The purpose of this paper is to study the oscillation of the second-order neutral differential equations of the form (a(t)[z^'(t)]^@c)^'+q(t)x^@b(@s(t))=0, where z(t)=x(t)+p(t)x(@t(t)). We explore properties of given equations by examining those of associated first-order delay equations. New comparison theorems essentially simplify the examination of the equations studied as they allow us to deduce the oscillation of the second-order delay differential equation by applying the oscillation criteria obtained to the first-order delay equations. The results obtained are easy to verify.

Oscillation theorems for second-order nonlinear neutral differential equations

Some oscillation theorems are established for the second-order linear neutral differential equations of mixed type $$ \left(r(t)[x(t)+p_1(t)x(t-\sigma_1)+p_2(t)x(t+\sigma_2)]'\right)'+q_1(t)x(t-\sigma_3) +q_2(t)x(t+\sigma_4)=0. $$ Several examples are also provided to illustrate the main results.

Oscillation results for second-order neutral differential equations of mixed type

Some oscillation criteria are established for the second-order nonlinear neutral differential equations of mixed type [(??(??)

https://downloads.hindawi.com/journals/aaa/2011/387483.pdf

Oscillation Criteria for Certain Second-Order Nonlinear Neutral Differential Equations of Mixed Type

Three comparison theorems are established for the oscillation of the second-order neutral dierential equations of mixed type ` r(t)(x(t)+p 1(t)x(t 1)+p2(t)x(t+ 2)) 0 ´0 +q 1(t)x(t 3)+q2(t)x(t+ 4) = 0. Our results are new even when p2(t) = q2(t) = 0. An example is provided to illustrate the main results.

Comparison theorems for second-order neutral differential equations of mixed type

In this work, some new oscillation criteria are established for a second-order nonlinear mixed neutral differential equation with distributed deviating arguments. Several examples are also provided to illustrate these results.

Oscillation of Second Order Nonlinear Mixed Neutral Differential Equations with Distributed Deviating Arguments

Oscillation Criteria for Second-Order Differential Equations of Neutral Type with Mixed Arguments

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Oscillation Criteria for Second-Order Differential Equations of Neutral Type with Mixed Arguments