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∈?. 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.
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