Nonoscillation and oscillation of second order half-linear differential equations

We study the oscillation problems for the second order half-linear differential equation [p(t)Φ(x )] + q(t)Φ(x) = 0, where Φ(u) = |u|r−1u with r > 0, 1/p and q are locally integrable on R+; p >0, q 0 a.e. on R+, and ∞0 p−1/r (t) dt =∞. We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p ≡ 1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J.Math. Anal. Appl. 210 (1997) 712–723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180–188] on linear equations to the half-linear case for all r > 0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39–47] on nonoscillation when 0 < r <1 and on oscillation when r > 1. The approach in this paper can also be used to fully extend Elbert’s criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363–373]. © 2006 Elsevier Inc. All rights reserved.