Comparison and oscillatory behavior for certain second order nonlinear dynamic equations

We present some new necessary and sufficient conditions for the oscillation of second order nonlinear dynamic equation $$bigl(abigl(x^{Delta }bigr)^{alpha }bigr)^{Delta }(t)+q(t)x^{beta }(t)=0$$ on an arbitrary time scale $mathbb{T}$ , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on $mathbb{T}$ . Comparison results with the inequality $$bigl(abigl(x^{Delta }bigr)^{alpha }bigr)^{Delta }(t)+q(t)x^{beta }(t)leqslant 0quad (geqslant 0)$$ are established and application to neutral equations of the form $$bigl(a(t)bigl(bigl[x(t)+p(t)x[tau (t)]bigr]^{Delta }bigr)^{alpha }bigr)^{Delta }+q(t)x^{beta }bigl[g(t)bigr]=0$$ are investigated.