Sufficient conditions for the oscillation and asymptotic beaviour of higher-order dynamic equations of neutral type

摘要

In this paper, we present sufficient conditions for the oscillation and asymptotic behaviour of solutions of delay dynamic equations of the formx(t)+A(t)xα(t)Δn+B(t)xβ(t)=0fort∈t0,∞T,where n∈N,T is a time scale unbounded above, t0∈T,A∈Crd([t0,∞)T,R) satisfies either A∈Crd([t0,∞)T,[0,1]R) with limsupt→∞A(t)<1 or A∈Crd([t0,∞)T,[-1,0]R) with liminft→∞A(t)>-1,B∈Crd([t0,∞)T,R0+) and α,β∈Crd([t0,∞)T,T) are unbounded nondecreasing functions such that α(t),β(t)⩽t for all t∈[t0,∞)T under the condition∫t0∞B(η)Δη=∞..