New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients

Abstract

We establish the oscillation and asymptotic criteria for the second-order neutral delay differential equations with positive and negative coefficients having the forms $\left[x\right(t)+{\sum }_{i\in R}{c}_i(t\left)x\right({\alpha }_i\left(t\right)){]}^{\mathrm{\prime \prime }}+r(t\left)\right[x\left(t\right)+{\sum }_{i\in R}{c}_i\left(t\right)x\left({\alpha }_i\right(t\left)\right){]}^{\prime }+{\sum }_{i\in P}{p}_i\left(t\right)x\left({\beta }_i\right(t\left)\right)-{\sum }_{i\in Q}{q}_i\left(t\right)x\left({\gamma }_i\right(t\left)\right)=0$ and $\left[x\right(t)+{\sum }_{i\in R}{c}_i(t\left)x\right({\alpha }_i\left(t\right)){]}^{\mathrm{\prime \prime }}+r(t\left)\right[x\left(t\right)+{\sum }_{i\in R}{c}_i\left(t\right)x\left({\alpha }_i\right(t\left)\right){]}^{\prime }+{\sum }_{i\in P}{p}_i\left(t\right)x\left({\beta }_i\right(t\left)\right)-{\sum }_{i\in Q}{q}_i\left(t\right)x\left({\gamma }_i\right(t\left)\right)=f\left(t\right)$. The obtained new oscillation criteria extend and improve the recent results given in the paperof B. Karpuz et al. (2009).