Differential Equations Of Neutral Type Research Articles

A theorem on the spectral radius of the sum of two operators and its application

In the present paper a theorem on the spectral radius of the sum of linear operators is established. The application of this theorem to a functional differential equation of neutral type is also given.

Mathematical controllability theory of capital growth of nations

In this report, we derive a nonlinear functional differential equation of neutral type as the dynamics of the growth of capital stock of n firms linked up in solidarity in a region. These equations are generalizations of ordinary differential systems studied by Takayama [21, p. 685–706], Arrow [2, p. 184], Knowles [17, p. 3], and Isaacs [13], and delay systems studied by Kalecki [14], Allen [1, p. 254], and Cooke and Yorke [9]. A neutral version (in which there is delay in the derivative) is implicit in the suggestion by Arrow [2, p. 184]. Given an initial capital endowment φ and an initial investment r, we investigate what conditions on the systems parameters and on the external interventions will guarantee that φ can be steered to some fixed target in some finite time. The controls, u, ( m-vectors) are investments(if 0⩽ u j(t) ⩽ 1) and consumption (if −1 ⩽ u j(t) ⩽ 0). From the controllability theorems proved for the hereditary systems, we derive universal laws for the control of the growth of capital stock. These laws provide broad, startling, but obvious policy prescription for the growth of companies and for national economies. For example, it describes how big external or governmental interventions on the firms (in the form of taxation or subsidy) must be, relative to the firms' investment and consumption to ensure the growth of capital stock to the given target. It also states when this growth is impossible. We use fixed point theorems and arguments from the theory of games of pursuit [10] to establish constrained controllability of our dynamics. Problems of optimality and interpretations are pursued in a subsequent communication.

Asymptotic stability of linear stochastic differential equations of neutral type

Asymptotic stability of linear stochastic differential equations of neutral type

Existence of almost periodic solutions of functional differential equations of neutral type

In this paper, we establish the theorems of existence of an almost periodic solution of an almost periodic neutral functional differential equation and its disturbed systems by Liapunov functional.

Oscillation of solutions of second-order nonlinear functional differential equations of neutral type

The oscillation condition of solutions to nonlinear differential equations of the second order of neutral type on the semiaxis t > 0 and the properties of nonoscillatory solutions are investigated.

Periodic solutions for some nonlinear differential equations of neutral type

where f: IR x iR + IR is a Caratheodory function, f( *, x) is 2n-periodic for every x, CI E R and r E IO, 2n[. This problem has been studied by Sadovskii in [ 151, by Hale and Mawhin in [6], in the case Ial 1 to the previous one. Our interest is particularly motivated by the fact that (*) represents a simple and meaningful model for a general phenomenon which often arises in the study of functional differential equations. In fact, as is well known, in some cases a delayed differential equation can be seen as a perturbation of an ordinary differential equation. In particular this occurs when the delayed term in the equation can be considered smaller than the ordinary one, or dominated by it. When such domination is available many good properties can be proved, and several results (such as those of [6] and [16]) take advantage of this fact. On the contrary, when no domination is present, the situation is much more complicated and in general very little is known. In problem (*), the delayed term can be considered dominated by the ordinary one whenever Ial < 1. As a matter of fact, in this case, one can prove the invertibility of the differential operator L,x = x’ + a~‘(. r) + ,ux, the compactness of its inverse and even some a priori estimates for the solutions. Consequently, one can reduce (*) to a fixed point problem and solve it by means of some topological degree argument. However, when ((11 = 1, so that the weight of the two terms x’(t) and x’(t T) is the same, the above properties fail, and moreover (for some delays) the kernel of the differential operator Lx = x’ + a~‘(* r) is infinite dimensional. Therefore one must look for different methods. Problem (*) is thus a simple but nevertheless significant example of the considerations illustrated by Hale in [5] about the difficulty of undertaking a unified approach in the study of functional differential equations. At the same time, it provides an analogy between neutral equations and the wave equation, in the line suggested by Hale in [5].

Necessary and sufficient conditions for oscillation of the solutions of linear functional differential equations of neutral type with distributed delay

For various classes of equations of the general form, d dt (x(t) + δ 1 ∝ 0 τ1 x(t − s) dr 1(s)) + δ 2 ∝ 0 τ2 x(t − s) dr 2(s) = 0 , where δ 1, δ 2 = ± 1 and r 1, r 2 are non-decreasing functions, necessary, sufficient, as well as necessary and sufficient, conditions have been obtained for the oscillation of all nonvanishing solutions in terms of the properties of the roots of the respective characteristic or quasi-characteristic equation.

On a Numerical‐Analytic Method of Solving of Boundary Value Problem for Functional Differential Equation of Neutral Type

On a Numerical‐Analytic Method of Solving of Boundary Value Problem for Functional Differential Equation of Neutral Type

On a class of first order nonlinear functional differential equations of neutral type

On a class of first order nonlinear functional differential equations of neutral type

Neutral functional integro-differential equations with weakly singular kernels

The well-posedness of a general functional differential equation of neutral type is considered, where the difference operator does not have an atom at zero. the results are applicable to certain classes of singular integro-differential equations that occur in aeroelasticity. The equations are considered as dynamical systems on product spaces

On approximation of the solutions of delay differential equations by using piecewise constant arguments

By using the Gronwall Bellman inequality we prove some limit relations between the solutions of delay differential equations with continuous arguments and the solutions of some related delay differential equations with piecewise constant arguments(EPCA). EPCA are strongly related to some discrete difference equations arising in numerical analysis, therefore the results can be used to compute numerical solutions of delay differential equations. We also consider the delay differential equations of neutral type by applying a generalization of the Gronwall Bellman inequality.

Sufficient conditions for oscillations in higher order linear functional differential equations of neutral type

Sufficient conditions for oscillations in higher order linear functional differential equations of neutral type

Oscillation of the solutions of parabolic differential equations of neutral type

Sufficient conditions are obtained for the oscillation of the solutions of linear parabolic differential equations of neutral type of the form ∂ ∂t [u - Σ i=1 m λ i(t)u(x,t-τ i]=a(t)▵u+ Σ i=1 s a i(ft)▵u(x,t-p i)-p(x,t)u- Σ i=1 k p i(x,t)u(x,t-σ i) , ( x, t)ϵΩ×(0,∞), where Ω is a bounded domain in R n with a piecewise smooth boundary.

Exponential dichotomy and stability of neutral type equations

A linear functional differential equation of neutral type with unbounded delay Lx( d ds )Dx+Bx=0 , where D and B are linear bounded retarded operators with exponentially fading memory, is considered. It is shown that if operator L is interpreted as operator from the space C into the special space C −1 of distributions, then its invertibility is equivalent to the presence of exponential dichotomy of the solutions of this equation. As applications, we prove the theorems on stability and instability in the first approximation for neutral functional differential equations of a general form.

The time control theory of linear differential equations of neutral type

The paper tackles the time optimal control theory of linear neutral equations. Controllability questions in function space are answered. The existence and forms of an optimal control are established both in function space and in Euclidean space.

Solutions for a class of neutral differential equations with nonatomicD-operator

This paper is concerned with the existence and uniqueness of solutions to first-order linear differential equations of neutral type with a nonatomicD-operator. We use an algebraic approach, in the context of the theory of convolution operators. Necessary and sufficient conditions are given for the initial value problem to have unique solutions for a class of equations.

A Boundary Value Problem for Functional Differential Equations of Neutral Type

A Boundary Value Problem for Functional Differential Equations of Neutral Type

Uniform asymptotic stability of perturbed systems of neutral differential equations

IN THIS work we discuss the effect of a certain type of perturbations on systems for which the zero solution is uniformly asymptotic stable, using the second method of Lyapunov. With this objective, we give a new alternative for the converse theorems for uniform asymptotic stability of Hale and Cruz [l]. We define a Lyapunov’s functional which together with its ‘derivative’ satisfies some inequalities involving the solution of the differential equation of neutral type and not only the uniformly stable operator D(t, .), defined in the sense of Section 1. On the other hand, our Lemma 1, of very simple demonstration, shows that the hypothesis that Ix,(c, 9) x,(5 491 < eL(+14 YI, t 2 0, where L is a constant and xr(g,& is a solution of a neutral functional differential equation, is unnecessary in Theorem 1 of Iti and dos Reis [2] ; and we can obtain the same and new results. We think that these results are important for studying the stability of perturbed systems of neutral differential equations and applications.

Bounded solutions of functional differential equations of the neutral type with infinite delays

SynopsisIn this paper the authors obtain sufficient conditions for the existence and uniqueness of the initial value problem of functional differential equations of neutral type with infinite delays, making use of some earlier results of the present authors.

Some remarks on existence of optimal controls for Mayer problems with singular components

This paper proves an existence theorem for optimal controls for systems governed by ordinary differential equations and a large class of functional differential equations of neutral type. Extensions beyond earlier work are made as a result of employing a new closure theorem originally used by Cesari and Suryanarayana in their study of Pareto optima and which is, in turn, based on the Fatou lemma for vector-valued functions as proved by Schmeidler. The use of these techniques simplifies the standard arguments for existence in the presence of singular components and allows the use of very weak semi-normality conditions. It also permits the consideration of a significantly larger class of hereditary systems than has been treated in the existing literature.