Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities (Q395398)

The paper considers homoclinic solutions for the periodic difference equation NEWLINE\[NEWLINE Lu_n - \omega u_n = \sigma g_n(u_n), NEWLINE\]NEWLINE where \((Lu)_n = a_nu_{n+1} + b_nu_n + a_{n-1}u_{n-1}\) is the Jacobi operator, \(\{a_n\}_n\), \(\{b_n\}_n\), \(\{g_n(\cdot)\}_n\) are \(T\)-periodic sequences. The nonlinear functions \(g_n:\mathbb R\to \mathbb R\) are subject to the following assumptions:NEWLINENEWLINEi) they are continuous;NEWLINENEWLINEii) the following holds NEWLINE\[NEWLINE\lim_{|u|\rightarrow\infty}\left(g_n(u)u - 2G_n(u)\right) = +\infty\;,\;G_n(u) = \int_0^ug_n(s)ds; NEWLINE\]NEWLINE iii) \(g_n(u)/u\) is strictly increasing on \((0,\infty)\) and strictly decreasing on \((-\infty,0)\), also NEWLINE\[NEWLINE\lim_{|u|\rightarrow 0}{{g_n(u)}\over{u}} = 0\;;\;\lim_{|u|\rightarrow\infty} {{g_n(u)}\over{u}} = d_n<+\infty. NEWLINE\]NEWLINE The main results are as follows:NEWLINENEWLINETheorem 1. Let \(\sigma =1\) and \(\omega\in(\alpha,\beta)\), where \((\alpha,\beta)\) is a spectral gap of the operator \(L\), \(\beta<+\infty\). If \(g_n(u)\) satisfies i)--iii) with \(d_n\equiv d_*>\beta-\omega\), then the system has at least a non-trivial solution in \(l^2\). This solution is such that there exists \(C>0\), \(\tau>0\) such that NEWLINE\[NEWLINE|u_n|\leq Ce^{-\tau|n|}\;,\;n\in \mathbb Z. NEWLINE\]NEWLINE Theorem 2. The system has no non-trivial solutions in \(l^2\) under one of the following three conditions:NEWLINENEWLINE(1) \(\sigma = 1\), \(\beta = +\infty\);NEWLINENEWLINE (2) \(\sigma = -1\), \(\alpha=-\infty\);NEWLINENEWLINE (3) \(d_*<\min\{ |\alpha-\omega|,|\beta-\omega|\}\).