Abstract
We study the second-order neutral delay half-linear differential equation , where , and . We use the method of Riccati type substitution and derive oscillation criteria for this equation. By an example of the neutral Euler type equation we show that the obtained results are sharp and improve the results of previous authors. Among others, we improve the results of Sun et al. (Abstr. Appl. Anal. 2012:819342, 2012) and discuss also the case when . MSC:34K11, 34K40.
Highlights
In the paper we study the equation r(t) z (t) + q(t) x σ (t) =, z(t) = x(t) + p(t)x τ (t), ( )where (t) = |t|α– t is the power type nonlinearity, α ≥, which ensures that the function (·) is a convex function on (, ∞)
We study the second-order neutral delay half-linear differential equation [r(t) (z (t))] + q(t) (x(σ (t))) = 0, where (t) = |t|α–1t, α ≥ 1 and z(t) = x(t) + p(t)x(τ (t))
By an example of the neutral Euler type equation we show that the obtained results are sharp and improve the results of previous authors
Summary
The paper [ ] suggests to imagine a child, which begins to grow more rapidly at the age of about years, growing more and more rapidly until a certain height is approached, at which time there is a rapid slowing of the growth, stopping at the adult height dictated by genes This process can be modeled by neutral equation. Section contains main results of the paper and examples which prove that we provide sharp oscillation constant for Euler type differential equation. These criteria are expressed in terms of positive mutually conjugate numbers l and l∗, the multiplicative factor φ, and a function Q(t)
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