Some necessary and sufficient conditions for oscillation of neutral differential equations

1996 ◽  
Vol 60 (1-2) ◽  
pp. 133-148
Author(s):  
L. Erbe ◽  
Qinckai Kong
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Neutral Differential Equations ◽  
Necessary And Sufficient

scholarly journals Oscillations of Third Order Half Linear Neutral Differential Equations

2015 ◽  
Vol 12 (3) ◽  
pp. 625-631
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Oscillation Criterion ◽  
Necessary And Sufficient Conditions ◽  
Third Order ◽  
Neutral Differential Equations ◽  
The Third ◽  
All Solutions ◽  
Necessary And Sufficient

In this paper the oscillation criterion was investigated for all solutions of the third-order half linear neutral differential equations. Some necessary and sufficient conditions are established for every solution of (a(t)[(x(t)±p(t)x(?(t) ) )^'' ]^? )^'+q(t) x^? (?(t) )=0, t?t_0, to be oscillatory. Examples are given to illustrate our main results.

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

2020 ◽  
Vol 75 (1) ◽  
pp. 121-134 ◽  
Author(s):  
Shyam Sundar Santra
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Neutral Delay ◽  
Neutral Differential Equations ◽  
Several Delays ◽  
Necessary And Sufficient

AbstractIn 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 \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} > 0,}under the assumption ∫∞(r(n))−1/αdη=∞. Our main tool is Lebesque’s dominated convergence theorem. Further, some illustrative examples showing the applicability of the new results are included.

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