On stability of a class of neutral type functional differential equations
Introduction
The aim of this paper is to investigate stability of certain types of dynamical systems, described by integral or integro-difference equations. Such equations can be often interpreted as retarded or neutral type functional differential equations 10, 11. But the possibility to write them as an integro-difference equation provides new approaches to study them. We shall denote this class of equations by NIDE.
Certain types of such equations are often used in modelling of biological systems. Cooke et al. in papers 4, 5, 6studied the following model for population dynamics or epidemics. Let N(t) denote the number of individuals in an isolated population at time t. The life-span of every individual is assumed to be a fixed constant L. Assume that the number of birth per unit time is some function of N(t), say f(N(t)), or more generally f(t, N(t)). Under these assumptions, the growth of the population is governed by the equationIn compartmental analysis of biological systems Gyori 7, 8and Arino et al. 1, 2used the equationas a model of fluid stream (for example, blood) throughout different organs assimilated to some boxes connected by pipes. The quantities of fluid entering or running out the boxes depend in nonlinear manner on quantities of fluids inside boxes. For all these models, it is natural to suppose that the function f(t, x(t)) is nondecreasing with respect to the second argument and that also f(t, x(t))=0. For population dynamics model (1), these assumptions mean that, firstly, the greater is the population, the greater is the number of births per unit time and, secondly, if there is no individual in the population, there is no birth also.
The stability of such models was studied by the Lyapunov direct methods in 6, 10, 11. Arino et al. 1, 2used Lyapunov functions based on an order relation for stability study of such equations. Existence of positive periodic, almost periodic or pseudo almost periodic solutions of , have been established under different assumptions in 1, 4, 5, 9, 12, 14.
We shall study stability and asymptotic properties of NIDEs using a new approach – the pattern equation method. This method is based on the construction of special scalar equations (called pattern equations) whose solutions are upper bounds of the solutions of the initial many-dimensional NIDE. It permits to give explicit conditions on the coefficients of the initial NIDE which imply required stability properties or desired asymptotic behaviour. For discrete Volterra equations, the pattern equation method is developed in 3, 13
Section snippets
Problem statement
Let n be the n-dimensional real vector space with a norm |.| and C=C([−h, 0], n) the space of all continuous functions mapping [−h, 0] onto n. The norm of a function ϕ(θ)∈C is .
For an (n×n)-matrix A(t), we denote by |A(t)| the operator norm of this matrix corresponding to the norm |.|.
Let us consider a nonlinear neutral type integro-difference equation (NIDE) of the formSuppose
Pattern equation method
Along with NIDE (3), (7) we shall consider a scalar NIDE of the same form:Here ai(t)>0, i=1,…,k are continuous functions, α(u) is a continuous monotone nondecreasing function such that α(0)=0, α(u)>0, u>0, the kernel k(t, s) is continuous and positive, the initial function ψ(θ) is also continuous and positive and verifies condition (4). Under these assumptions, the solution x(t) of problem (8) is positive, x(t)>0, t≥0.
The
Stability
Here we study stability properties of NIDE (3), (7). NIDE (3) under the assumption (6) has a trivial solution y(t)=0, t≥0. We consider the usual definition of stability 10, 11.
Definition 3 The trivial solution y(t)=0, t≥0 of NIDE (3) is called Lyapunov stable if for any ε>0 there exists a δ(ε)>0 such that |y(t)|≤ε, t≥0 if ‖ϕ‖≤δ, ϕ∈Cf.
The following theorem gives sufficient conditions for stability in terms of corresponding pattern equations and coefficients of the initial NIDE.
Theorem 4 Suppose
Asymptotic stability
Definition 6 The trivial solution y(t)=0 of NIDE (3) is called asymptotically stable if it is stable andThe set Q of such initial functions ϕ is called the attraction domain of the trivial solution. If Q=Cf, then the trivial solution is globally asymptotically stable.
Theorem 7 Suppose there exist
- 1.
a bounded pattern function p(t) such that
- 2.
numbers γi, i=1,…, k+1, satisfying conditions (11) and a number δ0>0, δ0P≤a such that for all 0<δ≤δ0, conditions (14)
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Cited by (9)
Stability analysis for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays
2010, NeurocomputingCitation Excerpt :Recently, neutral type systems have been intensively studied due to the reason that many practical processes can be modeled as general neutral type descriptor systems, such as circuit analysis, computer aided design, real time simulation of mechanical systems, power systems, chemical process simulation, population dynamics and automatic control. For example, Arino and Nosov [1] studied the stability and asymptotic properties of a class of neutral type functional differential equations based on the pattern equation method; by employing a new way on functional analysis of semi-group of operators, Hadd [6] investigated a class of singular functional differential equations of neutral type in Banach spaces; Han et al. [7] discussed robust absolute stability criteria for uncertain Lur’e systems of neutral type; Rabah et al. [15] investigated the asymptotic stability properties of neutral type systems in Hilbert space, and [16] studied the problem of strong stabilizability of linear systems of neutral type; Sahiner [19] discussed the oscillatory behavior of the second order neutral type delay differential equations; Wang and Liu [22] studied the existence, uniqueness and global attractivity of periodic solution for a kind of neutral functional differential systems with delays; Xu et al. [26] considered the problem of guaranteed cost control for uncertain neutral stochastic systems. Moreover, it is inspiring that the neutral type phenomenon has been already taken into account in delayed neural networks; see, for instance Lien et al. [4], Lou and Cui [10], Park et al. [13,14], Rakkiyappan and Balasubramaniam [17,18], and so on.
Ovide Arino: Friend and maestro - 24th April 1947-29th September 2003
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