Equi-Attraction and The Continuous Dependence of Attractors on Time Delays

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX EQUI-ATTRACTION AND THE CONTINUOUS DEPENDENCE OF ATTRACTORS ON TIME DELAYS P. E. Kloeden1, P. Mar´ ın-Rubio2 1FB Mathematik, Johann Wolfgang Goethe Universit¨at, D-60054 Frankfurt am Main, Germany 2Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain (Communicated by Aim Sciences) Abstract. Under appropriate regularity conditions it is shown that the continuous dependence of the global attractors Aτof semi dynamical systems S(τ)(t) in C([−τ, 0]; Z) with Za Banach space and time delay τ∈[T∗, T ∗], where T∗>0, is equivalent to the equi-attraction of the attractors. Examples and counter examples posed in this right framework are provided. 1. Introduction. The upper semi continuous dependence of attractors on a parameter is a standard result in dynamical systems theory, see e.g. [5, 11, 13, 15, 16]. In general, lower semi continuous, and hence continuous, dependence does not hold without additional assumptions, which usually are given in terms of the structure of the attractor, such as its being Morse-Smale. In another approach, Li and Kloeden [14] showed recently that continuous dependence in a parameter is equivalent to the equi-attraction of the parametrized attractors. These results also apply to attractors of delay differential equations (DDE) with a fixed time delay. On the other hand very little has appeared in the literature about the dependence of attractors of DDE on the time delay itself, a difficulty being that the attractors belong to different state spaces. An early paper on upper semi continuity for a concrete retarded nonlinear PDE is [1] (see also [2, 4] for the same question about inertial manifolds to deterministic and stochastic problems). Kloeden [11] showed how the upper semi continuous dependence of attractors in the time delay can be formulated by embedding the different semi dynamical systems and their attractors in a common state space. See also [3, 9] for other results. Our aim in this paper is to find an analogue of the equivalence of continuous dependence and equi-attraction in [14] (see also [12]) for the dependence of attractors 2000 Mathematics Subject Classification. Primary: 34D45, 37C70, 34K25. Key words and phrases. Semiflows for delay differential equations, parametric attractors, extended semiflows and attractors, continuity of attractors and equi-attraction. Partially supported by Ministerio de Educaci´on y Ciencia (Spain) and FEDER (European Community) grant MTM2005-01412 and Acci´on integrada Hispano-Alemana Ref. HA2005-0082. 1 2 P. E. KLOEDEN AND P. MAR´ IN-RUBIO of semi dynamical systems (SDS) generated DDE on the time delay. For this we first summarize the main ingredients from [14]. Let λbe a parameter in a compact metric space (Λ, DΛ) and let {S(λ) t, t ∈ R+}λ∈Λbe a family of SDS on a complete metric space (X, d). Define d(x, A) = infy∈Ad(x, y) for any x∈Xand A⊂X, and let BX(a, r) denote the open ball of Xwith center aand radius r; and P(X) and C(X) the classes of all nonempty and nonempty and closed subsets of X, respectively. In addition, denote the Hausdorff semidistance and Hausdorff distance on X, respectively, by H∗ X(A, B) = sup x∈A d(x, B), HX(A, B) = max {H∗ X(A, B), H∗ X(B, A)} for any closed nonempty subsets Aand Bof X. Definition 1. A nonempty compact subset Aof Xis called a global attractor of an SDS {St, t ∈R+}on X(i.e. a semi-group of mappings with St:X→X continuous for each fixed t≥0) if it is invariant, i.e. St(A) = Afor all t∈R+, and attracts bounded subsets Bof X, i.e. H∗(St(B),A)→0as t→+∞. Definition 2. Let {S(λ) t, λ ∈Λ}be a family of SDS on X. It is said to be (i) equi-dissipative on Xif there exists a bounded subset Uof Xso that for any bounded subset B⊂X, there exists a TB∈R+independent of λ∈Λsuch that S(λ) t(B)⊂ U, t ≥TB; (ii) eventually equi-compact (or uniformly compact for large tin [14]) if for any bounded subset Bof X, there exists a TB∈R+independent of λ∈Λsuch that Sλ∈ΛS(λ) t(B)is relatively compact in Xfor any t≥TB. Theorem 3. [14, Th.2.9] Suppose that a family of SDS {S(λ) t, λ ∈Λ}on Xis equi-dissipative and eventually equi-compact and that Aλis the global attractor of S(λ) tfor λ∈Λ. In addition, suppose that (A1) for any t∈R+fixed, S(λ) t(x)is jointly continuous in (x, λ)on X×Λ. (A2) S(λ) t(x)is equi-continuous in λfor (t, x)in any bounded subset of R+×X. Then {Aλ}is equi-attracting if and only if Aλis continuous in λwith respect to the Hausdorff distance. Remark 4. The above equivalence also holds if (A2) is replaced by: (A2’) S(λ) t(x)is equi-continuous in λfor tin any bounded subset of R+and xin any bounded subset of Sλ∈ΛAλ. Theorem 5. [14, Th.2.7] Suppose that {S(λ) t, λ ∈Λ}is equi-dissipative and eventually equi-compact and that the assumptions (A1) and (A3) For any bounded subset Bof Xand T > 0, S(λ) txis uniformly continuous in x∈Buniformly w.r.t. λ∈Λand t≤T, i.e. ∀ε > 0,∃δ > 0 : x, y ∈B, d(x, y)< δ ⇒dS(λ) t(x), S(λ) t(y)< ε, ∀t∈[0, T ], λ ∈Λ. hold. Then, if Aλis continuous in λ, the family {Aλ}is uniformly Lyapunov stable, i.e. for any ε > 0,there exists δ > 0(independent of λ) such that for all λ∈Λ,if d(x, Aλ)< δ, then d(S(λ) tx, Aλ)< ε for all t∈R+. EQUI-ATTRACTION AND CONTINUOUS DEPENDENCE FOR DDE 3 In Section 2 we reinterpret the above “equi” concepts for SDS generated by DDE with finite delay, where the finite delay is considered as the parameter and extend the SDS and their attractors (which are assumed to exist, see [11] for existence results) to a common state space. In Section 3 we interpret the above “equi” concepts for the SDS in their original state spaces. Finally, in Section 4 an example is given of a scalar DDE with attractors which are continuous and discontinuous in the time delay at different time delays. 2. Extension to a common state space. A delay differential equation in a Banach space (Z, |·|) with time delay τ > 0 generates an SDS in the function space Cτ:= C([−τ, 0]; Z) of continuous functions φ: [−τ, 0] →Z, which is a Banach space with the supremum norm k· kτ. We denote this SDS in Cτby S(τ)and consider a family of such SDS for different, fixed values of the time delay τ∈[T∗, T ∗], with 0< T∗< T∗<∞. In addition, we assume that each SDS S(τ)possesses a global attractor Aτin its state space Cτ. Theorem 3 cannot be applied directly to this family, but can be after we represent then as SDS on the common state space CT∗. In order to translate the different SDS to the common space we project a solution S(τ) tφin function space C([−τ, 0]; Z) onto the base space Zand then reconstitute it as a time dependent function taking values in the function space C([−T∗,0]; Z). Let φ∈CT∗and let φ|[−τ,0] be its truncation in Cτ. Hence S(τ) tφ|[−τ,0] is well defined for all t≥0. Define its projection x: [−T∗,∞)×CT∗→Zin Zby x(t, φ) := φ(t)t∈[−T∗,0], S(τ) t(φ|[−τ,0])(0) t > 0, where S(τ) tφ|[−τ,0](0) is the value that takes the function S(τ) t(φ|[−τ,0]) in Zat time 0. Finally, define b S(τ) t(φ)∈CT∗for each t≥0 by b S(τ) t(φ)(s) := x(t+s, φ), s ∈[−T∗,0]. Theorem 6. If S(τ)be an SDS on Cτ, then {b S(τ) t, t ∈R+}defines an SDS on CT∗. Moreover, if S(τ):R+×Cτ→Cτis jointly continuous in (t, φ)∈R+×Cτ, then b S(τ):R+×CT∗→CT∗is jointly continuous in (t, φ)∈R+×CT∗. Proof. The initial condition property of an SDS follows directly from the definition for t= 0, specifically b S(τ) 0(φ)(s) = x(s, φ) = φ(s), s ∈[−T∗,0], so b S(τ) 0(φ) = φfor all φ∈C([−T∗,0]; Rd). To check the semi-group property, that is, b S(τ) t1+t2(φ) = b S(τ) t1b S(τ) t2(φ),for all t1, t2≥0,and φ∈CT∗, we consider two cases: 4 P. E. KLOEDEN AND P. MAR´ IN-RUBIO Case 1: t1+s > 0. We use the semi-group property of the SDS S(τ)several times: b S(τ) t1+t2(φ)(s) = x(t1+t2+s, φ) =S(τ) t1+t2+s(φ|[−τ,0])(0) =S(τ) t1+s(S(τ) t2(φ|[−τ,0]))(0) =x(t1+s, b S(τ) t2(φ)) = b S(τ) t1(b S(τ) t2(φ))(s). Case 2: t1+s≤0 (since s∈[−T∗,0],this case only holds if T∗> t1). By the definitions we have b S(τ) t1+t2(φ)(s) = x(t1+t2+s, φ),(1) as well as b S(τ) t1(b S(τ) t2(φ))(s) = x(t1+s, b S(τ) t2(φ)) =b S(τ) t2(φ)(t1+s) = x(t1+t2+s, φ).(2) Comparing (2) with (1) we obtain the desired semi-group property. The continuity of b S(τ) tfrom CT∗into CT∗for each fixed t∈R+and the second assertion of the theorem can be proved similarly, so we prove just the latter. Suppose that φ(n)→¯ φin CT∗and tn→tin R+.Then φ(n)|[−τ,0] →¯ φ|[−τ,0] in Cτand hence S(τ) tn(φ(n)|[−τ,0])→S(τ) t(¯ φ|[−τ,0]) in Cτfor each tn→tin R+, which means that x(tn+s, φ(n)) = S(τ) tn(φ(n)|[−τ,0])(s) →S(τ) tn(¯ φ|[−τ,0])(s) = x(t+s, ¯ φ) for all s∈[−τ, 0] (not only punctually, but uniformly in [−τ, 0]). Concatenating as many intervals as necessary, we obtain in a finite number of steps that b S(τ) tn(φ(n))(s) = x(tn+s, φ(n))→x(t+s, ¯ φ) = b S(τ) t(¯ φ)(s) for all s∈[−T∗,0], i.e. b S(τ) tn(φ(n))→b S(τ) t(¯ φ) in CT∗as tn→tin R+and φ(n)→¯ φin CT∗. Hence the mapping (t, φ)7→ b S(τ) t(φ) is continuous. This completes the proof that b S(τ)is an SDS on CT∗. The next step in our goal is to extend the attractors to the common state space and to ensure that the extended objects are indeed attractors for the extended SDS. Observe that it is not enough to have H∗ CτS(τ) t−jτ (Bτ),Aτ< ε for j= 0,...,n∗−1,(3) where n∗is the first integer with n∗τ≥T∗.This does not ensure that there exists a corresponding concatenated set in CT∗satisfying the corresponding inequality EQUI-ATTRACTION AND CONTINUOUS DEPENDENCE FOR DDE 5 there. To show this we use the compactness of the attractors and the continuity of the SDS. Theorem 7. Suppose that an SDS S(τ):R+×Cτ→Cτhas a global attractor Aτ. Then, the extended SDS b S(τ)in CT∗given in Theorem 6 possesses a global attractor b Aτin CT∗, which is characterized by b Aτ:= ψ∈CT∗:∃entire trajectory ¯ Φ(τ) tof S(τ)in Aτ(4) with ψ(s) = ¯ φ(s)∀s∈[−T∗,0], where ¯ φ(t)is the projection in Zof the entire solution ¯ Φ(τ) tdefined by ¯ φ(t) := ¯ Φ(τ) t(0) for all t∈R. Proof. ¿From the strict invariance of Aτit is known that for each φ∈ Aτthere exists at least one entire solution ¯ Φ(τ) tof the SDS S(τ)in Aτwith ¯ Φ(τ) 0=φ, so the set b Aτis well defined. The invariance of b Aτunder the extended SDS b S(τ)follows immediately from the definitions. The compactness of Aτin CT∗follows from the definitions and the fact that the backward extension of an SDS in a compact invariant set generates a multivalued semi-group with compact attainability sets [10]. It remains to prove that b Aτis the global attractor for the extended SDS b Sτ. Let ε > 0 be arbitrary and let n∗be the first integer such that n∗τ≥T∗. For each χ∈ Aτ,define δ(χ) := min{δ1(χ),...,δn∗(χ)}, where δj(χ) for j= 1,...,n∗ are such that the continuous maps S(τ) jτ satisfy H∗ Cτ(S(τ) jτ (χ), S(τ) jτ (φ)) ≤εfor all φ∈BCτ(χ, δj(χ)). Since Aτis compact it has finite cover of open balls Aτ⊂ k [ i=1 BCτ(x(i), δ(x(i))). There thus exists an ρ > 0 with BCτ(Aτ, ρ)⊂ k [ i=1 BCτ(x(i), δ(x(i))).(5) Now consider a bounded set Bin CT∗.By the attraction of Aτthere exists T= T(ρ, B|[−τ,0])≥0 such that H∗ CτS(τ) t(B|[−τ,0]),Aτ≤ρ∀t≥T=T(ρ, B|[−τ,0]). Consider an arbitrary element ϕ∈Band a x(i0)∈Cτsuch that, by (5),   S(τ) jτ S(τ) T(ϕ)−S(τ) jτ (x(i0))  τ≤εfor j= 1,...,n∗. This implies that H∗ CT∗b S(τ) n∗τ+T(B),b Aτ≤ε. Thus b Aτis the global attractor of b S(τ)in CT∗. 6 P. E. KLOEDEN AND P. MAR´ IN-RUBIO Finally, following [11], we recall that the continuous convergence of the attractors for different time delays is understood as HCT∗b Aτ0,b Aτ→0 as τ0→τ. 3. The equi-properties for the original SDS. We will now translate the concepts of equi-attraction, equi-dissipative and eventually equi-compact of the family of extended SDS {b S(τ) t, τ ∈[T∗, T ∗]}on the space CT∗in terms of the original SDS S(τ) ton their state spaces Cτ. This is important as the properties will be verified here, especially when the SDS are generated by specific DDE. 3.1. Equi-dissipativity. The concept of equi-dissipativity in Definition 2, (i), in terms of the extended SDS reads: there exists a bounded subset Uof CT∗and for every bounded subset Bof CT∗there exists a TB∈R+, which is independent of τ, such that b S(τ) t(B)⊂ U for all t≥TBand τ∈[T∗, T ∗].(6) In terms of the original SDS and state space this implies that S(τ) t(B|[−τ,0])⊂ U|[−τ,0] for all t≥TBand τ∈[T∗, T∗], where the previous notation is used for the restricted sets, i.e. B|[−τ,0] ={φ|[−τ,0] :φ∈ B},U|[−τ,0] ={ψ|[−τ,0] :ψ∈ U}. The definition of equi-dissipativity has an equivalent form in terms of the underlying base space Z, namely: Lemma 8. A family of SDS {b S(τ), τ ∈[T∗, T∗]}is equi-dissipative if and only if there exists a bounded subset Uof Zsuch that for every bounded subset Bof Z there exists a TB∈R+, which is independent of τ, such that S(τ) t(B|[−τ,0])(0) ⊂Ufor all t≥TBand τ∈[T∗, T∗], where B:= {φ∈CT∗:φ(s)∈B∀s∈[−T∗,0]}.(7) Proof. Starting with (6), we simply define U={φ(s)∈Z:φ∈ U, s ∈[−T∗,0]} with TB:= TBcorresponding to the bounded subset Bof CT∗defined in (7). In the other direction, following (7), we define U:= {φ∈CT∗:φ(s)∈U, s ∈[−T∗,0]}.(8) Given a bounded subset Bof CT∗we define TB:= TB+T∗corresponding to the set B={φ(s)∈Z:φ∈ B, s ∈[−T∗,0]}. (Note that the new set Bdefined by (7) in terms of this Bwill contain and in general be larger than the original set B). EQUI-ATTRACTION AND CONTINUOUS DEPENDENCE FOR DDE 7 3.2. Eventual equi-compactness. We first note that the property of eventual equi-compactness in Definition 2, (ii), can be rewritten as: for any bounded subset Bof X, there exists a TB∈R+independent of λ∈Λand a family of compact subsets {K(t), t ≥TB}of Xsuch that S(λ) t(B)⊂K(t)for every t≥TB. We simply take U(t) to be the closure of Sλ∈ΛS(λ) t(B) in X. The compact sets U(t) here need not be uniformly bounded in t– if they were then we would also have equi-dissipativity. In our situation this definition takes the form: for every bounded subset Bof CT∗ there exists a TB∈R+, which is independent of τ, and a family of compact subsets {U(t), t ≥TB}of CT∗such that b S(τ) t(B)⊂ U(t)for all t≥TBand each τ∈[T∗, T∗]. In terms of the original dynamical systems this translates to S(τ) t(B|[−τ,0])⊂ U(t)|[−τ,0] for all t≥TBand each τ∈[T∗, T∗]. Remark 9. For DDE with finite delay, when Zis finite dimensional, compactness follows from the existence of a bounded absorbing family, thanks to Ascoli-Arzel`a Theorem, if the right hand side of the DDE is a bounded map (i.e. it maps bounded sets onto bounded sets). 3.3. Joint and equi-continuity. Theorem 3 requires that the family of SDS satisfies the continuity properties (A1) and (A2), i.e. (A1) For any t∈R+fixed, S(λ) t(x) is jointly continuous in (x, λ) on X×Λ. (A2) S(λ) t(x) is equi-continuous in λfor (t, x) in any bounded subset of R+×X. In our context the joint continuity property (A1) becomes: for any t∈R+fixed, b S(τ) t(φ)is jointly continuous in (τ, φ) in [T∗, T∗]×CT∗. Thus, if (τn, φ(n))→(τ, φ) in [T∗, T∗]×CT∗as n→ ∞, then so too does b S(τn) t(φ(n))→b S(τ) t(φ) as n→ ∞. Recalling the projection notation introduced before Theorem 6 x(τ)(t+s, φ) := b S(τ) t(φ)(s), s ∈[−T∗,0], joint continuity means that x(τn)(t, φ(n))→x(τ)(t, φ) as n→ ∞, in the base space Zuniformly on the interval [t−T∗, t] for each fixed t≥0. Thus it will also be uniform on all finite time intervals [−T∗, T] with T > 0. This uniform joint convergence in Zimplies the function space joint continuity of condition (A1) above. Similarly, the equi-continuity property (A2) becomes: b S(τ) t(φ)is equi-continuous in τfor (t, φ)in any bounded subset [T1, T2]×B of R+×CT∗,which is essentially 8 P. E. KLOEDEN AND P. MAR´ IN-RUBIO uniform continuity in τ, i.e. for every ε > 0 and bounded subset [T1, T2]× B of R+×CT∗there exists δ=δ(T1, T2,B, ε)>0 such that |τ0−τ|< δ =⇒  b S(τ0) t(φ)−b S(τ) t(φ)  T∗< ε ∀(t, φ)∈[T1, T2]×B. In terms of the projections in the base space Zthis reads as |τ0−τ|< δ =⇒x(τ0)(t, φ)−x(τ)(t, φ)< ε ∀(t, φ)∈[T1−T∗, T2]×B, which implies the function space equi-continuity condition (A2) above. 3.4. Equi-attraction. Suppose that each SDS S(τ) ton Cτhas global attractor Aτ in Cτfor τ∈[T∗, T∗]. Then, by Theorem 7, each extended SDS b S(τ) ton CT∗has an attractor b Aτin CT∗, where b Aτis defined in terms of Aτthrough (4). These extended attractors are equi-attracting if for every ε > 0 and bounded subset Bof CT∗there exists Tε,B∈R+independent of τ∈[T∗, T∗] such that H∗ CT∗b S(τ) t(φ),b Aτ< ε for all t≥Tε,B, φ ∈ B, τ ∈[T∗, T ∗] (9) which obviously implies that H∗ CτS(τ) t(φ|[−τ,0]),Aτ< ε for all t≥Tε,B, φ ∈ B, τ ∈[T∗, T∗].(10) Of course, one would like that (9) and (10) to be equivalent (perhaps with a slightly larger Tε,B). However, the value ρappearing in the proof of Theorem 7 depends on τin a not necessarily uniform way. We will use property (A3) and borrow some ideas from Theorem 5 to obtain an equivalence. Remark 10. Condition (A3) in Theorem 5 for the extended SDS b S(τ)is equivalent to the following condition for the original semi dynamical systems S(τ): (A3’) For any bounded subset Bof CT∗and T > 0, S(τ) t(χ|[−τ,0])is uniformly continuous in χ|[−τ,0] ∈ B|[−τ,0] uniformly w.r.t. τand t≤T, i.e. ∀ε > 0,∃δ > 0 : χ, φ ∈ B,kχ|[−τ,0] −φ|[−τ,0]kτ< δ ⇒ kS(τ) t(χ|[−τ,0])−S(τ) t(φ|[−τ,0])kτ< ε, ∀t∈[0, T ], τ ∈[T∗, T∗].(11) Theorem 11. Let S(τ):R+×Cτ→Cτfor τ∈[T∗, T∗]be a family of SDS with attractors Aτ, which is equi-dissipative and equi-attracting in the sense of (10) and also satisfies condition (A3’). Then the extended attractors b Aτare equi-attracting. Proof. By the equi-dissipativeness there exists a bounded subset Uof Zsuch that Aτ⊂ U|[−τ,0] for all τ, where the subset Uof CT∗is defined from Uthrough (8). Consider any ε > 0 and the bounded set B={φ∈CT∗:φ(s)∈BZ(U, ε), s ∈[−T∗,0]}. It is enough to check (9) only with this bounded set. By the equi-attraction of {Aτ}τ,there exists Tε,Bindependent of τ, such that (10) holds. In particular, this implies that B|[−τ,0] is positively invariant for any S(τ) twith t≥Tε,B, i.e. S(τ) t(B|[−τ,0])⊂ B|[−τ,0] ∀t≥Tε,B.(12) EQUI-ATTRACTION AND CONTINUOUS DEPENDENCE FOR DDE 9 For the bounded set B,by (A3’), there exists δ > 0 depending on εsuch that (11) holds for T=n∗T∗,with n∗the first integer such that n∗T∗≥T∗.We will use (11) for t=jT∗with j= 1,...,n∗, which ensures that we can cover any interval of length T∗by delays of length τ∈[T∗, T ∗]. Let ρ= min(δ, ε).By the equi-attraction again, analogously to (10), there exists a time Tρ,B(which we can take w.l.o.g. larger than Tε,B) such that H∗ Cτ(S(τ) t(B|[−τ,0]),Aτ)< ρ for all t≥Tρ,B, τ ∈[T∗, T∗].(13) To finish the proof, take ψ∈ B.By (13), for any t≥Tρ,B,there exists ξ∈ Aτsuch that kS(τ) t(ψ|[−τ,0])−ξkτ< ρ ≤δ. Using (11) for T=n∗T∗we have kS(τ) jT∗S(τ) t(ψ|[−τ,0])−S(τ) jT∗(ξ)kτ< ε for j= 1,...,n∗. This means that H∗ CT∗b S(τ) n∗T∗+t(ψ),b Aτ< ε ∀t≥Tρ,B, which is the equi-attraction property (9) with Tε,Breplaced by n∗T∗+Tρ,B. Remark 12. There is an equivalent condition to Assumption (A3’) in the above result, though apparently is less restrictive, in which the uniform continuity for tin bounded intervals can be substituted by uniform continuity at a single time instant t∗, namely, (A3”) There exists t∗∈(0, T∗]such that for any bounded subset Bof CT∗,the SDS S(τ) t∗(χ)is uniformly continuous in χ∈ B|[−τ,0] uniformly w.r.t. τ, i.e. ∀ε > 0,∃δ > 0 : χ, φ ∈ B,kχ|[−τ,0] −φ|[−τ,0]kτ< δ (14) ⇒ kS(τ) T∗(χ|[−τ,0])−S(τ) T∗(φ|[−τ,0])kτ< ε, ∀τ∈[T∗, T∗]. Actually, by (12), the above uniform continuity of S(τ) t∗in B|[−τ,0] holds for all S(τ) jt∗with j= 1,...,n∗,where n∗now denotes the first integer such that n∗t∗≥T∗. Indeed, since S(τ) t∗is uniformly continuous in B|[−τ,0],for an arbitrary ε > 0there exists δ1such that kS(τ) t∗(χ)−S(τ) t∗(φ)kτ≤εif kχ−φkτ< δ1.For S(τ) 2t∗,choose δ2 associated with ε2=δ1(property (12) plays an essential role here). Recursively, we conclude the claim in n∗steps, with δ= min{δ1,...,δn∗}. 4. An example. Li and Kloeden [14, Sec.3,Ex.3.2] gave the following example of a scalar ordinary differential equation to illustrate their results. Let λ0= 2√3/9 and Λ = [0, λ0] and let f: Λ ×R→Rbe given by f(λ, x) = −x3+x+ 4√3/9−λ, which is illustrated in Figure 1 below. In particular, for λ < λ0it has a single zero x(λ+)>0 and for λ=λ0a new zero x(λ− 0)appears.