A general comparison result for higher order nonlinear difference equations with deviating arguments

Abstract

The authors consider m-th order nonlinear difference equations of the form Dmpxn + δhj(n,xsj(n))=0, j= 1,2, (Ej) where m ≥ 1, n ∈ ℕ = {0,1,2,...}, D0pxn = xn, Dipxn = pniΔ(Di-1pxn), i = 1,2,...,m, Δxn = xn+1 - xn, {Pn1},..., {Pnm are real sequences, pni > 0, and pnm ≡ 1. In Eq. (E1), p = a and pni = ani, and in Eq. (E2), p = A and pni = Ani = 1, 2,..., m. Here, {sj(n)} are sequences of nonnegative integers with sj(n) → ∞ as n → ∞, and hj: ℕ × ℝ → ℝ is continuous with uhj(n, u) > 0 for u ≠ 0. They prove a comparison result on the oscillation of solutions and the asymptotic behavior of nonoscillatory solutions of Eq. (Ej) for j = 1, 2. Examples illustrating the results are also included.

Source or Periodical Title

Journal of Difference Equations and Applications

ISSN

10236198

Page

1033-1052

Document Type

Article

Subject

Comparison theorem, Difference equations with deviating arguments, Nonlinear difference equations, Oscillation

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