakshmikantham (ed.), 1978, 387-404.
  • [14] N. S. Papageorgiou, Kneser's theorems for differential equations in Banach spaces, Bull. Austral. Math. Soc. 33 (1986), 419-434.
  • [15] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304.
  • [16] A. Szep, Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6 (1971), 197-203.
  • [17] S. Szufla, Some remarks on ordinary differential equations in Banach spaces, Bull. Acad. Polon. Sci. Math. 16 (1968), 795-800.
  • [18] S. Szufla, Kneser's theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Bull. Acad. Polon. Sci. Math. 26 (1978), 407-413.
Typ dokumentu
Bibliografia
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