Summation inequalities and half-linear difference equations

The aim of this paper is to present new summation inequalities and to show that the half-linear difference equation Δ(an|Δxn|αsgnΔxn)=bn|xn+1|αsgnxn+1 exhibits not only similarities but also some discrepancies with the linear case, especially as regards the possible coexistence of the so-called extremal solutions. Some of our results are new also for the linear difference equation.