Asymptotic Behavior of Solutions of Emden-Fowler Difference Equations with Oscillating Coefficients

It is known that if ∑∞j |pj| < ∞ then the Emden-Fowler difference equation (A) Δ2yn−1 = pnyγn (γ > 0) has a positive solution {yn}, defined for n sufficiently large, such that limn→∞yn = c > 0, while if ∑∞jγ |pj| < ∞ then (A) has a positive solution }yn}, defined for n sufficiently large, such that limn→∞Δyn = c > 0. Here it is shown that these conclusions hold if the series converge (perhaps conditionally) and satisfy secondary conditions which do not imply absolute convergence. Estimates of }yn} and }Δyn} as n → ∞ are also given. Moreover, γ can be any real number other than 0 or 1.