Oscillation of second-order nonlinear neutral delay dynamic equations on time scales

Abstract

In this paper, some sufficient conditions for oscillation of the second-order nonlinear neutral delay dynamic equation ( r ( t ) ( [ y ( t ) + p ( t ) y ( t - τ ) ] Δ ) γ ) Δ + f ( t , y ( t - δ ) ) = 0 , on a time scale T are established; here γ ⩾ 1 is an odd positive integer with r ( t ) and p ( t ) are rd -continuous functions defined on T . Our results as a special case when T = R and T = N , involve and improve some well-known oscillation results for second-order neutral delay differential and difference equations. When T = h N and T = q N = { t : t = q k , k ∈ N , q > 1 } , i.e., for generalized neutral delay difference and q -neutral delay difference equations our results are essentially new and also can be applied on different types of time scales, e.g., T = N 2 = { t 2 : t ∈ N } and T = T n = { t n : n ∈ N 0 } where { t n } is the set of harmonic numbers. Some examples illustrating our main results are given.