Geodesic
Necessary and Sufficient Condition for Oscillatory and Asymptotic Behaviour of Second-Order Functional Differential Equations
Kragujevac Journal of Mathematics, Tome 44 (2020) no. 3, p. 459

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In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behaviour of solutions of second-order neutral delay differential equations of the form \begin{equation} \frac{d}{dt}eft[r(t)\frac{d}{dt}[x(t)+p(t)x(au(t))]\right]+q(t)Geft(x(igma(t))\right)=0, \quadext{for} t\geq{}t_{0},otag \end{equation} under the assumption $\int^{\infty}\frac{1}{r(\eta)}d\eta=\infty$ for various ranges of the bounded neutral coefficient $p$. Our main tools are Lebesgue's dominated convergence theorem and Banach's contraction mapping principle. Further, an illustrative example showing the applicability of the new results is included.
Classification : 34C10 34C15, 35K40
Keywords: Oscillation, nonoscillation, neutral, delay, nonlinear, Lebesgue's dominated convergence theorem, Banach's contraction mapping principle 
@article{KJM_2020_44_3_a10,
     author = {Shyam Sundar Santra},
     title = {Necessary and {Sufficient} {Condition} for {Oscillatory} and {Asymptotic} {Behaviour} of {Second-Order} {Functional} {Differential} {Equations}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {459 },
     publisher = {mathdoc},
     volume = {44},
     number = {3},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2020_44_3_a10/}
}
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Shyam Sundar Santra. Necessary and Sufficient Condition for Oscillatory and Asymptotic Behaviour of Second-Order Functional Differential Equations. Kragujevac Journal of Mathematics, Tome 44 (2020) no. 3, p. 459 . http://geodesic.mathdoc.fr/item/KJM_2020_44_3_a10/