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1996 | 16 | 2 | 171-177
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Existence theorem for the Hammerstein integral equation

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EN
Abstrakty
EN
In this paper we prove an existence theorem for the Hammerstein integral equation
$x(t) = p(t) + λ ∫_I K(t,s)f(s,x(s))ds$, where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.
Twórcy
  • Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
  • Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
Bibliografia
  • [1] J. Appell, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl. 83 (1981), 251-263.
  • [2] J. Appell, Misure di non compattezza in spazi ideali, Rend. Sc. Instituto Lombardo A 119 (1985), 175-186.
  • [3] O. Arino, S. Gautier and J. P. Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkcialaj Ekvac. 27 (1984), 273-279.
  • [4] J.M. Ball, Weak continuity properties of mappings and semi-groups Proc. Royal Soc. Edinbourgh Sect. A 72 (1979), 275-280.
  • [5] M. Cichoń, On bounded weak solutions of a nonlinear differential equation in Banach spaces, Functiones et Approximatio 21 (1992), 27-35.
  • [6] M. Cichoń, Weak solutions of differential equations in Banach spaces, Discuss. Math. Diff. Incl. 15 (1995), 5-14.
  • [7] J. Diestel and J.J. Uhl Jr., Vector Measures, Math. Surveys, Amer. Math. Soc., Providence, Rhode Island (15) (1977).
  • [8] I. Kubiaczyk, On a fixed point for weakly sequentially continuous mappings, Discuss. Math. - Diff. Incl. 15 (1995), 15-20.
  • [9] A.R. Mitchell and Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, in: 'Nonlinear Equations in Abstract Spaces', ed. V. Lakshmikantham, Academic Press (1978), 387-404.
  • [10] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlin. Anal. Th. Meth. Appl. 4 (1980), 985-999.
  • [11] D. O'Regan, Integral equations in reflexive Banach spaces and the weak topologies, Proc. AMS 124 (1996), 607-614.
  • [12] S. Szufla, On the application of measure of noncompactness to existence theorems, Rend Sem. Mat. Univ. Padova, 75 (1986), 1-14.
  • [13] S. Szufla, On the Hammerstein integral equation with weakly singular kernel, Funkcialaj Ekvac. 34 (1991), 279-285.
  • [14] M. Talagrand, Pettis integral and measure theory, Memoires Amer. Math. Soc., Amer. Math. Soc., Providence, Rhode Island 51 (307) (1984).
Typ dokumentu
Bibliografia
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