Retarded Functional Differential Equations Research Articles

A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

Conley index for manifold-valued retarded functional differential equations without uniqueness of solutions

We develop a Conley index theory for retarded functional differential equations $\dot x=f(x_{t})$ with values in a differentiable manifold and (merely) continuous nonlinearities f. We use this index to establish an existence result for nonconstant full solutions of such equations.

Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay

In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then we apply these estimates to stochastic convolutions incurred by distributed delay to study their regularity property. Last, we present some easily-verified results by considering the regularity of a class of systems whose delay operators have the same order derivatives as those in instantaneous ones.

On polynomial forms of nonlinear functional differential equations

<p style='text-indent:20px;'>In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.</p>

On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

Floquet Theory and Stability of Periodic Solutions of Renewal Equations

We prove the validity of a Floquet theory and the existence of Poincare maps for periodic solutions of renewal equations, also known as Volterra functional equations. Our approach is based on sun–star perturbation theory of dual semigroups and relies on a spectral isolation property and on the regularity of the semiflow. This contributes a new chapter to the stability analysis, in analogy with ordinary and retarded functional differential equations as well as the case of equilibria.

Conley index continuation for some classes of RFDEs on manifolds

We establish Conley index continuation results for manifold-valued retarded functional differential equations which are close to ordinary differential equations.

Some regularity results on stochastic convolutions in point delay differential equations perturbed by noise

In this work, we obtain some regularity results on stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded coefficient operators driven by white noise. We first establish some estimates on fundamental solutions which play an important role in the subsequent theory. Then we apply these estimates to some stochastic convolutions arising in stochastic delay differential equations to obtain the desired regularity property.

On regularity of stochastic convolutions of functional linear differential equations with memory

In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $ C_0 $-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates to stochastic convolutions of our equations to obtain the desired regularity property.

Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion

In this paper, we discuss a class of neutral retarded stochastic functional differential equations driven by a fractional Brownian motion on Hilbert spaces. We develop a $$C_0$$ -semigroup theory of the driving deterministic neutral system and formulate the neutral time delay equation under consideration as an infinite-dimensional stochastic system without time lag and neutral item. Consequently, a criterion is presented to identify a strictly stationary solution for the systems considered. In particular, the ergodicity of the strictly stationary solution is studied. Subsequently, the ergodicity behavior of non-stationary solution for the systems considered is also investigated. We present an example which can be explicitly determined to illustrate our theory in the work.

Multiplicity of forced oscillations for scalar retarded functional differential equations

We find multiplicity results for forced oscillations of a periodically perturbed autonomous second‐order equation, the perturbing term possibly depending on the whole history of the system. The techniques that we use are topological in nature, but the technical details are hidden in the proofs and completely transparent to the reader only interested in the results.

On Anti-Periodic Solutions for a Class of Impulsive Retarded Functional Differential Equations

In this paper, some new results on the existence and uniqueness of anti-periodic solution for a first-order impulsive functional differential equation with delay are obtained. The main tool to be employed here is the coincidence degree theory.

A necessary and sufficient condition for well-posedness of initial value problems of retarded functional differential equations

We introduce the retarded functional differential equations (RFDEs) with general delay structure to treat various delay differential equations (DDEs) in a unified way and to clarify the delay structure in those dynamics. We are interested in the question as to which space of histories is suitable for the dynamics of each DDE, and investigate the well-posedness of the initial value problems (IVPs) of the RFDEs. A main theorem is that the IVP is well-posed for any “admissible” history functional if and only if the semigroup determined by the trivial RFDE x˙=0 is continuous. We clarify the meaning of the Hale–Kato axiom (Hale & Kato [12]) by applying this result to RFDEs with infinite delay. We also apply the result to DDEs with unbounded time- and state-dependent delays.

Identification problems of retarded differential systems in Hilbert spaces

This paper deals with the identification problem for the $L^1$-valued retarded functional differential equation. The unknowns are parameters and operators appearing in the given systems. In order to identify the parameters, we introduce the solution semigroup and the structural operators in the initial data space, and provide the representations of spectral projections and the completeness of generalized eigenspaces. The sufficient condition for the identification problem is given as the so called rank condition in terms of the initial values and eigenvectors of adjoint operator.

On stationarity of stochastic retarded linear equations with unbounded drift operators

ABSTRACTThis article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.

A new continuous dependence result for impulsive retarded functional differential equations

We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impulsive RFDEs in the above setting, with convergent right-hand sides, convergent impulse operators and uniformly convergent initial data. We assume that the limiting equation is an impulsive RFDE whose initial condition is the uniform limit of the sequence of the initial data and whose solution exists and is unique. Then, for sufficient large indexes, the elements of the sequence of impulsive retarded initial value problem admit a unique solution and such a sequence of solutions converges to the solution of the limiting Cauchy problem.

Impulsive retarded fractional differential equations with variable moments

Integer order impulsive functional differential equations with fi xed and variable times and the fractional order ones with fi xed times have been studied in the related literature several times, so far. To the extent we know, the fractional order ones with variable times have not yet been considered. In this work, we deal with establishing some existence results for solutions of impulsive retarded functional differential equations of fractional order with variable moments.

A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

SummaryThis paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations, which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams for RFDEs. Thorough numerical case studies show that the rightmost characteristic roots of the ordinary differential equation approximation converge to the rightmost characteristic roots of the original RFDE. Application of the method to time‐periodic RFDEs is also demonstrated, and the convergence of the stability boundaries is verified numerically. The method is compared with recently developed highly efficient numerical methods: the pseudospectral collocation (also called Chebyshev spectral continuous‐time approximation), the spectral Legendre tau method, and the spectral element method. The comparison is based on the stability analysis of three linear autonomous RFDEs. The efficiency of the methods is measured by the convergence rate of stability boundaries in the space of system parameters, by the convergence rate of the rightmost characteristic exponent and by the computation time of the stability charts. Copyright © 2016 John Wiley & Sons, Ltd.

Existence and uniqueness of periodic solution for a class of impulsive second-order retarded functional differential equations

Existence and uniqueness of periodic solution for a class of impulsive second-order retarded functional differential equations

Converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations

In this article we give a collection of converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations. We show that the existence of a weakly-degenerate Lyapunov–Krasovskii functional is a necessary and sufficient condition for the global exponential stability of linear retarded functional differential equations. This is carried out using a switched system representation approach.