Characterization of shadowing for linear autonomous delay differential equations

Consider a linear neutral integrodelay differential equation of the form $$\dot{x}(t)+\sum_{j=1}^{m}b_{j}\dot{x}(t-\sigma_{1})+\beta \int_{0}^{\infty}K_{2}(s)\dot{x}(t-s)ds+a_{0}x(t)+\sum_{j=1}^{n}a_{j}x(t-\tau_{j})+\alpha \int_{0}^{\infty}K_{1}(s)x(t-s)ds=0$$ (5.1.1) in which ẋ(t) denotes the right derivative of x at t. (Throughout this chapter we use an upper dot to denote right derivative and this is convenient in writing neutral differential equations systematically). Asymptotic stability of the trivial solution of (5.1.1) and several of its variants have been considered by many authors. There exists a well developed fundamental theory for neutral delay differential equations (e.g. existence, uniqueness, continuous dependence of solutions on various data; see, for instance, the survey article by Akhmerov et al. [1984]); however, there exist no “easily verifiable” sufficient conditions for the asymptotic stability of the trivial solution of (5.1.1). By the phrase “easily verifiable” we mean a verification which is as easy as in the case of Routh-Hurwitz criteria, the diagonal dominance condition or the positivity of principal minors of a matrix etc. Certain results which are valid for linear autonomous ordinary and delay-differential equations cannot be generalized (or extended) to neutral equations. It has been shown by Gromova and Zverkin [1986] that a linear neutral differential equation can have unbounded solutions even though the associated characteristic equation has only purely imaginary roots (see also Snow [1965], Gromova [1967], Zverkin [1968], Brumley [1970], and Datko [1983]); such a behavior is not possible in the case of ordinary or (non-neutral) delay differential equations. It is known (Theorem 6.1 of Henry [1974]) that if the characteristic equation associated with a linear neutral equation has roots only with negative real parts and if the roots are uniformly bounded away from the imaginary axis, then the asymptotic stability of the trivial solution of the corresponding linear autonomous equation can be asserted. However, verification of the uniform boundedness away from the imaginary axis of all the roots of the characteristic equation is usually difficult. An alternative method for stability investigations is to resort to the technique of Lyapunov-type functional and functions; this will be amply illustrated in this chapter.

Certain linear autonomous delay as well as neutral delay difference equations are considered. A class of linear autonomous delay difference equations with continuous variable is also considered. Some results on the behavior of the solutions are established via two distinct positive roots of the corresponding characteristic equation.

One of types of stochastic retarded systems is under consideration. Our scheme of analysis is applicable for investigation of linear and nonlinear differential difference equations with single and multiple constant delays, linear differential equations with variable delays, linear neutral delay differential equations, and separate linear differential difference equations. In addition, a problem of sensitivity estimation for linear dynamic systems described by stochastic differential difference equations can be explored too. All these schemes are based on extensions of phase spaces.

One of types of stochastic retarded systems is under consideration. Our scheme of analysis is applicable for investigation of linear and nonlinear differential difference equations with single and multiple constant delays, linear differential equations with variable delays, linear neutral delay differential equations, and separate linear differential difference equations. In addition, a problem of sensitivity estimation for linear dynamic systems described by stochastic differential difference equations can be explored too. All these schemes are based on extensions of phase spaces.

Accuracy of the Laplace transform method for linear neutral delay differential equations

Oscillation and Stability of Linear Impulsive Delay Differential Equations

List of figures. Preface. Contributing Authors. Introduction.- 1. History of Delay Equations J.K. Hale.- Part I General Results and Linear Theory of Delay Equations in Finite Dimensional Spaces. 2. Some General Results and Remarks on Delay Differential Equations E. Ait Dads. 3. Linear Autonomous Functional Differential Equations F. Kappel.- Part II Hopf Bifurcation, Centre Manifolds and Normal Forms for Delay Differential Equations. 4. Variation of Constant Formula for Delay Differential Equations M.L. Hbid, K. Ezzinbi. 5. Introduction to Hopf Bifurcation Theory for Delay Differential Equations M.L. Hbid. 6. An Algorithmic Scheme for Approximating Center Manifolds and Normal Forms for Functional Differential Equations M. Ait Babram. 7. Normal Forms and Bifurcations for Delay Differential Equations T. Faria.- Part III Functional Differential Equations in Infinite Dimensional Spaces. 8. A Theory of Linear Delay Differential Equations in Infinite Dimensional Spaces O. Arino, E. Sanchez. 9. The Basic Theory of Abstract Semilinear Functional Differential Equations with Non-Dense Domain K. Ezzinbi, M. Adimy.- Part IV More on Delay Differential Equations and Applications. 10. Dynamics of Delay Differential Equations H.O. Walther. 11. Delay Differential Equations in Single Species Dynamics Sh. Ruan. 12. Well-Posedness, Regularity and Asymptotic Behaviour of Retarded Differential Equations by Extrapolation Theory L. Maniar.- References.- Index.

Preliminaries. Introduction. Initial Value Problem, Oscillation and Nonoscillation. Continuability and Boundedness. Some Basic Results for Second Order Linear Ordinary Differential Equations. Some Useful Criteria for First Order. Some Useful Results from Analysis and Fixed Point Theorems. Notes and General Discussions. References. Oscillations of Differential Equations with Deviating Arguments. Oscillation Theorems (I). Oscillation Theorems (II). Comparison Theorems for Second Order Functional Differential Equations. Oscillation of Functional Equations with a Damping Term. Oscillation of Second Order Linear Delay Differential Equations. Oscillation of Forced Functional Differential Equations. Oscillation of Functional Equations with Damping and Forcing Terms. Necessary and Sufficient Conditions for the Oscillation of Forced Equations. Oscillation for Perturbed Differential Equations. Asymptotic Behavior of Oscillatory Solutions of Functional Equations. Notes and General Discussions. References. Oscillation of Neutral Functional Differential Equations. Oscillation of Nonlinear Neutral Equations. Oscillation of Neutral Equations with Damping. Oscillation of Forced Neutral Equations. Oscillation of Neutral Equations with Mixed Type. Necessary and Sufficient Conditions for Oscillations of Neutral Equations with Deviating Arguments. Comparison and Linearized Oscillation Theorems for Neutral Equations. Existence of Nonoscillatory Solutions of Neutral Delay Differential Equations. Asymptotic Behavior of Nonoscillatory Solutions of Neutral Nonlinear Delay Differential Equations. Notes and General Discussions. References. Conjugacy and Nonoscillation for Second Order Differential Equations. Conjugacy of Linear Second Order Ordinary Differential Equations. Nonoscillation Theorems. Integral Conditions and Nonoscillations. Notes and General Discussions. References. Oscillation of Impulsive Differential Equations. Oscillation Criteria for Impulsive Delay Differential Equations. Oscillation of Second Order Linear Differential Equations with Impulses. Notes and General Discussions. References. Subject Index with Deviating Arguments. Oscillation of Neutral Functional Differential Equations. Conjugacy and Nonoscillation for Second Order Differential Equations. Oscillation of Impulsive Differential Equations.

We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. Our result is obtained by the use of a positive root with an appropriate property of the so called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equation. More precisely, we show that, for any solution, the limit of a specific integral transformation of it, which depends on a suitable positive root of the characteristic equation, exists as a real number and it is given explicitly in terms of the positive root of the characteristic equation and the initial function.

PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

Exponential stability of the exact solutions and the numerical solutions for a class of linear impulsive delay differential equations

Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

Moment boundedness of linear stochastic delay differential equations with distributed delay

A wide class of scalar first order linear autonomous neutral delay differential equations with distributed type delays is considered. By the use of two distinct real roots of the corresponding characteristic equation, a new result on the behavior of the solutions is established.

Exact numerical schemes have previously been obtained for some linear retarded delay differential equations and systems. Those schemes were derived from explicit expressions of the exact solutions, and were expressed in the form of perturbed difference systems, involving the values at previous delay intervals. In this work, we propose to directly obtain expressions of the same type for the fundamental solutions of linear delay differential equations, by considering vector equations with vector components corresponding to delay-lagged values at previous intervals. From these expressions for the fundamental solutions, exact numerical schemes for arbitrary initial functions can be proposed, and they may also facilitate obtaining explicit exact solutions. We apply this approach to obtain an exact numerical scheme for the first order linear neutral equation x′(t)−γx′(t−τ)=αx(t)+βx(t−τ), with the general initial condition x(t)=φ(t) for −τ≤t≤0. The resulting expression reduces to those previously published for the corresponding retarded equations when γ=0.