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Electronic Journal of Differential Equations, Vol. 2001(2001), No. 32, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp) Cauchy problem for derivors in finite dimension ∗ Jean-François Couchouron, Claude Dellacherie, & Michel Grandcolas Abstract In this paper we study the uniqueness of solutions to ordinary differential equations which fail to satisfy both accretivity condition and the uniqueness condition of Nagumo, Osgood and Kamke. The evolution systems considered here are governed by a continuous operators A defined on RN such that A is a derivor; i.e., −A is quasi-monotone with respect to (R+ )N . 1 Introduction For T > 0, we study the Cauchy Problem (CP) u̇(t) + Au(t) = f (t), t ∈ [0, T ] u(0) = u0 , (1.1) where A is a continuous operator on RN and f belongs to L1 ([0, T ] : RN ). We require in addition that A be a derivor on RN (or equivalently that −A be quasi-monotone with respect to the cone (R+ )N ) and has an additional order property (see Assumption H1T in Section 2). The existence of local solutions of (1.1) is proved by standard arguments (see [17] and Lemma 4.2). For instance, in the continuous case, this local existence comes from the Peano’s Theorem. So the problem is essentially to prove the uniqueness of a local solution and the existence of global solutions. An important remark is that the identity operator minus the limit of infinitesimal generators of increasing semigroups is a derivor on the domains of the operators (see remark 2.1.d). The aim of this paper consists of giving a special converse of this previous property. General studies of evolution problems governed by derivors can be found in [2, 8, 9, 17] (for existence of extremal solutions of differential inclusions in RN ) and in [13] for the behavior of the flow (stability, etc.) in the regular case: A is C 1 . This work establishes uniqueness for the Cauchy Problem and complements previous studies. ∗ Mathematics Subject Classifications: 34A12, 34A40, 34A45, 34D05. Key words: derivor, quasimonotone operator, accretive operator, Cauchy problem, uniqueness condition. c 2001 Southwest Texas State University. Submitted December 4, 2000. Published May 8, 2001. 1 2 Cauchy problem for derivors EJDE–2001/32 Let us point out that derivors often occur in the theory of production processes in Economics (for cooperative systems, see [10, 14]), in Chemistry [12], and in Biology [12]. Our uniqueness result given in the sequel applies to these situations. Notice also that the additional order property; namely, existence of uniform ascents, (see Definition 2.2) has obvious interpretations in applications and may be considered as a special extension of the submarkovian property (see remark 2.2.3) and [1]). Nevertheless the notion of uniform ascents is a new concept built from the concept of progressions in [10]. This ascent notion which extends the usual submarkovian property seems to lead naturally to the maximum principle worked in [10]. Finally, we emphasize that the ascent notion is the key to obtain a suitable increasing resolvent (see Proposition 2.4 and Theorem 3.2). In this paper, the operator A does not satisfy either uniqueness conditions such as those given by Nagumo, Osgood and Kamke [4, 15, 16] nor accretivity conditions, even in a generalized sense as in [4, 6, 16]. We will exhibit in Section 5 a simple example of operator on R2 which satisfies all our conditions and none of the uniqueness conditions quoted above. Consequently our framework is not included in the submarkovian case, since a continuous submarkovian derivor is accretive in (RN , k · k∞ ). Moreover based in our analysis, it appear that a simple natural-order property can replace a classical Lipschitz condition about uniqueness in the Cauchy Problem. Uniqueness and order-preserving dependence with respect to the initial value u0 are stated in Theorem 3.1. In the case f = 0, Theorem 3.2 guarantees the existence of a global solution and a special form of the Crandall-Ligget exponential formula [7, p. 319] involving suitable selections of the multi-valued operators (I + λA)−1 (while in [8] (I + λA)−1 is single valued and Lipschitz). This paper is organized as follows. Section 2 is devoted to general definitions and preliminaries. The main results are stated in Section 3, while the proofs are given in the next section. Section 5 gives an example in R2 which demonstrates the need for Theorems 3.1 and 3.2. Some remarks about the asymptotic behavior follow in Section 6. 2 Generalities We supply RN with the usual partial order relation u ≤ v if ui ≤ v i for all i = 1, . . . , N , where ui is the i-th component. The vector in RN whose components are C, . . . , C is denoted by C. The symbol k · k stands for any norm in RN . The symbol N∗ denotes the set of integers greater than zero. Definition 2.1 condition We say that the map A is a derivor on RN if it satisfies the (i) For each (u, v) ∈ (RN )2 and each i ∈ {1, . . . , N } (u ≤ v and ui = v i ) implies that Ai u ≥ Ai v (2.1) EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 3 We say that the map A is a moderate derivor (resp. a strong derivor) if, in addition to (i), it satisfies (ii) For each u ∈ RN , there exist u1 , u2 ∈ RN (resp. two sequences (uk )k → +∞, (vk )k → −∞) such that u1 ≤ u ≤ u2 and Au1 ≤ 0 ≤ Au2 (see [4]). (resp. limk→+∞ Auk = +∞ and limk→+∞ Avk = −∞). The previous notation limk→+∞ wk = +∞ in RN may be interpreted to mean that limk→+∞ wkj = +∞ in RN for each j ∈ {1, . . . , N }. The derivor notion coincides with the notion of quasimonotone operator on RN , except the sign (see [2], [8, p. 91], [15]). In these references, −A is quasimonotone with respect to (R+ )N if (u ≤ v and x∗ (u) = x∗ (v)) implies that x∗ (−Au) ≤ x∗ (−Av) (2.2) for any linear positive form x∗ on RN . Hence, A is a derivor, because if x∗ is a linear positive form on RN , x∗ is a linear combination with positive coefficients of coordinate forms on RN . Remark 2.1 a) Condition (i) in the definition of derivor is automatically fulfilled for any operator A from R to R, but it is not in the case of Condition (ii). A special case where (ii) holds for an operator A from R to R is the case where A is a non-decreasing operator such that there is v ∈ R satisfying Av = 0. b) When A is a linear derivor, the reader can check that Condition (ii) is equivalent to: there is u ≥ 1 satisfying Au ≥ 0. c) An equivalent form of definition 2.1.(i) is: Ai is decreasing with respect to xj for each i 6= j with i, j ∈ {1, . . . , N } (see [14]). d) If P is an increasing operator on RN , then A = I − P is a derivor. Therefore, t when (Pt ) is an increasing semi-group on RN , then At = I−P with t > 0 is a t derivor and so is A0 , defined by A0 u = limt↓0 At u (on the domain where this limit exists). Ascents We denote by VK (u0 ) the set of compact neighborhoods of u0 . Definition 2.2 We say that a derivor A has a (strict) uniform ascent at u0 if there are V ∈ VK (u0 ) and a sequence (vk ) in RN convergent to 0 such that (vki )k∈N∗ is strictly decreasing for all i = 1, . . . , N and min i∈{1,...,N } for each k ∈ N∗ and each u ∈ V . (Ai (u + vk ) − Ai u) > 0 (2.3) 4 Cauchy problem for derivors EJDE–2001/32 Remark 2.2 1) In terms of production operator (Ai u is the production of the i-th input of the product u), the uniform ascent property at u0 means that in a neighborhood of u0 it is possible to increase the level of production by means of small uniform augmentations around u0 . 2) The notion of uniform ascent plays a crucial part in this work. In our opinion, this concept is new, but it was inspired from the progression notion carried out in [10]. 3) The uniform ascent property may be connected to the submarkovian property namely, A(u + C) − Au ≥ 0 (2.4) for all u ∈ RN and all C ∈ R+ . Notice that a submarkovian derivor in RN is accretive in (RN , k · k∞ ); the verification of this claim is left to the reader. 4) The following dual notion of uniform ascent at u0 provides again the results of Section 3: There are V ∈ VK (u0 ) and a strictly increasing sequence (vk ) in RN convergent to 0 such that (vki )k∈N∗ is strictly increasing for all i = 1, . . . , N and sup (Ai (u + vk ) − Ai u) < 0 i∈{1,...,N } for each k ∈ N∗ and each u ∈ V . 5) In the case A = I − P with an increasing operator P , (see Remark 2.1.d)), Definition 2.2 means that the required sequence (vk ) satisfies P i (u + vk ) − P i (u) < vki for all i = 1, . . . , N , for each k ∈ N∗ and all u ∈ V . Assumptions In the sequel, by hypothesis H1T stands for the following three conditions: • A is a continuous derivor on RN • A is a moderate derivor with uniform ascent at each u0 • f ∈ L1 ([0, T ]; RN ). When necessary, we will make precise the arguments involved for the Cauchy Problem (1.1) as follows: CP (A, f, u0 ) or CP (A, f, u0 , T ) for the domain [0, T ], and CP (A, f, u0 , +∞) for [0, +∞[. The hypothesis H2T stand for the following condition (cf Section 6). • For each u0 ∈ RN , each local solution of CP (A, f, u0 ) can be extended to a solution on [0, T ]. It is well-known that sublinearity at infinity (kAuk ≤ a(kuk + 1)) guarantees H2T, [13]. Moreover we will see in Theorem 3.2 that in the autonomous case f = 0, H1T implies H2T. EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 5 Resolvents for a moderate continuous derivor In this section, we assume that A is a continuous moderate derivor. When B is a continuous derivor, we have the following Theorem [10]. Theorem 2.4 Let u and w fixed in RN , if the system v≥u Bv ≥ w (2.5) with v as unknown quantity has a solution then it has a smallest solution. Analogously, the system v≤u (2.6) Bv ≤ w has a largest solution v whenever it has a solution. In addition, in these two systems the constraints are optimal, i.e. ∀i ∈ {1, . . . , N }, v i = ui or Bv i = wi . In the case where A is accretive, the resolvent operators (I + λA)−1 , λ > 0 are single-valued contractions. But in our case (I + λA)−1 is a priori multi valued. Nevertheless it is possible to define suitable selectors Jλ of (I + λA)−1 as claimed in the following lemma. Lemma 2.4 Let A be a moderate continuous derivor on RN . Let u in RN , λ ∈ R+ and v a solution of v≤u (2.7) Av ≤ 0. Then the system w≥v (I + λA)w ≥ u (2.8) has a smallest solution denoted by Jλ,v u. Moreover we have Jλ,v u ∈ (I + λA)−1 (u). (2.9) Proof. According to (ii) in definition 2.1, Systems (2.7) and (2.8) have solutions. Let v be a solution of (2.7). Since B = I + λA is a continuous derivor, the existence of the smallest solution Jλ,v u of (2.8) is guaranteed by the Theorem 2.4. It remains to prove (I + λA)Jλ,v u = u. (2.10) Since the constraints are optimal in (2.8), we have for each i ∈ {1, . . . , N }, ((I +λA)Jλ,v u)i = ui or (Jλ,v u)i = v i . Thus we have to prove ((I +λA)Jλ,v u)i = 6 Cauchy problem for derivors EJDE–2001/32 ui when (Jλ,v u)i = v i . So assume (Jλ,v u)i = v i for some i ∈ {1, . . . , N }, Relation (2.1) and Jλ,v u ≥ v yield (AJλ,v u)i ≤ (Av)i . (2.11) Now (2.11) and (2.7) provide (Jλ,v u)i + λ(AJλ,v u)i = v i + λ(AJλ,v u)i ≤ ui + λ(AJλ,v u)i ≤ ui . Therefore, ((I + λA)Jλ,v u)i ≤ ui . But from (2.8) we have ((I + λA)Jλ,v u)i ≥ ui . Finally (2.10) is proved. In the same way, let v be a solution of v≥u Av ≥ 0. (2.12) w≤v (I + λA)w ≤ u (2.13) Then the system has a largest solution w = J˜λ,v u. Moreover J˜λ,v u satisfies again (2.9). Set Jλ u = Jλ,v u (resp. Jλ u = J˜λ,v u) for an arbitrary v satisfying (2.7) (resp. (2.12)). Let us notice that Jλ is defined on Dv = {u ∈ RN , u ≥ v} (resp. Dv = {u ∈ RN , u ≤ v}). The family of selectors (Jλ )λ≥0 of (I + λA)−1 is said to be the resolvent of A. Definition 2.4 For u given, the notation u (resp. û ) stands for the largest solution of (2.7) (resp. the smallest solution of (2.12)). Thanks to Theorem 2.4, such extremal elements u and û exist. Furthermore we have clearly u ≤ v =⇒ (u ≤ v and û ≤ v̂) (2.14) and u=u ˆ = û. and û (2.15) The resolvent operators satisfy the following properties. Proposition 2.4 For a given u ∈ RN , let v, v 0 ∈ RN satisfying (2.7) and w, w0 ∈ RN satisfying (2.12). Then (a) The map Jλ is single-valued and increasing on Dv . (b) We have v ≤ Jλ u ≤ w (2.16) u ≤ Jλ u ≤ û (2.17) In particular EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 7 (c) If Au ≥ 0 (resp. Au ≤ 0), then AJλ u ≥ 0 (resp. AJλ u ≤ 0) for each λ ≥ 0. (d) If Au ≥ 0, then λ → Jλ u is decreasing on RN (and increasing if Au ≤ 0). (e) We have Jλ,v u ≤ J˜λ,w u. In particular Jλ,u u ≤ J˜λ,û u. (f) Jλ,u u = u and J˜λ,û û = û. (g) If v ≤ v 0 and w ≤ w0 , then Jλ,v u ≤ Jλ,v0 u and J˜λ,w u ≤ J˜λ,w0 u. Proof. We prove only results (a),(b),(c),(d) in the case Jλ = Jλ,v . (a) Let u ≥ w. Then Jλ,v u satisfies Jλ,v u ≥ v (I + λA)Jλ,v u ≥ u ≥ w. Hence we get (a) from minimality of Jλ,v w for the previous system. (b) Inequality Jλ,v u ≥ v is required in the definition of Jλ,v . Since w satisfies w≥v (I + λA)w ≥ w ≥ u, we get (b) from minimality of Jλ,v u in the previous system. (c) Let Au ≥ 0 and λ ≥ 0. We have (I + λA)u ≥ u u≥v so Jλ,v u ≤ u. Hence u = Jλ,v u + λAJλ,v u ≤ u + λAJλ,v u and so AJλ u ≥ 0. (d) Let 0 ≤ λ ≤ µ. Then u = (I + λA)Jλ,v u ≤ (I + µA)Jλ,v u. Since we have Jλ,v u ≥ v, from minimality of Jµ u for these two constraints, it comes Jµ,v u ≤ Jλ,v u. The proof is similar when Au ≤ 0. (e) Since Av ≤ 0 and Aw ≥ 0, from (c) it follows AJλ,v u ≥ 0 and AJ˜λ,w u ≤ 0. Hence (e) results from Jλ,v u + λAJλ,v u = u = J˜λ,w u + λAJ˜λ,w u. Properties (f) and (g) result immediately from the definitions. Solution of (1.1) We recall that a (local) strong solution of (1.1) is a continuous function u Rt defined on [0, θ) ⊂ [0, T ], θ > 0 such that u(t) = u0 + 0 (−Au(τ ) + f (τ ))dτ for t ∈ [0, θ). In the sequel we only look for (local) strong solutions of (1.1). A maximal (resp. minimal) solution of (1.1) is the strong solution max min u = SA,f (t)u0 (resp. u = SA,f (t)u0 ) of (1.1) defined as follows: max min (i) The interval of definition [0, θ) of SA,f (.)u0 (resp. SA,f (.)u0 ) is maximal on [0, T ], i.e. there is no solution v 6= u, such that v = u on [0, θ]. max (ii) For each solution v of (1.1) on [0, T1 ) ⊂ [0, T ], we have v(t) ≤ SA,f (t)u0 min (resp. v(t) ≥ SA,f (t)u0 ) on [0, inf(θ, T1 )). 8 3 Cauchy problem for derivors EJDE–2001/32 Main results For the following results, we assume the hypothesis H1T defined in Section 2. Theorem 3.1 The problem CP (A, f, u0 ) has a unique local solution denoted by SA,f (t)u0 (or SA (t)u0 if f = 0) and defined on a maximal interval [0, Tmax ) ⊂ [0, T ]. Moreover if u0 ≤ u1 in RN and if f ≤ g in L1 ([0, T ], RN ) then SA,f (t)u0 ≤ SA,g (t)u1 on the common domain of existence of these two solutions. The next result concerns the autonomous case, for which we have global solutions. Theorem 3.2 Assume that f ≡ 0. Then SA (.)u0 is defined on the whole interval [0, T ] and SA (t)u0 = lim Jt/n,n (u0 ), (3.1) n→+∞ for t ∈ [0, T ], where Jλ = Jλ,u0 is as defined in Section 2. This is an exponential Crandall-Liggett’s type formula, but here (I + λA)−1 is a priori multi-valued. In the non-autonomous case f 6≡ 0, it is possible to exhibit a formula as (3.1) which gives the solution of (1.1) as a limit of a discrete scheme. But such a formula is more complicated than (3.1) and thus, is not of a particular interest. When f ∈ L∞ ([0, T ], RN ), from Theorem 3.1 and Theorem 3.2, we can deduce that CP (A, f ) has solution on [0, T ] if A is a strong continuous derivor (see def. 2.1). Unfortunately, we do not know what happens in the general case f ∈ L1 ([0, T ], RN ) without extra assumptions. 4 Proofs The proof of Theorem 3.1 follows immediately from the three lemmas below. Lemma 4.1 Let A be a continuous derivor. Let V be an element of VK (u0 ). Then the operator B defined by B(v) := inf [A(w + v) − A(w)] w∈V (4.1) is a continuous derivor. Proof. 1.) Let us show that B is a derivor on RN . If u ≤ v and ui = v i for some i ∈ {1, . . . , N }, we have u + w ≤ v + w and (u + w)i = (v + w)i for each w ∈ V . Since A is a derivor, it follows Ai (u + w) − Ai w ≥ Ai (v + w) − Ai w. Thus inf (Ai (u + w) − Ai w) ≥ inf (Ai (v + w) − Ai w). w∈V w∈V So B i u ≥ B i v for u ≤ v and ui = v i . 2.) At this stage we will show that B is continuous on RN . According to (4.1), EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 9 B is clearly upper semi-continuous (see [3, pp. 132-137]). So it is enough to prove that for each i ∈ {1, . . . , N }, B i is lower semi-continuous on RN . Fix i ∈ {1, . . . , N }. For each u ∈ RN , thanks to the compactness of V , there exists χ(u) (which depends on i) in V satisfying B i u = Ai (u + χ(u)) − Ai (χ(u)) (4.2) We have to prove now that B i is lower semi-continuous, that is (B i )−1 (]−∞, α]) is closed for all α ∈ R. In this goal, consider α ∈ R and a sequence (uk )k∈N∗ of elements of RN such that lim uk = u∞ and B i (uk ) ≤ α. It suffices to prove B i (u∞ ) ≤ α. By contradiction, let us suppose B i (u∞ ) > α. Without loss of generality, thanks to the compactness of V , we can suppose lim χ(uk ) = v∞ ∈ V. k→+∞ Equation (4.1) yields α < B i (u∞ ) = Ai (χ(u∞ ) + u∞ ) − Ai (χ(u∞ )) ≤ Ai (v∞ + u∞ ) − Ai (v∞ ). (4.3) From the continuity of Ai , it results Ai (v∞ +u∞ )−Ai (v∞ ) = lim Ai (χ(uk )+uk )−Ai (χ(uk )) = B i (uk ) ≤ α. (4.4) k→+∞ Equations (4.3) and (4.4) lead to a contradiction. Lemma 4.2 Let A be a continuous derivor. max (t)u0 defined on its (a) Problem (1.1) has a local unique maximal solution SA,f 1 min (t)u0 maximal interval of existence [0, T ) (resp. a unique minimal solution SA,f 2 on [0, T )). (b) If v0 ≤ u0 and if v(t) satisfies v(0) = v0 and v 0 (t) ≤ −Av(t) + f (t) a.e. on max [0, T̃ ) with T̃ < T 1 , then for t ∈ [0, T̃ ) we have v(t) ≤ SA,f (t)u0 . 0 (c) if v0 ≥ u0 and if v(t) satisfies v(0) = v0 and v (t) ≥ −Av(t) + f (t) a.e. on min [0, T̃ ) with T̃ < T 2 , then for t ∈ [0, T̃ ) we have v(t) ≥ SA,f (t)u0 . The previous lemma will be proved by standards arguments in an analogous way as the Kamke’s Lemma [15] and the arguments given in [13, 17]. Proof. We shall prove only parts (a) and (b). The proof of part (c) can be obtained in an analogous way. Let v be a solution on [0, T̃ ) ⊂ [0, T ] of v̇(t) ≤ −Av(t) + f (t), v(0) = v0 . t ∈ [0, T̃ ) For each n ∈ N∗ , Problem CP (A, f + n1 , u0 ) has at least a local solution un (see [17]) defined on a maximal interval of [0, Tn ]. 10 Cauchy problem for derivors EJDE–2001/32 1) Let us show v ≤ un on [0, T̃ ∧ Tn ), where T̃ ∧ Tn means min(T̃ , Tn ). One has un (t) − v(t) ≥ un (t0 ) − v(t0 ) + Z t (n (τ ) + t0 1 )dτ n . (4.5) n (τ ) = −Aun (τ ) + Av(τ ) for all t0 , t ∈ [0, T̃ ∧ Tn ), t0 ≤ t. Let E = {t ∈ [0, T̃ ∧ Tn ), v(τ ) ≤ un (τ ) for all τ ∈ [0, t]} First, remark that E is (not empty and) closed on [0, T̃ ∧ Tn ). Second, if t0 ∈ E, t0 < T̃ ∧ Tn and (v(t0 ))i = (un (t0 ))i , for some i ∈ {1, . . . , N } then the derivor property of Definition 2.1 (i) yields in (t0 ) ≥ 0 (4.6) Consequently, relations (4.5), (4.6) and the definition of t0 provide some η > 0 such that v i (τ ) ≤ uin (τ ) for τ ∈ [t0 , t0 + η] ⊂ [0, T̃ ∧ Tn ). Finally E is open in [0, T̃ ∧ Tn ) and thus E = [0, T̃ ∧ Tn ). 2) We have un+1 ≤ un on [0, Tn+1 ∧ Tn ). Indeed the proof is the same as 1) if we replace v by un+1 and −Aun (τ ) + Av(τ ) by −Aun (τ ) + Aun+1 (τ ). 3) We have T̃ ∧ Tn ≥ T̃ ∧ T1 . Indeed, from parts 1) and 2), for each n ∈ N∗ we have v ≤ un+1 ≤ un ≤ u1 (4.7) on the common interval of existence of these solutions. Then the extension principle of solutions implies T̃ ∧ Tn+1 ≥ T̃ ∧ Tn since a bounded solution is extendable. 4) The sequence (un ) converges uniformly to u∞ on each compact sub-interval of [0, T̃ ∧T1 ) thanks to (4.7) and the Lebesgue’s Dominated Convergence Theorem. Furthermore u∞ is solution of CP (A, f, u0 , T̃ ∧ T1 ) on [0, T̃ ∧ T1 ). Moreover, clearly u∞ is the maximal solution of CP (A, f, u0 , T̃ ∧ T1 ) (see Section 2). Let F be the set of S ∈ [0, T ] such that u∞ is extendable into a continuous function on [0, S) which is the maximal solution of CP (A, f, u0 , S). One has T̃ ∧ T1 ∈ F . By considering S∞ = sup F , we obtain a maximal extension of u∞ as a local solution of CP (A, f, u0 , T ) which is by construction the maximal solution of CP (A, f, u0 , T ). The next lemma makes use of the ascent assumption. Lemma 4.3 With the notation in Lemma 4.2, if H1T holds, we have min max SA,f (t)u0 = SA,f (t)u0 on [0, T 1 ∧ T 2 ) = [0, T 1 ). EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 11 max Proof. Thanks to Lemma 4.2(a), (1.1) has a maximal solution SA,f (t)u0 de1 min fined on a sub-interval [0, T ) of [0, T ] and a minimal solution u(t) = SA,f (t)u0 2 1 2 defined on a sub-interval [0, T ) of [0, T ]. Set T3 = T ∧ T and max min w(t) := SA,f (t)u0 − SA,f (t)u0 (4.8) for t ∈ [0, T3 ). We have to prove w = 0 on [0, T3 ), that is E = [0, T3 ) where E = {t ∈ [0, T3 ), w(τ ) = 0, ∀τ ∈ [0, t]}. Since E = w−1 (0) is closed in [0, T3 ) (w being continuous), it just remains to show that E is open to the right. Let t0 ∈ E, t0 < T3 . We have to prove that there exists h > 0 such that w = 0 on [t0 , t0 + h]. Eventually, by changing w into w(t0 + .) and f into f (t0 + .), we will suppose t0 = 0. Let V ∈ VK (u0 ) and B as in (4.1), in view of the continuity of u at 0, there exists T4 ∈]0, T3 [ such that, for each t ∈ [0, T4 ], u(t) ∈ V , hence w satisfies a.e.: w0 (t) = −(A(u(t) + w(t)) − Au(t)) ≤ −Bw(t) w(0) = 0, (4.9) a.e. t ∈ [0, T4 ]. By using Lemma 4.2 (b) with B instead of A, we have max w(t) ≤ SB (t)(0) (4.10) for each t ∈ [0, T4 ∧ T5 ], where [0, T5 ] is the maximal interval of existence of max max (t)(0) satisfies (t)(0). The function x(t) = SB SB x0 (t) = −Bx(t) x(0) = 0. (4.11) Let (vk )k∈N∗ be a sequence which defines a uniform ascent at the point u0 for the operator A on the setV (see section 2). B i (vk ) = Ai (vk + v̂k (i)) − Ai (v̂k (i)) > 0 (4.12) for k ∈ N∗ and i ∈ {1, . . . , N } where v̂k (i) is a vector minimizing v → Ai (vk + v) − Ai (v) on V . Let k ∈ N be fixed, then due to Lemma 4.2(b) there exists sk > 0 such that sk ≤ T4 ∧ T5 and max max SB (t)(0) ≤ SB (t)(vk ) (4.13) for each t ∈ [0, sk ]. Equation (4.12) and the continuity of B give the existence of tk > 0 and tk ≤ sk such that: max B(SB (t)(vk )) ≥ 0 max for t ∈ [0, tk ]. Thus t → SB (t)(vk ) is decreasing on [0, tk ]. Consequently, from (4.10) and (4.13), it results max max w(t) ≤ SB (t)(vk ) ≤ SB (0)(vk ) = vk (4.14) 12 Cauchy problem for derivors EJDE–2001/32 for each t ∈ [0, tk ]. In particular, we have w(tk ) ≤ vk . If we put y(t) = w(tk + t), we get y 0 (t) ≤ −By(t) y(0) = w(tk ) ≤ vk (4.15) for a.e. t ∈ [0, tk ]. Hence, according to (4.14) and (4.15), one has max w(tk + t) ≤ SB (t)(vk ) ≤ vk for t ∈ [0, tk ]. So w(t) ≤ vk for t ∈ [0, 2tk ∧ T4 ]. Whence by induction, we get 0 ≤ w(t) ≤ vk (4.16) ∗ for t ∈ [0, T4 ]. Since (4.16) is valid for each k ∈ N and lim vk = 0, it follows w(t) = 0 for each t ∈ [0, T4 ]. Hence for h = T4 > 0, we have [0, h] ⊂ E which completes the proof. Proof of Theorem 3.2 In this subsection, we assume that A satisfies H1T, and f ≡ 0 on [0, T ]. First, let us recall some basic facts about the discretization (1.1) in the Theory of Nonlinear Semigroups. It is known [7] that a strong solution of (1.1) is a mild solution, i.e. a continuous function which is a uniform limit of Euler’s implicit discrete schemes. Such discrete schemes are defined as follows. Let > 0 be fixed. Then an -discretization on [0, T ] of u̇ + Au = 0 on [0, T ] consists of a partition 0 = t0 ≤ t1 ≤ · · · ≤ tn of the interval [0, tn ] and a finite sequence (f1 , f2 , . . . , fn ) in RN such that (a) ti − ti−1 < for i = 1, . . . , n and T − < tn ≤ T . (b) Σni=1 (ti − ti−1 )kfi k ≤ . We will indicate these data by writing DA (0 = t0 , t1 , . . . , tn : f1 , . . . , fn ). A solution of a discretization DA (0 = t0 , t1 , . . . , tn : f1 , . . . , fn ) is a piecewise constant function v : [0, tn ] → RN whose values vi on (ti−1 , ti ] satisfy vi − vi−1 + Avi = fi ti − ti−1 v0 = u0 (4.17) for i ∈ {1, . . . , n}. An -approximate solution of CP (A, 0, u0 ) is a solution v of an -discretization DA (0 = t0 , t1 , . . . , tn : f1 , . . . , fn ). A mild solution of CP (A, 0, u0 ) on [0, T ] is a continuous function u on [0, T ] with the property that for each > 0 there is an -approximate solution v of CP (A, 0, u0 ) on [0, T ] such that kv(t) − u(t)k ≤ for t in the domain of v. Now, for n ∈ N∗ , let J = JT /n,u0 , and define the function un by un (0) = u0 and un (t) = J i (u0 ) for (i − 1)T /n < t ≤ iT /n where J i is the ith power of J. Then, thanks to (2.9), (4.17) holds with vi = un (iT /n), ti = iT /n and fi = 0 (Lemma 4.4 below guarantees u0 ≤ vi−1 for all i ≥ 1). In other words un is a T /n-approximate solution of CP (A, 0, u0 ). Then Theorem 3.2 results immediately from the following two lemmas. EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 13 Lemma 4.4 With the previous notations, for each t ∈ [0, T ], we have u0 ≤ un (t) ≤ û0 . Proof. (4.18) We set vi for un (iT /n). By Proposition 2.4 (b), we have u0 ≤ v1 = J(u0 ) ≤ û0 (4.19) Then (4.18) results by induction from (4.19) and Proposition 2.4 parts (a) and (f). The following lemma studies the continuous and discrete approach and gives an exponential formula such as the Crandall-Liggett’s formula (for the accretive autonomous case in [7]). Lemma 4.5 The sequence of approximate solutions (un ) defined in Lemma 4.4 converges uniformly on [0, T ] to SA (.)u0 . Moreover, for all t ∈ [0, T ], u0 ≤ SA (t)u0 ≤ û0 (4.20) and n SA (t)u0 = lim Jt/n (u0 ) n where Jt/n = Jt/n,u0 . Proof. The approximate solutions un satisfy an Ascoli-Arzel’s type condition A on [0, T ] [11, p. 260-268], namely: for each > 0 there exists N ∈ N and η > 0 such that (n ≥ N and |t−s| ≤ η ) implies kun (t)−un (s)k∞ ≤ . Indeed, relations (4.17) lead to un (tnj ) − un (tni ) = − Z tn j Aun (t)dt. (4.21) T kun (t) − un (s)k ≤ M (|t − s| + 2 ), n (4.22) tn i Using (4.18), Relation (4.21) yields where M = supu0 ≤w≤û0 kAwk. Consequently (see [11, p. 260]) the sequence (un ) is relatively compact in the Banach space B([0, T ], RN , k k∞ ) of bounded functions on [0, T ] with values in RN . So there exists a subsequence (unk ) converging to a continuous function u∞ which is a mild solution of CP (A, 0, u0 ). Then, passing to the limit in (4.21) (or from [7, p. 314]), we see that u∞ is a strong (even a classical) solution of CP (A, 0, u0 ) on [0, T ]. From Theorem 3.1, it results u∞ = SA (.)u0 (4.23) 14 Cauchy problem for derivors EJDE–2001/32 on [0, T ]. Thus (4.20) follows from (4.23) and (4.18) on [0, T ]. Then, taking T = t, (4.23) yields n SA (t)u0 = lim Jt/n (u0 ), n→+∞ where Jt/n = Jt/n,u0 . The proof is complete. 5 An example in R2 Let A0 be the operator defined on R2 by A0 x x + (x − 2y)1/3 := y y + (2y − x)1/5 (5.1) Lemma 5.1 The operator A0 satisfies H1T and H2T for all T > 0. The proof is left to the reader. In particular, the relation A0 x 2t x 2t + = A0 + y t y t (5.2) for t ∈ R+ , provides uniform ascents at each point. The sublinearity at infinity implies H2T. Therefore we can apply the results of Section 3 to the operator A0 for any T > 0. Hence CP (A0 , f, u0 , +∞) has a unique global solution, on [0, +∞[. Now, our task is to prove that no condition of Nagumo-OsgoodKamke and no accretivity condition (even in a generalized sense) can be applied to obtain the uniqueness of solutions of CP (A0 , f, u0 ). Generalized accretivity conditions Let k · kp , p ∈ [1, +∞], be the classical lp -norm in R2 . As usual (see [7, 8]), we set ku + λvk − kuk [u, v] = lim (5.3) λ↓0 λ for u, v ∈ R2 . For p ∈ [1, +∞], the notation [u, v]p , p ∈ [1, +∞] means [u, v], with k · kp instead of k · k in (5.3). In the sequel, φ stands for a continuous function φ : R → R+ satisfying the following condition U: For each T0 , the function x ≡ 0 is the unique positive solution on [0, T0 ] of ẋ(t) = φ(x(t)) x(0) = 0. EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 15 Definition 5.2 We will say that an operator B defined on R2 is φ-accretive in (R2 , k · k) if −[u − v, Bu − Bv] ≤ φ(ku − vk) (5.4) for all u, v ∈ R2 . We will say that B satisfies a φ-Osgood condition if kBu − Bvk ≤ φ(ku − vk) for all u, v ∈ R2 . Remark a) The condition B + ωI is accretive (ω ≥ 0) means B is φ-accretive with φ(x) = ωx. General studies of φ-accretive conditions can be found in [6, 16]. b) A φ-Osgood condition is a particular case of φ-accretivity. Lemma 5.2 Let p ∈ [1, +∞]. Then, there is no φ, such that A0 is φ-accretive in (R2 , k · kp ). Moreover, there are no φ and no norm k · k such that A0 satisfies a φ-Osgood condition in (R2 , k · k). Proof. a) Suppose first p = +∞. By contradiction, suppose that A0 is φ2 accretive in(R , k · k∞ ) for some φ. Let x ∈ [0, 1[. A direct computation yields 0 0 A0 = and 0 0 [ x x ,A ] = x1/3 (x2/3 + (−1 + x)1/3 ). x − 12 x2 ∞ x − 12 x2 (5.5) So, thanks to the φ-accretivity, (5.5) implies 1 1/3 x ≤ φ(x), 2 (5.6) for x ≥ 0 sufficiently small. Set z(t) = H −1 (t) , H(σ) = Z 0 σ dξ . φ(ξ) (5.7) From (5.6), H is defined for σ ≥ 0 sufficiently small and z(t) > 0 (5.8) on some interval ]0, T0 ] with T0 > 0. By using (5.7), a straightforward computation gives z 0 (t) = φ(z(t)) and z(0) = 0. Then U provides z≡0 on [0, T0 ]. Hence there is a contradiction between (5.8) and (5.9). (5.9) 16 Cauchy problem for derivors EJDE–2001/32 b) Suppose now p ∈ [1, +∞[. By contradiction again, suppose that A0 is φaccretive in (R2 , k · kp ). In this case, for x ∈ [0, 1], by setting u= x 1 1 2 x − 2 2x a direct computation gives 1 [u, A0 u]p = 2 p−1 2/5 xp− 3 − ( x−x x 2 ) 1+ p kukp ! kukp . (5.10) According to (5.10), the reader can check that the φ-accretivity property implies φ(kukp ) ≥ −[u, A0 u]p ≥ 2p1+1 x2/5 for x ∈ [0, 1] sufficiently small. Then we can deduce that for some x0 ∈]0, 1], there is C > 0 (for instance C = that Ckuk2/5 ≤ φ(kukp ) p e−1/5 2(2p +1) ), such for all x ∈ [0, x0 ]. Finally, there exists ξ0 > 0 such that φ(ξ) ≥ Cξ 2/5 for ξ ∈ [0, ξ0 ]. Now, as in step a), using the function H defined in (5.7), we can easily derive a contradiction. c) Let k · k be a norm in R2 and suppose that A0 satisfies a φ-Osgood condition 0 in (R2 , k · k). Then, by taking u = , in the φ-Osgood property we obtain x φ(ξ) ≥ cξ 1/5 for a constant c > 0, ξ1 > 0 and all ξ ∈ [0, ξ1 ]. So we can conclude as before and the lemma is proved. 6 Asymptotic behavior Figure 5.1 motivates the following remarks about asymptotic behavior of solutions of (1.1). Hypothesis H3 stands for following three conditions • f ≡0 • The assumption H2T holds for all T > 0 • A is a continuous derivor on RN . We do not assume the uniqueness of solutions of CP (A, 0, u0 , +∞). We set A+ = {u; Au ≥ 0} and A− = {u; Au ≤ 0}. Definition 6.1 A derivor A is absorbent if u0 ∈ A+ (resp. u0 ∈ A− ) implies u(t) ∈ A+ (resp. u(t) ∈ A− ) for all t ≥ 0 and each solution u(.) of the autonomous problem CP (A, 0, u0 , +∞). We say that A is u∞ -absorbent if B defined by Bu = Au − Au∞ is absorbent. EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 17 10 9 8 7 y 6 5 4 3 2 1 0 0 1 2 3 4 5 x 6 7 8 9 10 Figure 1: Flow relative to A0 Proposition 6.2 Assume H3. Let u0 , v0 , w0 be in RN such that Av0 ≤ 0, Aw0 ≥ 0, v0 ≤ w0 and v0 ≤ u0 ≤ w0 . Suppose A is a continuous u∞ -absorbent derivor on RN such that the equation Av = 0 has a unique solution u∞ in [v0 , w0 ]. Then every solution u of CP (A, 0, u0 , +∞) satisfies lim u(t) = u∞ t→+∞ max min Proof. It is sufficient to prove the result for SA (t)w0 and SA (t)v0 since min from Lemma 4.2 such extremal solutions exist and satisfy SA (t)v0 ≤ u(t) ≤ max max SA (t)w0 , t ∈ [0, +∞[. If w(t) = SA (t)w0 we have Z t w(t) − w0 = − Aw(x)dx. (5.11) 0 Consequently t → w(t) is decreasing because from the absorbent property min w0 (t) = −Aw(t) ≤ 0. In an analogous way v(t) = SA (t)u is increasing because 0 v (t) = −Av(t) ≤ 0 for each t ∈ [0, +∞[. Hence we get v0 ≤ v(t) ≤ w(t) ≤ w0 . Then l1 = limt→+∞ w(t) and l2 = limt→+∞ v(t) exist in RN . Hence, according R +∞ R +∞ to (5.11), 0 Av(τ )dτ and 0 Aw(τ )dτ converge. Since limt→+∞ Aw(t) = Al1 and limt→+∞ Av(t) = Al2 , we have necessarily Al1 = Al2 = 0. So by hypothesis limt→+∞ w(t) = limt→+∞ v(t) = u∞ . 18 Corollary 6.3 Cauchy problem for derivors EJDE–2001/32 For the operator A0 introduced in (5.1), we have 0 lim SA0 (t)(u0 ) = . 0 t→∞ 0 0 Proof. We can show that A0 u = holds if and only if u = . Moreover 0 0 A0 is absorbent. Indeed, with the notation of Lemma 4.4, let u0 ∈ A+ 0 (resp. i A− ) and u (t) = J (u ) for (i − 1)T /n < t ≤ iT /n. Then, owing to Propon 0 T /n 0 + − sition 2.4.(c), un (t) ∈ A0 (resp. un (t) ∈ A0 ). Consequently, Lemma 4.5 yields − SA0 (t)u0 ∈ A+ 0 (resp. SA0 (t)u0 ∈ A0 ) for all t ≥ 0. So Corollary 6.3 is a direct consequence of Proposition 6.2. References [1] J. Bliedtner, W. Hansen: Potential Theory, Springer-Verlag. [2] D. Bothe: Minimal solutions of multivalued differential equations, Differential and Integral Equations, Vol.4,No. 2, p. 445-447, 1991. [3] G. Choquet: Cours d’analyse, tome 2, Topologie, Masson, 1973. [4] J-F. Couchouron: Equations d’évolution, le problème de Cauchy, Thèse de Rouen. [5] J-F. Couchouron: Equations d’évolution, application la théorie discrète du potentiel (C.R. Acad. Sci. Paris, t. 315, Série 1, p. 275-278, 1992). [6] J-F. Couchouron: Equations d’évolution, le problème de Cauchy (C.R. Acad. Sci. Paris, t.319, Série 1, p. 133-136, 1994). [7] M. G. Crandall: Nonlinear Semigroups and Evolution Governed by Accretive Operators, Proceedings of Symposia in pure Mathematics, vol. 45, 1986, part 1. [8] K. Deimling: Multivalued Differential Equations, de Gruyter. [9] K. Deimling, V. Lakshmikantham: On existence of extremal solutions of differential equations in Banach spaces. (Nonlinear Analysis, Theory, Methods and Applications, Vol.3, n.5, p. 563-568, 1979). [10] C. Dellacherie: Thorie des processus de production, Sm. Proba. 24, Lectures Notes in Math, Springer Verlag, Vol. 1426, p. 52-104, 1990. [11] N. Dunford, J.T. Schwartz: Linear Operators Part 1, John Wiley and Sons, 1957. EJDE–2001/32 J-F Couchouron, C. Dellacherie, & M. Grandcolas 19 [12] L. S. Hal: An introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, vol. 41, Amer. Math. Soc., Providence, RI, 1995. [13] P. Hartmann: Ordinary differential equations, John Wiley and Sons, 1964. [14] M. W. Hirsch: Systems of Differential equations that are competitive or cooperative, Convergence almost everywhere, Siam J. Math. Anal., vol.16, n. 3, 1985. [15] E. Kamke: Zur Theorie der Systeme gewohnlicher Differentialgleichungen, II, Acta Math. 58 (1932) 57-85 (II 4,III 4). [16] Y. Kobayashi and K. Tanaka: Nonlinear semigroups and evolution governed by ’generalized’ dissipative operators, Advances in mathematical Sciences and Applications, Gakkotosho, Tokyo, Vol.3, 93/94, p. 401-426. [17] W. Walter: Differential and Integral Equations, Springer Verlag, Berlin, Heidelberg, New York, 1970. Jean-François Couchouron (e-mail: couchour@loria.fr) Michel Grandcolas (e-mail: grandcol@poncelet.univ-metz.fr) UFR MIM Departement de Mathématiques, Université de Metz Ile du Saulcy 57045 Metz Cedex 01 France Dellacherie Claude Departement de Mathématiques UFR Sciences, Site Colbert, Université de Rouen 76821 Mont Saint Aignan, france e-mail: dellache@univ-rouen.fr