Pullback attractors for differential equations with multiple variable delays in Lipschitz nonlinearities

June 26, 2013 15:10 142IJBCaccepted International Journal of Bifurcation and Chaos c World Scientific Publishing Company Pullback attractors for differential equations with multiple variable delays in Lipschitz non-linearities TOM´ AS CARABALLO Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160 41080 Sevilla, Spain. carabal[email protected] G´ ABOR KISS Department of Mathematics University of Durham Durham United Kingdom DH1 3LE gab[email protected] TAKESHI TANIGUCHI Division of Mathematical Sciences, Graduate School of Comparative Culture Kurume University, Miimachi, Kurume Fukuoka 839-8502, Japan takeshi [email protected] Received (to be inserted by publisher) We establish results on the existence of pullback attractors for non-autonomous functional differential equations with multiple delays appearing within non-linear Lipschitz terms. The results are complementary to recently presented findings in [Caraballo & Kiss,2013b,a], and they extend the class of non-linearities to which existence results can be established by improving on a condition presented in [Caraballo & Kiss,2013b]. Keywords: Delay differential equation, non-autonomous dynamics, pullback attractors 1. Introduction Our knowledge on equations with time-varying delays is relatively patchy; this fact is also mentioned in [Mallet-Paret & Nussbaum,2013] where the analyticity of solutions of equations of this type is investigated. The objective of the present work is to contribute to the analysis of the long-time behaviour of differential equations of the form d dtD(t, xt) = F0(t, x(t)) + m X i=1 Fi(x(t−ρi(t))),(1) mainly focusing on the existence of pullback attractors. Throughout the paper, for r > 0, C=C([−r, 0]; Rn) denotes the Banach space of continuous functions with the sup-norm. We assume that the difference operator D:R×C→Rnis continuous together with its first and second Fr´echet derivatives with respect 1 June 26, 2013 15:10 142IJBCaccepted 2Caraballo et al to its second variable; and the first and the second derivatives of Dwith respect to the second variable are continuous at zero. In this paper, we consider two particular family of difference operators. For the delay terms, we assume that ρi:R→[0, r] is a measurable function, and ris a finite number. Lastly, our standing assumption will be the following: (A1) For i= 1, . . . , m, there exists Li>0, such that for any x, y ∈Rn |Fi(x)−Fi(y)| ≤ Li|x−y|, and Fi(0) = 0. Under these and some difference-operator specific assumptions on F0, we generalise very recent results presented in [Caraballo & Kiss,2013b] and [Caraballo & Kiss,2013a] which respectively deals with the existence of attractors for delay differential equations and neutral differential equations with time varying delays. In the proof of our results we will use the following technical result proved as Lemma 3.1 in [Chen, 2010] and which is formulated as follows. Lemma 1. Let y: [t0−t, ∞)→R+be a function such that for some γ > 0there are constants κ, κ0>0 κ0 γ<1and y(t)≤   κe−γt +κ0Rt t0e−γ(t−s)sup θ∈[−r,0] y(s+θ)ds, t > t0, κe−γt, t ∈[t0−r, t0]. Then y(t)≤κe−µt, t ≥t0−r(2) where µis the positive solution of κ0 γ−µeµr = 1. 2. Processes It is known that (see [Hale & Verduyn Lunel,1993]) if an initial function φ∈Cis prescribed at the initial time s∈R, the basic theory of delay differential equations implies the existence of the unique solution x(·;s, φ) of (1) on [s−r, ∞), which satisfies, in addition, the initial condition xs(·) = φ, in other words, xs(θ)=x(s+θ)=φ(θ) for all θ∈[−r, 0]. The unique solution of the initial value problems associated to (1) defines the solution map U(t, s) : C3φ7→ xt(·;s, φ)∈Cfor t≥s, which is, in fact, a process (also called a two-parameters semigroup), i.e. •U(t, s) : C→Cis a continuous map for all t≥s; •U(s, s) = idC, the identity on C, for all s∈R, •U(t, s) = U(t, τ)U(τ, s) for t≥τ≥s. As in the autonomous case, we look for invariant attracting sets. First, we introduce the Hausdorff semi–distance between subsets Aand Bin a metric space (X, d) as dist(A, B) = sup a∈A inf b∈Bd(a, b). Definition 2.1. Let Ube a process on a complete metric space X. A family of compact sets {A(t)}t∈Ris said to be a (global) pullback attractor for Uif, for all s∈R, it satisfies •U(t, s)A(s) = A(t) for all t≥s, and •lims→∞ dist(U(t, t −s)D, A(t)) = 0, for all bounded subsets Dof X. Definition 2.2. {B(t)}t∈Ris said to be absorbing with respect to the process Uif, for t∈Rand D⊂X bounded, there exists TD(t)>0 such that for all τ≥TD(t) U(t, t −τ)D⊂B(t). June 26, 2013 15:10 142IJBCaccepted Attractors for multiple variable delays 3 The following result (see, e.g. [Schmalfuss,1992]) shows that the existence of a family of compact absorbing sets implies the existence of a pullback attractor. Theorem 1. Let U(t, s)be a process on a complete metric space X. If there exists a family {B(t)}t∈Rof compact absorbing sets then, there exists a pullback attractor {A(t)}t∈Rsuch that A(t)⊂B(t)for all t∈R. Furthermore, A(t) = [ D⊂X bounded ΛD(t) where ΛD(t) = \ n∈N[ s≥n U(t, t −s)D. The next result will be crucial in our analysis. Theorem 2. (see, e.g., Caraballo & Kiss [Caraballo & Kiss,2013b]) Suppose that U(t, s)maps bounded sets into bounded sets and there exists a family {B(t)}t∈Rof bounded absorbing sets for U. Then there exists a pullback attractor for problem (1). 3. Main results In this section we will establish the main results of our paper. First we will consider the case of retarded equations, and we will finally analyze some neutral delay equations. 3.1. Retarded equations In this subsection we assume that Dφ =φ(0) for all φ∈C, that is, (1) becomes a delay differential equation of the form ˙x(t) = F0(t, x(t)) + m X i=1 Fi(x(t−ρi(t))).(3) In addition to (A1), we assume (A2) F0:Rn+1 →Rnis continuous and there exist α0>0,and a non-negative measurable function β(·) such that hF0(t, x), xi≤−α0|x|2+β(t), t ∈R, x ∈Rn(4) In [Caraballo & Kiss,2013b] the following result was proved: Theorem 3. (Caraballo and Kiss [Caraballo & Kiss,2013b, Theorem 3.4]) Assume that hypotheses (A1) and (A2) are satisfied and that there exists λ>α0such that Zt −∞ eλsβ(s)ds < +∞,∀t∈R.(5) Then, if ρi, i = 1, . . . , m, is measurable and Li<α0 m, there exists a family of bounded absorbing sets, {B(t)}t∈R, and consequently, there exists a pullback attractor for the process generated by (3). Remark 3.1. Notice that the assumption λ>α0in Theorem 3can be replaced by: there exists λ > 0 such that Zt −∞ eλsβ(s)ds < +∞,∀t∈R. Indeed, it is easy to see that if (5) holds for certain λ0, then it also holds true for any λ>λ0, and then we can pick one value bigger than α0. Consequently, a first improvement is that in Theorem 3we can assume λ > 0. June 26, 2013 15:10 142IJBCaccepted 4Caraballo et al However, in case that the constant λbelongs to the interval (0, α0), then it is possible to prove a complementary result by imposing a different smallness condition on the Lipschitz terms. This is established in the next theorem. Theorem 4. Assume that hypotheses (A1) and (A2) are satisfied and that there exists λ∈(0, α0)such that Zt −∞ eλsβ(s)ds < +∞,∀t∈R.(6) Then, if ρi, i = 1, . . . , m, is measurable and Pm i=1 L2 i<λα0 m, there exists a family of bounded absorbing sets, {B(t)}t∈R, and consequently, there exists a pullback attractor for the process generated by (3). Proof. Let us consider the number λ > 0 from (6), pick ε=α0 m, and denote x(τ) = x(τ;t0−t, ψ), τ ∈ [t0−t, t0], for any ψ∈Csuch that kψk ≤ d, and t0∈R. Then, applying the Young inequality in the delay terms below, taking into account that λ<α0, d dτ eλτ |x(τ)|2=λeλτ |x(τ)|2+ 2eλτ hx(τ), F0(τ, x(τ))i + 2eλτ m X i=1 hx(τ), Fi(x(τ−ρi(τ)))i ≤(λ−2α0)eλτ |x(τ)|2+ 2eλτ β(τ) + eλτ |x(τ)|2mε +eλτ ε−1 m X i=1 |Fi(x(τ−ρi(τ)))|2 ≤λ−α0)eλτ |x(τ)|2+ 2eλτ β(τ) +eλτ ε−1 m X i=1 L2 i|x(τ−ρi(τ))|2 ≤2eλτ β(τ) + eλτ ε−1 m X i=1 L2 i|x(τ−ρi(τ))|2. Integration on the interval [t0−t, τ] yields that eλτ |x(τ)|2−eλ(t0−t)|x(t0−t)|2≤2Zτ t0−t eλsβ(s)ds +ε−1 m X i=1 L2 iZτ t0−t eλs|x(s−ρi(s))|2ds. The integrand in the third sum can be estimated as follows Zτ t0−t eλs|x(s−ρi(s))|2ds ≤Zτ t0−t eλs sup θ∈[−r,0] |x(s+θ)|2ds. We thus have |x(τ)|2≤eλ(t0−t−τ)|x(t0−t)|2(7) +e−λτ 2Zτ t0−t eλsβ(s)ds +ε−1 m X i=1 L2 iZτ t0−t eλs sup θ∈[−r,0] |x(s+θ)|2ds!. June 26, 2013 15:10 142IJBCaccepted Attractors for multiple variable delays 5 Now, (7) reads as |x(τ)|2≤eλ(t0−t)d2+ 2 Zt0 −∞ eλsβ(s)dse−λτ + ε−1 m X i=1 L2 i!Zτ t0−t e−λ(τ−s)sup θ∈[−r,0] |x(s+θ)|2ds. (8) Let κ=eλ(t0−t)d2+ 2 Rt0 −∞ eλsβ(s)ds,κ0=ε−1Pm i=1 L2 iand γ=λ. Then, κ0 γ=ε−1Pm i=1 L2 i λ<1. Thus, Lemma 1implies that |x(τ)|2≤κe−µτ =eλ(t0−t)d2+ 2 Zt0 −∞ eλsβ(s)dse−µτ .(9) where µis the positive solution of ε−1Pm i=1 L2 i λ−µeµr = 1. This allows us to obtain sup θ∈[−r,0] |x(t0+θ)|2≤d2e(λ−µ)t0e−λt + 2 Zt0 −∞ eλsβ(s)dseµr ≤1+2Zt0 −∞ eλsβ(s)dseµr provided t≥TD=λ−1log d2e(λ−µ)t0. Consequently, the family of bounded sets {B(t)}t∈Rin C([−r, 0]; Rn) given by B(t) := B(0; R(t)), for all t∈R, where B(0; R(t)) denotes the ball centered at zero with radius R(t) = 1+2Rt −∞ eλsβ(s)dseµr, is absorbing. Taking into account that the associated process maps bounded sets into bounded sets of C([−r, 0]; Rn), the existence of the pullback attractor is ensured by Theorem 2(see also Theorem 2.4 in [Caraballo & Kiss,2013b]).  3.2. Neutral equations This section focuses on equations with difference operators of the form Dφ =φ(0) −g(φ(−σ)) where function gmaps Rninto itself. In other words, here we consider the family of neutral functional differential equations of the form: d dt [x(t)−g(x(t−σ))] = F0(t, x(t)) + m X i=1 Fi(x(t−ρi(t))),(10) where σ > 0 and finite, and without loss of generality, we assume σ < r. In this section, we assume that our non-delay term satisfies a non-autonomous dissipativity condition, i.e. we impose the following assumption on F0: (A3) F0:Rn+1 →Rnis continuous and there exist α0>0,and a non-negative measurable function β(·) such that June 26, 2013 15:10 142IJBCaccepted 6Caraballo et al hF0(t, φ(0)), φ(0) −g(φ(−σ))i≤−α0|φ(0) −g(φ(−σ))|2+β(t), t ∈R, x ∈Rn.(11) In addition, we assume that for any continuous function x:R→Rn, we have (Ag)|g(x(τ−σ))|2≤c2|x(τ)|2+δ(τ),0< c < 1, τ ∈Rsuch that δ(τ)∈[0, Mδ] for some Mδ∈R+. In [Caraballo & Kiss,2013a], the following result is shown. Theorem 5. Assume that conditions (A1) and (A3) are satisfied and that there exists λ∈(0, α0)such that Zt −∞ eλsβ(s)ds < +∞,∀t∈R. Then, if ρi, i = 1, . . . , m, is measurable, and λ(1 −c)2α0≥2m m X i=1 L2 i,(12) there exists a family of bounded absorbing sets, {B(t)}t∈R, and consequently, there exists a pullback attractor for the process generated by (10). The following theorems improve on (12) and complement Theorem 5. Before formulating them, we include a lemma from [Caraballo et al.,2007] which will be useful in subsequent computations. Lemma 2. Let 0< ξ < 1. Then |u|2≤1 1−ξ|u−v|2+1 ξ|v|2 for any u, v ∈Rn. Now we can establish our main result in this section. Theorem 6. Assume that assumptions (A1) and (A3) are satisfied and that there exists λ>α0such that Zt −∞ eλsβ(s)ds < +∞,∀t∈R.(13) Assume that ρi, i = 1, . . . , m, is measurable and that the following conditions hold: λ<α0(c+1 1−cc+1 1−c−(1 −c)2−1),(14) m X i=1 L2 i<α0 mλ(1 −c)2−c+1 1−c+c+1 1−cα0.(15) Then, there exists a family of bounded absorbing sets, {B(t)}t∈R, and consequently, there exists a pullback attractor for the process generated by (10). Remark 3.2. It is assumed in [Caraballo & Kiss,2013b] that λ∈(0, α0). Thus, in that sense, Theorem 6is complementary to Theorem 5. Proof. Let us consider the number λ > 0 from (13), pick ε=α0 m, and denote x(τ) = x(τ;t0−t, ψ), τ ∈ [t0−t, t0], for any ψ∈Csuch that kψk ≤ d, and t0∈R. Then, applying again the Young inequality in the June 26, 2013 15:10 142IJBCaccepted Attractors for multiple variable delays 7 delay terms below, we get d dτ eλτ |x(τ)−g(x(τ−σ))|2=λeλτ |x(τ)−g(x(τ−σ))|2 + 2eλτ hx(τ)−g(x(τ−σ)), F0(τ, x(τ))i + 2eλτ m X i=1 hx(τ)−g(x(τ−σ)), Fi(x(τ−ρi(τ)))i ≤(λ−2α0)eλτ |x(τ)−g(x(τ−σ))|2 + 2eλτ β(τ) + eλτ |x(τ)−g(x(τ−σ))|2mε +eλτ ε−1 m X i=1 |Fi(x(τ−ρi(τ)))|2 ≤(λ−α0)eλτ |x(τ)−g(x(τ−σ))|2+ 2eλτ β(τ) +eλτ ε−1 m X i=1 L2 i|x(τ−ρi(τ))|2 ≤(λ−α0)c+1 1−ceλτ |x(τ)|2+ 2eλτ β(τ) +(λ−α0)Mδeλτ c+eλτ ε−1 m X i=1 L2 i|x(τ−ρi(τ))|2. Integration on the interval [t0−t, τ] yields that eλτ |x(τ)−g(x(τ−σ))|2−eλ(t0−t)|x(t0−t)−g(x(t0−t−σ))|2(16) ≤(λ−α0)c+1 1−cZτ t0−t eλs|x(s)|2ds + 2 Zτ t0−t eλsβ(s)ds +(λ−α0)Mδ λc eλτ +ε−1 m X i=1 L2 iZτ t0−t eλs|x(s−ρi(s))|2ds. (17) The integrand in the third sum can be estimated Zτ t0−t eλs|x(s−ρi(s))|2ds ≤Zτ t0−t eλs sup θ∈[−r,0] |x(s+θ)|2ds. Thus we have |x(τ)−g(x(τ−σ))|2≤eλ(t0−t−τ)|x(t0−t)|2 1−c+c|x(t0−t)|2+Mδ c + (λ−α0)c+1 1−cZτ t0−t eλ(s−τ)|x(s)|2ds +(λ−α0)Mδ λc +e−λτ 2Zτ t0−t eλsβ(s)ds +ε−1 m X i=1 L2 iZτ t0−t eλs sup θ∈[−r,0] |x(s+θ)|2ds!. June 26, 2013 15:10 142IJBCaccepted 8Caraballo et al Now, we obtain |x(τ)|2≤1 1−ceλ(t0−t)d2 1−c+d2c+Mδ c +2 Zt0 −∞ eλsβ(s)ds +(λ−α0)Mδeλt0 λc e−λτ + (λ−α0)c+1 1−c+ε−1 m X i=1 L2 i!Zτ t0−t e−λ(τ−s)sup θ∈[−r,0] |x(s+θ)|2ds) +c|x(τ)|2+Mδ c. And so, |x(τ)|2≤1 (1 −c)2eλ(t0−t)d2 1−c+d2c+Mδ c +2 Zt0 −∞ eλsβ(s)ds +(λ−α0)Mδeλt0 λc +(1 −c)Mδ ceλτ e−λτ + (λ−α0)c+1 1−c+ε−1 m X i=1 L2 i!Zτ t0−t e−λ(τ−s)sup θ∈[−r,0] |x(s+θ)|2ds).(18) Let κ= eλ(t0−t)d2 1−c+d2c+Mδ c+ 2 Rt0 −∞ eλsβ(s)ds +(λ−α0)Mδeλt0 λc +(1−c)Mδ ceλτ (1 −c)2, κ0= (λ−α0)c+1 1−c+ε−1Pm i=1 L2 i (1 −c)2 and γ=λ. Then, using (14) and (15), it follows that κ0 γ= (λ−α0)c+1 1−c+ε−1Pm i=1 L2 i λ(1 −c)2 < (λ−α0)c+1 1−c+λ(1 −c)2−c+1 1−c+c+1 1−cα0 λ(1 −c)2 = 1. Thus, Lemma 1implies that |x(τ)|2≤κe−µτ =eλ(t0−t)d2 1−c+d2c+Mδ c+ 2 Rt0 −∞ eλsβ(s)ds +(λ−α0)Mδeλt0 λc +(1−c)Mδ ceλτ  (1 −c)2e−µτ ,(19) where µis the positive solution of κ0 λ−µeµr = 1. June 26, 2013 15:10 142IJBCaccepted Attractors for multiple variable delays 9 Thus we obtain that sup θ∈[−r,0] |x(t0+θ)|2 ≤d2 1−c+d2c+Mδ ce(λ−µ)t0e−λt + 2 Rt0 −∞ eλsβ(s)ds +(λ−α0)Mδeλt0 λc +(1−c)Mδ ceλτ  (1 −c)2eµr ≤1+2Rt0 −∞ eλsβ(s)ds +(λ−α0)Mδeλt0 λc +(1−c)Mδ ceλτ  (1 −c)2eµr, provided t≥TD=λ−1log  d2 1−c+d2c+Mδ ce(λ−µ)t0. Consequently, the family of bounded sets {B(t)}t∈Rin C([−r, 0]; Rn) given by B(t) := B(0; R(t)), for all t∈R, where B(0; R(t)) denotes the ball centred at zero with radius R(t) = „1+2 Rt −∞ eλsβ(s)ds+(λ−α0)Mδeλt λc +(1−c)Mδ ceλτ « (1−c)2eµr, is absorbing. Taking into account again that the associated process maps bounded sets into bounded sets of C([−r, 0]; Rn), the existence of the pullback attractor is ensured again by Theorem 2. The next result improves on condition (12). Theorem 7. Assume that hypotheses (A1) and (A3) are satisfied and that there exists λ∈(0, α0)such that Zt −∞ eλsβ(s)ds < +∞,∀t∈R.(20) Then, if ρi, i = 1, . . . , m, is measurable, and m X i=1 L2 i<λ(1 −c)2α0 m, there exists a family of bounded absorbing sets, {B(t)}t∈R, and consequently, there exists a pullback attractor for the process generated by (10). Proof. Let us consider the number λ∈(0, α0) from (20), pick ε=α0 m, and denote x(τ) = x(τ;t0−t, ψ), τ ∈ [t0−t, t0], for any ψ∈Csuch that kψk ≤ d, and t0∈R. Then, applying once again the Young inequality