Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients

Abstract

Positive decreasing solutions of the nonlinear difference equation
$$
\Delta(p_n|\Delta x_n|^{\alpha-1}\Delta x_n) = q_n|x_{n+1}|^{\beta-1}x_{n+1} , \quad n\geq1,
$$
are studied under the assumption that $p,\,q$ are regularly varying sequences. Necessary and sufficient conditions are established for the existence of regularly varying strongly decreasing solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique formula.