Attractivity for Neutral Functional Differential Equations

DISCRETE AND CONTINUOUS doi:xxxxxxx DYNAMICAL SYSTEMS Volume xx, Number 0x, xxxx xxxx pp. xxxx–xxxx ATTRACTIVITY FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS Tom´ as Caraballoaand G´ abor Kissb aDpto. Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla Campus Reina Mercedes, Apdo. de Correos 1160 41080 Sevilla, Spain and bDepartment of Mathematical Sciences, University of Durham, UK DH1 3LE. (Communicated by Associate Editor) Abstract. We study the long term dynamics of non-autonomous functional differential equations. Namely, we establish existence results on pullback attractors for non-linear neutral functional differential equations with time varying delays. The two main results differ in smoothness properties of delay functions. 1. Introduction. From an application point of view, differential equations relate state variables and their derivative(s); however, it is not always possible to formulate such a relation. In some cases, we can only establish connection between the rate of change of a difference operator and the state variables. In those situations, the difference operator uses the state variables at different time instants, and neutral functional differential equations (NFDE) are the appropriate mathematical models. Neutral functional differential equations are of form d dtD(t, xt) = f(t, xt) (1) where f:R×C→Rnis continuous and maps bounded sets into bounded sets, where, for r > 0, C=C([−r, 0]; Rn) denotes the Banach space of continuous functions with the sup-norm. Furthermore, the difference operator D:R×C→Rn is continuous together with its first and second Fr´echet derivatives with respect to its second variable; and the first and the second derivatives of Dwith respect to the second variable are continuous at zero. Lastly, for a given continuous function x(·) : R→Rnand t∈R, we denote by xt(·) an element in Cgiven by xt(θ) = x(t+θ), θ ∈[−r, 0]. 2010 Mathematics Subject Classification. Primary: 37C30; Secondary: 34D45, 34K20. Key words and phrases. Non-autonomous dynamics; Pullback Attractor; Neutral equation. T. Caraballo has been partly supported by FEDER and Ministerio de Econom´ıa y Competitividad (Spain) under grant MTM2011-22411. G. Kiss has been partially supported by the EPSRC Mathematics Platform grant EP/I019111/1 and by the grant MTM2011-24766 of MICINN (Spain). 1 2 TOM´ AS CARABALLO AND G´ ABOR KISS When x(·) is a solution of (1), then xt(·) is said to be the solution segment at time t. When Dφ =φ(0) for all φ∈Cthen (1) becomes ˙x(t) = f(t, xt),(2) a delay differential equation. The knowledge about delay differential equations, in particular those of the form ˙x(t) = f(t, x(t), x(t−1)), has advanced substantially during the last half of a century, see [16,21]. When the memory function on the right-hand side is more complicated because of the presence of delay distribution, our knowledge is not so advanced. Recently, stability results, existence results on periodic solutions to equations with distributed delays have been reported in [9,10] and [11], respectively. The study of non-autonomous attracting sets of dynamical systems was initiated in [19,20]. The notion of pullback attractors for non-autonomous dynamical systems was introduced [2,12]. Furthermore, the asymptotic behaviour of non-autonomous ordinary differential equations is studied in [17]. Ideas for non-autonomous functional differential equations are presented in [1,3,4,5]. Because of various possible reasons, the development of ideas for the family of equations complementary to (2) is much slower. Thus, every fact which might even be valid on a relatively small subset of the phase space Cis of great value. For instance, [22] establishes existence of periodic solutions on BC, the Banach space (with the supremum norm) of bounded continuous functions from R to Rn. We adumbrate that Theorem 3.2 of the present paper might suffer from similar limitation since some of our assumptions are fulfilled when solutions are bounded, although they can be satisfied also in other situations. The present work focuses on difference operators of form Dφ =φ(0) −g(φ(−σ)) where function gmaps Rninto itself. In other words, the family of NFDEs that we consider here is d dt [x(t)−g(x(t−σ))] = F0(t, x(t)) + m X i=1 Fi(x(t−ρi(t))).(3) Here σ > 0, and ρi:R→[0, h] are functions representing the variable delays of the model; additional restrictions are imposed on them in Sections 3 (3.1) and (3.2), as well as on the terms Fi:Rn→Rn,i= 0, . . . , m. The inner product in Rn is denoted by h·,·i. Throughout the paper, 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(t)∈[0, Mδ] for some Mδ∈R+. The aim of this work is to derive results on the asymptotic behaviour of solutions to NFDEs. Namely, it intends to extend the findings of [1] and [5] on the existence of pullback attractors for delay differential equations to NFDEs of form (3). The rest of the paper is organised as follows. Section 2summarises the necessary theory of processes. In section 3, we state and prove our main results. ATTRACTIVITY FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS 3 2. Preliminaries. To start with, recall that, for r= max{σ, h}>0, we denote by C=C([−r, 0]; Rn) the Banach space of continuous functions φ: [−r, 0] →Rnwith the usual kφk= sups∈[−r,0] |φ(s)|supremum norm. The basic theory of neutral functional differential equations (see [8]) implies, under standard assumptions, the existence of the unique solution of (3) on [s−r, ∞). That is, if an initial function φ∈ Cis prescribed at the initial time s∈Rthen there is an x(·;s, φ) which satisfies (3) and, in addition, the initial condition xs(·) = φ, in other words, xs(θ)=x(s+θ)=φ(θ) for all θ∈[−r, 0]. Now, we present the necessary background on the theory of processes. For more details on the topic we refer to [7]. The unique solution of the initial value problem associated to (2) 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⊂Xbounded, there exists TD(t)>0 such that for all τ≥TD(t) U(t, t −τ)D⊂B(t). The following results (see [13,18]) shows that the existence of a family of compact absorbing sets implies the existence of a pullback attractor. Theorem 2.3. 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. Theorem 2.4. 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 (3). We emphasize that it is possible to consider a more general definition of pullback attractor which attracts family of sets in a universe instead of only bounded sets (see [2],[15] for a detailed analysis of this theory). However, the present concept is enough for our interests. 4 TOM´ AS CARABALLO AND G´ ABOR KISS 3. Main results. Before formulating our main results, we include a lemma from [5] which will be useful in subsequent computations. Lemma 3.1. Let 0< ξ < 1. Then |u|2≤1 1−ξ|u−v|2+1 ξ|v|2 for any u, v ∈Rn. 3.1. Continuously differentiable delay functions ρi.In this section we impose conditions on the non-linearities of (3) as follows (A1): F0:Rn+1 →Rnis continuous and there exist α0>0, β0≥0 such that for any φ∈C, hF0(t, φ(0)), φ(0) −g(φ(−σ))i≤−α0|φ(0) −g(φ(−σ))|2+β0, t ∈R. (A2): Fi, i = 1, . . . , m is sublinear, i.e., there exist ki>0, i = 1, . . . , m, such that |Fi(x)|2≤k2 i(1 + |x|2), x ∈Rn. Remark 1. As it was already mentioned in the Introduction, (A1) might impose relatively strong limitations on the solutions of (3) and on function g. Namely, if g is continuous and the solutions of (3) are bounded then (A1) is satisfied. However, these properties are not necessary for satisfying (A1). Theorem 3.2. Assume that assumptions (A1) and (A2) are satisfied. Furthermore, suppose that each delay function ρi(·)is continuously differentiable with ρ0 i(t)≤ ρi∗<1for all t∈R. Then, if m2k2 i< α2 0(1 −ρi∗), for all i= 1, . . . , m then there exists a family of bounded absorbing sets, {B(t)}t∈R, and consequently, there exists a pullback attractor for this problem. Proof. Let λ > 0 be a constant to be determined later on, and denote (for the sake of simplicity) ε=α0 m. Denote x(τ) = x(τ;t0−t, ψ), τ ∈[t0−t, t0], for any ψ∈Csuch that kψk ≤ d. Then, by a suitable application of the Young inequality (2ab ≤εa2+ε−1b2), it follows that ATTRACTIVITY FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS 5 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 β0+eλτ |x(τ)−g(x(τ−σ))|2 m X i=1 ε +eλτ m X i=1 ε−1|Fi(x(τ−ρi(τ)))|2 ≤(λ−α0)eλτ |x(τ)−g(x(τ−σ))|2 + 2eλτ β0+eλτ ε−1 m X i=1 k2 i +eλτ ε−1 m X i=1 k2 i|x(τ−ρi(τ))|2. Using Lemma 1 with u:= x(τ)−g(x(τ−σ)) and v:= g(x(τ−σ)) and assumption (Ag), we obtain |x(τ)−g(x(τ−σ))|2≤|x(τ)|2 1−c+|g(x(τ−σ))|2 c ≤|x(τ)|2 1−c+c|x(τ)|2+δ(τ) c ≤|x(τ)|2 1−c+c|x(τ)|2+Mδ c. Thus we derive that d dτ eλτ |x(τ)−g(x(τ−σ))|2≤(λ−α0)eλτ |x(τ)|2 1−c+c|x(τ)|2 + 2β0+ε−1 m X i=1 k2 i+ (λ−α0)Mδ c!eλτ +eλτ ε−1 m X i=1 k2 i|x(τ−ρi(τ))|2. 6 TOM´ AS CARABALLO AND G´ ABOR KISS 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(4) ≤c+1 1−c(λ−α0)Zτ t0−t eλs|x(s)|2ds +(λ−α0)Mδ c+ 2β0+ε−1Pm i=1 k2 i λheλτ −eλ(t0−t)i +ε−1 m X i=1 k2 iZτ t0−t eλs|x(s−ρi(s))|2ds. (5) Now we compute the integrals for the addends in the last sum. Zτ t0−t eλs|x(s−ρi(s))|2ds ≤1 1−ρi∗Zτ t0−t−h eλu+λh|x(u)|2du ≤eλh 1−ρi∗Zt0−t t0−t−h eλu|x(u)|2du +Zτ t0−t eλu|x(u)|2du ≤eλh 1−ρi∗Zt0−t t0−t−h eλu|ψ(u)|2du +Zτ t0−t eλu|x(u)|2du ≤d2eλh λ(1 −ρi∗)heλ(t0−t)−eλ(t0−t−h)i +eλh 1−ρi∗Zτ t0−t eλu|x(u)|2du. It follows that eλτ |x(τ)−g(x(τ−σ))|2 ≤eλ(t0−t)d2 1−c+d2c+Mδ c +(λ−α0)Mδ c+ 2β0+ε−1Pm i=1 k2 i λheλτ −eλ(t0−t)i +d2eλh λheλ(t0−t)−eλ(t0−t−h)im X i=1 k2 iε−1 1−ρi∗ +(λ−α0)c+1 1−c +eλhε−1 m X i=1 k2 i 1−ρi∗!Zτ t0−t eλs|x(s)|2ds. ATTRACTIVITY FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS 7 Now, observe that ε−1 m X i=1 k2 i 1−ρi∗ =m α0 m X i=1 k2 i 1−ρi∗ =1 mα0 m X i=1 m2k2 i 1−ρi∗ <1 mα0 mα2 0 =α0. Consequently, since 0 < c < 1 we can choose a positive, but small enough, λsuch that (λ−α0)c+1 1−c+eλhε−1 m X i=1 k2 i 1−ρi∗ <0. This implies that |x(τ)−g(x(τ−σ))|2≤"d2 1−c+d2c+Mδ c+d2eλhε−1 λ m X i=1 k2 i 1−ρi∗#eλ(t0−t−τ) +(λ−α0)Mδ c+ 2β0+ε−1Pm i=1 k2 i λ. Setting τ=t0+θ, θ ∈[−r, 0] we obtain |x(t0+θ)−g(x(t0+θ−σ))|2 ≤"d2 1−c+d2c+Mδ c+d2eλhε−1 λ m X i=1 k2 i 1−ρi∗#e−λ(t+θ) +(λ−α0)Mδ c+ 2β0+ε−1Pm i=1 k2 i λ. Since (1 −c)|u|2−1−c c|v|2≤ |u−v|2, |x(t0+θ)|2≤1 1−c(" d2 1−c+d2c+Mδ c+d2eλhε−1 λ m X i=1 k2 i 1−ρ∗ i#e−λ(t+θ) +(λ−α0)Mδ c+ 2β0+ε−1Pm i=1 k2 i λ)+c|x(t0+θ)|2+Mδ c. And so |x(t0+θ)|2≤1 (1 −c)2(" d2 1−c+d2c+Mδ c+d2eλhε−1 λ m X i=1 k2 i 1−ρ∗ i#e−λ(t+θ) +(λ−α0)M c+ 2β0+ε−1Pm i=1 k2 i λ)+Mδ c(1 −c). Thus 8 TOM´ AS CARABALLO AND G´ ABOR KISS sup θ∈[−r,0] |x(t0+θ))|2≤1 (1 −c)2(" d2 1−c+d2c+Mδ c+d2eλhε−1 λ m X i=1 k2 i 1−ρi∗#e−λt+λr +(λ−α0)Mδ c+ 2β0+ε−1Pm i=1 k2 i λ)+Mδ c(1 −c) ≤1 + 1 (1 −c)2 (λ−α0)Mδ c+ 2β0+ε−1Pm i=1 k2 i λ+Mδ c(1 −c) provided that t≥TD=λ−1log 1 (1 −c)2(" d2 1−c+d2c+Mδ c+d2eλhε−1 λ m X i=1 k2 i 1−ρi∗#eλr). Consequently, the family of bounded sets {B(t)}t∈Rgiven by B(t) := B, for all t∈R, where Bdenotes the ball in C([−r, 0]; Rn) centred at zero with radius R= 1 + 1 (1−c)2 (λ−α0)Mδ c+2β0+ε−1Pm i=1 k2 i λ+Mδ c(1−c), is absorbing. On the other hand, as the associated process maps bounded sets of C([−r, 0]; Rn) into bounded sets, then Theorem 2.4 (see also Theorem 4.1 in [4]) ensures the existence of the pullback attractor. 3.2. Measurable delay functions. In the previous section, the differentiability of the delay functions played an important role. Now we prove a similar result on the existence of pullback attractor when the aforementioned differentiability condition is relaxed. Our analysis will be carried out assuming that the delay functions ρi(·) are only measurable. But we would like to point out that there exists another technique which can be used when the variable delays are continuous, the so-called Razumikhin method (see, for instance [14]). This will be analysed in a subsequent paper. 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: (A1’): F0:Rn+1 →Rnis continuous and there exist α0>0,and a non-negative measurable function β(·) such that hF0(t, φ(0)), φ(0) −g(φ(−σ))i ≤ −α0|φ(0) −g(φ(−σ))|2+β(t), t ∈R, x ∈Rn.(6) As for Fi, we assume Lipschitz continuity, i.e., (A2’): There exists Li>0, i = 1, . . . , m such that for any x, y ∈Rn |Fi(x)−Fi(y)| ≤ Li|x−y|, and Fi(0) = 0. Furthermore, we shall use Lemma 3.1 of [6] formulated as follows. Lemma 3.3. Let y: [t0−t, ∞)→R+be a function such that, there are constants γ > 0,κ, κ0>0with 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(7) ATTRACTIVITY FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS 9 where µis the positive solution of κ0 γ−µeµr = 1. Now we can establish our main result in this section. Theorem 3.4. Assume that assumptions (A1’) and (A2’) are satisfied and that there exists λ∈(0, α0)such that Zt −∞ eλsβ(s)ds < +∞,∀t∈R.(8) Then, if ρi, i = 1, . . . , m, is measurable, and λ(1 −c)2α0≥2m m X i=1 L2 i, 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, α0) from (8), pick ε=2Pm i=1 L2 i λ(1 −c)2, 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 delay terms below, we obtain 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 ≤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)−g(x(τ−σ))|2−eλ(t0−t)|x(t0−t)−g(x(t0−t−σ))|2 ≤2Zτ t0−t eλsβ(s)ds +ε−1 m X i=1 L2 iZτ t0−t eλs|x(s−ρi(s))|2ds. (9) The integrand in the last 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