Nonlinear oscillation of a sublinear delay equation of arbitrary order

Abstract

The equations considered generalize \[ x ( n ) ( t ) + p ( t ) | x ( g ( t ) ) | α sgn ⁡ x ( g ( t ) ) = 0 , 0 > α > 1. {x^{(n)}}(t) + p(t)|x(g(t)){|^\alpha }\operatorname {sgn} x(g(t)) = 0,\quad 0 > \alpha > 1. \] A necessary and sufficient condition is established that all solutions are oscillatory when n n is even and are either oscillatory or strongly monotone when n n is odd. The result makes clear a difference in oscillatory property between sublinear delay equations and the corresponding ordinary differential equations.