Existence of Solutions for Second Order Partial Neutral Functional Differential Equations

We establish existence of mild solutions for a class of abstract second-order partial neutral functional differential equations with unbounded delay in a Banach space. The work of the second author was supported by FONDECYT-CONICYT, Grants 1050314 and 7050034.     

RAZUMIKHIN-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.     

On stationarity of stochastic retarded linear equations with unbounded drift operators

This 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.     

Asymptotically almost periodic solutions of abstract retarded functional differential equations of second order

We establish existence of asymptotically almost periodic mild solutions for a class of semi-linear second-order abstract retarded functional differential equations with infinite delay.     

Asymptotically almost periodic solutions of abstract retarded functional differential equations of first order

We establish the existence of asymptotically almost periodic mild solutions for a class of semi-linear first-order abstract retarded functional differential equations with infinite delay.     

Representation of solutions and stability of linear differential-difference equations in a Banach space

In this paper the theory of linear delay differential equations is extended in three directions. One, the underlying phase space is allowed to be a Banach space so that equations with unbounded operators may be considered. Two, the delay is permitted to be effective over an infinite interval and a connection is made between this type of system and neutral systems whose delay is effective over a finite interval. Three, a theory of uniform asymptotic stability for linear delay differential equations in a Hilbert space is developed.     

Asymptotically almost periodic solutions of abstract retarded functional differential equations of second order

We establish existence of asymptotically almost periodic mild solutions for a class of semi-linear second-order abstract retarded functional differential equations with infinite delay. Research supported in part by FONDECYT, grant 1050314.     

Pseudo-almost periodic solutions for non-autonomous neutral differential equations with unbounded delay

In this note we establish the existence of pseudo-almost periodic solutions for a non-autonomous partial neutral functional differential with unbounded delay. We apply our abstract results to establish the existence of this type of solutions for a neutral differential equation which arises in the theory of heat conduction in materials with fading memory.     

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

A semigroup method for delay equations with relatively bounded operators in the delay term

We consider well-posedness and stability of abstract partial differential equations with unbounded operators in their delay terms. We show that the problem is equivalent to an abstract Cauchy problem in a product Banach space, give sufficient conditions on the well-posedness, make a complete spectral analysis and give robust stability criteria. The results are applied to the problem of small delays and to damped plate equations.     

Evolution Systems for Differential Equations with Variable Time Lags

We consider nonautonomous retarded functional differential equations under hypotheses which are designed for the application to equations with variable time lags, which may be unbounded, and construct an evolution system of solution operators which are continuously differentiable. These operators are defined on manifolds of continuously differentiable functions. The results apply to pantograph equations and to nonlinear Volterra integro-differential equations, for example. For linear equations we also provide a simpler evolution system with solution operators on a Banach space of continuous functions.     

A Semigroup Method for Delay Equations with Relatively Bounded Operators in the Delay Term

We consider well-posedness and stability of abstract partial differential equations with unbounded operators in their delay terms. We show that the problem is equivalent to an abstract Cauchy problem in a product Banach space, give sufficient conditions on the well-posedness, make a complete spectral analysis and give robust stability criteria. The results are applied to the problem of small delays and to damped plate equations. September 12, 2000     

Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect

We study functional differential equations with unbounded operator coefficients in Hilbert spaces such that the principal part of the equation is an abstract hyperbolic equation perturbed by terms with delay and terms containing Volterra integral operators. The well-posed solvability of initial boundary-value problems for the specified problems in weighted Sobolev spaces on the positive semi-axis is established.     

An existence criterion of positive solutions of p-type retarded functional differential equations

The conditions of existence of a positive solution (i.e., a solution with positive coordinates on a considered interval) of systems of retarded functional equations in the case of unbounded delay with finite memory are discussed. A general criterion for nonlinear case is given as well as its application to a linear system. Illustrative special cases are considered too.     

GLOBAL EXISTENCE FOR SEMILINEAR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

This paper is concerned with the global existence of solutions for abstract retarded functional differential equations (RFDEs) with infinite delay,via a fixed point approach.Some sufficient conditions are established under which the existence of a globally mild solution are obtained by using Leray-Schauder.     

State space theory of linear time-invariant systems with delays in state,control, and observation variables,II

We study a coupled mathematical system which provides a good model for important families of linear time-invariant hereditary systems: delay-differential equations, integrodifferential equations, Volterra-Stieltjes integral equations, functional differential equations of retarded and neutral types, etc. Appropriate states are constructed and associated semigroups and abstract differential equations are obtained. We emphasize the structural operator approach as in Delfour and Manitius. Control operators are added to the coupled mathematical system allowing delays in the control variables. Again structural operators are introduced to define the state and obtain abstract differential equations without delays in the control variable as in the work of Vinter and Kwong. Finally observation operators are added which allow for delays in the observation variable. Again a new state and a state equations are constructed in such a way that no delay appear in the new observation operator as in the recent work of D. Salamon.     

Periodic solutions of neutral fractional differential equations

We characterize the existence of periodic solutions for some abstract neutral functional fractional differential equations with finite delay when the underlying space is a UMD space.     

Periodic solutions of abstract neutral functional differential equations

We characterize the existence of periodic solutions of some abstract neutral functional differential equations with finite and infinite delay when the underlying space is a UMD space.     

A neutral FDE with stable D-operator is retarded

A neutral functional differential equation with a linear, autonomous, and stable D-operator can be rewritten as a retarded functional differential equation with infinite delay. This fact can be used, e.g., in the study of the stability of the neutral equation and to simplify the proofs of some standard asymptotic results.     

Spectral analysis and solvability of abstract hyperbolic equations with aftereffect

We analyze functional-differential equations with unbounded operator coefficients in a Hilbert space whose leading part is an abstract hyperbolic equation perturbed by terms with a retarded argument and by terms with Volterra integral operators.We consider spectral problems for the operator functions that are the symbols of abovementioned equations in the autonomous case.