Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales

Abstract

Abstract In this paper, oscillation criteria are obtained for higher-order half-linear delay difference equations involving generalized difference operator of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mtext>Δ</m:mtext> <m:mi>b</m:mi> </m:msub> <m:mo>(</m:mo> <m:msub> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:msubsup> <m:mtext>Δ</m:mtext> <m:mi>b</m:mi> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:msub> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> <m:mi>α</m:mi> </m:msup> <m:mo>)</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>q</m:mi> <m:mi>n</m:mi> </m:msub> <m:msubsup> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>σ</m:mi> </m:mrow> <m:mi>β</m:mi> </m:msubsup> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mtext> </m:mtext> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:msub> <m:mi>n</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> </m:mrow> </m:math> $${\Delta _b}({p_n}{(\Delta _b^{m - 1}{x_n})^\alpha }) + {q_n}x_{n - \sigma }^\beta = 0,\qquad n \geq {n_0},$$ where ∆ b is defined by ∆ b yn = y n+1 - by n , b ∈ ℝ - {0}, p: ℕ → ℝ+, α, β are the ratio of odd positive integers with β ≤ α; m, n, n 0, σ are non-negative integers, q: ℕ → ℝ. The cases of b negative and positive and qn ≥ 0, which has important role for oscillation of this equation, are considered. Also we provide some examples to illustrate our main results.