GLOBAL ASYMPTOTIC STABILITY IN A CLASS OF NONLINEAR DIFFERENTIAL DELAY EQUATIONS

. An essentially nonlinear differential equation with delay serving as a mathematical model of several applied problems is considered. Sufficient conditions for the global asymptotic stability of a unique equilibrium are de rived. An application to a physiological model by M.C. Mackey is treated in detail.

1. Introduction. This paper concerns the qualitative analysis of a class of scalar essentially nonlinear differential delay equations. The equations are given by x'(t) = F(x(t-r ))-G(x(t)) (1) where F and G are continuous real-valued functions .
Equations of this type have attracted a. significant interest in recent years due to their frequent appearance in a wide range of applications. They serve as mathematical models describing various real life phenomena in mathematical biology, population dynamics and physiology, electrical circuits and laser optics, economics and life sciences, others. See papers[:;, I , X, 10 , II , 1:;, \.-,] and references therein for a partial list of applications and further details.
There is a significant body of theoretical mathematical research on differential delay equation ( I) done in the past 20-30 years. They address various aspects of the dynamics in such equations including global asymptotic stability of equilibria, existence of periodic solutions, complicated behavior· and chaos, among others.
However, most of it deals with the case of linear function G, i.e. G(x) = bx, b > 0.
Papers [ 1 , I , I l, I.~] represent a pru·tial list of related references.
The principal problem of our interest in this paper is the global asymptotic stability of equilibria in equation (I) . We treat the case when both functions F and G are nonlinear, which we brand as "essentially nonlinear". For the simpler case of linear G there are several papers that deal with the global asymptotic stability, see for example [7, ' ] and further references therein. Several methods and approaches have been used to tackle this problem and a good variety of results has been derived.
Very little research was done for the case of nonlinear G. However, namely equations with the non-linear G have appeared recently in several important applications. We refer the reader to paper [12] where a model of hematopoietic cell replication and control is considered, and to paper [11] for additional models where both F and G are nonlinear, as well as to monograph ['-'] for additional applications and references. There are only a few publications on the global asymptotic stability in such equations; the ones we are aware of are [.i, '-'] . The new mathematical results of this paper mainly concern the case of nonlinear G. We address problems of global asymptotic stability of unique trivial (zero) or positive equilibrium. We further extend and improve some of the above mentioned results in [\ ~'>], including the case of non-monotone G.
The paper consists of three principal parts. The first one concerns necessary preliminaries and general properties of solutions of equation ( I) and the corresponding difference equations and one-dimensional maps. In the second part, t he main aspect under discussion is the global asymptotic stability of a unique positive equilibrium. We establish a global asymptotic stability result for such equations which makes use of the global attractivity in a limiting one-dimensional map. The limiting map can in general be a multi-valued one. This way we extend the known global stability results earlier proved in [· \ 7] to this new class of differential delay equations. T he third part concerns applications of the mathematical theory of these equations to several real life models. One of the applications is a class of physiological population models proposed by M.C. Mackey in a series of papers. We prove in particular global stability results for the model. Other applications are in economics and related fields.
2. Assumpt io ns and Preliminaries. In this subsection we present some basic properties and mathematical results on differential delay equation ( 1). The following hypotheses on t he nonlinearities F and G will be assumed in different combinations throughout the paper. It is a standard form of singularly perturbed differential delay equations with the normalized delay T = 1 [7]. Therefore, we will be considering the differential delay as an equivalent form of equation (I).  (2) is rather trivial then, as it will be shown later {in particular for some applied models). The significance of the general assumptions contained in hypothesis A is seen from the following simple statement. Proof. This is a well known property of equation ( 1) j (1) which is contained in various forms in several other papers, in particular in paper [i] for the case of linear . It is easily established by simple reasoning, which we provide here for the sake of completeness and subsequent references in the paper.
Indeed, if on the contrary x(to) = O,x(t) > 0 Vt < t 0 , t 0 > 0 then there exists a sequence {tn} t to such that x'(tn) < 0. Since G{O) = 0, F(x(to-1)) > 0 and /.lX'(tn ) = F(x(tn-1))-G(x(tn)) one arrives to a contradiction by taking the limit in t he latter as n --+ oo. As it is seen from the proof any solut ion x(t) with arbitrary init ial function <1> E C = C(JR+, JR+) not only cannot become negative but also cannot reach the zero level in a finite time. 0 The limiting case /.l --+ 0+ ( r --+ oo) in equation (?) corresponds to the implicit difference equation which can also be written in the form Note t hat in the case of monotone G, when the inverse function c-1 exists, the latter can be explicitly resolved for Xn+l (4) In the case of non-monotone G equation {:l) implicitly defines a multi-valued difference equation. We shall denote it by (5) where scalar fu nction <l > is generally multi-valued. In this paper we shall restrict our considerations to the case when <l> can assume only a finite number of values. This restriction results from the case of G being piecewise monotone in JR+ with a finite number of t he monotonicity branches.
In the case of monotone increasing G this definition coincides with the corresponding notion of the attracting interval for t he map <l> = c -1 oF (see related details in [7, J 7] for the case G(x) = x). Note that one can easily deduce that the point x = x. in the above definition is a fixed point of the map <l>.
3. Principal Results. This section contains main mathematical results on the dynamical behavior of solutions of differential delay equation (; )/(l.) . They primarily concern the global asymptotic stability of equlibria.
Assume that map <l> has an invariant interval I C JR+, and int roduce a subset Proof. The proof is similar to that of Proposition 1. Indeed, assume that¢ E C 1 and let t 0 be the first exit point of the solution x(t) from the inter val I. To be definite, We shall provide a proof that is essentially different from those found elsewhere. It is based on the squizzing property of a p-invariant interval based on Definition 2.:~. The proof uses the following Lemma, which is also a crucial statement in the proof of t he main global stability result (see Theorem :u below) . The Lemma represents a significant independent interest on its own. Proof. Let the initial function ¢ satisfies ¢ E CK. Then, due to Proposition :! , x(t , </>) E K Vt ~ 0. Next we shall show t hat t here exists t 1 = t 1 (¢) ~ 0 such t hat x(t,</>) E L Vt ~ t 1 . Two cases are possible: (a) . ¢(0) E L. Then x(t, ¢) E L for all t ~ 0. This is proved exactly the same way as the invariance property above (Proposition :!). By assuming there exists a first exit time of the solution from interval L one arrives at a contradiction. We leave the related details to the reader. To prove the permanence (see Proposition :1 above) we now observe the following. Since the assumption B is satisfied, and G is monotone increasing in some where the nonlinear function /3 is a monotone Hill function 1 {3(x) = /3o 1 + xn (7) and /3o, k = 2e-v, n, o are all positive constants defined by the physiological process behind. In this subsection we provide a detailed analysis of model (ii) based on the given nonlinearities F and G X F (x) = kf3o -- and values of the parameters /3o, k, n, o. We establish conditions for the global asymptotic stability of the equilibria. Our results are complementary to those recently obtained in [1:2] .
We first make several simple observations about t he involved nonlinearities F and G. Next statement describes cases when the unique trivial equilibri um x(t) = 0 is globally asymptotically stable. that the corresponding map <I> is globally attracting on JR+ . However, the construction of a squizzing sequence of embedded p-invariant intervals does not look possible due to the fact that both functions F and G are decreasing on the interval [max{x1, Xcr }, x2] .
Next statement describes cases when the unique positive equilibrium x(t) = x. is globally asymptotically stable. Those cases are based on the existence of a squizzing sequence of p-invariant intervals for the corresponding multi-valued map <I>.
Pro p ositio n 6. Suppose that k > 1+8/fJo holds. The positive equilibrium is globally asymptotically stable if either one of the following three conditions is satisfied: ( Proof. We briefly describe the main initial steps of the construction of the squizzing sequence of embedded p-invariant intervals in each of the cases, leaving remaining details to the reader. One can further show that equation (li) can be globally asymptotically stable on a larger set than C1, , depending on the mutual shape of nonlinearities F and G.
In particular, when either F (xcr ) < G(x2) or Xcr > X2 holds, t here always exists a p-invariant interval Io := [0, b] such that ~(Io) ~ h , where b can be chosen such that b 2 max{xcr,x 2 }. This leads to the global asymptotic stability of equa tion (C1) on the entire set C. Concluding Re marks and a n Ope n Prob le m. This work initiates a study of the global asymptotic stability in t he essentially nonlinear differential delay equation (l )/ (:2) in terms of its limiting difference equation (:l), or equivalently, -in terms of the multi-dimensional map~. The new results extend t hose previously obtained in [7] for the case of linear G(x) = x. The price of dealing with the case of general nonlinear G(x) is a more strong assumption on t he attractive nature of t he fixed points 0 and x. of t he limi ting map~. which is the existence of imbedded sequence of p-invariant squizzing intervals. In the prior studied linear case G(x) = x the mere global attractivity of the fixed point x. was sufficient for the global asymptotic stability of the steady state x(t) = x. of equation ( ! )/ (:~) (for all J.L = 1/ T > 0). These considerations motivate the following open problem. Ope n Proble m. Investigate whether it is true or not t hat the global attractivity in t he li mit ing multi-valued one-dimensional map ~ is sufficient fo r the global asymptotic stability of the unique equilibrium in equation (l)/(:2). Derive sharper conditions for the global asymptotic stability t han those given by Theorem .l.:2 in terms of t he existence of the squizzing sequence of embedded p-invariant intervals.
A cknowledge m e nts. This research was supported in part by t he Aust ralian Research Council and by the Centre for Informatics and Applied Optimization (CIAO) of t he University of Ballarat. This work has begun when A. Ivanov visited t he CIAO/ GSITMS during July -December 2009. He is grateful to the University of Ballarat for t he hospitality and support extended to him during his stay. The paper was completed in the summer 2010 during h·anov's stay at the University of Giessen, Germany, under the support from t he Alexander von Humboldt Stiftung.