Abstract
We study the second-order neutral half-linear differential equation and formulate new oscillation criteria for this equation, which are obtained through the use of the modified Riccati technique. In the first statement, the oscillation of the equation is ensured by the divergence of a certain integral. The second one provides the condition of the oscillation in the case where the relevant integral converges, and it can be seen as a Hille–Nehari-type criterion. The use of the results is shown in several examples, in which the Euler-type equation and its perturbations are considered.
Highlights
We study the oscillatory properties of the second-order half-linear neutral differential equation
We suppose that the coefficients of the equation satisfy the usual conditions: r ∈ C ([t0, ∞), R+ ), b ∈ C1 ([t0, ∞), R0+ ), c ∈ C ([t0, ∞), R0+ ), c is not identically equal to zero in any neighborhood of infinity, and Academic Editor: Jozef Džurina b(t) ≤ 1
Concerning the deviating arguments, we assume that τ, σ ∈ C1 ([t0, ∞), R), lim τ (t) = ∞, Accepted: 20 January 2021
Summary
We study the oscillatory properties of the second-order half-linear neutral differential equation. 0 r (t)Φ(z0 (t)) + c(t)Φ( x (τ (t))) = 0, Citation: Pátíková, Z.; Fišnarová, S. Use of the Modified Riccati Technique for Neutral Half-Linear Differential. We suppose that the coefficients of the equation satisfy the usual conditions: r ∈ C ([t0 , ∞), R+ ), b ∈ C1 ([t0 , ∞), R0+ ), c ∈ C ([t0 , ∞), R0+ ), c is not identically equal to zero in any neighborhood of infinity, and Academic Editor: Jozef Džurina b(t) ≤ 1. Concerning the deviating arguments, we assume that τ, σ ∈ C1 ([t0 , ∞), R), lim τ (t) = ∞, Accepted: 20 January 2021. We suppose that the coefficients of the equation satisfy the usual conditions: r ∈ C ([t0 , ∞), R+ ), b ∈ C1 ([t0 , ∞), R0+ ), c ∈ C ([t0 , ∞), R0+ ), c is not identically equal to zero in any neighborhood of infinity, and Academic Editor: Jozef Džurina b(t) ≤ 1. (2)
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