Classification schemes for positive solutions to a second-order nonlinear difference equation (Q1300794)

The authors classify difference equations of the form \(\Delta(r_ng(\Delta x_n))+f(n,x_n)=0\), \(n=K, K+1,\ldots\), for all eventually positive solutions under the assumptions that \(R_K=\sum_{n=K}^\infty g^{-1}\left(1/r_n\right)<\infty\) or \(=\infty\). Necessary and sufficient conditions for the existence of these solutions are also provided. The following types of functions are used. The function \(f\) is superlinear or sublinear if, for each integer \(n\), \(f(n,x)/x\) is nondecreasing or nonincreasing, respectively, as \(x>0\) increases. The function \(g\) satisfies the \(\omega\)-condition if \(g^{-1}(uv)=\omega g^{-1}(u)g^{-1}(v)\) for some \(\omega>0\) and all \(u\) and \(v\).