Summation Characterization of the Recessive Solution for Half-Linear Difference Equations

Abstract

We show that the recessive solution of the second-order half-linear difference equation Δ(rkΦ(Δxk))+ckΦ(xk+1)=0,   Φ(x):=|x|p−2x,  p>1, where r,c are real-valued sequences, is closely related to the divergence of the infinite series ∑∞(rkxkxk+1|Δxk|p−2)−1.