On unbounded positive solutions of second-order difference equations with a singular nonlinear term

In this paper, several necessary and/or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equa tions of the form Delta(r(n)Delta x(n)) + f(n, x(n)) = 0, n is an element o f N(n(0)), where N(n(0)) = {n(0), n(0) + 1,...,}(n(0) is a fixed nonnegativ e integer), r(n)> 0 for n greater than or equal to n(0) such that Sigma(n=n 0)(infinity) 1/r(n) = infinity, and f: N(n(0)) X R --> R is a real-valued c ontinuous function such that f(n, x) greater than or equal to 0 for all (n, x) is an element of N(n(0)) x (0, infinity). (C) 2000 Academic Press.