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In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behavior of solutions to second-order nonlinear neutral delay differential equations of the form d d t [ r ( t ) [ d d t ( x ( t ) + p ( t ) x ( t - τ ) ) ] α ] + ∑ i = 1 m q i ( t ) H ( x ( t - σ i ) ) = 0       for   t ≥ t 0 &gt; 0 , {d \over {dt}}\left[ {r\left( t \right){{\left[ {{d \over {dt}}\left( {x\left( t \right) + p\left( t \right)x\left( {t - \tau } \right)} \right)} \right]}^\alpha }} \right] + \sum\limits_{i = 1}^m {{q_i}\left( t \right)H\left( {x\left( {t - {\sigma _i}} \right)} \right) = 0\,\,\,{\rm{for}}\,t \ge {t_0} &gt; 0,} under the assumption  ∫ ∞ (r(n)) −1/α dη=∞ . Our main tool is Lebesque’s dominated convergence theorem. Further, some illustrative examples showing the applicability of the new results are included.

tatra mountains mathematical publications

Necessary and Sufficient Conditions for Oscillatory and Asymptotic Behaviour of Solutions to Second-Order Nonlinear Neutral Differential Equations with Several Delays