Oscillations of retarded differential equations

1974 ◽  
Vol 75 (1) ◽  
pp. 95-101 ◽  
Author(s):  
Y. G. Sficas ◽  
V. A. Staikos
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Half Line ◽  
Retarded Differential Equation ◽  
Image Position ◽  
Retarded Differential Equations

In this paper we are dealing with the oscillatory and asymptotic behaviour of the nth order, (n > 1), retarded differential equationwhere g is differentiable on the half-line (t0,∞) with (I) g(t) ≤ t for every t ≥ t0 (II) g'(t) ≥ 0 for every t ≥ t0 (III) = ∞.

Strongly Monotone Solutions of Retarded Differential Equations

1979 ◽  
Vol 22 (4) ◽  
pp. 403-412 ◽  
Author(s):  
Y. G. Sficas
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Strongly Monotone ◽  
Retarded Differential Equation ◽  
Image Position ◽  
Monotone Solutions ◽  
Retarded Differential Equations

Let us consider the retarded differential equation1for which the following assumptions are made:(i) p: [t0, ∞) → [0, ∞) is continuous and not identically zero for all large t.(ii) σ [t0, ∞)→ ℝ is continuous, strictly increasing,(iii) φ: ℝ → ℝ is continuous, non-decreasing and

scholarly journals Existence, uniqueness, and regularity of solutions to semilinear retarded differential equations

2004 ◽  
Vol 2004 (3) ◽  
pp. 213-219 ◽  
Author(s):  
D. Bahuguna
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Regularity Of Solutions ◽  
Retarded Differential Equation ◽  
Retarded Differential Equations

In the present work, we consider a semilinear retarded differential equation in a Banach space. We first establish the existence and uniqueness of a mild solution and then prove its regularity under different additional conditions. Finally, we consider some applications of the abstract results.

scholarly journals On the non-existence of L2 solutions of a class of non-linear differential equations

1965 ◽  
Vol 14 (4) ◽  
pp. 257-268 ◽  
Author(s):  
J. Burlak
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Half Line ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equationdoes not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if

scholarly journals Oscillation of differential equations with non-monotone retarded arguments

2016 ◽  
Vol 19 (1) ◽  
pp. 98-104 ◽  
Author(s):  
George E. Chatzarakis ◽  
Özkan Öcalan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Oscillation Criterion ◽  
Retarded Argument ◽  
Real Numbers ◽  
Positive Real ◽  
First Order ◽  
Retarded Differential Equation ◽  
Oscillation Condition ◽  
Image Position

Consider the first-order retarded differential equation $$\begin{eqnarray}x^{\prime }(t)+p(t)x({\it\tau}(t))=0,\quad t\geqslant t_{0},\end{eqnarray}$$ where $p(t)\geqslant 0$ and ${\it\tau}(t)$ is a function of positive real numbers such that ${\it\tau}(t)\leqslant t$ for $t\geqslant t_{0}$, and $\lim _{t\rightarrow \infty }{\it\tau}(t)=\infty$. Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving $\liminf$, is established when the well-known oscillation condition $$\begin{eqnarray}\liminf _{t\rightarrow \infty }\int _{{\it\tau}(t)}^{t}p(s)\,ds>\frac{1}{e}\end{eqnarray}$$ is not satisfied. An example illustrating the result is also given.

Subexponential solutions of linear integro-differential equations and transient renewal equations

2002 ◽  
Vol 132 (3) ◽  
pp. 521-543 ◽  
Author(s):  
John A. D. Appleby ◽  
David W. Reynolds
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Renewal Equation ◽  
Asymptotically Stable ◽  
Renewal Equations ◽  
Integro Differential Equation ◽  
All Solutions ◽  
Image Position

This paper studies the asymptotic behaviour of the solutions of the scalar integro-differential equation The kernel k is assumed to be positive, continuous and integrable.If it is known that all solutions x are integrable and x(t) → 0 as t → ∞, but also that x = 0 cannot be exponentially asymptotically stable unless there is some γ > 0 such that Here, we restrict the kernel to be in a class of subexponential functions in which k(t) → 0 as t → ∞ so slowly that the above condition is violated. It is proved here that the rate of convergence of x(t) → 0 as t → ∞ is given by The result is proved by determining the asymptotic behaviour of the solution of the transient renewal equation If the kernel h is subexponential, then

scholarly journals Approximation of solutions to retarded differential equations with applications to population dynamics

2005 ◽  
Vol 2005 (1) ◽  
pp. 1-11 ◽  
Author(s):  
D. Bahuguna ◽  
M. Muslim
Keyword(s):  
Differential Equation ◽  
Population Dynamics ◽  
Differential Equations ◽  
Unique Solution ◽  
Existence And Uniqueness ◽  
Finite Dimensional ◽  
Retarded Differential Equation ◽  
Retarded Differential Equations

We consider a retarded differential equation with applications to population dynamics. We establish the convergence of a finite-dimensional approximations of a unique solution, the existence and uniqueness of which are also proved in the process.

scholarly journals LINEAR ASYMPTOTIC BEHAVIOUR OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

2007 ◽  
Vol 49 (1) ◽  
pp. 105-120 ◽  
Author(s):  
MATS EHRNSTRÖM
Keyword(s):  
Differential Equation ◽  
Initial Data ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Ordinary Differential Equations ◽  
Growth Conditions ◽  
Second Order ◽  
Half Line ◽  
Semilinear Differential Equation

Abstract.We study the semilinear differential equation u″ + F(t,u,u′)=0 on a half-line. Under different growth conditions on the function F, equations with globally defined solutions asymptotic to lines are characterized. Both fixed initial data and fixed asymptote are considered.

Asymptotic Behaviour of Disconjugate nth Order Differential Equations

1971 ◽  
Vol 23 (2) ◽  
pp. 293-314 ◽  
Author(s):  
D. Willett
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Linear Differential Equation ◽  
Ordered Set ◽  
Minimal Solution ◽  
Linear Differential ◽  
Image Position

An ordered set (u1, …, un) of positive Cn(a, b)-solutions of the linear differential equation1.1will be called fundamental principal system on [a, b) provided that1.2and1.3A system (u1, …, un) satisfying just (1.2) will be called a principal system on [a, b). In any principal system (u1, …, un) the solution u1 will be called a minimal solution.

scholarly journals Method of semidiscretization in time to nonlinear retarded differential equations with nonlocal history conditions

2004 ◽  
Vol 2004 (37) ◽  
pp. 1943-1956 ◽  
Author(s):  
S. Agarwal ◽  
D. Bahuguna
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Strong Solution ◽  
Existence And Uniqueness ◽  
Retarded Differential Equation ◽  
Retarded Differential Equations

This paper deals with the applications of the method of semidiscretization in time to a nonlinear retarded differential equation with a nonlocal history condition. We establish the existence and uniqueness of a strong solution. Finally, we consider some applications of the abstract results.

scholarly journals Oscillation Criteria for Second Order Neutral Differential Equations

1996 ◽  
Vol 48 (4) ◽  
pp. 871-886 ◽  
Author(s):  
Horng-Jaan Li ◽  
Wei-Ling Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Image Position ◽  
Delay Differential

AbstractSome oscillation criteria are given for the second order neutral delay differential equationwhere τ and σ are nonnegative constants, . These results generalize and improve some known results about both neutral and delay differential equations.

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