On stability of a class of neutral type functional differential equations

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Abstract

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.

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 equationN(t)=∫t−Ltf(s,N(s))dsIn compartmental analysis of biological systems Gyori 7, 8and Arino et al. 1, 2used the equationx(t)−cx(t−h)=∫t−htf(s,x(s))dsas 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 Rn be the n-dimensional real vector space with a norm |.| and C=C([−h, 0], Rn) the space of all continuous functions mapping [−h, 0] onto Rn. The norm of a function ϕ(θ)∈C is ‖ϕ‖=sup{|ϕ(θ)|,−h≤θ≤0}.

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 formy(t)=∑i=1kAi(t)y(t−hi)+∫t−1tf(s,y(s))ds,y(t)∈Rn, t≥0,y(θ)=ϕ(θ),−h≤θ≤0,h=max{h1,…,hk,1},hi>0Suppose

Pattern equation method

Along with NIDE (3), (7) we shall consider a scalar NIDE of the same form:x(t)=∑i=1kai(t)x(t−hi)+∫t−1tk(t,s)α(|x(s)|)ds,t≥0,x(θ)=ψ(θ),−h≤θ≤0,x(t)∈R1Here 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 andy(t,ϕ)→0,t→∞, ϕ∈Q⊂Cf.The 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 that0<p(t)≤P,t≥−h,p(t)→0,t→∞

  • 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)

References (14)

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