Linear Functional Differential Equations Research Articles

Unique solvability of the Darboux problem for linear hyperbolic functional differential equations

Abstract We obtain new conditions sufficient for the unique solvability of the Darboux problem for linear partial functional differential equations of hyperbolic type. The main results are applied to the hyperbolic differential equations with argument deviations.

Multi-point boundary value problems for linear functional-differential equations

Abstract Efficient conditions guaranteeing the solvability of multi-point boundary value problems for linear functional-differential equations are established in this paper. The results are proved using the theorems on functional-differential inequalities.

On positive periodic solutions of linear second order functional differential equations

Abstract The periodic boundary value problem for linear second order functional differential equations is considered. Sharp sufficient conditions for the positiveness of solutions are obtained.

On exponential stability of linear non-autonomous functional differential equations of neutral type

ABSTRACTGeneral linear non-autonomous functional differential equations of neutral type are considered. A novel approach to exponential stability of neutral functional differential equations is presented. Consequently, explicit criteria are derived for exponential stability of linear non-autonomous functional differential equations of neutral type. A brief discussion to the obtained results and illustrative examples are given.

Pontryagin principle for a Mayer problem governed by a delay functional differential equation

We establish Pontryagin principles for a Mayer's optimal control problem governed by a functional differential equation. The control functions are piecewise continuous and the state functions are piecewise continuously differentiable. To do that, we follow the method created by Philippe Michel for systems governed by ordinary differential equations, and we use properties of the resolvent of a linear functional differential equation.

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.

Almost linear functional differential equations with Properties A and B

For the general functional differential equation u(n)(t)+F(u)(t)=0, where F:C(R+;R)→Lloc(R+;R) is a continuous operator, the sufficient conditions in order to have Property A (Property B) are established. As a particular case, we consider the ordinary differential equation with a deviating argument (0.1)u(n)(t)+p(t)|u(σ(t))|μ(t)signu(σ(t))=0, where p∈Lloc(R+;R), σ∈C(R+;R+), μ∈C(R+;(0,+∞)) and limt→+∞σ(t)=+∞. Eq. (0.1) is called almost linear if limt→+∞μ(t)=1. For Eq. (0.1), the sufficient conditions are obtained for the solutions to be oscillatory. These criteria cover the well-known results for the linear differential equation (μ(t)≡1).

A note on stability of functional difference equations

In this note, we consider perturbed linear functional difference equations with discrete and distributed delays which play a fundamental role in several stability and stabilizability problems of time-delay systems. Linear and nonlinear perturbations are considered. Sufficient conditions for exponential stability of the perturbed solutions are given.

Input-to-State Stability of Linear Stochastic Functional Differential Equations

The purpose of the paper is to show how asymptotic properties, first of all stochastic Lyapunov stability, of linear stochastic functional differential equations can be studied via the property of solvability of the equation in certain pairs of spaces of stochastic processes, the property which we call input-to-state stability with respect to these spaces. Input-to-state stability and hence the desired asymptotic properties can be effectively verified by means of a special regularization, also known as “theW-method” in the literature. How this framework provides verifiable conditions of different kinds of stochastic stability is shown.

A numerical method for solving systems of higher order linear functional differential equations

Abstract Functional differential equations have importance in many areas of science such as mathematical physics. These systems are difficult to solve analytically.In this paper we consider the systems of linear functional differential equations [1-9] including the term y(αx + β) and advance-delay in derivatives of y .To obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. For this purpose, to obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. This method transform the problem into a system of linear algebraic equations. The solutions of last system determine unknown co-efficients of original problem. Also, an error estimation technique is presented and the approximate solutions are improved by using it. The program of method is written in Matlab and the approximate solutions can be obtained easily. Also some examples are given to illustrate the validity of the method.

On solvability of periodic boundary value problems for second order linear functional differential equations

On solvability of periodic boundary value problems for second order linear functional differential equations

Investigation of the Structure of the Set of Continuous Solutions for Systems of Linear Functional-Difference Equations

We establish the conditions of existence of continuous solutions for a certain class of systems of linear functional-difference equations and study the properties of these solutions.

On the Structure of the Set of Continuous Solutions of Linear Functional-Difference Equations

We establish conditions for the existence of continuous solutions of systems of linear equations and study the structure of the set of these solutions.

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.

Resolvent of nonautonomous linear delay functional differential equations

AbstractThe aim of this paper is to give a complete proof of the formula for the resolvent of a nonautonomous linear delay functional differential equations given in the book of Hale and Verduyn Lunel [9] under the assumption alone of the continuity of the right-hand side with respect to the time,when the notion of solution is a differentiable function at each point, which satisfies the equation at each point, and when the initial value is a continuous function.

Solvability of the periodic problem for higher-order linear functional differential equations

We obtain necessary and sufficient conditions for the unique solvability of the periodic boundary value problem for a family of nth-order linear functional differential equations with pointwise constraints on the functional operators.

Finite-dimensional guides for conflict-controlled linear systems of neutral type

We develop approximations to systems of functional-differential equations of the neutral type by ordinary differential equations as applied to conflict control problems. We develop and justify a procedure of mutual feedback tracking between the motion of the original conflict-controlled plant described by linear functional-differential equations of neutral type and the motion of the modeling guide plant described by ordinary differential equations.

Almost periodic solutions of partial differential equations with delay

In this paper we study the existence of almost periodic solutions for linear retarded functional differential equations with finite delay and values in a Banach space. We relate the existence of almost periodic solutions with the stabilization of distributed control systems. We apply our results to transport models and to the wave equation.

Hyers–Ulam stability of linear functional differential equations

In this paper, the stability of some classes of linear functional differential equations was discussed by direct method, iteration method, fixed point method and open mapping theorem. It is shown that the Hyers–Ulam stability holds true for y(n)=g(t)y(t−τ)+h(t). The stability of functional differential equations with multiple delays of first order and general delay differential equations also have been discussed.

Stability and determinacy conditions for mixed-type functional differential equations

This paper analyzes the solution of linear mixed-type functional differential equations with either predetermined or non-predetermined variables. Conditions characterizing the existence and uniqueness of a solution are given and related to the local stability and determinacy properties of the steady state. In particular, it is shown that the relationship between the uniqueness of the solution and the stability of the steady-state is more subtle than the one that holds for ordinary differential equations, and gives rise to new dynamic configurations.