Nonlinear oscillation of a sublinear delay equation of arbitrary order

Kusano, Takaŝi; Onose, Hiroshi

doi:10.1090/S0002-9939-1973-0324177-5

Nonlinear oscillation of a sublinear delay equation of arbitrary order
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Proc. Amer. Math. Soc. 40 (1973), 219-224 Request permission

Abstract:

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

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