Functional Differential Equations Of Neutral Type Research Articles

Norm estimates for fundamental solutions of neutral type functional differential equations

This paper deals with a vector autonomous neutral type functional differential equation. Estimates for various norms of the fundamental solution are established. By the derived estimates we obtain the stability conditions for equations with non-linear causal mappings. Equations with causal mappings include differential-delay, integro-differential, and other equations. Our stability conditions are explicitly formulated in terms of the entries of the characteristic matrices and Lipschitz constants of non-linearities. The suggested approach is based on a combined use of the recent estimates for the Euclidean norm of matrix-valued functions with some properties of the fundamental solutions.

Lyapunov functionals of constant signs in the stability problem for a functional-differential equation of neutral type

We present a generalization of the Lyapunov functional method and use it to study the stability of nonautonomous functional-differential equations of neutral type with finite delay. This generalization is based on constructing the limit equations and the limit functionals.

Oscillatory Behaviour of a Higher Order Nonlinear Neutral Delay Type Functional Differential Equation with Oscillating Coefficients

In this paper we are concerned with the oscillation of solutions of a certain more general higher order nonlinear neutral type functional differential equation with oscillating coefficients. We obtain two sufficient criteria for oscillatory behaviour of its solutions.

Topological index for condensing maps on Finsler manifolds with applications to functional-differential equations of neutral type

The topological index for maps of infinite-dimensional Finsler manifolds, condensing with respect to internal Kuratowski's measure of non-compactness, is constructed under the hypothesis that the manifold can be embedded into a certain Banach linear space as a neighbourhood retract so that the Finsler norm in tangent spaces and the restriction of the norm from enveloping space on the tangent spaces are equivalent. It is shown that the index is an internal topological characteristic, i.e. it does not depend on the choice of enveloping space, embedding, etc. The total index (Lefschetz number) and the Nielsen number are also introduced. The developed machinery is applied to investigation of functional-differential equations of neutral type on Riemannian manifolds. A certain existence and uniqueness theorem is proved. It is shown that the shift operator, acting in the manifold of $C^1$-curves, is condensing, its total index is calculated to be equal to the Euler characteristic of (compact) finite-dimensional Riemannian manifold where the equation is given. Some examples of calculating the Nielsen number are also considered.

Delay dependent estimates for waveform relaxation methods for neutral differential-functional systems

In this paper, the problem of delay dependent error estimates for waveform relaxation methods applied to Volterra type systems of functional-differential equations of neutral type including systems of differential-algebraic equations is discussed. Under a Lipschitz condition (with delay dependent right-hand side) imposed on the so-called splitting function it is shown how the error estimates depend on the character of delays and that for this reason they are better than the known error estimates for relaxation methods. It is proved that under some assumptions the exact solution can be obtained after a finite number of steps of the iterative process, i.e., we prove that the waveform relaxation methods have the same property as the classical method of steps for solving delay-differential equations with nonvanishing delays. We also show the convergence of the waveform relaxation method without assuming that the spectral radius of the corresponding matrix related to the Lipschitz coefficients for the neutral argument is less than one.

Convergence of the spline function for functional-differential equation of neutral type

The existence, uniqueness and stability for the functional-differential equation of neutral type using spline of deficiency 3 with stepsize 3h spline function of degree four are presented in Ref. [1]. In this paper, we extend the study to the convergence of our proposed spline method. We prove that, if the local error is of order p, then the global error is of order p as well. Numerical examples are presented to illustrate the convergence of the method.

Stability analysis of numerical methods for systems of functional-differential and functional equations

This paper is concerned with the numerical solution of functional-differential and functional equations which include functional-differential equations of neutral type as special cases. The adaptation of linear multistep methods, one-leg methods, and Runge-Kutta methods is considered. The emphasis is on the linear stability of numerical methods. It is proved that A-stable methods can inherit the asymptotic stability of underlying linear systems. Some general results of stability on explicit and implicit methods are also given.

Existence, Uniqueness and Applications of the Spline Method for Functional-Differential Equation of Neutral Type

This paper presents a class of numerical method for the approximate solution for the functional-differential equation of neutral type. These methods are essentially based on the spline functions. The study of existence and uniqueness are considered of 3h-step spline function of degree m =4. Numerical examples and comparisons with other methods are given.

Controlling linear functional differential equation

Necessary conditions for the optimal control of a linear system of neutral type functional differential equations are obtained. We show that a success in reducing the initial problem to the problem of solvability of a certain boundary value problem is, in principle, a question of one's ability to construct adjoint operators to the operators appearing in the initial control problem.

Existence and stability of periodic solutions of semilinear neutral type systems

Forced periodic oscillations of coupled systems of semilinear neutral type functional differential equations are investigated. Explicit conditions for existence, uniqueness and stability of periodic solutions are derived. These conditions are formulated in terms of the roots of characteristic polynomials. In addition, estimates for periodic solutions and their derivatives are established.

The method of Lyapunov functionals in stability analysis of functional-differential equations

We develop the method of Lyapunov functionals in the stability analysis of linear nonautonomous functional-differential equations of neutral type. The approach is based on the construction of limit equations and limit Lyapunov functionals.

On stability of a class of neutral type functional differential equations

This paper deals with the study of stability and estimation of stability domain for a class of nonlinear integro-difference equations, which coincide with special class of neutral type functional differential equations. The new approach for stability study called the pattern equation method is proposed.

Solubility and properties of solutions of functional-differential equations in Hilbert space

A class of functional-differential equations of neutral type with coefficients that are functions taking values in the set of (in general, unbounded) operators in separable Hilbert space is considered. For such equations results on the well-posed solubility of initial-boundary-value problems on the semiaxis in Sobolev weighted spaces are established.

A MULTIVALUED GOURSAT PROBLEM FOR FUNCTIONAL-DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

A MULTIVALUED GOURSAT PROBLEM FOR FUNCTIONAL-DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

Stability estimates of solutions of systems of functional-differential equations of neutral type

A system of linear functional-differential equations of neutral type is considered. Investigation of stability is carried out by the method of Lyapunov functions of quadratic form. Sufficient conditions for the asymptotic stability and estimates of the exponential decay of solutions for an arbitrary and a small deviation of the argument are obtained.

Oscillation properties of first order nonlinear functional-differential equations of neutral type

Oscillation properties of first order nonlinear functional-differential equations of neutral type

Neutral functional differential equations subject to mild integral conditions

This paper is concerned with a neutral type functional differential equation of the form .Under appropriate mild integral conditions on f and g there are derived asymptotic properties for the solutions of(E)

Oscillation of the solutions of a certain class of first-order functional-differential equations of neutral type

Oscillation of the solutions of a certain class of first-order functional-differential equations of neutral type

On some oscillation criteria for a class of neutral type functional differential equations

AbstractThe present paper proves criteria for oscillation of the soultions of functional differential equations of the typewhere λ, τ > 0.

Equivalence of functional-differential equations of neutral type and abstract Cauchy problems

Equivalence of functional-differential equations of neutral type to well-posed abstract Cauchy problems in the state spaces ℝ n }L p ,L p resp.C is investigated. The assumptions allow non-atomic difference operators.