Interval oscillation of second-order half-linear functional differential equations

Abstract

By using an inequality due to Hardy, Littlewood and Polya and averaging functions, new interval oscillation criteria are established for the half-linear functional differential equation r(t) y ′(t) α−1y ′(t) ′+q(t) y(τ(t)) α−1y(τ(t))=0, that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [ t 0,∞), rather than on the whole half-line. Our results extend and improve some previous oscillation criteria and handle the cases which are not covered by known results and implies that the delay τ( t)= t± τ does not effect the oscillation, where τ>0 is a constant. In particular, several examples that dwell upon the sharp conditions of our results are also included.