scholarly journals Iterative criteria for bounds on the growth of positive solutions of a delay differential equation

1978 ◽  
Vol 25 (2) ◽  
pp. 195-200
Author(s):  
Raymond D. Terry
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Sufficient Condition ◽  
All Solutions ◽  
Image Position ◽  
Delay Differential

AbstractFollowing Terry (Pacific J. Math. 52 (1974), 269–282), the positive solutions of eauqtion (E): are classified according to types Bj. We denote A neccessary condition is given for a Bk-solution y(t) of (E) to satisfy y2k(t) ≥ m(t) > 0. In the case m(t) = C > 0, we obtain a sufficient condition for all solutions of (E) to be oscillatory.

A Necessary and Sufficient Condition for the Oscillation of an Even Order Nonlinear Delay Differential Equation

1973 ◽  
Vol 25 (5) ◽  
pp. 1078-1089 ◽  
Author(s):  
Bhagat Singh
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Oscillatory Behavior ◽  
Necessary And Sufficient Condition ◽  
Even Order ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Nonlinear Delay Differential Equation ◽  
Nonlinear Delay

In this paper we study the oscillatory behavior of the even order nonlinear delay differential equation(1)where(i) denotes the order of differentiation with respect to t. The delay terms τi σi are assumed to be real-valued, continuous, non-negative, non-decreasing and bounded by a common constant M on the half line (t0, + ∞ ) for some t0 ≧ 0.

scholarly journals Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-28 ◽  
Author(s):  
Zeqing Liu ◽  
Ling Guan ◽  
Sunhong Lee ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Higher Order ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper is concerned with the higher order nonlinear neutral delay differential equation[a(t)(x(t)+b(t)x(t-τ))(m)](n-m)+[h(t,x(h1(t)),…,x(hl(t)))](i)+f(t,x(f1(t)),…,x(fl(t)))=g(t),for allt≥t0. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

scholarly journals A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1332 ◽  
Author(s):  
Ábel Garab
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Continuous Functions ◽  
Oscillation Criterion ◽  
Variable Delay ◽  
All Solutions ◽  
Linear Differential ◽  
Delay Differential ◽  
Additive Form ◽  
Condition B

We consider linear differential equations with variable delay of the form x ′ ( t ) + p ( t ) x ( t − τ ( t ) ) = 0 , t ≥ t 0 , where p : [ t 0 , ∞ ) → [ 0 , ∞ ) and τ : [ t 0 , ∞ ) → ( 0 , ∞ ) are continuous functions, such that t − τ ( t ) → ∞ (as t → ∞ ). It is well-known that, for the oscillation of all solutions, it is necessary that B : = lim sup t → ∞ A ( t ) ≥ 1 e holds , where A : = ( t ) ∫ t − τ ( t ) t p ( s ) d s . Our main result shows that, if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions on p and τ , condition B > 1 / e implies that all solutions of the above delay differential equation are oscillatory.

Oscillation and Global Attractivity in a Periodic Delay Equation

1996 ◽  
Vol 39 (3) ◽  
pp. 275-283 ◽  
Author(s):  
J. R. Graef ◽  
C. Qian ◽  
P. W. Spikes
Keyword(s):  
Global Attractivity ◽  
Positive Periodic Solution ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
All Solutions ◽  
Periodic Delay ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractConsider the delay differential equationwhere α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.

scholarly journals Positive Solutions and Mann Iterations of a Fourth Order Nonlinear Neutral Delay Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22
Author(s):  
Zeqing Liu ◽  
Jingjing Zhu ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Approximate Solutions ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given.

scholarly journals New Oscillation and Nonoscillation Criteria for a Class of Linear Delay Differential Equation

2019 ◽  
Vol 8 (10) ◽  
pp. 308-313
Keyword(s):  
Differential Equation ◽  
Continuous Function ◽  
Delay Differential Equation ◽  
Sufficient Condition ◽  
First Order ◽  
Linear Delay ◽  
Piecewise Continuous ◽  
Delay Differential ◽  
Oscillation And Nonoscillation Criteria ◽  
Oscillation And Nonoscillation

In this article the authors established sufficient condition for the first order delay differential equation in the form , ( ) where , = and is a non negative piecewise continuous function. Some interesting examples are provided to illustrate the results. Keywords: Oscillation, delay differential equation and bounded. AMS Subject Classification 2010: 39A10 and 39A12.

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.

New stability theorem for uncertain pantograph differential equations

2021 ◽  
Vol 40 (5) ◽  
pp. 9403-9411
Author(s):  
Zhifu Jia ◽  
Xinsheng Liu ◽  
Yu Zhang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Stability Theorem ◽  
Sufficient Condition ◽  
Stability In Distribution ◽  
Delay Differential

Uncertain pantograph differential equation (UPDE for short) is a special unbounded uncertain delay differential equation. Stability in measure, stability almost surely and stability in p-th moment for uncertain pantograph differential equation have been investigated, which are not applicable for all situations, for the sake of completeness, this paper mainly gives the concept of stability in distribution, and proves the sufficient condition for uncertain pantograph differential equation being stable in distribution. In addition, the relationships among stability almost surely, stability in measure, stability in p-th moment, and stability in distribution for the uncertain pantograph differential equation are also discussed.

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