Functional Differential Systems Research Articles

Global attractivity of periodic solution for neutral functional differential system with multiple deviating arguments

In this paper, a kind of neutral functional differential system with multiple deviating arguments is considered. By means of Mawhin's coincidence degree theory and Lyapunov method, a sufficient condition is obtained for guaranteeing the existence and global attractivity of periodic solution for the system. It is interesting that the result is related to the multiple deviating arguments τi(i = 1, 2, …, m). An example is given to show the feasibility of the result in the last section. Copyright © 2011 John Wiley & Sons, Ltd.

New properties of the [formula omitted]-operator and its applications on the problem of periodic solutions to neutral functional differential system

In this paper, the authors first investigate some new properties of the D -operator which is associated with neutral functional differential equations. Then, by using these new properties, the problem of existence of periodic solutions of the D -operator neutral functional differential system is studied. The interesting thing is that the D -operator is allowed to be unstable, which is generalized by the corresponding studies in the past under the crucial assumption that the D -operator is stable.

Generalized cauchy problem for hyperbolic functional differential systems

Generalized cauchy problem for hyperbolic functional differential systems

On the Pathwise Growth Rates of Large Fluctuations of Stochastic Population Systems with Infinite Delay

In general, time delay and system uncertainty are commonly encountered for all population systems. To examine whether the presence of environmental noise affects infinite delay population systems significantly, this paper perturbs the functional Kolmogorov-type system with infinite delay into the infinite delay stochastic functional differential system By the M-matrix technique, this paper examines the global positive solution and its pathwise estimation for this stochastic functional Kolmogorov-type population system. To illustrate the applications of our theory more clearly, this paper also discusses a Lotka–Volterra system with mixed delays as a special case.

On the solvability of the periodic problem for systems of linear functional differential equations with regular operators

Systems of two linear functional differential equations of the first order with regular operators are considered. General necessary and sufficient conditions for the unique solvability of the periodic problem are obtained. For one system with monotone operators we get effective necessary and sufficient conditions for the unique solvability of the periodic problem. We consider some classes of two-dimensional systems of first order linear func- tional differential equations with regular operators. General necessary and sufficient conditions for the solvability of the periodic problem for such classes are obtained. These conditions mean that some function on a set in a finite-dimensional space is positive (this functions is quadratic with respect to all variables). Moreover, in terms of norms of the operators appearing in the functional differential system, we get the necessary and sufficient conditions for the unique solvability of the periodic problem for one case of two-dimensional system with monotonic operators. It is found there exist two domains of parameters corresponding to the unique solvability. These result do not have analogues for systems. Non-improvable results for periodic problem are known only for cyclic first order functional differential systems (32). Necessary and sufficient conditions for the unique solvability of two-dimensional

On Asymptotic Behaviour of Solutions to n‐Dimensional Systems of Neutral Differential Equations

This paper presents the properties and behaviour of solutions to a class of n‐dimensional functional differential systems of neutral type. Sufficient conditions for solutions to be either oscillatory, or limt→∞yi(t) = 0, or limt→∞|yi(t)| = ∞, i = 1,2, …, n, are established. One example is given.

Positive solutions of functional-differential systems via the vector version of Krasnoselskii’s fixed point theorem in cones

We study the existence of positive solutions of the functional-differential system ... (0 < t < 1), subject to linear boundary conditions. We prove the existence of at least one positive solution by using the vector version of Krasnoselskii’s fixed point theorem in cones.

EXISTENCE AND CONTROLLABILITY RESULTS FOR INFINITE DELAY PARTIAL FUNCTIONAL DIFFERENTIAL SYSTEMS WITH MULTI-VALUED IMPULSES IN BANACH SPACES

This paper is concerned with the existence and controllability of solutions for infinite delay functional differential systems with multi-valued impulses in Banach space. Sufficient conditions for the existence are obtained by using a fixed point theorem for multi-valued maps due to Dhage. An example is also given to illustrate our results.

Preservation of exponential stability for linear non-autonomous functional differential systems

We consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation. As particular cases, the system includes models with concentrated delays and systems of integrodifferential equations. Our method is based on Bohl–Perron type theorems.

Stochastic suppression and stabilization of functional differential equations

Without the linear growth condition or the one-sided linear growth condition, this paper discusses whether or not stochastic noise feedback can stabilize a given unstable nonlinear functional system x ̇ ( t ) = f ( x t , t ) . Since f may defy the linear growth condition or the one-sided linear growth condition, this system may explode in a finite time. To stabilize this system, this paper stochastically perturbs this system into the stochastic functional differential system d x ( t ) = f ( x t , t ) d t + q x ( t ) d w 1 ( t ) + σ | x ( t ) | β x ( t ) d w 2 ( t ) by two independent Brownian motions w 1 ( t ) and w 2 ( t ) . This paper shows that the Brownian motion w 2 ( t ) may suppress the potential explosion of the solution for appropriate β . Moreover, for sufficiently large q , this stochastic functional system is exponentially stable. These results can be used to examine stochastic stabilization.

Stability of linear functional differential systems with a multivalued feedback with respect to impulse disturbances

We consider a linear controlled functional differential system with a linear feedback channel. The system is assumed to be exponentially stable in the closed state. The feedback is assumed to be nonideal, which consists in an uncertain delay either distributed or not. The only assumption is that this delay is sufficiently small. Such a nonideal system is described by a functional differential inclusion of special type. A generalized derivative of a function of locally bounded variation is admitted to the input of the system as a free term. We obtain exponential estimates for solutions of the resulting system.

Global exponential stability of impulsive stochastic functional differential systems

In this paper, based on the Razumikhin techniques and Lyapunov functions, several criteria on the global exponential stability and instability of impulsive stochastic functional differential systems are obtained. Our results show that stochastic functional differential systems may be exponentially stabilized by impulses. Two illustrative examples are given to show the effectiveness of the results.

Razumikhin-type theorems for asymptotic stability of impulsive stochastic functional differential systems

In this paper, we investigate the pth moment uniformly asymptotic stability of impulsive stochastic functional differential systems by extending some Razumikhin-type theorems. Based on the Lyapunov functions and Razumikhin techniques, some criteria are established and their applications to impulsive stochastic delay systems are proposed. An illustrative example shows the effectiveness of our results.

Approximate Controllability of Neutral Stochastic Functional Differential Systems with Infinite Delay

In this article, we consider a class of control systems governed by the neutral stochastic functional differential equations with unbounded delay and study the approximate controllability of the system. An example is given to illustrate the result.

Implicit difference methods for nonlinear first order partial functional differential systems

Initial problems for nonlinear hyperbolic functional differential systems are considered. Classical solutions are approximated by solutions of suitable quasilinear systems of difference functional equations. The numerical methods used are difference schem

Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations

The method to compare only one component of the solution vector of linear functional differential systems, which does not require heavy sign restrictions on their coefficients, is proposed in this paper. Necessary and sufficient conditions of the positivity of elements in a corresponding row of Green's matrix are obtained in the form of theorems about differential inequalities. The main idea of our approach is to construct a first order functional differential equation for the th component of the solution vector and then to use assertions about positivity of its Green's functions. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. It should be also noted that the sufficient conditions, obtained in this paper, cannot be improved in a corresponding sense and does not require any smallness of the interval , where the system is considered.

Nontrivial periodic solutions of nonlinear functional differential systems with feedback control

This paper examines the existence of nontrivial periodic solutions for the nonlinear functional differential system with feedback control: \{\aligned x'(t)=x(t)a(t)-\big[\sum_{i=1}^n a_i(t)\int_0^{+\infty} f(t, x(t-\theta)) d}\varphi_i(\theta) +\sum_{j=1}^m b_j(t) \int_0^{+\infty} f(t,x'(t-\theta))\,d}\phi_j(\theta)+\sum_{\mu=1}^p c_\mu(t) \int_0^{\infty} u(t-\theta)\,d}\delta_\mu(\theta)\big], u'(t)=-\rho(t)u(t)+\sum_{\nu=1}^q \beta_\nu(t) \int_0^{\infty} f(t, x(t-\theta))\,d}\psi_\nu(\theta).\endaligned Under certain growth conditions on the nonlinearity f, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative.

Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces

In this paper, we investigate the controllability for a class of abstract impulsive neutral functional differential systems with infinite delay where the linear part is non-densely defined and satisfies the Hille–Yosida condition. The approach used is the Schauder fixed point theorem combined with the operator semigroups. Particularly, the compactness of the operator semigroups is not needed in this article.

Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay

We regard the stochastic functional differential equation with infinite delay d x ( t ) = f ( x t ) d t + g ( x t ) d w ( t ) as the result of the effects of stochastic perturbation to the deterministic functional differential equation x ̇ ( t ) = f ( x t ) , where x t = x t ( θ ) ∈ C ( ( − ∞ , 0 ] ; R n ) is defined by x t ( θ ) = x ( t + θ ) , θ ∈ ( − ∞ , 0 ] . We assume that the deterministic system with infinite delay is exponentially stable. In this paper, we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable.

Approximate controllability of semilinear partial functional differential systems

In this paper, we study the approximate controllability of semilinear neutral functional-differential systems and impulsive functional-differential systems with finite delay. Since the considered equations admit nonlinear terms involving spatial derivatives, the fraction power theory and ?-norm is used to discuss the problem so that the established results can be applied to them. An example is provided to illustrate the application of the obtained results.