Asymptotic Behaviour of Monotone Multi-Valued Dynamical Systems

Asymptotic behaviour of monotone multi-valued dynamical systems T. Caraballo1,J.A. Langa1and J. Valero2 1Departamento de Ecuaciones Diferenciales Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. e-mails: [email protected]; [email protected] 2Centro de Investigación Operativa Universidad Miguel Hernández, Avda. Universidad, s/n. - Elche - 03202 e-mail: [email protected] Abstract We introduce the concept of a monotone multi-valued semi-‡ow as an order-preserving map. This de…nition is motivated by the applications in the theory of di¤erential equations without uniqueness of solutions. For an order preserving multi-valued semi-‡ow we prove several results on the structure of the global attractor. Some applications to models governed by ordinary di¤erential equations and delay equations with continuous right-hand side are presented. In particular, the abstract results are applied to a biochemical control circuit. Key words: Monotone dynamical systems, order-preserving semi-‡ows, multi-valued systems, global attractor, ordinary di¤erential equations, equations with delay AMS Subject Classi…cation (2000): 34C12, 34D05, 34D23, 34D45, 34K12 1 Introduction There exists a wide literature on the theory of monotone methods and comparison principles for di¤erential equations; that is, ordered initial states remain ordered for subsequent times (among many others, see, for example, the monographs of Smith [20] or Hirsch [13]). Actually, this theory has become an important branch in the theory of dynamical systems. In particular, monotonicity leads to some important properties of the asymptotic behaviour of the models; among them, the forward evolution of intervals, the existence of …xed points for monotone dynamical systems, and some properties related to the structure of attracting sets. But none of these concepts is well de…ned when we have a multi-valued dynamical system related, for example, to a di¤erential equation without uniqueness of solutions, or to a di¤erential inclusion. In these cases, the concept of an order, and even those of sub, super-equilibria, and equilibria becomes unclear. In this paper we contribute to the theory of order-preserving multi-valued dynamical systems, by proposing a suitable de…nition of an order-preserving multi-valued semi-‡ow G, which has a direct application to a variety of models. If we suppose that the multi-valued dynamical system has a global attractor in the sense of Melnik and Valero [17], we then prove our main theorem, which gives the existence of maximal and minimal equilibria on the attractor in which all the asymptotic behaviour of the system is then con…ned, providing also some information on the stability properties of these equilibria and on the structure of the global attractor. In particular, under some conditions we show the existence of two equilibria which attract all the bounded sets allocated, respectively, above or below these values. To our knowledge, this is the …rst time in which the well developed techniques related to monotone dynamical systems (see, for example, Smith [20]) have been applied to analyse the asymptotic behaviour of multi-valued semi-‡ows. 1 Some applications to di¤erential equations without uniqueness of solutions are developed in the following section. In all of them, we prove that Gis order preserving, so that our main theorem can be applied. Also, under some conditions we prove that the set G(t; x0), corresponding to a …xed initial data and a …xed t, is a totally ordered set. The results are applied to cooperative systems of di¤erential equations and to scalar delay equations. In particular, we consider a model of a biochemical control circuit. Finally, we note that these results could be extended to other situations, as di¤erential inclusions, or non-autonomous and stochastic di¤erential equations and inclusions (see [1], [10], where monotone systems are studied for stochastic equations with uniqueness of solutions). This is part of our future work. 2 Monotone multi-valued dynamical systems Let (X; )be a complete metric space and let P(X)be the set of all non-empty subsets of X. Denote B(X) = fA2P(X) : Ais boundedg; C(X) = fA2P(X) : Ais closedg; K(X) = fA2P(X) : Ais compactg; dist (A; B) = sup x2A inf y2B(x; y);for A; B X: De…nition 1 A map G:R+X!P(X)is said to be a multi-valued semi-‡ow (MSF) on Xif: i) G(0;) = Id; i.e., the identity map; ii) G(t+s; x)G(t; G(s; x)), for all x2X; s; t 2R+: It is called a strict multi-valued semi-‡ow (SMSF) if, moreover, G(t+s; x) = G(t; G(s; x)). Multi-valued semi-‡ows are a natural generalization of the semigroups of operators for equations without uniqueness (see [4], [15], [14], [17], [18], [19], [22]). However, other approaches, such as generalized semigroups ([5], [6], [11]), have also been used (see [8] for a comparison of the two methods with respect to the theory of attractors). Order preserving semigroups have been used widely and fruitfully in the literature. Among many other applications, this property has enabled the proof of some nice properties on the asymptotic behaviour of solutions for ordinary and partial di¤erential equations (see [1], [2], [10], [20]). The question about the possible generalization of this rich concept to the multi-valued case arises naturally. In the following subsections we give such a generalization concerning the asymptotic behaviour of these systems. 2.1 Order preserving multi-valued semi-‡ows We now introduce the concept of order-preserving (or monotone) multi-valued systems. De…nition 2 A MSF fG(t;) : X!C(X)gis said to be order-preserving if there exists an order relation ‘’in Xsuch that, if x0y0;then G(t; x0)G(t; y0);for all t0;in the sense that a) There exists x(t)2G(t; x0)such that x(t)y(t);for all y(t)2G(t; y0); b) There exists y(t)2G(t; y0)such that x(t)y(t);for all x(t)2G(t; x0): In particular, it follows from the de…nition the existence of maximal and minimal elements in G(t; x0), so that x(t)(resp. y(t)) can be chosen independent of y0(resp. x0) if y06=x0. It is important to assume that the order is compatible with the topology in the following sense: 2 1. Any bounded subset Bof Xis contained in some “interval”, i.e., there exist absuch that B [a; b] = fx2X:axbg. 2. If xn!x; yn!yand xnyn, then xy. The second condition implies that any interval [a; b]is closed. 2.2 Global attractors for order-preserving multi-valued semi-‡ows Now we brie‡y recall some concepts and results from the theory of attractors for MSF. De…nition 3 A set A  Xis said to be a global attractor associated to Gif: i) A  G(t; A), for all t2R+;that is, it is negatively semi-invariant; ii) Aattracts any bounded subset DX, i.e. lim t!+1dist(G(t; D);A) = 0: In applications it is desirable to prove also that Ais compact and strictly invariant, i.e. A=G(t; A);for all t2R+: We now establish some su¢ cient conditions that guarantee the existence of a global attractor (see [17]). Suppose the following conditions for the MSF G: (H1) Gis point dissipative, i.e., there exists a bounded subset B0such that for every x2Xthere is txfor which G(t; x)B0, for all ttx: (H2) The set + 0(B) = [t0G(t; B)is bounded for every B2 B(X)and Gis asymptotically compact, i.e., any sequence n2G(tn; B), where tn!+1, is pre-compact for all B2 B(X). (H3) G(t;) : X!C(X)is upper semi-continuous, that is, for all t2R+;given x2Xand a neighbourhood O(G(t; x)) of G(t; x), there exists  > 0such that if (x; y)<  then G(t; y) O(G(t; x)): The following theorem is a direct consequence of Lemma 1, Theorem 3 and Remark 8 in [17]: Theorem 4 If (H1) (H3) hold, then Ghas a global compact attractor A. If, in addition, Gis a strict semi-‡ow, then Ais strictly invariant. De…nition 5 A point x2Xis said to be an equilibrium (or a …xed point) of Gif x2G(t; x), for all t0. The following result provides su¢ cient conditions for the existence of upper and lower asymptotically stable equilibria, as well as some information on the structure of the global attractor. Theorem 6 Let Gbe an order-preserving SMSF satisfying (H1)(H3) and Aits associated global compact invariant attractor. Then, there exist equilibria x; y2 A such that: 1. xyand A  [x; y]:(1) 3 2. x(resp. y)is minimal (resp. maximal) in the sense that any other …xed points are contained in the interval [x; y]. If the solutions corresponding to the initial conditions xand yare unique (i.e. if G(t; x) = xand G(t; y) = yfor all t2R), then: 3. xis globally attracting from below, that is, for all v2Xwith vx;we have that lim t!+1dist(G(t; v); x) = 0: 4. yis globally attracting from above, that is, for all v2Xwith yvwe have that lim t!+1dist(G(t; v); y) = 0: Proof. We note that Ais bounded, and then by the condition of compatibility of the order with the topology there exist xysuch that A  [x; y]. Hence, xuy, for all u2 A: Since Gis order preserving, we have G(t; x)G(t; u)G(t; y); so that, there exist x(t; u)2G(t; x)and y(t; u)2G(t; y)such that x(t; u)zy(t; u);for all z2G(t; u),u2 A: But note also that G(t; x)G(t; x)implies that G(t; x)has a minimal and a maximal element, and then there exist x(t)2G(t; x)and y(t)2G(t; y)(not depending on u) such that x(t)zy(t);for all z2G(t; u),u2 A: Thus, since A  G(t; A)we have x(t)uy(t);for all u2 A: The compact set Aattracts xand y, so that dist (x(t);A)!0,dist (y(t);A)!0, as t!+1, and then we can choose converging subsequences x(tn)!x2 A,y(tn)!y2 A. The compatibility of the order with the topology gives xuy, for all u2 A:(2) In order to check that x; yare equilibria we …rst note that x; y2 A implies that these points are uniquely determined by (2). Indeed, if x1 ; x2 2 A are two points satisfying (2), then x1 x2 and x1 x2 , so that x1 =x2 . Since the attractor Ais negatively semi-invariant and the MSF Gorder preserving, for any t0 there exist x(t)2G(t; x),y(t)2G(t; y)such that x(t)uy(t), for all u2 A: But Ais also positively semi-invariant, so that x(t); y(t)2 A, and then x(t) = x,y(t) = y. It is clear that any equilibrium has to be contained in the global attractor (by the attractivity property), so that point 2 of the theorem is also proved. Finally, if the solution corresponding to xis unique, then for any vxwe have G(t; v)G(t; x) = x; dist (G(t; v);A)!0, as t!+1, so that the only possibility is dist (G(t; v); x)!0. The same argument is valid for y. 4 3 Applications In this section we apply our theory to some systems modeled by ordinary di¤erential equations and delay equations without uniqueness of solutions, so that a multi-valued semi-‡ow has to be de…ned. Under some conditions, we show that the corresponding semi-‡ow is monotone, so that our previous results can be applied. 3.1 Ordinary di¤erential equations with continuous right hand-side 3.1.1 An N-dimensional case: a cooperative system Consider the usual order in RN, i.e. xy() xiyifor all i= 1;2; :::; N: We say that x < y if xyand there exists j2 f1;2; :::; Ngsuch that xj< yj. Let us now consider the system x0(t) = f(x(t)); x(0) = x0;(3) with f:RN! RNcontinuous and satisfying (f(x); x)c1kxk2+c2,c12R; c2>0;(4) where (,),kkare respectively the scalar product and the norm in RN. Suppose also the following monotonicity hypothesis on f: fi(x)fi(y);if x; y 2RN; xi=yi;and xy: (5) Condition (4) implies easily that any solution is globally de…ned in time, and for any T > 0the set of solutions is uniformly bounded in bounded sets of initial data, so that, for all BRNbounded and T > 0;there exists C(B; T)such that supt2[0;T ]jx(t)j  C(B; T );where x(t)denotes any solution of (3) corresponding to x02B. Denote by S(x0)C[0;+1);RN +the set of all solutions with initial condition x0. De…ne also the set S(x0; T) = fx()2C([0; T];RN) : x()is a solution of (3) and x(0) = x0g; which is compact in C([0; T];RN)by the Ascoli-Arzelà theorem, and the map G:R+RN! K(RN)by G(t; x0) = fx(t) : x()2S(x0)g: It is easy to check that Gis a strict multi-valued semi-‡ow. The inclusion G(t+s; x)G(t; G (s; x)) follows from the fact that if x()2S(x0; t +s), then y() = x(s+)2S(x(s); t). The converse inclusion G(t; G (s; x)) G(t+s; x)follows from the concatenation property: if x()2S(x0; s)and y()2 S(x(s); t), then z(r) = x(r), if 0rs; y(rs), if srs+t; belongs to S(x0; t +s). Hence, we have the SMSF G:R+RN! K(RN). We also note that the set of …xed points coincides with the set of zeros of f. Indeed, if x0is a …xed point and for some i, say, fi(x0)>0, then for any x()2S(x0)we have x0 i(t)>0for t2[0; ], so that xi(t)> x0i in (0; ], and x062 G(t; x0). Hence, x0is not an equilibrium. Conversely, if f(x0)=0, then x(t) = x0, for all t, is a solution and then x02G(t; x0). In many physical and biological applications the variables xihave to be non-negative. Hence, we need to de…ne a multi-valued semi-‡ow in the phase space RN +instead of RN. 5 Lemma 7 Suppose that fi(x)0;for all iand x2RN +such that xi= 0.(6) Then, for any x00(i.e. x0i0for all i) there exists at least one global solution such that x(t)0, for any t0: Proof. De…ne the approximate functions f"(x) = f(x) + "d," > 0, where d= (1; :::; 1), which satis…es f" i(x)", for all iand x2RN +such that xi= 0. Consider an arbitrary solution x"(t)of the equation x0=f"(x)corresponding to x00. Suppose x"(t)6 0, for some t. Let the icomponent of this solution be negative in some interval (t0; t1); xi(t)0, for tt0, with ithe …rst component which becomes negative. Thus, by the continuity of f"we have d dtx" i(t) = f" i(x"(t)) >0, for t2(t0; t0+); which is a contradiction. Hence, x"(t)0, for all t0. Fix T > 0. Then by (4) and the continuity of fthe functions x"and d dtx"are uniformly bounded in C[0; T];RN, so that by the Ascoli-Arzelá theorem there exists a converging subsequence x"n. The limit x()is a solution of (3) in [0; T]and x(t)0, for all t2[0; T], as can be easily checked. Repeating the same in the intervals [T; 2T],[2T; 3T], etc., and using a diagonal argument we obtain a globally de…ned solution x()such that x(t)0, for t0, as desired. Remark 8 This lemma does not guarantee that all the solutions are non-negative, as happens in the case of equations with uniqueness. This can be checked in the following simple example: x0=sign (x)pjxj; x (0) = 0; where sign (x) = 1 (resp. 1;0);if x > 0(resp. <0;= 0), We have the solutions x1(t) = 0; x2(t) = t2 4; x3(t) = t2 4among many others. Denote by U(x0)C[0;+1);RN +the set of all solutions with initial condition x0such that x(t)0 for all t0. De…ne also the set D(x0; T) = fx()2C([0; T];RN) : x()is a solution of (3) and x(0) = x0,x(t)0g and the map U:R+RN +! K(RN +) : U(t; x) = fx(t) : x()2D(x0)g: Uis a SMSF. We note that D(x0; T)is a closed subset of S(x0; T ), so that it is compact. Therefore, U(t; x0) is compact as well. We are going to study whether the SMSF Gand Uare order preserving. Let us begin with the map G. Lemma 9 Let conditions (4) and (5) hold. Then for any x0y0there exists two solutions y()2S(T; y0); x()2S(T; x0), such that y(t)x(t), for all t2[0; T]and x()2S(T; x0); x(t)y(t), for all t2[0; T]and y()2S(T; y0): Hence, Gis order preserving. 6 Proof. De…ne the functions g"(x) = f(x) + "d," > 0, where d= (1; :::; 1). It is clear from (5) that g" i(y)fi(x) + ", if xi=yiand xy. (7) Let y"(); y"(0) = y0;be a solution of (3) where fis replaced by g", and let x()2S(T; x0)be arbitrary. We claim that y"(t)x(t), for all t2[0; T ]. If not, then there exist a time 0t0< T and i;  > 0such that xi(t0) = yi(t0); y(t0)x(t0); y"i(t)< xi(t);for all t0< t t0+: Then (7) gives g" i(y(t0)) "+fi(x(t0)) and the continuity of the functions g"; f implies that d dt(y"i(t)xi(t)) = g" i(y(t)) fi(x(t)) >0;for all t2(t0; t0+~ ); for some ~ ; and thus y"i(t)> xi(t);which is a contradiction. Further, by (4) and the continuity of fwe obtain that the functions y";d dty"are uniformly bounded in C[0; T];RN, so that by the Ascoli-Arzelà theorem there exists a converging subsequence y"n. The limit y(t)is a solution of (3) and clearly y(t)x(t), for all t2[0; T]. The existence of the function x()can be proved in a similar way by de…ning the function g"(x) = f(x)"d. This means that Gis order preserving. We have the following result: Theorem 10 Let conditions (4) and (5) hold. Assume also that (H1)(H2) are satis…ed. Then Gsatis…es also (H3) and the statement of Theorem 6 holds true. Remark 11 In this case (H2) is equivalent to the fact that the set + 0(B)is bounded for any Bbounded. Remark 12 In the scalar case, that is N= 1, the order preserving property is satis…ed without assuming any condition of the type (5) [12, p.27, Corollary 4.2]. Proof. We have already seen that G(t; x)is a compact set, so that in order to get (H3) it remains to prove that the map G(t; )is upper semi-continuous. Arguing again by contradiction, take a sequence yn2G(t; xn),xn!x0, and a neighborhood Nof G(t; x0);such that yn=2 N. If xn()2S(xn; t)are such that xn(0) = xn,yn=xn(t), then, by (4), we have that xn(t)and x0 n(t)are uniformly bounded in [0; t]. The Ascoli-Arzelà theorem gives (up to a subsequence) xn()!x()in C[0; t];RN. Hence, xn(t) = xn+Rt 0f(xn()) d # # # x(t) = x0+Rt 0f(x()) d: Therefore, x()2S(x0; t)and x(t)2G(t; x0), which is a contradiction. (H3) is then proved. Since Gis order preserving by Lemma 9, we can apply Theorem 6. Consider now the map U. We note …rst that if we try to repeat the same proof of Lemma 6, then one can get the existence of the solution y(). However, the proof for the solution x()fails because, when we take the function g"(x) = f(x)"d; the solutions of (3) are not in general non-negative. We will be able to prove the order preserving property for Uunder more restrictive conditions. We shall use the following condition: fi(x)< fi(y);if x; y 2RN; xi=yi;and x < y: (8) 7 Theorem 13 Let conditions (4), (6) and (8) hold and let the set U(t; x)have a minimal and a maximal element for any (t; x). Then Uis order preserving. If we assume that (H1) (H2) are satis…ed, then (H3) also holds and the statement of Theorem 6 holds true. Proof. Denote by xand xthe maximal and the minimal element in U(T; x0);and let x(t)be a solution corresponding to the maximal element x2U(T; x0): Take y0< x0and y()2D(T; y0). If y(t0) = x(t0)for some t0, then we de…ne x()2D(x0; T)as x(t) = x(t);if 0tt0; y(t);if t0tT: Since x(T)2U(T; x0)and x(T)is the maximal element of U(T; x0), we have x(T) = y(T)x(T) = x. Assume then that y(T)xis not true and y(t0)6=x(t0)for all t0. Then there exists i2 f1; : : : ; Ng,t0< T and  > 0such that xi(t0) = yi(t0); x (t0)6=y(t0); y(t0)x(t0); yi(t)> xi(t)for all t0< t t0+: Thus, by (8), we have that fi(y(t0)) fi(x(t0)) <0:Then, the continuity of fimplies that d dt(yi(t)xi(t)) = fi(y(t)) fi(x(t)) <0;for all t2(t0; t0+~ ); for some ~ ; and thus yi(t)xi(t)<0;which is a contradiction. Therefore, repeating a similar argument for x, it follows that Uis order preserving. We conclude by repeating the same proof of Theorem 10. In order to prove that the set U(T; x)always has maximal and minimal points we need to assume an additional condition on the function f. Let us now consider the case N= 2 and assume the additional condition: fi(x)< fi(y), if x; y 2R2; xiyi; xj< yj;for j6=i: (9) Clearly, (9) implies (8). We have: Theorem 14 Let N= 2 and let conditions (4), (6) and (9) hold. Then Uis order preserving. If (H1) (H2) hold, then Ualso satis…es (H3) and the statement of Theorem 6 is valid. Proof. Condition (8) gives us that, when two solutions with the same initial data become di¤erent, they do that simultaneously in all directions. Indeed, suppose that, for the two solutions x();~x(), and t02[0; T ]; we have that xi(t0) = ~xi(t0);for i= 1;2;but x1(t)>~x1(t)and x2(t) = ~x2(t)for t2(t0; t0+):Then, d dt(x2(t)~x2(t)) = f2(x(t)) f2(~x(t)) >0, for t2(t0; t0+); so that x2(t)>~x2(t);which is a contradiction. The stronger condition (9) implies that it is not possible that xi(t0) = ~xi(t0);for i= 1;2; x1(t)>~x1(t); x2(t)<~x2(t), for t2(t0; t0+); since we then obtain d dt(x2(t)~x2(t)) = f2(x(t)) f2(~x(t)) >0, for t2(t0; t0+); 8 which is a contradiction. Using again (9) or (8) one can see that after t0either the two components xi(t)remain di¤erent for all t, or for some t1> t0we have again x(t1) = ex(t1). Indeed, if x1(t1) = ex1(t1); x2(t1)>ex2(t1)and xi(t)>exi(t);for t2[t1; t1),i= 1;2(for instance), then d dt(x1(t)~x1(t)) = f1(x(t)) f1(~x(t)) >0, for t2[t1; t1); so that x1(t1)>ex1(t1), a contradiction. We repeat again the argument in the following interval and conclude that for any given moment of time tthe set U(t; x0)is totally ordered. Moreover, as this set is compact, then we have the existence of a maximal and minimal elements in U(t; x0). We conclude by applying Theorem 13. Remark 15 We have proved here the additional property that the set U(t; x)is totally ordered. This is valid also for the map Gif we assume condition (9). It is also clear that in the case of the map Uit is su¢ cient to assume (9) for x; y 2R2 +: 3.1.2 A biochemical control circuit As an application of the previous results we consider a model on the control of protein synthesis in the cell. Consider the two dimensional system (see Arnold and Chueshov [1], Chueshov [10] for the random case and Smith [20] for the deterministic case): 8 < : x0 1(t) = g(x2(t)) 1x1(t) x0 2(t) = x1(t)2x2(t) xi(0) = xi 0; (10) where gsatis…es g(0) 0and is a continuous strictly increasing sub-linear function, so that there exist a; b > 0such that jg(x)j  a+bjxj;(11) and i>0; i = 1;2;(12) 12>(b+ 1)2 4or f1>1and 2> bg:(13) The unknowns represent concentrations of macromolecules in a cell, so they are supposed to be non-negative (see Smith [20] or Chueshov [10]). It is clear that (4) and (6) hold, which means that the SMSF Uis well de…ned. This system satis…es (H1) (H2), so that it is dissipative. Indeed, according to (11)-(13) we have to consider two cases: Case I: 12>(b+1)2 4. We obtain that 1 2 d dt kxk2=f1(x)x1+f2(x)x2=g(x2)x11x2 1+x1x22x2 2 (a+bx2)x11x2 1+x1x22x2 2 ax1+ ((b+ 1) 2(1 ")p1p2)x1x2 (1 ")(p1x1p2x2)2"(1x2 1+2x2 2)  " 21x2 1" 22x2 2+c1  c2jjxjj2+c1; for some ci>0,i= 1;2. Hence by Gronwall’s Lemma kx(t)k2e2c2tkx0k2+c1 c2, and then (H1) (H2) follow. 9 References [1] Arnold L. and Chueshov I.D., Order-preserving random dynamical systems: equilibria, attractors, applications. Dynam. Stability Systems 13 (1998), no. 3, 265–280. [2] Arrieta J.M., Rodríguez-Bernal A. and Vidal-López A., Regiones atractoras delimitadas por puntos de equilibrio extremales para ecuaciones de reacción-difusión, Proceedings in CDRom of the XVIII CEDYA-VIII CMA, Tarragona (Spain), 2003. [3] Aubin J.P. and Frankowska H., Set-Valued Analysis, Birkhäuser, Boston, 1990. [4] Babin A.V., Attractors of the generalized semigroup generated by an elliptic equation in a cylindrical domain, Russian Acad. Sci. Izv. Math.,44 (1995), 207-223. 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