Asymptotic behaviour of solutions of nonlinear delay difference equations in Banach spaces

We consider the second-order nonlinear difference equations of the form ∆(rn−1∆xn−1) + pn f (xn−k) = hn. We show that there exists a solution (xn), which possesses the asymptotic behaviour ‖xn − a ∑n−1 j=0 (1/r j) + 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.