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New Oscillation Results of Second-Order Mixed Nonlinear Neutral Dynamic Equations with Damping on Time Scales |
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PP: 137-145 |
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doi:10.18576/jant/050208
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Author(s) |
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H. A. Agwa,
Ahmed M. M. Khodier,
Heba M. Arafa,
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Abstract |
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| This paper is concerned with the oscillatory behavior of all solutions of the second-order mixed nonlinear neutral dynamic
equation with damping
(r(t)f (zD (t)))D + p(t)f (zD (t))+ f (t, x(t1(t)))+g(t, x(t2(t))) = 0,
where f (s) = |s|g−1 and z(t) = x(t)+ p1(t)x(h1(t))+ p2(t)x(h2(t)). Our results complement and improve the results reported in
[[1],[2], [6]-[9], [12], [13], [15]-[20]] because our results can applied to the case where p2(t) 6= 0, p(t) 6= 0,g(t, x(t2(t))) 6= 0 and
a 6=b 6=g , specially the results obtained in [12] and [9] are considered as special cases of our results when taking a =b =g , p(t) =
p2(t) = 0 and g(t, x(t2(t))) = 0 or either g(t, x(t2(t))) = 0 or f (t, x(t1(t))) respectively. An example is given to illustrate our main
result. |
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