# Asymptotic behaviour of solutions of difference equations in Banach spaces

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

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

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topAnna 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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- [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] 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] I. Kubiaczyk, On a fixed point theorem for weakly sequentially continuous mapping, Discuss. Math. Diff. Incl. 15 (1995), 15-20. Zbl0832.47046
- [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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