Oscillation Of Delay Differential Equations Research Articles

Simplified and improved criteria for oscillation of delay differential equations of fourth order

An interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

Oscillation, convergence, and stability of linear delay differential equations

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation $$ x^{\prime}(t)+c(t)x(\tau(t))=0,~~t\geq 0, $$ where $c$ is locally integrable of any sign, $\tau(t)\leq t$ is Lebesgue measurable, $\lim_{t\rightarrow\infty}\tau(t)=\infty$, we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the $3/2-$stability criterion to the case of measurable parameters, improving $1+1/e$ to the sharp $3/2$ constant.

Oscillations of delay differential equations with variable coefficients

Oscillations of delay differential equations with variable coefficients

Bifurcations in Delay Differential Equations: An Algorithmic Approach in Frequency Domain

In this work we study local oscillations in delay differential equations with a frequency domain methodology. The main result is a bifurcation equation from which the existence and expressions of local periodic solutions can be determined. We present an iterative method to obtain the bifurcation equation up to a fixed arbitrary order. It is shown how this method can be implemented in symbolic math programs.

On forced fast oscillations for delay differential equations on compact manifolds

We prove an existence result for forced oscillations of delay differential equations on compact manifolds with nonzero Euler–Poincaré characteristic. When the period is smaller than the delay we need the asymptotic fixed point index theory for C 1 maps due to Eells and Fournier, and Nussbaum.

New results on oscillation for delay differential equations with piecewise constant argument

We introduce a new technique to obtain some new oscillation criteria for the oscillating coefficients delay differential equation with piecewise constant argument of the form x′( t) + a( t) x( t) + b( t) x({ t − k}) = 0, where a( t) and b( t) are right continuous functions on [− k, ∞), k is a positive integer, and [·] denotes the greatest integer function. Our results improve and generalize the known results in the literature. Some examples are also given to demonstrate the advantage of our results.

Oscillation of delay differential equation with several positive and negative coefficients

Some Sufficient conditions for oscillation of first order nonautonomuous delay differential equations with several positive and negative coefficients are obtained. Summary: English

Oscillation of delay differential equations on time scales

Consider the following equation: ▪, where t is in a measure chain. We apply the theory of measure chains to investigate the oscillation and nonoscillation of the above equation on the basis of some well-known results. And in some sense, we show a method to unify the delay differential equation and delay difference equation.

Oscillation of delay differential equations with piecewise constant argument

The delay differential equation with piecewise constant argument x′(t)+a(t)x(t)+b(t)x ([t—k])=0 is considered, where a(t) and b(t) are continuous functions on [—k, ∞), b(t)≥0, k is a positive integer and [·] denotes the greatest integer function. Some new oscillation and nonoscillation conditions are obtained.

A Note on General Oscillation for Delay Differential Equations

In this note we demonstrate the existence of linear delay differential equations whose solutions are neither generically oscillatory nor generically nonoscillatory.

On the oscillation of delay differential equations with real coefficients

A set of necessary conditions and another set of sufficient conditions for the oscillation of all the solutions of urn:x-wiley:01611712:media:ijmm539509:ijmm539509-math-0001 are proved.

Linearized oscillations of delay differential equations and applications

The necessary and sufficient conditions for the oscillations of every solution of the nonlinear delay equation x(t)+f(itx(t−l))+g(itx([t− k])) = 0 are oblained.

Generic Oscillations for Delay Differential Equations

In this article we study oscillation and nonoscillation of the solutions of linear delay differential equations. We define generic oscillation and generic nonoscillation as a topological assessment of the extent of oscillatory or nonoscillatory solutions of a delay differential equation. We provide sufficient conditions for both phenomena based on the roots of the characteristic equation and concrete conditions for some special cases.

Multiple Integral Average Conditions for Oscillation of Delay Differential Equations

We obtain some multiple integral average conditions for oscillation and nonoscillation of the first order nonautonomous delay equations. Our theorems improve several previous well-known results.

Oscillations of Delay Differential Equations with Variable Coefficients

Consider the delay differential equationx′(t)+p(t)x(t−τ)=0, wherep(t)∈C([t0,∞),R+) and τ is a positive constant. We obtain a sharp sufficient condition for the oscillation of this equation, which improves previously known results.

Necessary and sufficient conditions for the oscillation of delay differential equation with a piecewise constant argument

The characteristic equation for an equation with continuous and piecewise constant argument in the form urn:x-wiley:01611712:media:ijmm659532:ijmm659532-math-0001 is presented, which when q = 0 reduces to urn:x-wiley:01611712:media:ijmm659532:ijmm659532-math-0002 and when p = 0 reduces to urn:x-wiley:01611712:media:ijmm659532:ijmm659532-math-0003 . Also, the necessary and sufficient conditions for oscillation are obtained.

Oscillations of Delay Differential Equations with Variable Coefficients

Consider the delay differential equation ẋ( t) + p( t) x( t − τ) = 0, where p( t) ∈ C[[ t 0, ∞), R +] and τ is a positive constant. We show that every solution of this equation oscillates if ∫ t t−τ p( t) dt ≥ 1/ e for sufficiently large t and ∫ ∞ t 0+τ p( t)[exp(∫ t t−τ p( s) ds − 1/ e) − 1] dt = ∞.

A Comparison Result for the Oscillation of Delay Differential Equations

We obtain a comparison result for the oscillation of all solutions of the equation \[ \dot y(t) + \sum \limits _{i = 1}^n {{q_i}(t)y(t - {\sigma _i}(t)) = 0} \] in terms of the oscillation of all solutions of the equation \[ \dot x(t) + \sum \limits _{i = 1}^n {{p_i}(t)x(t - {\tau _i}(t)) = 0} \] under appropriate hypotheses on the asymptotic behavior of the quotients ${p_i}(t)/{q_i}(t)$ and ${\tau _i}(t)/{\sigma _i}(t)$ for $i = 1,2, \ldots ,n$. An application to the oscillation of the nonautonomous delay-logistic equation is given.

A comparison result for the oscillation of delay differential equations

We obtain a comparison result for the oscillation of all solutions of the equation\[y˙(t)+∑i=1nqi(t)y(t−σi(t))=0\dot y(t) + \sum \limits _{i = 1}^n {{q_i}(t)y(t - {\sigma _i}(t)) = 0}\]in terms of the oscillation of all solutions of the equation\[x˙(t)+∑i=1npi(t)x(t−τi(t))=0\dot x(t) + \sum \limits _{i = 1}^n {{p_i}(t)x(t - {\tau _i}(t)) = 0}\]under appropriate hypotheses on the asymptotic behavior of the quotientspi(t)/qi(t){p_i}(t)/{q_i}(t)andτi(t)/σi(t){\tau _i}(t)/{\sigma _i}(t)fori=1,2,…,ni = 1,2, \ldots ,n. An application to the oscillation of the nonautonomous delay-logistic equation is given.

Oscillations of delay differential equations

AbstractSufficient conditions are established for all solutions of the linear systemto be oscillatory, where qij ∈(−∞, ∞), τij ∈ (0, ∞), i, j = 1, 2, …, n.