Dynamics of Nonautomous Impulsive Multivalued Processes

Set-Valued and Variational Analysis https://doi.org/10.1007/s11228-023-00667-2 Dynamics of Nonautomous Impulsive Multivalued Processes Tom´ as Caraballo1 ·Jos´ e M. Uzal2 Received: 9 April 2022 / Accepted: 24 January 2023 ©The Author(s) 2023 Abstract In this paper we study the asymptotic behaviour of multivalued processes which are under the influence of impulsive action. We provide conditions to guarantee the existence of a pullback attractor and we illustrate the results with several examples. Keywords Generalized process ·Impulses ·Pullback attractors ·Multivalued process Mathematics Subject Classification (2010) 35B41 ·34A37 ·34D45 ·35R12 1 Introduction The theory of attractors for nonautonomous dynamical systems has been well-studied in the last few years (see for example [1–7]). An attractor usually gives us information about the long-term behaviour of the solutions of the system. Here we will focus on the study of the multivalued situation, that is, when we may have more than one solution for a given initial data. These dynamical systems include, for example, some reaction-diffusion equations, other models of physics or differential inclusions. Some results can be found on [1,4,7–11]. Furthermore, solutions can experience discontinuities, that is, continuous trajectories could have perturbations or changes in their state when they reach a certain set in the phase space. These changes can be interpreted as jumps or some forced corrections in order to avoid undesirable situations. The analysis of systems with impulsive perturbations has been studied during the last 30 years, see for example the monographs [12,13]. The theory of Tom´ as Caraballo [email protected] Jos´ e M. Uzal [email protected] 1Departmento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Facultad de Matem´ aticas, Universidad de Sevilla, c/Tarfia s/n, Sevilla, 41012, Spain 2Departamento de Estat´ ıstica, An´ alise Matem´ atica e Optimizaci´ on, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, R Lope G´ omez de Marzoa s/n, Santiago de Compostela, 15782, Spain T. Caraballo, J.M. Uzal impulsive dynamical systems can be traced back to the 1970’s, see [14,15]. In [16,17]it was studied the evolution of such systems and in [18,19] some results were obtained related to the continuity of the impact time map (see (2) for the definition). More recently, there have been results on the existence of different types of attractors and some properties for different types of systems, most of them in the autonomous situation, see for example [20– 24]. There are also several results regarding the stability and control theory of these systems, see for example [25–29] In this paper we focus on the study of the dynamics of systems in the nonautonomous and impulsive multivalued situation. For example, in [23,30,31] they consider autonomous multivalued dynamical systems. We study the nonautonomous situation and we give conditions in order to guarantee the existence of the pullback attractor. In order to do that, we define the notion of impulsive generalized process, we state some properties, and later we prove the existence of the pullback attractor. In Section 2we recall some facts about generalized processes and multivalued processes and we define the notion of impulsive generalized process (which consists of a generalized process G, an impulsive family of sets ˆ M and a collection of multifunctions I). Then we study some properties of these processes. In Section 3we define the notion of pullback attractor for these systems and we give conditions to guarantee its existence. In Section 4we present some applications of the results. Finally, in an appendix we include the proofs of some results of Section 3. 2 Impulsive Generalized Processes Let (X, d) be a complete metric space. The following definitions can be found, for example, in [6]. Definition 1 A generalized process G={G(t)}t∈Rin Xis a family of sets G(t) consisting of functions ϕ:[t,+∞)−→ Xsatisfying: (G1) (Existence) For each t∈Rand x∈X, there exists at least one ϕ∈G(t) such that ϕ(t) =x. (G2) (Translation) If ϕ∈G(t) and s≥0, then the map ϕ+s∈G(t +s), with ϕ+s= ϕ|[t+s,+∞). (G3) (Upper semicontinuity with respect to initial data) If {ϕn}n⊂G(s) and ϕn(s) −→ x, then there exist a subsequence {ϕnk}kof {ϕn}nand ϕ∈G(s) with ϕ(s) =xsuch that ϕnk(t) −→ ϕ(t) as k→∞for each t≥s. In this work we will assume that: (G4) (Continuity) Every map ϕ:[τ,+∞)−→ Xin G(τ) is continuous. Definition 2 We say that a generalized process G={G(t)}t∈Ris exact (or strict) if it satisfies the following condition: (G5) (Concatenation) If ϕ∈G(τ),ψ∈G(r) and ϕ(s) =ψ(s) for some s≥r≥τ,then θ∈G(τ), with θdefined as θ(t) := ϕ(t), t ∈[τ,s], ψ(t), t > s. Dynamics of Nonautomous Impulsive Multivalued Processes Definition 3 Let Gbe a generalized process. A multivalued process {U(t,s)}t≥sis a family of multivalued operators U(t,s) :P(X) −→ P(X) defined as U(t,s)D := {ϕ(t) :ϕ∈G(s), ϕ(s) ∈D}. This multivalued process satisfies: 1. U(t,t)x =xfor all t∈Rand x∈X 2. U(t,s)x ⊂U(t,τ)(U(τ,s)x)for all s≤τ≤tand x∈X. Furthermore, if we have an exact generalized process, then on the second property we have an equality, that is, U(t,s) =U(t,τ)U(τ,s). We recall for completeness the following result. Its proof can be seen, for example, in [32, Theorem 2.2], for the autonomous case. Proposition 1 Let Gbe an exact generalized process, s∈Rand {ϕn}n,ϕelements of G(s) such that ϕn(t) converges to ϕ(t) for all t>s.Thenϕn(t) converges to ϕ(t) uniformly for tin compact subsets of (s, ∞). In particular, we have the following property: ⎧ ⎪ ⎨ ⎪ ⎩ If {ϕn}n⊂G(s) and ϕn(s) −→ x, then there exist a subsequence {ϕnk}kand ϕ∈G(s) with ϕ(s) =xand ϕnk(t) −→ ϕ(t) uniformly for tin compact subsets of (s, ∞). This result, in general, is not valid for compact subsets of [s,∞). Examples can be found in [33]orin[32, Section 6.2]. Definition 4 Let Gbe a generalized process and ˆ D={D(t)}t∈Ra family of sets. We say that: •ˆ Dis positively invariant if U(t,s)D(s) ⊂D(t) for all t≥s. •ˆ Dis negatively invariant if D(t) ⊂U(t,s)D(s) for all t≥s. •ˆ Dis invariant if ˆ Dis both positively and negatively invariant. Definition 5 Let Dbe a collection of non-empty families of sets. We say that Dis inclusion-closed if for any ˆ D={D(t)}t∈R∈Dand any ˆ D1={D1(t)}t∈Rwith ∅ = D1(t) ⊂D(t) for all t∈R,then ˆ D1∈D. Any collection of non-empty family of sets which is inclusion-closed is called a universe. In applications, the two usual examples of universes are the “bounded universe” DB, which consists of all the families {D(t)}t∈Rsuch that there exists Ba bounded set with D(t) ⊂Bfor all t∈R; and the “tempered universe”, consisting of families {D(t)}t∈Rsuch that the map t−→ sup{x:x∈D(t)} grows subexponentially when t→−∞.Seeforexample[3,34]. From now on, Dwill denote an arbitrary universe. We need to give some sense to the word “attraction”. In order to do that, we consider the following definition of pullback attraction, see [3, Chapter 1] for more information. T. Caraballo, J.M. Uzal Definition 6 Let ˆ Aand ˆ Bbe two families of sets. We say that ˆ Apullback attracts ˆ Bif lim s→−∞ dH(U(t, s)B(s), A(t)) =0 for each t∈R, where dHdenotes the Hausdorff semidistance, which is given by dH(C, D) := sup c∈C inf d∈Dd(c,d). We remark that dH(C, D) =0 only implies that C⊂D. We recall here the definition of upper semicontinuity of multifunctions, as well as a known result in set-valued analysis which is useful in the study of the upper semicontinuity. The proof of this result can be found in [35] Definition 7 Let Xand Ybe two metric spaces. A multifunction F:X−→ P(Y ) is upper semicontinuous at x∈Xif for every open neighborhood Vof F(x) thereexistsan open neighborhood Uof xsuch that F(U) ⊂V. Proposition 2 Let Xand Ybe two metric spaces. A multifunction F:X−→ P(Y ) is upper semicontinuous and compact valued at x∈Xif and only if for every sequence xn−→ xand every sequence yn∈F(x n), there exist a subsequence {ynk}kand y∈F(x) such that {ynk}kconverges to y. Definition 8 A family of sets ˆ D={D(t)}t∈Rwill be called collectively closed if for any tn−→ tand xn∈D(tn)with xn−→ xwe have x∈D(t). It will be called collectively compact if for any tn−→ tand xn∈D(tn), the sequence {xn}nhas a convergent subsequence with limit in D(t). After all these definitions, we are in position to define the notion of impulsive generalized process. The goal of this paper is to study these type of processes. Definition 9 An impulsive generalized process (G,ˆ M,I) consists of a generalized process G, a collectively closed family of sets ˆ M={M(t)}t∈Rsuch that for every s∈R,x∈M(s) and ϕ∈G(s) with ϕ(s) =x, ∃ε=ε(ϕ, s) > 0 such that  r∈(0,ε) {ϕ(s +r)}∩M(s +r) =∅,(1) and collection of collectively upper semicontinuous multifunctions which are compactvalued I={It:M(t) −→ P(X)}t∈R,thatis: for every sequences tn−→ t,xn−→ xand yn∈Itn(xn), there exists a convergent subsequence {yn}nwith limit in It(x). Remark 1 Condition (1) is different from some previous papers (cf. [24,30]), which also include a condition on ϕ“backwards” in time, that is, before “touching” the set ˆ M.See also [36, Remark 2]. Let (G,ˆ M,I) be an impulsive generalized process. For each s∈Rand ϕ∈G(s),we define the impact time map by φ(ϕ,s) := inf{t>0:ϕ(s +t) ∈M(s +t)},(2) Dynamics of Nonautomous Impulsive Multivalued Processes and we denote φ(ϕ,s) =∞if ϕ(s +t) /∈M(s +t) for all t>0. Proposition 3 The map φ(ϕ,s) > 0for all s∈Rand ϕ∈G(s). Proof Fix s∈Rand ϕ∈G(s).Ifϕ(s) ∈M(s),thenφ(ϕ,s) ≥ε, with εgivenby(1). If ϕ(s) /∈M(s) and φ(ϕ,s) =0, then there exists a sequence {rn}nof positive numbers convergent to 0 such that ϕ(s +rn)∈M(s +rn).Asϕis continuous and ˆ Mis collectively closed, then ϕ(s) ∈M(s), a contradiction. Remark 2 If φ(ϕ,s) =∞,thenϕ(s +φ(ϕ,s)) ∈M(s +φ(ϕ,s)). This positive number, if it exists, is the smallest number such that ϕ(s +t) ∈M(s +t), meaning that if ϕ(r) ∈M(r) for some r>s,thens+φ(ϕ,s) ≤r. We note that this is a generalization of the impact time map in the single-valued case (see [24]), which was defined as a function from X×R. The properties of this map will help us understand better the evolution of the impulsive trajectories, which will be defined next. This definition of impulsive trajectories is a generalization from the single-valued case. Definition 10 Given s∈R,amap ˜ϕ:[s,ω) −→ X, with ω∈(s, +∞), will be called an impulsive trajectory of (G,ˆ M,I) if there exists a division of [s,ω) into a family of subintervals [s,ω) =[t0,t 1)∪[t1,t 2)∪··· with t0=s,tk<t k+1and the union could be finite or not finite. Furthermore, for each k, there exists ϕk∈G(tk)satisfying: (i) φ(ϕk,t k)=∞or φ(ϕk,t k)=tk+1−tk, (ii) ˜ϕ(t) =ϕk(t) for t∈[tk,t k+1), (iii) if φ(ϕk,t k)=∞,then ˜ϕ(tk+1)∈Itk+1(ϕk(tk+1)). The times tkwill be called jump times of ˜ϕ, the family of impulsive trajectories starting at swill be denoted by ˜ G(s), and we will also denote ˜ G={ ˜ G(s)}s∈R. From the previous definition, our first result is existence of (local) impulsive impulsive trajectories. It follows from the existence property (G1) in the definition of generalized processes (see Definition 1). Proposition 4 For each s∈Rand x∈X, there exists ˜ϕ∈˜ G(s), defined on an interval [s,ω), with ω>s, such that with ˜ϕ(s) =x. Proof By definition of impulsive trajectory and (G1), there exists ϕ0∈G(s) with ϕ0(s) = x.Ifφ(ϕ0,s) =∞,then ˜ϕ(t) =ϕ0(t) for all t≥s. On the other hand, if φ(ϕ0,s) =∞, then φ(ϕ0,s) > 0. Denote t1:= s+φ(ϕ0,s).Wehavethatϕ0(t1)∈M(t1).Takex1∈ It1(ϕ0(t1)) and ϕ1∈G(t1)with ϕ1(t1)=x1.Ifφ(ϕ1,t 1)=∞,thenwedefine ˜ϕ(t) =ϕ0(t), s ≤t<t 1, ϕ1(t), t1≤t. If φ(ϕ1,t 1)is finite, we denote t2=t1+φ(ϕ1,t 1),andthenϕ1(t2)∈M(t2).Takex2∈ It2(ϕ1(t2)) and ϕ2∈G(t2)with ϕ2(t2)=x2. We continue analogously. T. Caraballo, J.M. Uzal We introduce next a condition which will be used through the paper in order to prove the main results. Iτ(M(τ)) ∩M(τ) =∅ ∀τ∈R. (I) With this condition, we are able to prove the following useful result. Proposition 5 Let (G,ˆ M,I) be an impulsive generalized process satisfying Condition (I). Then, for each s∈R,˜ϕ∈˜ G(s) and t∈(s, ω), with the interval (s, ω) the domain of definition of the impulsive trajectory ˜ϕ, we have ˜ϕ(t) /∈M(t). From now on we will assume: Every impulsive trajectory is defined on [s,+∞).(3) From the definition of impulsive trajectories we can define a new family of multivalued maps {˜ U(t,s)}t≥s,givenby ˜ U(t,s) :P(X) −→ P(X) and defined as ˜ U(t,s)D := { ˜ϕ(t) :˜ϕ∈˜ G(s), ˜ϕ(s) ∈D}. Lemma 6 Let (G,ˆ M,I) be an impulsive generalized process. Then 1. ˜ Gsatisfies (G2) and (G5), 2. ˜ U(t,s) =˜ U(t,τ) ˜ U(τ,s) for any s≤τ≤t. The definitions of invariance and pullback attraction for ˜ Uare analogous, just replace U by ˜ U. 3 Existence of the Pullback Attractor Definition 11 Let (G,ˆ M,I) be an impulsive generalized process. We say that a family ˆ A∈Dis a pullback D-semi attractor if: (a) A(t) is compact for all t∈R, (b) ˆ Apullback attracts each ˆ D∈D. When ˆ Asatisfies (c) the family ˆ A\ˆ M={A(t) \M(t)}t∈Ris invariant we will say that ˆ Ais a pullback D-attractor. We remark that a pullback D-semi attractor may not satisfy (c). An example on the autonomous case can be found in [20]orin[30]. The following result tells us that the pullback D-attractor is unique “up to ˆ M”. Proposition 7 Let (G,ˆ M,I) be an impulsive generalized process. If ˆ Aand ˆ Bare two pullback D-attractors, then ˆ A\ˆ M=ˆ B\ˆ M. Proof Fix t∈R. We know that ˆ B∈D, which implies that ˆ B\ˆ Malso belongs to Dbecause Dis a universe. Using the invariance of ˆ B\ˆ Mwe have that dH(B(t) \M(t), A(t)) =dH(˜ U(t, s)(B(s) \M(s)), A(t)) Dynamics of Nonautomous Impulsive Multivalued Processes for any s≤t.Usingthat ˆ Ais a pullback D-attractor, taking s−→ − ∞ we have that lim s→−∞ dH(˜ U(t, s)(B(s) \M(s)), A(t)) =0 so dH(B(t) \M(t), A(t)) =0, which implies B(t) \M(t) ⊂A(t). Interchanging ˆ Aand ˆ B we get the desired result. We present the definition of impulsive pullback ω-limit and the related concepts of pullback D-asymptotically compactness and pullback D-dissipativeness. These definitions will turn out to be very important on the construction of the pullback D-semi attractors and pullback D-attractors, and they are a little bit different that the continuous case. Definition 12 Let ˆ Dbe a family of sets. The impulsive pullback ω-limit set of ˆ Dat time t∈R, denoted by ˜ω( ˆ D,t), is defined as the set of elements x∈Xsuch that there exist sn−→ − ∞ ,εn−→ 0and ˜ϕn∈˜ G(sn)with ˜ϕn(sn)∈D(sn)for each n∈Nsuch that ˜ϕn(t +εn)−→ x. The impulsive pullback ω-limit of ˆ Dis the family ˜ω( ˆ D) ={˜ω( ˆ D,t)}t∈R. Definition 13 We say that ˜ Gis pullback D-asymptotically compact if for each D∈D, t∈R,sn−→ − ∞ ,εn−→ 0and ˜ϕn∈˜ G(sn)with ˜ϕn(sn)∈D(sn), then the sequence {˜ϕn(t +εn)}nhas a convergent subsequence. Definition 14 We say that ˜ Gis pullback D-dissipative if there exists ˆ B0∈Dcollectively closed such that for all ˆ D∈D,t∈R,sn−→ − ∞ and εn−→ 0, there exists n0= n0(ˆ D,t) ∈Nsuch that if n≥n0,˜ϕ∈˜ G(sn)and ˜ϕ(sn)∈D(sn),then ˜ϕ(t+εn)∈B0(t+εn). The family ˆ B0is called pullback D-absorbing family. The main difference between these three definitions and the related ones in the continuous case is the presence of the sequence of {εn}n. 3.1 Existence of the Pullback semi Attractor We present some properties of the impulsive pullback ω-limit in combination with the previous definitions. Proposition 8 Let ˜ Gbe a pullback D-asymptotically compact impulsive generalized process, ˆ D∈Dand t∈R. Then the impulsive pullback ω-limit of ˆ D,˜ω( ˆ D), is non-empty, collectively compact and pullback attracts ˆ D. Proof The non-emptiness is trivial. First we prove the collective compactness. Take tn−→ tand yn∈˜ω( ˆ D,tn). We want to prove that the sequence {yn}nhas a convergent subsequence with limit in ˜ω( ˆ D,t). For each n∈N,wehavethatyn∈˜ω( ˆ D,tn). Then there exist sn≤t−n,|εn|<1/n and ˜ϕ∈˜ G(sn)with ˜ϕ(sn)∈D(sn)such that d( ˜ϕn(tn+εn), yn)<1/n.Asδn:= tn−t+εn−→ 0, we have that {˜ϕn(t +(tn−t+εn))}nhas a convergent subsequence by pullback Dasymptotical compactness. Thus we may assume that ˜ϕn(t +δn)=˜ϕn(tn+εn)−→ yfor some y∈X. But this implies that y∈˜ω( ˆ D,t).Furthermore, d(yn,y)≤d(yn,˜ϕn(tn+εn)) +d(˜ϕ(tn+εn), y) −→ 0, so we can say yn−→ y. T. Caraballo, J.M. Uzal Finally, we prove that ˜ω( ˆ D) pullback attracts ˆ D.If ˜ω( ˆ D) does not pullback attract ˆ D, there exist t∈R,ε>0, sn−→ − ∞ and ˜ϕn∈˜ G(sn)with ˜ϕn(sn)∈D(sn)for all n∈N such that d(˜ϕn(t), ˜ω( ˆ D,t)) ≥ε.But{˜ϕn(t)}nhas a convergent subsequence by the pullback D-asymptotical compactness, so we may assume that ˜ϕn(t) −→ xfor some x∈X.But this implies that x∈˜ω( ˆ D,t), a contradiction with d(˜ϕn(t), ˜ω( ˆ D,t)) ≥ε. Proposition 9 Let ˜ Gbe a pullback D-dissipative impulsive generalized process with ˆ B0a pullback D-absorbing family. Then for any ˆ D∈Dwe have that ˜ω( ˆ D) ⊂ˆ B0. Proof Fix t∈Rand x∈˜ω( ˆ D,t). Then there exist sn−→ − ∞ ,εn−→ 0and ˜ϕn∈˜ G(sn) with ˜ϕn(sn)∈D(sn)such that ˜ϕn(t +εn)−→ x. The definition of pullback D-dissipative implies that there exists n0∈Nsuch that if n≥n0we have that ˜ϕn(t +εn)∈B0(t +εn). As ˆ B0is collectively closed, this implies that x∈B0(t). Theorem 10 Let ˜ Gbe a pullback D-asymptotically compact impulsive generalized process and pullback D-dissipative. Then there exists a pullback D-semi attractor. Proof Take ˆ A=˜ω( ˆ B0), with ˆ B0a pullback D-absorbing family. The family ˆ Apullback D-attracts ˆ B0and ˆ A⊂ˆ B0, by Proposition 9. Then ˆ A∈Dbecause ˆ B0∈Dand Dis a universe. The family ˆ Ais collectively compact by Proposition 8, so A(t) is compact for all t∈R. We have to prove that ˆ Apullback attracts every ˆ D∈D. Fix t∈R,ˆ D∈Dand ε>0. We want to prove that there exists r≤tsuch that if s≤r and ˜ϕ∈˜ G(s) with ˜ϕ(s) ∈D(s),thend(˜ϕ(t), A(t)) < ε. We know that ˆ Apullback attracts ˆ B0, so there exists s0≤tsuch that if s≤s0and ˜ϕ∈˜ G(s) with ˜ϕ(s) ∈B0(s),thend(˜ϕ(t), A(t)) < ε. By pullback D-dissipativity, there exists s1≤s0such that if s≤s1and ˜ϕ∈˜ G(s) with ˜ϕ(s) ∈D(s),then ˜ϕ(s) ∈B0(s). Finally, take r:= s1.Ifs≤rand ˜ϕ∈˜ G(s) with ˜ϕ(s) ∈D(s), then we know that ˜ϕ(s0)∈B0(s0),so ˜ϕ|[s0,∞)∈˜ G(s0)and ˜ϕ(s0)∈B0(s0). This implies that d(˜ϕ(t), A(t)) < εbecause ˆ Apullback attracts ˆ B0. 3.2 Invariance In this subsection we find conditions to obtain the invariance of the impulsive pullback ωlimits. In particular we look for conditions to guarantee the invariance of ˆ A\ˆ Mwhen ˆ Ais a pullback D-semi attractor. First, we need a condition closely related to (G3) in Definition 1 and to Proposition 1, but a little stronger. (G3’) If τn−→ τ,ϕn∈G(τn)and ϕn(τn)−→ x, then there is a subsequence {ϕnk}kof {ϕn}nand ϕ∈G(τ) with ϕ(τ) =xand satisfying the following condition: For every {tk}kwith tk≥τnkand tk−→ t, we have ϕnk(tk)−→ ϕ(t). Dynamics of Nonautomous Impulsive Multivalued Processes Furthermore, we need to add some conditions in order to get the invariance. The first condition asks about the behavior of the trajectories near the impulsive family ˆ M. ⎧ ⎪ ⎨ ⎪ ⎩ Fix s∈R,x ∈X\M(s),{ϕn}na sequence in G(s) and ϕ∈G(s) such that ϕ(s) =xand ϕn(t) −→ ϕ(t) for each t≥s. Then lim inf n→∞ φ(ϕn,s)≤φ(ϕ,s). (NT) The second condition implies some restrictions on the jump times. There exists ξ>0suchthatφ(ϕ,s) ⩾2ξfor all s∈R and ϕ∈˜ G(s) with ϕ(s) ∈Is(M(s)).(H) Remark 3 Condition (NT) generalizes other conditions in the literature, for example the tubes conditions in [20–22] or Condition (T) in [30]. This means that if an impulsive generalized process satisfies the tubes conditions or Condition (T) in [30], then it satisfies Condition (NT). Remark 4 Condition (H) implies (3), that is, all impulsive trajectories are defined until +∞. It also implies that if ˜ϕ∈˜ G(s) and t1<t 2are two different jump times of ˜ϕ,then t2−t1≥2ξ. Theorem 11 Let ˜ Gbe a pullback D-asymptotically compact impulsive generalized process satisfying Conditions (G3’), (H),(I)and (NT).Then ˜ω( ˆ D) \ˆ Mis negatively invariant for any ˆ D∈D. The proof of this result is shown in the appendix. The following result tells us that, in the particular case that ˜ω( ˆ D) is a pullback D-semi attractor, then negative invariance implies positive invariance. Theorem 12 Let ˜ Gbe a pullback D-asymptotically compact impulsive generalized process and pullback D-dissipative, ˆ Aa pullback D-semi attractor such that ˆ A\ˆ Mis negatively invariant and ˜ Gsatisfies Condition (I).Then ˆ A\ˆ Mis also positively invariant. Proof Let t>s. The negative invariance of ˆ A\ˆ Mimplies that B(s) ⊂˜ U(s,s −n)B(s −n) for any n∈N, with B(r) =A(r)\M(r). This implies that ˜ U(t,s)B(s) ⊂˜ U(t,s−n)B(s − n), so we can say dH˜ U(t,s)B(s),A(t)≤dH˜ U(t,s −n)B(s −n), A(t). We have that ˆ A\ˆ M∈D, which implies that lim n→∞ dH˜ U(t,s −n)B(s −n), A(t)=0. As a consequence we can say that dH˜ U(t,s)(A(s) \M(s)),A(t) =0, T. Caraballo, J.M. Uzal We claim that ψ(r) /∈M(r) for r∈[s−ε/2,s +ξ].Ifψ(r) ∈M(r) for some r∈ [s−ε/2,s +ξ],thenφ(ψ,s −ε/2)≤r−(s −ε/2).But ˜ϕnhas no jump times on [s−ε/2,s +5ξ/4],soφ(ψn,s −ε/2)≥(s +5ξ/4)−(s −ε/2). Then Condition (NT) would imply that (s +5ξ/4)−(s −ε/2)≤φ(ψ,s −ε/2)≤r−(s −ε/2)<(s+ξ)−(s −ε/2), a contradiction. We take ˜α∈˜ G(s +ξ) with ˜α(s +ξ) =ψ(s +ξ) and define ˜ϕ(r) =ψ(r), s ≤r≤s+ξ, ˜α(r), s +ξ≤r. Then ˜ϕ∈˜ G(s),˜ϕ(s) =ψ(s) =xand finally ˜ϕn(t) =ψn(t) −→ ψ(t) =˜ϕ(t),so ˜ϕ(t) ∈˜ω( ˆ D,t) \M(t). CASE 2. Up to a subsequence (denoted the same), there exists ε∈(0,ξ/2)such that τn>s+ε. We know that τnis the only jump time of ˜ϕnin [s−ξ/2,s+5ξ/4]because of Condition (H), and we can assume that τn>s+εfor all n∈N. For each n∈Nthere exist ψn∈G(s−ξ/2) and θn∈G(τn)such that ˜ϕn(r) =ψn(r), s −ξ/2≤r<τ n, θn(r), τn≤r≤s+5ξ/4. We may assume that ˜ϕn(s −ξ/2)−→ yby pullback D-asymptotical compactness, and by definition of generalized process for ψnwe may assume that there exist a subsequence (denoted the same) and ψ∈G(s −ξ/2)such that ψn(r) −→ ψ(r) for r≥s−ξ/2. Furthermore we have that ˜ϕn(s +εn)=ψn(s +εn), which converges to ψ(s) by Proposition 1, so x=ψ(s). This implies that ˜ϕn(s) also converges to x. Subcase 1. Up to a subsequence (denoted the same), there exists δ>0suchthatτn> t+δ. We claim that ψ(r) /∈M(u) for r∈[s−ξ/2,t +δ).Ifψ(r) ∈M(r) for some r∈ [s−ξ/2,t +δ),thenφ(ψ,s −ξ/2)≤r−(s −ξ/2). But we know that τn>t+δ,so φ(ψn,s−ξ/2)≥(t +δ) −(s −ξ/2)and Condition (NT) implies that (t +δ) −(s −ξ/2)≤φ(ψ,s −ξ/2)≤r−(s −ξ/2)<(t+δ) −(s −ξ/2), a contradiction. We take ˜α∈˜ G(t +δ/2)with ˜α(t +δ/2)=ψ(t +δ/2)and define ˜ϕ(r) =ψ(r), s ≤r≤t+δ/2, ˜α(r), t +δ/2≤r. Then ˜ϕ∈˜ G(s),˜ϕ(s) =ψ(s) =xand ˜ϕn(t) =ψn(t), which converges to ψ(t) =˜ϕ(t),so ˜ϕ(t) ∈˜ω( ˆ D,t) \M(t). Subcase 2. Up to a subsequence (denoted the same), there exists δ>0suchthatτn< t−δ. As τn∈(s +ε, t −δ), we may assume that τnconverges to ¯τ∈[s+ε, t −δ].We have ψn(τn)∈M(τn),soψ(¯τ) ∈M(¯τ) by Proposition 1. We also have that θn(τn)= ˜ϕ(τn)∈Iτn(ψn(τn)). By the collective upper semicontinuity of I, there exist a subsequence Dynamics of Nonautomous Impulsive Multivalued Processes of {˜ϕn(τn)}n, still denoted the same, and z∈I¯τ(ψ( ¯τ))such that θn(τn)=˜ϕn(τn)converges to z,soz∈˜ω( ˆ D, ¯τ). Once again, we claim that ψ(r) /∈M(r) for r∈(s −ξ/2,¯τ).Ifψ(r) ∈M(r) for some r∈(s −ξ/2,¯τ),thenφ(ψ,s −ξ/2)≤r−(s −ξ/2). But we know that φ(ϕn,s−ξ/2)= τn−(s −ξ/2)and Condition (NT) implies that ¯τ−(s −ξ/2)≤φ(ψ,s −ξ/2)≤r−(s −ξ/2)< ¯τ−(s −ξ/2), a contradiction. We can assume by Condition (G3’) that there exists θ∈G(¯τ)such that θn(un)converges to θ(u) for any sequence {un}nwith un≥τnand unconverging to u.Furthermore,θ(r) /∈ M(r) for r∈[τn,s+5ξ/4]because θ(¯τ) ∈I¯τ(M( ¯τ)) and Condition (H) applies. We take ˜α∈˜ G(s +ξ) with ˜α(s +ξ) =θ(s +ξ) and define ˜ϕ(r) =⎧ ⎪ ⎨ ⎪ ⎩ ψ(r), s ≤r<¯τ, θ(r), ¯τ≤r<s+ξ, ˜α(r), s +ξ≤r. We have that ˜ϕ∈˜ G(s),˜ϕ(s) =xand ˜ϕn(t) =θn(t), which converges to θ(t) =˜ϕ(t),so ˜ϕ(t) ∈˜ω( ˆ D,t) \M(t). Subcase 3. τnconverges to t. We have that ψn(τn)∈M(τn),soψ(t) ∈M(t) using Proposition 1 and that ˆ Mis collectively closed. We also have that θn(τn)=˜ϕn(τn)∈Iτn(ψn(τn)). By the collective upper semicontinuity of I, there exist a subsequence of {˜ϕn(τn)}n, still denoted the same, and z∈It(ψ(t)) such that θn(τn)=˜ϕn(τn)converges to z,soz∈˜ω( ˆ D,t). We claim that ψ(r) /∈M(r) for r∈[s−ξ/2,t).Ifψ(r) ∈M(r) for some r∈[s− ξ/2,t),thenφ(ψ,s−ξ/2)≤r−(s −ξ/2). But we know that τnis the only jump time of ˜ϕn in [s−ξ/2,s+5ξ/4],soφ(ψn,s−ξ/2)=τn−(s −ξ/2), and Condition (NT) implies that t−(s −ξ/2)≤φ(ψ,s −ξ/2)≤r−(s −ξ/2)<t−(s −ξ/2), a contradiction. We take ˜α∈˜ G(t) with ˜α(t) =zand we define ˜ϕ(r) =ψ(r), s ≤r<t, ˜α(r), t ≤r. Then ˜ϕ∈˜ G(s),˜ϕ(s) =xand ˜ϕn(τn)=θn(τn)converges to z=˜α(t) =˜ϕ(t),so ˜ϕ(t) ∈ ˜ω( ˆ D,t) \M(t). CASE 3. τnconverges to s In this case τnis the only jump time of ˜ϕnin [s−ξ/2,s +5ξ/4]. Then there exist ψn∈ G(s −ξ/2)and θn∈G(τn)such that ˜ϕn(r) =ψn(r), s −ξ/2≤r<τ n, θn(r), τn≤r≤s+5ξ/4. By pullback D-asymptotical compactness we may assume ˜ϕn(s −ξ/2)converges to y,and by definition of generalized process we may assume that there exists ψ∈G(s −ξ/2)such that ψn(r) −→ ψ(r) for r≥s−ξ/2. Subcase 1. Up to a subsequence, still denoted the same, s+εn<τ n T. Caraballo, J.M. Uzal We have that ˜ϕn(s +εn)=ψn(s +εn), which converges to ψ(s) by Proposition 1, so x=ψ(s).Furthermore,ψn(τn)∈M(τn),soψ(s) ∈M(s) by Proposition 1 and the collective closedness of ˆ M. This implies that x∈M(s), a contradiction. As a consequence, this case cannot happen. Subcase 2. Up to a subsequence, still denoted the same, τn≤s+εn. We have ψn(τn)∈M(τn),soψ(s) ∈M(s) by Proposition 1 and the fact that ˆ Mis collectively closed. Furthermore, ˜ϕn(τn)=θn(τn)∈Iτn(ψn(τn)). Using the collective upper semicontinuity of I, there exist a subsequence {˜ϕn(τn)}n, still denoted the same, and z∈Is(ψ(s)) such that θn(τn)=˜ϕ(τn)converges to z,soz∈˜ω( ˆ D,s). By Condition (G3’) we assume that there exists θ∈G(¯τ) such that θn(rn)converges to θ(r)for any sequence {rn}nwith rn≥τnand rnconverging to r.Wehavethatθ(u) /∈M(u) for u∈[s, s +ξ]because of Condition (H). We take ˜α∈˜ G(s +ξ)with ˜α(s +ξ) =θ(s+ξ) and define ˜ϕ(r) =θ(r), s ≤r≤s+ξ, ˜α(r), s +ξ≤r. Then ˜ϕ∈˜ G(s),˜ϕn(s +εn)=θn(s +εn)converges to θ(s) =z,sox=z=˜ϕ(s). Finally ˜ϕn(t) =θn(t) converges to θ(t) =˜ϕ(t),so ˜ϕ(t) ∈˜ω( ˆ D,t) \M(t). Acknowledgements The first author has been partially supported by the Spanish Ministerio de Ciencia, Innovaci´ on y Universidades (MCIU), Agencia Estatal de Investigaci´ on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andaluc´ ıa (Consejer´ ıa de Econom´ ıa y Conocimiento) and FEDER under projects US-1254251 and P18-FR-4509. The second author was partially supported by grant BES-2017-082334, Agencia Estatal de Investigaci´ on (AEI). Funding Funding for open access publishing: Universidad de Sevilla/CBUA Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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