Asymptotic behaviour of solutions of nonlinear delay difference equations (Q2756999)

The two main results of this paper are concerned with the existence of asymptotically constant and asymptotically linear solutions of NEWLINE\[NEWLINE\Delta(r_{n-1} \Delta x_{n-1})+ a_nf(x_{n-k})= b_n,\;n=1,2,\dots, \tag{1}NEWLINE\]NEWLINE where \(k\geq 0\), \(\{r_n\}\) is a positive sequence, \(\{a_n\}\) and \(\{b_n\}\) are real sequences and \(f\) is a real continuous function. Roughly, let NEWLINE\[NEWLINER_n= \sum^{n-1}_{j=0} {1\over r_j},\;n=1,2, \dots,NEWLINE\]NEWLINE then under the conditions \(\sum^\infty_{n=1} R_n|a_n|<\infty\) and \(\sum^\infty_{n=1} R_n|b_n|<\infty\), equation (1) has a solution which tends to any given number. In addition, if \(f\) is uniformly continuous and bounded, then given any \(c\) and \(d\), there exists a solution \(\{x_n\}\) such that \(x_n=cR_n +d+o(1)\). Schauder's fixed point theorem is employed.