The asymptotic behaviour of fractional lattice systems with variable delay

THE ASYMPTOTIC BEHAVIOUR OF FRACTIONAL LATTICE SYSTEMS WITH VARIABLE DELAY Linfang Liu 1, Tom´as Caraballo 2, Peter E. Kloeden 3 Abstract The existence and uniqueness of global solutions for a fractional functional differential equation is established. The asymptotic behaviour of a lattice system with a fractional substantial time derivative and variable time delays is investigated. The existence of a global attracting set is established. It is shown to be a singleton set under a certain condition on the Lipschitz constant. MSC 2010: Primary 26A33; Secondary 37K40, 37L6, 39B72 Key Words and Phrases: fractional substantial derivative, fractional lattice systems, variable delay, Leray-Schauder theorem, global attracting sets 1. Introduction In this paper, we study the global existence and the long time behaviour of solutions for a fractional delay lattice system of the form Dα sui(t)−(ui−1(t)−2ui(t)+ui+1(t))+λui(t)+fi(t, uit)=0, t ≥τ, ui(t) = φi(t−τ),∀t∈[τ−h, τ], (1.1) where λ∈R, 0 < α < 1, i∈Zand the Caputo fractional substantial derivative Dα sis defined as Dµ sf(x)=Dm s[Iν sf(x)], ν =m−µ, m is the smallest integer which exceeds µ, where Iν sf(x) = 1 Γ(ν)Zx 0 (x−τ)ν−1e−β(x−τ)f(τ)dτ, ν > 0, is the fractional substantial integral [5, 7] and Γ is the Gamma function. β is a positive constant and c Year Diogenes Co., Sofia pp. xxx–xxx, DOI: ...................... 2 L. Liu, T. Caraballo, P.E. Kloeden Dm s=∂ ∂x +βm =D+βm =D+βD+β. . . D+β. The concept of fractional substantial derivative was introduced in 2006 by Friedrich et al. [8] to consider retardation effects in Kramers-Fokker- Planck type equations, where these are expressed mathematically by a fractional substantial derivative, which is more appropriate to characterize nonlocal coupling in time and space than the classic fractional derivative. The fractional substantial derivative is widely applied in physics, such as long jump of L´evy flights, anomalous diffusion, turbulence research, Fokker- Planck equations, forward and backward fractional Feynman-Kac equations etc [13, 6]. Moreover, the fractional substantial diffusion model is useful in applications to geophysics [11] as well as finance [4]. In finance, where the tempered stable process models price fluctuation which displays semiheavy tails, resembling a power law behavior at moderate time scales, but finally converging to a Gaussian at large time scales. See [8, 3] for more information about the application of fractional substantial derivative, [10] and related literatures for the classic fractional calculus. A main objective is to analyze the existence of solutions of fractional functional differential equations with the Caputo fractional substantial derivative, since there has been little published on this topic or on fractional lattice systems with delay. First we use Leray-Schauder’s fixed point theorem to prove the existence of local solutions for a fractional functional differential equation with α∈ (0,1). Then, by a priori estimates, we prove that the local solution is actually a global one. Our assumption here on delay termf(t, ut)is weaker than many available references. Finally, we apply this result to a fractional lattice system with variable delay, i.e., system(1.1), and obtain the existence and uniqueness of global solutions, as well as global attracting sets. Our result also applies to fractional lattice systems with distributed delay. The paper is structured as follows. Notation, some basic definitions and preliminary results are included in the next section. Additionally, Section 2 is also devoted to the global existence and uniqueness of solutions for an abstract fractional functional differential equation (2.1) in a separable Hilbert space. This theoretical result is applied to a fractional delay lattice system in Section 3. A priori estimates are obtained in Section 4 and used in Section 5 to establish the main result of this paper, namely, the existence of global attracting set while, which becomes a singleton under appropriate conditions. THE ASYMPTOTIC BEHAVIOUR OF FRACTIONAL . . . 3 2. Preliminaries Let Xbe a separable Hilbert space with norm k·k and inner product (·,·). For a fixed τ∈R,CT:= C([τ−h, T ]; X), h > 0, T ≥0, be the Banach space of continuous functions u: [τ−h, T]→X, endowed with the norm kukCT= supτ−h≤s≤Tku(s)k.For any u∈ CTand t∈[τ, T ], we denote by ut the element of C0=C([−h, 0]; X) defined by ut(θ)=u(t+θ), θ∈[−h, 0]. The dual coupling between Xand its dual X∗is denoted by h·,·i. Since Xis a Hilbert space, by the Riesz representation theorem, X∗is isomorphic to X. So the dual coupling h·,·i is the inner product in X. Let Xwbe the space Xendowed with the weak topology. We consider the space C0,w =C([−h, 0]; Xw) and say that un t→utin C0,w if for any θ∈[−h, 0] with un(t+θn)→u(t+θ) in Xwfor all θn→θ. We say that the function f: [0,∞)×C0→Xis weakly continuous in bounded sets for each t∈[0,∞), if un→uin C0,w, and kunkC0≤Mfor all nimply that f(tn, un t)→f(t, ut) in Xw. Similar concepts were introduced in [2] to establish the existence of solutions for a delay differential equation in a Banach space. We consider the global existence theorems for the following delay differential equations with Caputo fractional substantial time derivative in the separable Hilbert space X, Dα su(t) = f(t, ut), t ≥τ, u(t) = φ(t−τ),∀t∈[τ−h, τ],(2.1) where the nonlinear term fis weakly continuous in bounded sets. Definition 2.1.A function u∈ CTis called a solution of the initial value problem (2.1) if u(t) = φ(t−τ) for t∈[τ−h, τ] with φ∈ C0, and, for t∈[τ, T ], u(t) satisfies the integral equation u(t) = φ(0)e−β(t−τ)+1 Γ(α)Zt τ (t−r)α−1e−β(t−r)f(r, ur)dr, t ∈[τ, T ]. Hereafter, we assume that: (H1) The mapping f: [τ, ∞)× C0→Ris continuous, and there exists M1>0 such that kf(t, ϕ)−f(t, ψ)k ≤ M1kϕ−ψkC0for all ϕ, ψ ∈ C0and t∈[τ, ∞). (H2) The function f(t, 0) is L1+ 1 αlocally integrable, and there exists G > 0 such that Zt τ (t−r)α−1e−β(t−r)kf(r, 0)k2dr ≤G, ∀t≥τ. 4 L. Liu, T. Caraballo, P.E. Kloeden Now for any φ∈ C0, we define an operator Sφ:CT→ CTby Sφu(t) :=      φ(t−τ), t ∈[τ−h, τ], φ(0)e−β(t−τ)+1 Γ(α)Zt τ (t−r)α−1e−β(t−r)f(r, ur)dr, t ∈[τ, T]. (2.2) We can prove that the operator Sφis well-defined on CTby a similar method as Lemma 2.1, so the details are omitted. Since Xis a separable space, there exists a sequence such that the subset {e1, e2,··· , en,···} is dense in X, and by the Gram-Schmidt procedure, we can obtain an orthonormal subset of X, still denoted by {e1, e2,··· , en,···}. Taking Xn={e1, e2,··· , en}, and Pn:X→Xnan orthonormal projector, defined as Pnu=un, and Pnφ=φn, for every n, we can introduce the mapping Sn,φ :CT→ CTas, for any φn, Sn,φun(t):=     φn(t−τ), t ∈[τ−h, τ], φn (0)e−β(t−τ)+1 Γ(α)Zt τ (t−r) α−1e−β(t−r) Pnf(r, un,r)dr,t ∈[τ, T ]. We will need the following auxiliary theorem. Theorem 2.1.(Leray-Schauder’s fixed point theorem [9, 12]) Let T be a continuous and compact mapping of a Banach space X, such that the set {x∈X:x=λTx, for some 0≤λ≤1}is bounded. Then Tpossesses a fixed point. For any positive constant ρ, denote by A(ρ):=(un∈CT:un(t)=φn(t−h), t∈[τ−h, τ],and sup t∈[τ,T ]kun(t)k≤ρ). (2.3) Lemma 2.1.For any φ∈ C0, the operator Sn,φ maps the bounded subset A(ρ)⊂ CTinto another bounded set A(ρ0)⊂ CT. P r o o f. For our purpose, it is enough to show that for ∀ρ > 0, there exists a positive constant ρ0such that for each un∈A(ρ) we have supt∈[τ,T ]kSn,φun(t)k ≤ ρ0, where A(ρ) is defined in (2.3). Let un∈A(ρ) be arbitrary. From the definition of Sn,φ as in (2.2), we have for all t∈[τ, T], kSn,φun(t)k2≤2kφn(0)k2+2β−α Γ(α)Zt τ (t−r)(α−1)e−β(t−r) kf(r, un,r)k2dr. (2.4) THE ASYMPTOTIC BEHAVIOUR OF FRACTIONAL . . . 5 By (H1) and (H2), we derive kf(r, un,r)k2=kf(r, un,r)−f(r, 0)+f(r, 0)k2≤2M2 1kun,rk2 C0+2kf(r, 0)k2, which together with (2.4) and the Gronwall lemma implies that kSn,φun(t)k2 CT≤2kφk2 C0+2β−αG Γ(α)+ 2M2 1β−2αρ2:= ρ02<∞,∀t∈[τ, T ]. Hence, Sn,φ maps the bounded set A(ρ) into another bounded set A(ρ0). 2 Lemma 2.2.For any φ∈ C0, the mapping Sn,φ is continuous and completely continuous. P r o o f. Step 1: First, we verify that Sn,φA(ρ) is relatively compact. Since Xnhas finite dimension n, it suffices to prove that the family of continuous functions {Sn,φun:un∈A(ρ)}is uniformly bounded and is a family of equicontinuous functions of CTon [τ, T ]. As proved in Lemma 2.1, for any un∈A(ρ), we find kSn,φun(t)kCT≤ρ0, which implies that {Sn,φun:un∈A(ρ)}is uniformly bounded. On the other hand, for any t1, t2∈[τ, T ] with t1< t2, we have kSn,φu(t1)−Sn,φu(t2)k ≤ eβτ kφn(0)k|e−βt1−e−βt2|+A1+A2.(2.5) By (H2) and the H¨older inequality, A1can be bounded by A1=1 Γ(α)kZt1 τ [(t1−r)α−1e−β(t1 −r)−(t2−r)α−1e−β(t2−r)]f(r, un,r)drk ≤M0 1ρ0 Γ(α+1)[(t1−τ)α−(t2−τ)α+ (t2−t1)α]+ 1 Γ(α)( Zt1 τ (t1−r)α−1 −(t2−r)α−1)pdr)1 p×(Zt1 τkf(r, 0)kqdr)1 q+M1ρ0 βα(1−e−β(t2 −t1)) +(1 −e−β(t2 −t1)) pΓ(α)βα( Zt1 τ (t1−r)α−1e−β(t1 −r)kf(r, 0)k2dr)1 2. (2.6) Again, thanks to the H¨older inequality, A2=1 Γ(α)kZt2 t1 (t2−r)α−1e−β(t2−r)f(r, un,r)drk ≤M1ρ(t2−t1)α Γ(α+ 1) +(t2−t1)α 2 √αΓ(α)·( Zt2 t1 (t2−r)α−1e−β(t2 −r) kf(r, 0)k2dr)1 2. (2.7) From (2.5), (2.6) and (2.7), the right-hand side of (2.5) tends to zero as t1→t2. Therefore {Sn,φun:un∈A(ρ)}is equicontinuous on [τ, T], and therefore Sn,φA(ρ) is relatively compact in Xn. 6 L. Liu, T. Caraballo, P.E. Kloeden Step 2: Now we show that Sn,φ is continuous on CT. Let ui n∈ CTbe a sequence such that ui n→uin CTas i→ ∞. We can prove that, for any t∈[τ, T ], as igoes to ∞, kui n,t −utkCT= sup θ∈[−h,0] kui n,t(t+θ)−u(t+θ)k≤ sup t∈[τ,T ]kui n,t(t)−u(t)k → 0. Then from Eq. (2.2), (H4) and the H¨older inequality, we have for all t∈[τ, T ], kSn,φui n(t)−Sn,φu(t)k2 =1 Γ2(α)   Zt τ (t−r)α−1e−β(t−r)Pnf(r, ui n,r)−Pnf(r, ur)dr    2 ≤M2 1β−α Γ(α)Zt τ (t−r)(α−1)e−β(t−r)kui n,r −urk2 C0dr. Hence, kSn,φui n(t)−Sn,φu(t)k2≤M2 1β−2αkui n−uk2 CT→0 as i→ ∞. Therefore the operator Sn,φ is continuous on CT. By the arbitrariness of ρ and Lemma 2.1, we have that Sn,φui nis uniformly bounded on CT, while Step 1 implies that Sn,φui nis a family of equi-continuous functions. Therefore, by the Arzel`a-Ascoli theorem, we obtain that Sn,φui n→Sn,φuin CT. And hence, from Lemma 2.1, Step 1, Step 2 and the Arzel`a-Ascoli theorem, we obtain that Sn,φ :C([τ−h, T ]; Xn)→C([τ−h, T ]; Xn) is continuous and completely continuous. 2 Lemma 2.3.Assume that (H1)−(H2)hold true, then for any φ∈ C0, the fractional functional differential equation (2.1) has at least one local solution in Xn. P r o o f. We use the Leray-Schauder fixed point theorem to prove this conclusion, i.e., there exists an open set O ⊂ C([τ−h, T]; Xn) with un6= λSn,φunfor λ∈(0,1) and un∈∂O. We need to prove that if un=λSn,φunwith λ∈(0,1) on O, then un∈intO. Let un∈C([τ−h, T]; Xn) and un=λSn,φunfor some 0 <λ<1. Then for every t∈[τ, T ], un(t) = λφn(0)e−β(t−τ)+1 Γ(α)Zt τ (t−r)α−1e−β(t−r)Pnf(r, un,r)dr. THE ASYMPTOTIC BEHAVIOUR OF FRACTIONAL . . . 7 From (H1)−(H2) and the H¨older inequality with p= 1+αand q= 1+ 1 α, we find for all t+θ≥τ, kun(t+θ)k =λ   φn(0)e−β(t+θ−τ)+1 Γ(α)Zt+θ τ (t+θ−r)α−1e−β(t+θ−r)Pnf(r, un,r)dr    ≤kφn(0)e−β(t+θ−τ)k+β−αG Γ(α)+T0Zt+θ τ e−qβ(t+θ−r) 2kur−vrkq C0dr1 q , (2.8) where 0 < G(t+θ) = β−α Γ(α)(Rt+θ τ(t+θ−r)α−1e−β(t+θ−r)kf(r, 0)k2dr), and T0=M1Γ(α2)1 1+α (pβ 2)α2 1+αΓ(α) . Note that if τ−h≤t≤τ, then kun(t)k=kφn(t−τ)k ≤ e−qβ(t+2θ−2τ) 2kφnkC0. Then for any t≥τ, eqβt 2kun,tkq C0≤C∗e−qβ(t−2τ) 2kφnkq C0+G1eqβt 2+T1Zt τ eqβr 2kun,rkq C0dr, where C∗= 3qeqβh,G1=β−qα Γq(α)3q−1Gqand T1= 3q−1Tq 0eqβh 2. Applying Gronwall’s inequality, we find for all t≥τ, kun,tkq C0≤Ce(T1−qβ 2)(t−τ)+C. Thus, there exists a positive constant B∗such that kun,tkC0≤B∗for all t∈[τ, T ]. Set O={un∈C([τ−h, T]; Xn) : kunkCT≤B∗+ 1}. Note that Sn,φ :O → C([τ−h, T]; Xn) is continuous and completely continuous. From the choice of O, there is no un∈∂Osuch that un=λSn,φ(un) for λ∈(0,1), on the other hand 0 ∈ O is obvious. Therefore, as a consequence of Theorem 2.1, we deduce that Sn,φ has a fixed point unin O, which is a local solution of (2.1) in C([τ−h, T]; Xn). 2 Lemma 2.4.Under the assumptions of Lemma 2.3, for any φ∈C0, problem (2.1) has a local solution in C([τ−h, T ]; X). P r o o f. In Lemma 2.3, we have proved the existence of local solutions of problem (2.1) in C([τ−h, T]; Xn). By taking the limit, we prove the 8 L. Liu, T. Caraballo, P.E. Kloeden existence of local solutions of problem (2.1) in C([τ−h, T ]; X). For this, we form the approximating equations: Dα sun(t) = Pnf(t, un,t), t ≥τ, un(t) = φn(t−τ),∀t∈[τ−h, τ].(2.9) Similarly as in Lemma 2.3, there exists a solution unto problem (2.9) on [τ, T ] such that sup t∈[τ,T ]kun(t)k ≤ C. (2.10) Since Xis a separable Hilbert space, from (2.10) we can conclude that for any t∈[τ, T ], {un(t)}∞ n=1 is weakly convergent to u(t) in X. Moreover for any tm, t ∈[τ−h, τ] with tm→t, we have when n, m go to ∞, kun(tm)−u(t)k≤kφn(tm−τ)−φ(tm−τ)k+kφ(tm−τ)−φ(t−τ)k → 0. On the other hand, for any tm, t ∈[τ, T] with tm→t, and any v∈X∗, we have as n, m go to ∞, |hun(tm)−u(t), vi| ≤ |hun(tm)−u(tm), vi|+|hu(tm)−u(t), vi| → 0. Therefore, {un,t}∞ n=1 is weakly convergent to utin C0. Now we prove that u(·) is a solution of (2.1). For this we need to pass to the limit in the following integral un(t):=     φn(t−τ), t ∈[τ−h, τ], φn(0)e−β(t−τ)+1 Γ(α)Zt τ (t−r)α−1e−β(t−r)Pnf(r, un,r)dr, t ∈[τ, T]. (2.11) Since fis Lipschitz continuous, for any t∈[τ, T], we obtain f(t, un,t)→f(t, ut),weakly in X. (2.12) Using the Riesz representation theorem, we obtain that for any v∈X∗, there exists a unique w∈Xsuch that hu, vi= (u, w) for all u∈X, and therefore |hPnf(t, un,t), vi−hf(t, un,t), vi| =|(Pnf(t, un,t)−f(t, un,t), w)| =|(f(t, un,t),(I−Pn)w)| ≤ Ck(I−Pn)wk → 0, n → ∞. (2.13) Then by (2.12)-(2.13) and Lebesgue’s dominated convergence theorem we obtain for any v∈X∗, hZt τ (t−r)α−1e−β(t−r)Pnf(r, un,r)dr, vi→hZt τ (t−r)α−1e−β(t−r)f(r, ur)dr, vi. (2.14) Hence, (2.11) and (2.14) imply that, for any v∈X∗, THE ASYMPTOTIC BEHAVIOUR OF FRACTIONAL . . . 9 hu(t), vi=hu(τ)e−β(t−τ), vi+1 Γ(α)hZt τ (t−r)α−1e−β(t−r)f(r, ur)dr, vi, which means that u(t) = u(τ)e−β(t−τ)+Zt τ (t−r)α−1e−β(t−r)f(r, ur)dr, for all t∈[τ, T ]. 2 Theorem 2.2.(Global existence and uniqueness and continuous dependence on the initial values of solutions of Caputo substantial fractional differential equation) Suppose that (H1)and (H2)hold. Then, (i) for any φ∈ C0, the initial value problem (2.1) with the initial condition u(t) = φ(t−τ),t∈[τ−h, τ], has a unique global solution u(·, φ)on the whole interval [τ−h, ∞). (ii) on any bounded time interval [τ, T ], where T > 0, the solution u(·, φ)depends continuously on φ, i.e., lim φ→ϕsup ku(t, φ)−v(t, ϕ)kCT= 0. P r o o f. Part (i): From Lemmas 2.4, we know that (2.1) has a local solution on [τ−h, T]. So we need to prove that the local solution is a global one. Namely, we prove that the local solutions will not blow up on any bounded interval. By a similar procedure as we used in Lemma 2.3, we can prove that for any solution u(t), we have for all t≥τ, kutkq C0≤Ce(T1−qβ 2)(t−τ)+C, which implies that the local solution u(t) will not blow up on any finite interval [τ, T ]. Then it is actually a global solution on [τ−h, ∞). Finally, we show that the global solution is unique. Let u(t), v(t) be solutions of (2.1) with the same initial value φ. Then for t∈[τ, T], ku(t)−v(t)k=    1 Γ(α)Zt τ (t−r)α−1e−β(t−r)(f(r, ur)−f(r, vr)) dr   .(2.15) Replacing tby t+θin (2.15), note that ku(t+θ)−v(t+θ)k= 0 when t+θ≤τwith θ∈[−h, 0]. On the other hand, taking p= 1 + α,q= 1 + 1 α, then by H¨older’s inequality, we have, for t+θ≥τ, ku(t+θ)−v(t+θ)k ≤ T0Zt+θ τ e−qβ(t+θ−r) 2kur−vrkq C0dr1 q , which means (here T0is defined in (2.8)) 16 L. Liu, T. Caraballo, P.E. Kloeden Now we need to prove that un tnis relatively compact in C0,w. Since X=`2 is reflexive, from Lemma 4.1 we see that for fixed θ∈[−h, 0], {un tn(θ)} is relatively compact in Xw. On the other hand, we deduce that {un tn}is equicontinuous. In fact, by the method we used in Step 1, we could obtain that from Lemma 2.2, (C5) and (C6), if n∈N,θ1, θ2∈[−h, 0] and θ1≤θ2, then we have for all nsufficiently large, kun(tn+θ1)−un(tn+θ2)k ≤ C|e−βθ1−e−βθ2|+C(θ2−θ1)α. Thus we conclude that {un tn}is relatively compact in C0. Combining this with (5.2), we obtain that {un(·)}is relatively compact in C0, and thus A=[{ω(B) : B⊂ CT, B bounded} is compact in the topology of C0. From Lemma 4.1 and the conclusion (2), we see that Ais bounded in CT, and attracts all bounded subsets of CTin the topology of CT. Finally, we show that Ais the minimal closed set attracting set. Indeed, if there is another closed set A0which attracts any bounded set B⊂ CT, then by the definition of ω(B), we deduce that ω(B)⊂A0, and thus S{ω(B) : B⊂ CT, B bounded}belongs to A0. Since A0is closed, we have A=S{ω(B) : B⊂ CT, B bounded} ⊆ A0.The proof of Theorem 5.1 is completed. 2 5.2. Singleton case. To show that the forward attracting set is a singleton set, we need the following lemma, which is proved like Theorem 2.2. Lemma 5.1.Assume that e−βh 2β2α>2α+1 1+αp−α2 1+αq−α 1+αΓ1 1+α(α2)C11, where C11 := 8(16+λ2+A2 1+M2 1) Γ(α). Then for any two solutions u(t)and w(t)of problem (3.2) corresponding to initial values φand ψ, we have the strict contracting property kut−wtk2q CT≤Ckφ−ψk2q CTe−(qβ 2−C12)(t−τ)∀t≥τ, where C12 := 2α+1 αp−αΓ1 α(α2)Cq 11eqβh 2β−2α. Theorem 5.2.Assume the condition in Theorem 5.1 . If we additionally assume that, for each i∈Zand any s∈[−h, 0],F1,i(0) = bi(t, s, 0) = 0. Then the setA=S{ω(B):B⊂ CT, B bounded}is a singleton set. Moreover, it is the minimal set that exponentially attracts all bounded subsets of CT. Thus, every solution exponentially decays to zero in the topology of CT. THE ASYMPTOTIC BEHAVIOUR OF FRACTIONAL . . . 17 P r o o f. Since for each i∈Zand any θ∈[−h, 0], F1,i(0) = bi(t, θ, 0) = 0,it follows from Lemma 4.1 that for any solution u(·) of problem (3.2) with initial value φ, we have kutk2q CT≤Ckφk2q C0e−(qβ 2−c)t,∀t≥0.(5.3) Let Bbe a bounded subset of CTwith kBkCT= supψ∈BkψkCT≤d. We show that ω(B) is a singleton set. If not, then there would exist x, y ∈ω(B) such that x6=y. By the definition of ω(B), there exist sequences {τn}and {sm}, respectively, with τn→ ∞ (n→ ∞) and sm→ ∞ (m→ ∞), {φn}and {ψm}with φn, ψm∈B, solutions {un(·)}and {wm(·)} of problem (3.2) with un τ=φn,wm τ=ψmsuch that un τn→x(n→ ∞) and wm sm→y(m→ ∞) in CT. From (5.3) we deduce that kun τn−wm smkCT≤Ce−(qβ−2C12) 2q(τn−τ)+Ce−(qβ−2c) 2q(sm−τ), which implies that kun τn−wm smkCT→0 as n→ ∞ and m→ ∞. Hence, kx−ykCT= 0, this is a contradiction. Since ω(B) is the same singleton set for each bounded set B, we obtain from (5.3) that Ais a singleton set. Indeed, as ω(B1) = {x1}and ω(B2) = {x2}. Now B= B1∪B2, so ω(Bi)⊂ω(B) for i= 1,2, but as ω(B) only contains one point. Hence x1=x2. Moreover, every solution exponentially decays to zero in the topology of CT. Thus the proof is completed. 2 References [1] P. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems. Int. J. Bifur. Chaos 11 (2001), 143–153. [2] T. Caraballo, F. Morillas and J. 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Eule, Anomalous diffusion of inertial, weakly damped particles. Phys. Rev. Lett. 96 (2006), Art. # 230601. [9] A. Granas, J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York (2003). [10] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006). [11] M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusions in heterogeneous systems. Geophys. Res. Lett. 35 (2008), L17403–L17407. [12] D. O’Regan, Fixed point theorems for nonlinear operators, J. Math. Anal. Appl. 202 (1996), 413–432. [13] M. Zayernouri, M. Ainsworth and G. Karniadakis, Tempered fractional Sturm-Liouville eigenproblems. SIAM J. Sci. Comput. 37 (2015), A1777–A1800. 1Dept. Applied Mathematics, School of Mathematics and Statistics Xi’an Jiaotong Univeristy, Xianning West Road 28, 710049, Xi’an, Shaanxi, P.R. CHINA e-mail: [email protected]om 2Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla c/Tarfia s/n, 41012 Sevilla, SPAIN e-mail: carabal[email protected] 3Dpto. Mathematisches Institut, Universit¨at T¨ubingen D-72076 T¨ubingen, GERMANY e-mail: kloe[email protected]ankfurt.de Received: December 13, 2018