Asymptotic behaviour of solutions of difference equations in Banach spaces

Anna Kisiołek

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2008)

  • Volume: 28, Issue: 1, page 5-13
  • ISSN: 1509-9407

Abstract

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In this paper we consider the first order difference equation in a Banach space Δ x n = ∑ i = 0 ∞ a n i 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 Δ x n = ∑ i = 0 ∞ a n i g ( x n + i ) + ∑ i = 0 ∞ b n i 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.

How to cite

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Anna Kisiołek. "Asymptotic behaviour of solutions of difference equations in Banach spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 28.1 (2008): 5-13. <http://eudml.org/doc/271171>.

@article{AnnaKisiołek2008,
abstract = {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.},
author = {Anna Kisiołek},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {Banach space; difference equation; fixed point; measure of noncompactness; asymptotic behaviour of solutions; first-order difference equation},
language = {eng},
number = {1},
pages = {5-13},
title = {Asymptotic behaviour of solutions of difference equations in Banach spaces},
url = {http://eudml.org/doc/271171},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Anna Kisiołek
TI - Asymptotic behaviour of solutions of difference equations in Banach spaces
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2008
VL - 28
IS - 1
SP - 5
EP - 13
AB - 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.
LA - eng
KW - Banach space; difference equation; fixed point; measure of noncompactness; asymptotic behaviour of solutions; first-order difference equation
UR - http://eudml.org/doc/271171
ER -

References

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  6. [6] M. Dawidowski, I. Kubiaczyk and J. Morchało, A discrete boundary value problem in Banach spaces, Glasnik Mathematicki, 36 56(2001), 233-239. Zbl1011.39002
  7. [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. Zbl0365.46015
  8. [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. Zbl0962.39007
  9. [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. Zbl0997.39005
  10. [10] I. Kubiaczyk, On a fixed point theorem for weakly sequentially continuous mapping, Discuss. Math. Diff. Incl. 15 (1995), 15-20. Zbl0832.47046
  11. [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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