International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 17, Pages 2769-2774
doi:10.1155/IJMMS.2005.2769
Asymptotic behaviour of solutions of nonlinear delay difference equations in Banach spaces
Anna Kisiolek1
and Ireneusz Kubiaczyk2
1Institute of Mathematics, Poznan University of Technology, 5 Maria Sklodowska-Curie Square, Poznan 60-965, Poland
2Collegium Mathematicum, Adam Mickiewicz University, Umultowska 87, Poznan 61-614, Poland
Abstract
We consider the second-order nonlinear difference equations of the form Δ(rn−1Δxn−1)+pnf(xn−k)=hn. We show that there exists a solution (xn), which possesses the asymptotic behaviour ‖xn−a∑j=0n−1(1/rj)+b‖=o(1), a,b∈ℝ. In this paper, we extend the results of Agarwal (1992), Dawidowski et al. (2001), Drozdowicz and Popenda (1987), M. Migda (2001), and M. Migda and J. Migda (1988). We suppose that f has values in Banach space and satisfies some conditions with respect to the measure of noncompactness and measure of weak noncompactness.