Research Article Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities (original) (raw)
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Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities
Advances in Difference Equations, 2011
This paper is concerned with some oscillation criteria for the second order neutral delay dynamic equations with mixed nonlinearities of the form r t u t Δ q t |x τ t | α−1 x τ t n i 1 q i t |x τ i t | αi−1 x τ i t 0, where t ∈ T and u t | x t p t x δ t Δ | α−1 x t p t x δ t Δ with α 1 > α 2 > • • • > α m > α > α m 1 > • • • > α n > 0. Further the results obtained here generalize and complement to the results obtained by Han et al. 2010. Examples are provided to illustrate the results.
New oscillation criteria for second-order nonlinear neutral delay differential equations
2009
In this paper, several new oscillation criteria for the second-order nonlinear neutral delay differential equation [r(t)|(x(t)+p(t)x[@s(t)])^'|^m^-^1(x(t)+p(t)x[@s(t)])^']^'+q(t)f(x[@t(t)])=0,t>=t"0 are established. These oscillation criteria extend and improve some known results. An interesting example illustrating the importance of our results is also provided.
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In this paper, new sufficient conditions are obtained for oscillation of second-order neutral delay differential equations of the form d dt r(t) d dt x(t) + p(t)x(t − τ) + q(t)G x(t − σ 1) + v(t)H x(t − σ 2) = 0, t ≥ t 0 , under the assumptions ∞ 0 dη r(η) = ∞ and ∞ 0 dη r(η) < ∞ for |p(t)| < +∞. Two illustrative examples are included.
On oscillatory nonlinear second order neutral delay differential equations
Differential Equations and Applications, 2016
In this work, we investigate the oscillation criteria for second order neutral delay differential equations of the form (r(t)[y(t) + p(t)y(δ (t))]) + q(t)G(y(τ(t))) = 0 and (r(t)[[y(t) + p(t)y(δ (t))] ] α) + q(t)(y β (τ(t))) = 0, where α and β are the ratio of odd positive integers.
Computers & mathematics with applications, 2010
Interval oscillation criteria are established for second-order forced delay dynamic equations on time scales containing mixed nonlinearities of the form where T is a time scale, t0∈ T a fixed number;[Formula: see text] is a time scale interval; Φ∗(u)=| u|∗− 1u; the functions [Formula: see text] are right-dense continuous with r> 0 nondecreasing; τk: T→ T are nondecreasing right-dense continuous with τk (t)≤ t, limt→∞ τk (t)=∞; and the exponents satisfy All results are new even for T= R and T= Z. Analogous results for related advance ...
Abstract and Applied Analysis, 2011
We establish some new oscillation criteria for the second-order neutral delay dynamic equations of Emden-Fowler type, a t x t r t x τ t Δ Δ p t x γ δ t 0, on a time scale unbounded above. Here γ > 0 is a quotient of odd positive integers with a and p being real-valued positive functions defined on T. Our results in this paper not only extend and improve the results in the literature but also correct an error in one of the references.