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1995 | 15 | 1 | 5-14
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Weak solutions of differential equations in Banach spaces

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EN
In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential generalization of previous results.
Twórcy
  • Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
Bibliografia
  • [1] A. Alexiewicz, Functional Analysis, Monografie Matematyczne 49, Polish Scientific Publishers, Warsaw 1968 (in Polish).
  • [2] O. Arino, S. Gautier, J.P.Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkcialaj Ekvac. 27 (1984), 273-279.
  • [3] J. M. Ball, Weak continuity properties of mappings and semi-groups, Proc. Royal Soc. Edinbourgh Sect. A 72 (1979), 275-280.
  • [4] J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Marcel Dekker, New York - Basel 1980.
  • [5] J. Banas, J. Rivero, On measures of weak noncompactness, Ann. Mat. Pura Appl. 125 (1987), 213-224.
  • [6] M. Cichoń, On measures of weak noncompactness, Publicationes, Math. Debrecen 45 (1994), 93-102.
  • [7] E. Cramer, V. Lakshmikantham and A.R.Mitchell, On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlin. Anal. TMA 2 (1978), 169-177.
  • [8] F. S. DeBlasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R. S. Roumanie 21 (1977), 259-262.
  • [9] G. A. Edgar, Geometry and the Pettis integral, Indiana Univ. Math. J. 26 (1977), 663-677.
  • [10] G. A. Edgar, Geometry and the Pettis integral II, ibid. 28 (1979), 559-579.
  • [11] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Side, Moscow 1985 (in Russian).
  • [12] R. F. Geitz, Pettis integration, Proc. Amer. Math. Soc. 82 (1981), 81-86.
  • [13] R. F. Geitz, Geometry and the Pettis integral, Trans. Amer. Math. Soc. 269 (1982), 535-548.
  • [14] E. Hille, R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, Rhode-Island 1957.
  • [15] W. J. Knight, Solutions of differential equations in B-spaces, Duke Math. J. 41 (1974), 437-442.
  • [16] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 607-614.
  • [17] I. Kubiaczyk, S. Szufla, Kneser's theorem for weak solutions of ordinary differential equations in Banach spaces, Publ. Inst. Mat. Beograd 32 (1982), 99-103.
  • [18] A. R. Mitchell, Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, pp. 387-404 in: Nonlinear Equations in Abstract Spaces, ed. by V. Lakshmikantham 1978.
  • [19] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304.
  • [20] A. Szep, Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6 (1971), 197-203.
  • [21] M. Talagrand, Pettis integral and measure theory, Memoires Amer. Math. Soc. 307 Vol. 51, Amer. Math. Soc., Providence, Rhode-Island 1984.
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Bibliografia
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