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Tytuł artykułu

Asymptotic behaviour of solutions of difference equations in Banach spaces

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EN
In this paper we consider the first order difference equation in a Banach space
$Δx_{n} = ∑_{i=0}^∞ a^{i}_{n} f(x_{n+i})$.
We show that this equation has a solution asymptotically equal to a.
As an application of our result we study the difference equation
$Δx_{n} = ∑_{i=0}^∞ a^i_{n}g(x_{n+i}) + ∑_{i=0}^∞ b^{i}_{n}h(x_{n+i}) + y_{n}$
and give conditions when this equation has solutions.
In this note we extend the results from [8,9]. For example, in [9] the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure of noncompactness and measure of weak noncompactness.
Twórcy
  • Technical University of Poznań, Piotrowo 3, PL-60-965 Poznań, Poland
Bibliografia
  • [1] O. Arino, S. Gautier and J.P. Penot, A fixed point theorem for sequentially continous mappings with application to ordinary differential equations, Func. Ekvac. 27 (1984), 273-279.
  • [2] J.M. Ball, Properties of mappings and semigroups, Proc. Royal. Soc. Edinburg Sect. (A) 72 (1973/74), 275-280.
  • [3] J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, New York-Basel, 1980.
  • [4] J. Banaś and J. Rivero, Measures of weak noncompactness, Ann. Math. Pura Appl. 125 (1987), 213-224.
  • [5] G. Darbo, Punti uniti in transformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92.
  • [6] M. Dawidowski, I. Kubiaczyk and J. Morchało, A discrete boundary value problem in Banach spaces, Glasnik Mathematicki, 36 56(2001), 233-239.
  • [7] F.S. de Blasi, On a property of the unit sphere in Banach space, Bull. Math. Soc. Sci. Math. R.S. Raumannie 21 (1997), 259-262.
  • [8] C. Gonzalez and A. Jimenez-Melado, An application of Krasnoselskii fixed point theorem to the asymptotic behavior of solutions of difference equations in Banach spaces, J. Math. Anal. Appl. 247 (2000), 290-299.
  • [9] C. Gonzalez and A. Jimenez-Melado, Asymptitic behaviour of solutions of difference equations in Banach spaces, Proc. Amer. Math. Soc., 128 (6) (2000), 1743-1749.
  • [10] I. Kubiaczyk, On a fixed point theorem for weakly sequentially continuous mapping, Discuss. Math. Diff. Incl. 15 (1995), 15-20.
  • [11] A.R. Mitchell and C. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, Nonlinear equations in abstract spaces, V. Lakshmikantham, ed. 387-404, Orlando, 1978.
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1093
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