scholarly journals Necessary and sufficient conditions for the oscillation of a first-order neutral differential equation of Euler form

2011 ◽  
Vol 41 (5) ◽  
pp. 1457-1469
Author(s):  
Kaizhong Guan ◽  
Jianhua Shen
Keyword(s):  
Differential Equation ◽  
Sufficient Conditions ◽  
Neutral Differential Equation ◽  
Necessary And Sufficient Conditions ◽  
First Order ◽  
Euler Form ◽  
Necessary And Sufficient

scholarly journals Oscillation and nonoscillation of a delay differential equation

1994 ◽  
Vol 49 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Chunhai Kou ◽  
Weiping Yan ◽  
Jurang Yan
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
First Order ◽  
Oscillating Coefficients ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Oscillation And Nonoscillation

In this paper, some necessary and sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the formare established. Several applications of our results improve and generalise some of the known results in the literature.

scholarly journals Bounded Oscillation of a Forced Nonlinear Neutral Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-33 ◽  
Author(s):  
Zeqing Liu ◽  
Yuguang Xu ◽  
Shin Min Kang ◽  
Young Chel Kwun
Keyword(s):  
Differential Equation ◽  
Sufficient Conditions ◽  
Neutral Differential Equation ◽  
Necessary And Sufficient Conditions ◽  
Bounded Solutions ◽  
Necessary And Sufficient

This paper is concerned with thenth-order forced nonlinear neutral differential equation[x(t)-p(t)x(τ(t))](n)+∑i=1mqi(t)fi(x(σi1(t)),x(σi2(t)),…,x(σiki(t)))=g(t),  t≥t0. Some necessary and sufficient conditions for the oscillation of bounded solutions and several sufficient conditions for the existence of uncountably many bounded positive and negative solutions of the above equation are established. The results obtained in this paper improve and extend essentially some known results in the literature. Five interesting examples that point out the importance of our results are also included.

Exact regions of oscillation for a neutral differential equation

2000 ◽  
Vol 130 (2) ◽  
pp. 277-286
Author(s):  
S. S. Cheng ◽  
Y. Z. Lin
Keyword(s):  
Differential Equation ◽  
Sufficient Conditions ◽  
Neutral Differential Equation ◽  
Necessary And Sufficient Conditions ◽  
All Solutions ◽  
Constant Coefficients ◽  
Complete Set ◽  
Necessary And Sufficient

This paper is concerned with a neutral differential equation with four constant coefficients, one delay and one advancement. By means of the theory of envelopes, we consider all possible values of the parameters involved in the equation and obtain a complete set of necessary and sufficient conditions for all solutions to be oscillatory.

scholarly journals A note on a paper of Bowcock and Yu

1989 ◽  
Vol 40 (3) ◽  
pp. 421-424
Author(s):  
I. P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Order ◽  
All Solutions ◽  
Necessary And Sufficient

Consider the first order differential equation (1) , where pi, and τi, for i = 1,…,n, are positive constants. To find necessary and sufficient conditions, in terms of the coefficients and the delays only, under which all solutions of (1) oscillate, is a problem of great importence. In a recent paper, Bowcock and Yu claimed that is a necessary and sufficient condition for all solutions of (1) to be oscillatory. In this paper a counterexample shows that the above result is not valid and the error in this paper is indicated.

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

Analysis ◽  
2019 ◽  
Vol 39 (3) ◽  
pp. 97-105 ◽  
Author(s):  
Sandra Pinelas ◽  
Shyam S. Santra
Keyword(s):  
Differential Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
First Order ◽  
Several Delays ◽  
Necessary And Sufficient ◽  
T Tau ◽  
First Order Differential Equations

AbstractIn 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.

scholarly journals Noncanonical Neutral DDEs of Second-Order: New Sufficient Conditions for Oscillation

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2026
Author(s):  
Awatif A. Hindi ◽  
Osama Moaaz ◽  
Clemente Cesarano ◽  
Wedad R. Alharbi ◽  
Mohamed A. Abdou
Keyword(s):  
Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Neutral Differential Equation ◽  
First Order ◽  
Riccati Substitution

In this paper, new oscillation conditions for the 2nd-order noncanonical neutral differential equation (a0t((ut+a1tug0t)′)β)′+a2tuβg1t=0, where t≥t0, are established. Using Riccati substitution and comparison with an equation of the first-order, we obtain criteria that ensure the oscillation of the studied equation. Furthermore, we complement and improve the previous results in the literature.

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