# Asymptotic behaviour of solutions of difference equations in Banach spaces

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

top

## Abstract

top
In this paper we consider the first order difference equation in a Banach space $\Delta {x}_{n}={\sum }_{i=0}^{\infty }{a}_{n}^{i}f\left({x}_{n+i}\right)$. We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation $\Delta {x}_{n}={\sum }_{i=0}^{\infty }{a}_{n}^{i}g\left({x}_{n+i}\right)+{\sum }_{i=0}^{\infty }{b}_{n}^{i}h\left({x}_{n+i}\right)+{y}_{n}$ 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

top

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

top
1. [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. Zbl0599.34008
2. [2] J.M. Ball, Properties of mappings and semigroups, Proc. Royal. Soc. Edinburg Sect. (A) 72 (1973/74), 275-280.
3. [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. Zbl0441.47056
4. [4] J. Banaś and J. Rivero, Measures of weak noncompactness, Ann. Math. Pura Appl. 125 (1987), 213-224. Zbl0653.47035
5. [5] G. Darbo, Punti uniti in transformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92. Zbl0064.35704
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.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.