Linear Functional Differential Equations Research Articles

On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

Consider the homogeneous equation $$u'(t) = l(u)(t){\rm{ for a}}{\rm{.e}}{\rm{. }}t \in [a,b]$$ where l: C([a, b];ℝ) → L([a, b];ℝ) is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.

Stability property and essential spectrum of linear retarded functional differential equations

In this work, we shall study stability and stabilization of a class of retarded functional differential equations in Banach spaces. We present sufficient (and necessary, sometimes) conditions for weak, asymptotic and exponential stability properties. It is shown that retarded growth bounds are totally determined by retarded non-isolated spectra, isolated eigenvalues with infinite algebraic multiplicity and spectral bounds. Stabilization problem via compact perturbations, a topic which is practically useful in the context of control theory of dynamical systems, is considered as well.

On a Two-point Boundary Value Problem for Third-order Linear Functional Differential Equations. Part I.

Ecient conditions sucient for the solvability of the problem u 000 (t) = g(t)u( (t)) p(t)u( (t)) +q(t); u(a) = c1; u 0 (a) = c2; u(b) = c3 are derived using the general results obtained in our recent paper (1). Here, p;g2 L (a;b);R + , q 2 L (a;b);R , ; : (a;b) ! (a;b) are measurable functions, and ci2 R (i = 1; 2; 3). Sign-constant solutions are discussed as well. '(v)(t) = p(t)v( (t)) for a.e. t2 (a;b); v2 C (a;b);R ; (1.3) '(v)(t) = g(t)v( (t)) for a.e. t2 (a;b); v2 C (a;b);R ; (1.4)

Properties for Certain System of Linear Delay Functional Differential Equations

In this paper, we consider the properties for certain system of linear delay functional differential equations, and get some criteria for oscillation and non-oscillation by utilizing the matrix measures.

Non-Oscillatory Solutions of Odd-Dimensional System of Functional Differential Equations

In this paper, we consider the non-oscillatory problems of odd-dimensional system of linear delay functional differential equations. Based upon the corresponding characteristic equations, we get some criteria for non-oscillatory solutions by utilizing the matrix measures.

One class of singular linear functional differential equations

We consider one class of first-order functional differential equations with a singularity in the independent variable. We obtain conditions for the Fredholm property and the solvability of the mentioned equations.

The W-transform in stability analysis for stochastic linear functional difference equations

We apply the method of integral transforms to study stability properties of rather general linear functional difference equations of Itô-type. This approach called “the W-method” is based on integral transforms coming from auxiliary equations that already possess the desired asymptotic properties. Some sufficient conditions for different kinds of stability are found in this way. A general framework is offered as well that can be used to study asymptotic properties of other classes of stochastic difference equations.

Multipoint boundary value problems of Neumann type for functional differential equations

We obtain the expression of the explicit solution to a class of multipoint boundary value problems of Neumann type for linear ordinary differential equations and apply these results to study sufficient conditions for the existence of solution to linear functional differential equations with multipoint boundary conditions, considering the particular cases of equations with delay and integro-differential equations.

Study of Solutions to Some Functional Differential Equations with Piecewise Constant Arguments

We provide optimal conditions for the existence and uniqueness of solutions to a nonlocal boundary value problem for a class of linear homogeneous second‐order functional differential equations with piecewise constant arguments. The nonlocal boundary conditions include terms of the state function and the derivative of the state function. A similar nonhomogeneous problem is also discussed.

Functional Differential and Difference Equations with Applications

and Applied Analysis 3 solutions to a class of nonlocal boundary value problems for linear homogeneous secondorder functional differential equations with piecewise constant arguments are obtained. The last but not the least, this issue features a number of publications that report recent progress in the analysis of problems arising in various applications. In particular, dynamics of delayed neural network models consisting of two neurons with inertial coupling were studied, properties of a stochastic delay logistic model under regime switching were explored, and analysis of the permanence and extinction of a single species with contraception and feedback controls was conducted. Other applied problems addressed in this special issue regard Cohen-Grossberg BAM neural networks with time-varying delays, adaptive observer-based fault estimation for stochastic Markovian jumping systems, hematopoiesis models, Lotka-Volterra systems, and finite-time attractivity for diagonally dominant systems with off-diagonal delays. It is certainly impossible to provide in this short editorial note a more comprehensive description for all articles in the collection. However, the team of the Guest Editors believes that results included in this volume represent significant contemporary trends in the qualitative theory of ordinary, functional, partial, impulsive, dynamic, stochastic differential, and difference equations and in applications. We hope that this special issue will serve as a source of inspiration for researchers working in related areas providing specialists with a wealth of new ideas, techniques, and unsolved problems. Josef Diblik Elena Braverman

Approximation of Eigenvalues of Evolution Operators for Linear Retarded Functional Differential Equations

This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autonomous retarded functional differential equations follow at once. Numerical tests are provided.

Existence results for three-point boundary value problems for systems of linear functional differential equations

Conditions guaranteeing the solvability of certain three-point boundary value problems for a system of linear functional differential equations are obtained by using a special successive approximation scheme. We also establish some conditions necessary for a certain set belonging to the domain of the space variables to contain a point determining the initial value of the solution. An algorithm for selecting such points is also indicated.

Slowly growing solutions of singular linear functional differential systems

AbstractWe obtain new conditions under which a system of linear functional differential equations with singular coefficients has a unique solution possessing a given initial value and satisfying a certain growth restriction.

Almost automorphy for abstract neutral differential equations via control theory

In this paper we study the existence of almost automorphic solutions for a class of linear neutral functional differential equations with finite delay and values in a Banach space. We show that the existence of an almost automorphic mild solution is related to the approximate controllability of a distributed control system. We applied our results to establish the existence of an almost automorphic solution for a neutral wave equation with delay.

Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.

Hyperbolicity of solution semigroups for linear neutral differential equations

Consider the linear neutral functional differential equation of the form $$\left\{\begin{array}{l@{\quad }l}\frac{\partial}{\partial t}Fu_t=BFu_t+\Phi u_t, & t\ge 0,\\[3pt]u_0(s)=\varphi(s),& s\in [-r,0],\end{array}\right.$$ where the function u(⋅) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ we prove that the solution semigroup for this equation is hyperbolic provided that the differential operator B generates a hyperbolic semigroup on X.

The weighted Cauchy problem for linear functional differential equations with strong singularities

Abstract The sufficient conditions of well-posedness of the weighted Cauchy problem for higher order linear functional differential equations with deviating arguments, whose coefficients have nonintegrable singularities at the initial point, are found.

Bivariate delta-evolution equations and convolution polynomials: Computing polynomial expansions of solutions

Abstract This paper describes an application of Rota and collaborator’s ideas, about the foundation on combinatorial theory, to the computing of solutions of some linear functional partial differential equations. We give a dynamical interpretation of the convolution families of polynomials. Concretely, we interpret them as entries in the matrix representation of the exponentials of certain contractive linear operators in the ring of formal power series. This is the starting point to get symbolic solutions for some functional–partial differential equations. We introduce the bivariate convolution product of convolution families to obtain symbolic solutions for natural extensions of functional-evolution equations related to delta-operators. We put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations. We create an adequate framework to base theoretically some of the performed constructions and to get some existence and uniqueness results.

On Regularity Property of Retarded Ornstein–Uhlenbeck Processes in Hilbert Spaces

In this work, some regularity properties of mild solutions for a class of stochastic linear functional differential equations driven by infinite-dimensional Wiener processes are considered. In terms of retarded fundamental solutions, we introduce a class of stochastic convolutions which naturally arise in the solutions and investigate their Yosida approximants. By means of the retarded fundamental solutions, we find conditions under which each mild solution permits a continuous modification. With the aid of Yosida approximation, we study two kinds of regularity properties, temporal and spatial ones, for the retarded solution processes. By employing a factorization method, we establish a retarded version of the Burkholder–Davis–Gundy inequality for stochastic convolutions.

The linear theory of functional differential equations: existence theorems and the problem of pointwise completeness of the solutions

A boundary value problem and an initial-boundary value problems are considered for a linear functional differential equation of point type. A suitable scale of functional spaces is introduced and existence theorems for solutions are stated in terms of this scale, in a form analogous to Noether's theorem. A key fact is established for the initial boundary value problem: the space of classical solutions of the adjoint equation must be extended to include impulsive solutions. A test for the pointwise completeness of solutions is obtained. The results presented are based on a formalism developed by the author for this type of equation.Bibliography: 7 titles.