Báo Cáo Báo cáo hóa học NONOSCILLATORY HALF-LINEAR DIFFERENCE EQUATIONS AND RECESSIVE SOLUTIONS MARIELLA CEC

NONOSCILLATORY HALF-LINEAR DIFFERENCE EQUATIONS AND RECESSIVE SOLUTIONS MARIELLA CECCHI, ZUZANA DOSLA, AND MAURO MARINI Received 30 January 2004 and in revised form 26 May 2004 Recessive and dominant solutions for the nonoscillatory half-linear difference equation are investigated. By using a uniqueness result for the zero-convergent solutions satisfying a suitable final condition, we prove that recessive solutions are the “smallest solutions in a neighborhood of infinity,” like in the linear case. Other asymptotic properties of recessive and dominant solutions are treated too.

1. Introduction

Consider the second-order half-linear difference equation ∆ an Φ ∆xn + bn Φ xn+1 = 0, (1.1) where ∆ is the forward difference operator ∆xn = xn+1 ư xn , Φ(u) = |u| pư2 u with p 1, and a = {an }, b = {bn } are positive real sequences for n ≥ 1. It is known that there is a surprising similarity between (1.1) and the linear difference equation ∆ an ∆xn + bn