Convergence of Monotone Dynamical Systems with Minimal Equilibria Author(s): Jianhong Wu Source: Proceedings of the American Mathematical Society, Vol. 106, No. 4, (Aug., 1989), pp. 907911 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2047273 Accessed: 14/08/2008 13:29 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showpublisher?publishercode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a notforprofit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. http://www.jstor.org
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 106, Number 4, August 1989 CONVERGENCE OF MONOTONE DYNAMICAL SYSTEMS WITH MINIMAL EQUILIBRIA JIANHONG WU (Communicated by Kenneth R. Meyer) ABSTRACT. We show that each precompact orbit of strongly monotone dynamical systems on a Banach lattice X is convergent if there is a continuous map e: X + E, the set of equilibria, such that e(x) is the maximal element in E with e(x) < x. This result can be applied to study the convergence of a class of functional differential equations and partial differential equations. Monotone dynamical systems have a strong tendency to converge to the set of equilibria. Under some compactness assumptions, Hirsch [3] shows that generically positive semiorbits are quasiconvergent, i.e. asymptotic to the set E of equilibria. Therefore, when the set of equilibrium points has no accumulation point, almost every orbit is convergent, i.e. asymptotic to some equilibrium point. However, some application problems do generate dynamical systems whose equilibrium point sets have accumulation points, and therefore quasiconvergence does not imply convergence. Simple examples are the following functional differential equation (1) x(t) = f(x(t)) + f(x(t r)) describing a compartmental system with one pipe, the motion of a classically radiating electron, epidemics and population growth, where r > 0, f: R + R is continuous and increasing; or the following partial differential equation au = Au + g(x, u, Vu), t>o, xeq (2) u(x, 0) = v(x), x E Q, u(x, t) = 0 xeaq, t>o an where Q is a bounded smooth domain in Rn, A is a secondorder uniformly elliptic differential operator and g: Q x R x Rn + R is locally Lipschitz and satisfies g(x,c,o) = 0 for any x E Q and c E R, au/an is the derivative of u in the direction of the outward normal to Q. For these equations, each Received by the editors October 3, 1988 and, in revised form December 2, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 06F05, 34K20, 34K25, 34C35, 35K10, 58F10, 92A15. (?) 1989 American Mathematical Society 00029939/89 $1.00 + S.25 per page 907
908 JIANHONG WU constant function is a solution, the set of equilibria has accumulation points, and thus quasiconvergence can tell us nothing about asymptotic behaviors of solutions but boundedness. In this paper, we announce some convergence criteria for strongly monotone dynamical systems whose set of equilibrium points possesses a minimal or stable property. We will also present some examples to show how to apply our results to functional differential equations and partial differential equations which have infinitely many equilibrium points. Let X be a Banach lattice such that (X, <, 1111) has ordercontinuous norm. A dynamical system '1 = {ot}t>o on X is monotone, if y > x and t > 0 imply kt(y) > Ot(x). A monotone dynamical system 1P = {Ot}t>O on X is eventually strongly monotone if there exists a constant T > 0 such that y > x and t > T imply Ot(y) > qt(x). Let y+(x) = {qt(x);t > O} denote the positive semiorbit of x, co(x) be the colimit set and E be the set of all equilibrium points, that is, E = {x E X; qt5(x) = x for all t > O}. A point x E X (or equivalently, the orbit y+(x)) is convergent if lim 0(t, x) exists and the limit is in E. The following minimal property is essential throughout this paper. Definition 1. A closed subset M C E possesses minimal property with respect to a positive invariant subset U if, there exists a continuous function e: U M n U so that (1) xe(x) E X+IntX+ for any x E U, (2) if yem and y?<x, then y e(x). Theorem 1 (Convergence criterion with minimal property). Suppose that {0t}t>O is an eventually strongly monotone dynamical system, a closed subset M of E possesses minimal property with respect to U. Then any precompact orbit y+(x) through x E U is convergent, and limt o (t, x) E M. Sketch of Proof. Let a positive constant T be given so that 0(t, y) > q(t, z) for any y, z E U with y > z and for all t > T. Let ei = e(q(it, x)). Then ei < q(it,x). If ei :$ (it,x), then ei = q(t, ei) << (T, q(it, x)) = 0((i + 1)T, x) and thus ei < ej+j. Therefore by induction we get the following increasing sequence eo < e, < e2< e3 < Therefore limk_o, ek = e exists by the ordercontinuity of the norm. Let y E cow(x). Then we can find an unbounded increasing subsequence {kn}??= of nonnegative integers and a sequence {Tn}1n?=1 in [0, T] so that q(kn T+T n, x) ` y as n oo. Without loss of generality we may assume that Trn p TE [0, T] and q(kn T, x) p z as n poo. Therefore from continuity of semiflows, it follows y = (/(T, Z). Note that ek < q(kn T, x). By the closeness of order relation, we get e < z and thus e < Q(T, Z) = y. That is e < co(x). By invariance of c(x),
CONVERGENCE OF MONOTONE DYNAMICAL SYSTEMS 909 we can find an element w E co(x) so that q(t+ T, w) = z. If w # e, then by eventually strong monotonicity we have e = q(t + T, e) < q(t + T, w) = z. That is z e E Int X+ which implies that q(k, T, x) e(q(k, T, x)) E Int X+ for sufficiently large n, a contradiction to our assumption. Therefore w = e, and thus z = q(t5+ T, e) = e'. Hence y = q(t,5e) = e'. This shows co(x) = {e} C M C E. A simple example is the following twodimensional compartmental system with pipes d TtXI(t) = g1(x(t)) (3) d X2(t) = g21(xi(t r21)) g21(x1(t)) + g22(x2(t + g11(x1(t r22)) r11)) + g12(x2(t g12(x2(t)) g22(x2(t)) r12)) where gij: R p R are Lipschitz continuous and increasing with gij(o) = 0, and limjsj.00 Igjj(s)j = oo, rij > 0 are constants for i, j = 1,2. By emplying standard comparison technique and adopting a similar argument to that in Gyori [1] and Smith [5], we can prove the solution flow defined by equation (3) defines an eventually strongly monotone semiflow on P = C([rr,, 0,0, oo)) x C([r2, O], [O, oo)) where ri = max{r11, r2j}, i = 1, 2. It is clear that the set E of equilibrium points contains the following set of constant functions M = {(',c2); (c1,c2) E R x R and g21(c1) = g12(c2)} where ci is a constant function on [ri, 0] with the value ci E R. By constructing a function e: P M C E as following (gl g2 (mirn 2(i)n, min 2(5) 92\92SE[P2,0] S [m /SE[P2,0A l,~0 2 (S)[in? () > 2102 (S )) if mm Ik(s)? 921 912\ e(01, 02) = s g1 1g(min SE[mi,O0lS g12 921 SE>[r,,o]0() 02E(r if min q(s? (s) < ( min SE[r,O] g 921 g12 \SE[r2 01 02]( / We can verify that the set M possesses minimal property. Therefore by Theorem 1, any orbit y+(0, 02) through (0 X02) E P is convergent. As an immediate consequence of Theorem 1, we get the following sufficient condition for asympotic constancy: Corollary (Asymptotic constancy criterion). Suppose I is a compact topological space or a smooth compact ndimensionoal submanifold of Rn, Cr(I) denotes the Banach lattice of Cr maps U: I p Rk such that U and its derivatives of order < r are continuous, endowed with the usual Cr norm and the usualfunctional ordering. Suppose {0t}t>0 is an eventually strongly monotone dynamical system on Cr(I) and the set of equilibrium points contains all constant functions. Then each precompact orbit converges to a constant function as t poo.
910 JIANHONG WU Sketch of Proof. Let M = {Ic; c E R and c denotes the constant function on I with the value c}. By assumption M C E, the set of equilibria. Evidently, M possesses the minimal property with respect to Cr(I) since the map e: Cr(I) M defined e(u) = min u(x) XEI satisfies the conditions in Definition 1. Therefore our conclusion follows trivially from Theorem 1. According to the above criterion, each solution of the functional differential equation (1) or partial differential equation (2) is convergent. The following theorem, based on the general quasiconvergence criterion by Hirsch [3], provides another sufficient condition for convergence. Theorem 2 (Convergence criterion with stable equilibria). Suppose that {kt}t,o is an eventually strongly monotone dynamical system such that each orbit is precompact and upper stable, and that any equilibrium point is stable. Then any orbit is convergent. Here and hereafter, an equilibrium point x E X is stable if, for any given E there exists 3 = 3(e, x) > 0 such that y+ (y) C Ue (X) for any y E U,5 (x), where Ue is the Eneighborhood. An orbit y+(x) is upper stable iffor every E > 0 there exists a > 0 such that if y > x and d(y,x) < a then d(q(t,y), q(t,x)) < E, where d is a metric defining the order topology of X. Sketch of Proof. Let y+(x) be a fixed orbit. Employing the argument of [3, Theorem 8.3] we can prove that y+(x) is convergent to co(x) C E. Therefore our conclusion follows trivially from the stability of each equilibrium point. As an illustrating example, we consider an ncompartmental system with pipes described by the following functional differential equation dnn dtxi (t) Egi i(xi(0) + Egij (xj (t r, i=1 n j=1 j=l where gij(u) is monotone increasing, continuous and locally Lipschitz with gij(o) = 0. The phase space is chosen as C = C([r1,0],[0,oo)) x... x C([rn, 0], [O, oo)) where ri = maxj=l,n r.i I It has been proved that for any (.., qn) E C there is a unique nonnegative solution on [0, oo) which satisfies the following law of energy E i t) + E g1j(xj1(s)) ds] i= L j=1 tr,j i i ] const for all t > 0. Moreover, in Wu [6], we proved that the semiflow defined by the above equation is eventually strongly monotone and the law of energy implies the upper stability of any nonnegative orbit, the stability of equilibrium points and precompactness of any orbit in C. Therefore as an immediate consequence
CONVERGENCE OF MONOTONE DYNAMICAL SYSTEMS 911 of Theorem 2, we can assert that any orbit of compartmental systems with pipes is convergent. REFERENCES 1. I. Gyori, Connections between compartmental systems with pipes and integrodifferential equations, Math. Modelling 7 (1986), 12151238. 2. P. Hess, On stabilization of discrete strongly orderpreserving semigroups and dynamical processes, Proceedings of Trends in Semigroup Theory and Applications, Trieste, September 28October 2, 1987. 3. M. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew Math. 383 (1988), 153. 4. X. Mora, Semilinear problems define semiflows on Ck spaces, Trans. Amer. Math. Soc. 278 (1983), 155. 5. H. Smith, Monotone semiflows generated by functional differential equations, J. Differential Equations 66 (3) (1987), 420442. 6. Jianhong Wu, Monotone semiflows on product spaces and active compartmental systems with pipes (submitted). DEPARTMENT OF MATHEMATICAL SCIENCES, MEMPHIS STATE UNIVERSITY, MEMPHIS, TENNESSEE 38152 Current address: Canada T6G 2G1 Department of Mathematics, University of Alberta, Edmonton, Alberta,