Oscillations for Nonlinear Neutral Delay Differential Equations with Variable Coefficients

Ahmed, Fatima N.; Ahmad, Rokiah Rozita; Din, Ummul Khair Salma; Noorani, Mohd Salmi Md

doi:10.1155/2014/179195

Cited by 2 publications

(4 citation statements)

References 14 publications

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“…Hence, every solution of Equation oscillates on the half timescale interval

false[4, {\infty false)}_{T}

. On the other hand, we note that none of the existing results in the above literature can be applied to Equation because of an opposite sign of variable coefficient R ( t ) < 0 and the function

f false(x false(t false) false) ≇ x false(t false)

for t ∗ ≤ t , for instance, theorems 3, 5, and 8 in Ahmed et al, theorems 2.1 to 2.6 in Kubiaczyk, theorem 2.1 in Lin and Tang, and theorem 3.1 in Zhang and Yu . Therefore, it is right to speak that the method applied here is different than the ones employed in the literature.…”

Section: Resultsmentioning

confidence: 90%

“…The authors have proved that if there exists λ > η −1 ln ( ξ ) such that

{lim inf}_{t \to \infty} false[T false(t false) e x p false(- e^{λ t} false) false] > 0

, then every solution oscillates. Later in 2014, a similar nonlinear neutral delay differential equation has been considered by Ahmed et al of the form

{false(a false(t false) x false(t false) - R false(t false) x false(t - η false) false)}^{'} + T false(t false) true \prod_{i = 1}^{n} false| x false(t - τ_{i} false) |^{α_{i}} s i g n false(x false(t - τ_{i} false) false) = 0,

where a is a positive continuous function on

false[t_{0}, {\infty false)}_{T}

, and they have studied the oscillation criterion. Apart from all above oscillation criteria, there are also other oscillation criteria, which are known as “Kamenev‐type” and “Philos‐type” oscillation criteria.…”

Section: Introductionmentioning

confidence: 99%

“…Before delving into our contributions, we provide some background details that motivate our study. We refer the readers to the papers for the typical results, for instance, Ahmed et al, Graef et al, Erbe and Zhang, Graef et al, Kubiaczyk et al, Lin and Tang, Luo and Shen, Stavroulakis, Wang, and Zhang and Yu, and references cited therein, where several particular cases of Equation have been discussed for oscillation. Particularly, Zhang and Yu have considered the following first‐order nonlinear delay differential equation

{false(x false(t false) - R false(t false) x false(t - η false) false)}^{'} + T false(t false) true \prod_{i = 1}^{n} false| x false(t - τ_{i} false) |^{α_{i}} s i g n false(x false(t - τ_{i} false) false) = 0,

where

R false(t false), T false(t false) \in C false(false[t_{0}, \infty false), double-struckR false), τ_{i}, η \in false(0, \infty false), α_{i} \geq 0

and

\sum_{i = 1}^{n} α_{i} = 1

.…”

Section: Introductionmentioning

confidence: 99%

“…Before delving into our contributions, we provide some background details that motivate our study. We refer the readers to the papers for the typical results, for instance, Ahmed et al, 5 Graef et al, 6 Erbe and Zhang, 7 Graef et al, 8 Kubiaczyk et al, 9 Lin and Tang, 10 Luo and Shen, 11 Stavroulakis, 12 Wang, 13 and Zhang and Yu, 14 and references cited therein, where several particular cases of Equation 1 have been discussed for oscillation. Particularly, Zhang and Yu 14 have considered the following first-order nonlinear delay differential equation…”

mentioning

confidence: 99%

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Oscillation for a nonlinear neutral dynamic equations on time‐scales with variable exponents

Negi

Abbas

Malik

2019

Math Methods in App Sciences

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In the present manuscript, we study the oscillation theory for a first‐order nonlinear neutral dynamic equations on timescales with variable exponents of the form false(xfalse(σfalse(tfalse)false)−Rfalse(tfalse)xξfalse(t−ηfalse)false)Δ+Tfalse(tfalse)true∏m=1nfalse|fmfalse(xfalse(t−τmfalse)false)|αmfalse(tfalse)signfalse(xfalse(t−τmfalse)false)=0,2em∀0.1emt∈false[t∗,∞false)T, where ξ is a quotient of odd positive integers; t∗∈double-struckT be a fixed number; false[t∗,∞false)T is a timescale interval; η,τm > 0; fm∈Cfalse(double-struckR,double-struckRfalse) for m = 1,2,…,n such that xfmfalse(xfalse)>03.0235pt∀x∈double-struckR\false{0false};R,T∈Crdfalse(false[t∗,∞false)T,double-struckRfalse), and the variable exponents αm(t) satisfy ∑m=1nαmfalse(tfalse)=1. The principal goal of this paper is to establish some new succinct sufficient conditions for oscillation. Furthermore, we introduce a forcing term normalΞfalse(·,xfalse(·false)false)∈Crdfalse(double-struckT×double-struckR,double-struckRfalse) and then study the oscillation. Afterward, some interesting special cases are also studied to obtain similar sufficient conditions of oscillation under certain conditions. Moreover, the oscillatory behaviour of the solutions of a first‐order neutral dynamic equation on timescale with a nonlocal condition and a forced nonlinear neutral dynamic equation on time scale are studied. But the proofs are based on the prior estimates obtained in this paper. Some enthralling examples are constructed to show the effectiveness of our analytic results. These counterparts are quite different in the literature even when double-struckT=double-struckR. Finally, the Kamenev‐type and Philos‐type oscillation criterions are established.

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“…Hence, every solution of Equation oscillates on the half timescale interval

false[4, {\infty false)}_{T}

. On the other hand, we note that none of the existing results in the above literature can be applied to Equation because of an opposite sign of variable coefficient R ( t ) < 0 and the function

f false(x false(t false) false) ≇ x false(t false)

Section: Resultsmentioning

confidence: 90%

“…The authors have proved that if there exists λ > η −1 ln ( ξ ) such that

{lim inf}_{t \to \infty} false[T false(t false) e x p false(- e^{λ t} false) false] > 0

, then every solution oscillates. Later in 2014, a similar nonlinear neutral delay differential equation has been considered by Ahmed et al of the form

{false(a false(t false) x false(t false) - R false(t false) x false(t - η false) false)}^{'} + T false(t false) true \prod_{i = 1}^{n} false| x false(t - τ_{i} false) |^{α_{i}} s i g n false(x false(t - τ_{i} false) false) = 0,

where a is a positive continuous function on

false[t_{0}, {\infty false)}_{T}

Section: Introductionmentioning

confidence: 99%

{false(x false(t false) - R false(t false) x false(t - η false) false)}^{'} + T false(t false) true \prod_{i = 1}^{n} false| x false(t - τ_{i} false) |^{α_{i}} s i g n false(x false(t - τ_{i} false) false) = 0,

where

R false(t false), T false(t false) \in C false(false[t_{0}, \infty false), double-struckR false), τ_{i}, η \in false(0, \infty false), α_{i} \geq 0

and

\sum_{i = 1}^{n} α_{i} = 1

.…”

Section: Introductionmentioning

confidence: 99%

“…Before delving into our contributions, we provide some background details that motivate our study. We refer the readers to the papers for the typical results, for instance, Ahmed et al, 5 Graef et al, 6 Erbe and Zhang, 7 Graef et al, 8 Kubiaczyk et al, 9 Lin and Tang, 10 Luo and Shen, 11 Stavroulakis, 12 Wang, 13 and Zhang and Yu, 14 and references cited therein, where several particular cases of Equation 1 have been discussed for oscillation. Particularly, Zhang and Yu 14 have considered the following first-order nonlinear delay differential equation…”

mentioning

confidence: 99%

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Oscillation for a nonlinear neutral dynamic equations on time‐scales with variable exponents

Negi

Abbas

Malik

2019

Math Methods in App Sciences

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Necessary and sufficient conditions for oscillation of nonlinear first-order forced differential equations with several delays of neutral type

Pinelas

Santra

2019

Analysis

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In this work, necessary and sufficient conditions are obtained such that every solution of nonlinear neutral first-order differential equations with several delays of the form\bigl{(}x(t)+r(t)x(t-\tau)\bigr{)}^{\prime}+\sum_{i=1}^{m}\phi_{i}(t)H\bigl{(}% x(t-\sigma_{i})\bigr{)}=f(t)is oscillatory or tends to zero as {t\rightarrow\infty.} This problem is considered in various ranges of the neutral coefficient r. Finally, some illustrating examples are presented to show that feasibility and effectiveness of main results.

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Oscillations for Nonlinear Neutral Delay Differential Equations with Variable Coefficients

Abstract: A class of nonlinear neutral delay differential equations is considered. Some new oscillation criteria of all solutions are derived. The obtained results generalize and extend some of well known previous results in the literature.

Cited by 2 publications

References 14 publications

Oscillation for a nonlinear neutral dynamic equations on time‐scales with variable exponents

Oscillation for a nonlinear neutral dynamic equations on time‐scales with variable exponents

Necessary and sufficient conditions for oscillation of nonlinear first-order forced differential equations with several delays of neutral type

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