Retarded Differential Equation Research Articles

The Wong-Zakai approximations of invariant manifolds for retarded partial differential equations with multiplicative white noise

We focus on the dynamics and Wong-Zakai approximation for a class of retarded partial differential equations subjected to multiplicative white noise. We show that when restricted to a local region and under certain conditions, there exists a unique global solution for the truncated system driven by either the white noise or the approximation noise. Such solution generates a random dynamical system, and the solutions of Wong-Zakai approximations are convergent to solutions of the stochastic retarded differential equation. We also show that there exist invariant manifolds for the truncated system driven by either the white noise or the approximation noise, which are then the local manifolds for the untruncated systems, and prove that such invariant manifolds of the Wong-Zakai approximations converge to those of the stochastic retarded differential equation.

A Time-Delayed Hyperchaotic System Composed of Multiscroll Attractors With Multiple Positive Lyapunov Exponents

The paper proposes a time-delayed hyperchaotic system composed of multiscroll attractors with multiple positive Lyapunov exponents (LEs), which are described by a three-order nonlinear retarded type delay differential equation (DDE). The dynamical characteristics of the time-delayed system are far more complicated than those of the original system without time delay. The three-order time-delayed system not only generates hyperchaotic attractors with multiscroll but also has multiple positive LEs. We observe that the number of positive LEs increases with increasing time delay. Through numerical simulations, the time-delayed system exhibits a larger number of scrolls than the original system without time delay. Moreover, different numbers of scrolls with variable delay and coexistence of multiple attractors with a variable number of scrolls are also observed in the time-delayed system. Finally, we setup electronic circuit of the proposed system, and make Pspice simulations to it. The Pspice simulation results agree well with the numerical results.

On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation

Abstract In this paper, we investigate some new nonlinear retarded integral inequalities of Gronwall–Bellman–Pachpatte type. These inequalities generalize some former famous inequalities and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of some nonlinear retarded differential and integral equations. An application is also presented to illustrate the usefulness of some of our results in estimation of solution of certain retarded nonlinear differential equations with the initial conditions.

Note on the paper of Džurina and Stavroulakis

In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the form ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + p ( t ) | u [ τ ( t ) ] | α - 1 u [ τ ( t ) ] = 0 . The results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper.

Approximation of solutions to non-local history-valued retarded differential equations

In the present work we consider a retarded differential equation and prove the existence, uniqueness and convergence of approximate solutions. Also we consider the Faedo–Galerkin approximations of the solution and the convergence.

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

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.

Positive solutions of nonlocal boundary value problem for nonlinear retarded differential equation

By using Krasnoselskii’s fixed-point theorem on a suitable cone, several existence results and corresponding properties of positive solutions of nonlocal boundary value problem for nonlinear retarded differential equation are obtained.

Existence, uniqueness, and regularity of solutions to semilinear 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.

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

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.

Oscillation criteria for second-order delay differential equations

The aim of this paper is to establish some new oscillation criteria for the second-order retarded differential equation (r(t)|u ′(t)| α−1u ′(t)) ′+p(t)|u[τ(t)]| α−1u[τ(t)]=0. The results obtained essentially improve known results in the literature.

Designing stable ABR flow control with rate feedback and open-loop control: first-order control case

In this paper we present a control-theoretic approach to design stable rate-based flow control for ATM ABR services. The flow control algorithm that we consider has the most simple form among all the queue-length-based flow control algorithms, and is referred to as first-order rate-based flow control (FRFC) since the corresponding closed loop can be modeled as a first-order retarded differential equation. We analyze the equilibrium and the asymptotic stability of the closed loop for the case of multiple connections with diverse round-trip delays. We also characterize the asymptotic decay rate at which the stable closed loop tends to the equilibrium. The decay rate is shown to be a concave function of control gain with its maximum being the inverse of round-trip delay. We also consider an open loop control in which the queue control threshold is dynamically adjusted according to the changes in the available bandwidth and the number of connections. This open loop control is shown to be necessary and effective to prevent the closed loop from converging to an undesirable equilibrium point.

Extrapolation and inhornogeneous retarded differential equations on infinite dimensional spaces

Using the extrapolation spaces introduced by Da Prato-Grisvard and Nagel, we prove the well-posedness of a more general inhomogenous retarded differential equation on infinite dimensional spaces.

Oscillation criteria for second-order retarded differential equations

Some oscillation criteria for the second order retarded differential equation [r(t)¦u′(t)¦ α−1 u′(t)]′ + p(t)¦u(τ(t))¦ β−1u(τ(t)) = 0 are given, where α and β are positive constants, and p, τ, r ϵ C([t 0,∞), R satisfy some suitable conditions. Our results generalize those of Erbe, Ohriska and Lakshmikatham.

Oscillations of first order linear retarded differential equations

The oscillatory behavior of differential equations with deviating arguments has been studied by many authors. In particular, several authors have studied the oscillations caused by the deviating arguments. The study of such oscillations is a relatively new field and is very interesting in applications. For some contributions in this area see the papers by Arino, Gyori, and Jawhari [ 11, Bowcock and Yu [2], Gyiiri [4,5 J, Hunt and Yorke [6], KulenoviC, Ladas, and Meimaridou [7], Ladas [S], Ladas, Sficas, and Stavroulakis [9, lo], Ladas and Stavroulakis [ 111, Sugie [ 141, Tramov [15], and Yan [16]. We also refer to the recent book by Ladde, Lakshmikantham, and Zhang [123. The purpose of this paper is to investigate the oscillatory behavior of the solutions of first order linear retarded differential equations with variable coefftcients and nonconstants delays. Consider the first order retarded differential equation

Adaptive head control of a hydraulic open channel model

In this paper we are concerned with the head control of a reach of an open channel. We consider a control configuration where the head sensor and actuator (gate position) are located nearby. This configuration does not seem to have been explored in the literature since the control task is made considerably difficult by the aftereffect of the water reflecting wave. To design the controller, the open channel reach is modeled by a linear retarded differential equation, where the retarded part takes into account the aftereffects. Using the theory developed in Johansson ( Automatica, 22, 555–560 (1986)) we design an adaptive controller that compensates for the latter. Simulation results of the proposed controller applied to a partial differential equation model of the channel are given in Constantinides et al. ( Int. J. Syst. Sci., 7 (1976)) and Chow ( Open-channel Hydraulics, McGraw-Hill, Maidenhead (1973)).

A Lyapunov Functional for a Retarded Differential Equation

Consider the autonomous linear retarded differential equation \[\dot x(t) = A_0 x(t) + A_1 x(t - r) + \int_{ - r}^0 {A_{01} (\theta )x(t + \theta )d\theta .} \] A positive definite quadratic functional is given that yields an equivalent norm in the phase space, which measures the exact asymptotic behavior of its solutions, providing the best estimate for the rate of growth or decay of the said solutions.

On oscillation of solutions of nonlinear retarded differential equation of Emden-Fowler type

On oscillation of solutions of nonlinear retarded differential equation of Emden-Fowler type

Oscillation of linear retarded differential equation

Oscillation of linear retarded differential equation