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

  • Authors:

  • S. H. Saker

  • Affiliations:

  • Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2006

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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 = hN and T = qN = {t: t = qk, 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 = N2 = {t2 : t ∈ N} and T = Tn = {tn: n ∈ N0} where {tn} is the set of harmonic numbers. Some examples illustrating our main results are given.