Oscillation criteria for higher order quasilinear dynamic equations with Laplacians and a deviating argument

Abstract

In this paper, we deal with the oscillation of the solutions of the higher order quasilinear dynamic equation with Laplacians and a deviating argument in the form of
(x[n−1]
)(t) + p(t)φγ (x(g(t))) = 0
on an above-unbounded time scale, where n ≥ 2,
x[i]
(t) := ri(t)φαi
x[i−1]
(t)

, i = 1, 2, . . ., n − 1, with x[0] = x.
By using a generalized Riccati transformation and integral averaging technique, we establish some new
oscillation criteria for the cases when n is even and odd, and when α > γ , α = γ , and α < γ , respectively, with α = α1 ···αn−1 and without any restrictions on the time scale.

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