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.
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Acknowledgements
The first author has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andalucía (Consejería de Economía 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 Investigación (AEI).
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Appendix: Proofs of Theorems 11 and 14
Appendix: Proofs of Theorems 11 and 14
Theorem 1
Let \(\tilde {{\mathscr{G}}}\) be a pullback \(\mathfrak {D}\)-asymptotically compact impulsive generalized process satisfying Conditions (G3’), (H), (I) and (NT). Then \(\tilde {\omega }(\hat {D})\setminus \hat {M}\) is negatively invariant for any \(\hat {D}\in \mathfrak {D}\).
Proof
We restrict ourselves to the case where t > s and t − s ∈ (0, ξ]. By recursion the general case follows easily. Take \(x\in \tilde {\omega }(\hat {D},t)\setminus M(t)\). We want to prove that there exists \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\) with \(\tilde {\varphi }(s)\in \tilde {\omega }(\hat {D},s)\), \(\tilde {\varphi }(s)\notin M(s)\) and \(\tilde {\varphi }(t)=x\).
As \(x\in \tilde {\omega }(\hat {D},t)\), there exist \(s_{n}\longrightarrow -\infty \), εn→0 and \(\tilde {\varphi }_{n}\in \tilde {{\mathscr{G}}}(s_{n})\) with \(\tilde {\varphi }_{n}(s_{n})\in D(s_{n})\) such that \(\tilde {\varphi }_{n}(t+\varepsilon _{n})\longrightarrow x\). Each impulsive trajectory \(\tilde {\varphi }_{n}\), which is defined on \([s_{n},+\infty )\), has Nn ≥ 0 jump times. We consider τn the last jump time on the interval [sn, t + ξ/4]. If there are no jump times in that interval, we take τn = sn. We will split the proof into three different cases.
- Case 1.:
-
Up to a subsequence (denoted the same), there exists ε ∈ (0, ξ/2) such that τn < s − ε.
We have that there exist \(\psi _{n}\in {\mathscr{G}}(s-\varepsilon /2)\) such that \(\tilde {\varphi }_{n}(r)=\psi _{n}(r)\) for r ∈ [s − ε/2, t + ξ/4], because τn < s − ε and τn was the last jump time of the trajectory \(\tilde {\varphi }_{n}\) in [sn, t + ξ/4].
The sequence \(\{\tilde {\varphi }_{n}(s-\varepsilon /2)\}_{n}\) has a convergent subsequence by pullback \(\mathfrak {D}\)-asymptotical compactness, so we can assume \(\tilde {\varphi }_{n}(s-\varepsilon /2)\longrightarrow y\), or equivalently ψn(s − ε/2)→y. By the definition of generalized process, in particular (G3), there exist a subsequence (still denoted the same) and \(\psi \in {\mathscr{G}}(s-\varepsilon /2)\) such that ψ(s − ε/2) = y and ψn(r)→ψ(r) for each r ≥ s − ε/2. We claim that ψ(r)∉M(r) for r ∈ [s − ε/2, t]. Suppose that ψ(r) ∈ M(r) for some r ∈ [s − ε/2, t]. This implies that ϕ(ψ, s − ε/2) ≤ t − (s − ε/2). But \(\tilde {\varphi }_{n}\) has no jump times on [s − ε/2, t + ξ/4], so (t + ξ/4) − (s − ε/2) ≤ ϕ(ψn, s − ε/2). Then Condition (NT) would imply that
which is a contradiction.
We have that \(\tilde {\varphi }_{n}(t+\varepsilon _{n})=\psi _{n}(t+\varepsilon _{n})\) for n sufficiently large, so x = ψ(t). Consider \(\tilde {\theta }\in \tilde {{\mathscr{G}}}(t)\) such that \(\tilde {\theta }(t)=x\) and define \(\tilde {\psi }\in \tilde {{\mathscr{G}}}(s)\) as
We have that \(\tilde {\varphi }(s)=\psi (s)\notin M(s)\) and \(\tilde {\varphi }(s)\in \tilde {\omega }(\hat {D},s)\) because \(\tilde {\varphi }_{n}(s)=\psi _{n}(s)\) and ψn(s)→ψ(s). This implies that \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\), \(\tilde {\varphi }(s)\in \tilde {\omega }(\hat {D},s)\setminus M(s)\) and \(\tilde {\varphi }(t)=x\).
- Case 2.:
-
Up to a subsequence (denoted the same), there exists ε ∈ (0, ξ/2) such that s + ε < τn.
We have that τn ∈ (s + ε, t + ξ/4], so we may assume that \(\tau _{n}\longrightarrow \bar {\tau }\in [s+\varepsilon ,t+\xi /4]\). The sequence \(\{\tilde {\varphi }_{n}(t-3\xi /2)\}_{n}\) has a convergent subsequence by pullback \(\mathfrak {D}\)-asymptotical compactness, so we may assume \(\tilde {\varphi }_{n}(t-3\xi /2)\longrightarrow v\). Note that t − 3ξ/2 < s. We have that τn is the only jump time in [t − 3ξ/2, t + ξ/4], because \(\lvert (t+\xi /4) - (t-3\xi /2) \rvert = 7\xi /4<2\xi \) and Condition (H) applies (see Remark 4). This implies that there exist \(\psi _{n}\in {\mathscr{G}}(t-3\xi /2)\) and \(\theta _{n}\in {\mathscr{G}}(\tau _{n})\) such that
By definition of generalized process, there exist a subsequence (still denoted the same) and \(\psi \in {\mathscr{G}}(t-3\xi /2)\) such that ψ(t − 3ξ/2) = v and ψn(r)→ψ(r) for r ≥ t − 3ξ/2. This implies that \(\psi _{n}(\tau _{n})\longrightarrow \psi (\bar {\tau })\) by Proposition 1. The fact that \(\hat {M}\) is collectively closed implies that \(\psi (\bar {\tau })\in M(\bar {\tau })\), because ψn(τn) ∈ M(τn). We also have that \(\theta _{n}(\tau _{n})=\tilde {\varphi }_{n}(\tau _{n})\in I_{\tau _{n}}(\psi _{n}(\tau _{n}))\). By the collective upper semicontinuity of I, there exist a subsequence of \(\{\tilde {\varphi }_{n}(\tau _{n})\}_{n}\) (denoted the same) and \(y\in I_{\bar {\tau }}(\psi (\bar {\tau }))\) such that \(\theta _{n}(\tau _{n})=\tilde {\varphi }_{n}(\tau _{n})\) converges to y. In particular this implies that \(y\in \tilde {\omega }(\hat {D},\bar {\tau })\), because \(\tilde {\varphi }_{n}(\tau _{n})=\tilde {\varphi }_{n}(\bar {\tau }+(\tau _{n}-\bar {\tau }))\).
Using Condition (G3’), there exist a subsequence (denoted the same) and \(\theta \in {\mathscr{G}}(\bar {\tau })\) such that 𝜃n(rn) converges to 𝜃(r) for any sequence {rn}n with rn ≥ τn and rn converging to r. In particular, 𝜃n(τn) converges to \(\theta (\bar {\tau })\), so \(\theta (\bar {\tau })=y\).
We claim that ψ(r)∉M(r) for \(r\in (t-3\xi /2,\bar {\tau })\). If there exists \(r\in [t-3\xi /2,\bar {\tau })\) such that ψ(r) ∈ M(r), then
But τn was the only jump time of \(\tilde {\varphi }_{n}\) on [t − 3ξ/2, t + ξ/4], so ϕ(ψn, t − 3ξ/2) = τn − (t − 3ξ/2). Finally Condition (NT) implies
a contradiction.
First, take \(z=\psi (s)\in \tilde {\omega }(\hat {D},s)\) because \(\tilde {\varphi }_{n}(s)=\psi _{n}(s)\) and ψn(s) converges to ψ(s). Then z∉M(s) by the previous claim. Consider \(\tilde {\alpha }\in \tilde {{\mathscr{G}}}(t+\xi /4)\) with \(\tilde {\alpha }(t+\xi /4)=\theta (t+\xi /4)\) and define
Then \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\) with \(\tilde {\varphi }(s)\in \tilde {\omega }(\hat {D},s)\setminus M(s)\). We want to prove that \(\tilde {\varphi }(t)=x\).
- Subcase 1.:
-
For a subsequence (denoted the same), τn ≤ t + εn.
This implies that \(\bar {\tau }\leq t\). Then \(\tilde {\varphi }_{n}(t+\varepsilon _{n})=\theta _{n}(t+\varepsilon _{n})\longrightarrow \theta (t)\), so \(x=\theta (t)=\tilde {\varphi }(t)\).
- Subcase 2.:
-
For a subsequence (denoted the same), t + εn < τn.
This implies that \(t\leq \bar {\tau }\). On the one hand, if \(t=\bar {\tau }\), then \(\tilde {\varphi }_{n}(t+\varepsilon _{n})=\psi _{n}(t+\varepsilon _{n})\), which converges to \(\psi (t)=\psi (\bar {\tau })\in M(\bar {\tau })\) by Proposition 1, so \(x=\psi (\bar {\tau })\in M(t)\), a contradiction. This implies that t cannot be equal to \(\bar {\tau }\) in this Subcase. On the other hand, if \(t<\bar {\tau }\), then \(\tilde {\varphi }_{n}(t+\varepsilon _{n})=\psi _{n}(t+\varepsilon _{n})\), which converges to \(\psi (t)=\tilde {\varphi }(t)\), so \(x=\tilde {\varphi }(t)\).
- Case 3.:
-
τn converges to s.
We have that τn is the only jump time of \(\tilde {\varphi }_{n}\) in (t − 3ξ/2, t + ξ/4) for n sufficiently large. Then there exist \(\psi _{n}\in {\mathscr{G}}(t-3\xi /2)\) and \(\theta _{n}\in {\mathscr{G}}(\tau _{n})\) such that
The sequence \(\{\tilde {\varphi }_{n}(t-3\xi /2)\}_{n}\) has a convergent subsequence by pullback \(\mathfrak {D}\)-asymptotical compactness, so we may assume \(\tilde {\varphi }_{n}(t-3\xi /2)\longrightarrow v\), with v ∈ X. By definition of generalized process, there exist a subsequence (still denoted the same) and \(\psi \in {\mathscr{G}}(t-3\xi /2)\) such that ψ(t − 3ξ/2) = v and ψn(r)→ψ(r) for r ≥ t − 3ξ/2. This implies that ψ(s) ∈ M(s), because ψn(τn) ∈ M(τn), \(\hat {M}\) is collectively closed and Proposition 1 applies. Furthermore, \(\theta _{n}(\tau _{n})=\tilde {\varphi }_{n}(\tau _{n})\in I_{\tau _{n}}(\psi _{n}(\tau _{n}))\). By the collective upper semicontinuity of I, there exist a subsequence of \(\{\tilde {\varphi }_{n}(\tau _{n})\}_{n}\) (denoted the same) and y ∈ Is(ψ(s)) such that \(\theta _{n}(\tau _{n})=\tilde {\varphi }_{n}(\tau _{n})\) converges to y, so \(y\in \tilde {\omega }(\hat {D},s)\).
By Condition (G3’), there exist a subsequence (denoted the same) and \(\theta \in {\mathscr{G}}(s)\) such that 𝜃n(rn) converges to 𝜃(r) for any sequence {rn}n with rn ≥ τn and rn converging to r. This implies that 𝜃n(τn) converges to 𝜃(s) = y ∈ Is(ψ(s)), so y∉M(s) by Condition (I). Furthermore \(\tilde {\varphi }_{n}(\tau _{n})=\tilde {\varphi }_{n}(s+(\tau _{n}-s))\) converges to y, so \(y\in \tilde {\omega }(\hat {D},s)\).
We have that 𝜃(r)∉M(r) for r ∈ [s, t + ξ/4] because 𝜃(s) = y ∈ Is(ψ(s)) and Condition (H) applies. Take \(\tilde {\alpha }\in \tilde {{\mathscr{G}}}(t+\xi /4)\) with \(\tilde {\alpha }(t+\xi /4)=\theta (t+\xi /4)\) and define
Then \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\) and \(\tilde {\varphi }(s)=y\in \tilde {\omega }(\hat {D},s)\setminus M(s)\). Finally,
which implies that \(x=\tilde {\varphi }(t)\). □
Theorem 2
Let \(\tilde {{\mathscr{G}}}\) be a pullback \(\mathfrak {D}\)-asymptotically compact impulsive generalized process satisfying Conditions (G3’), (H), (I) and (NT). Then \(\tilde {\omega }(\hat {D})\setminus \hat {M}\) is positively invariant for any \(\hat {D}\in \mathfrak {D}\).
Proof
We restrict ourselves to the case where t > s and t − s ∈ (0, ξ]. By recursion the general case follows easily. Let \(x\in \tilde {\omega }(\hat {D},s)\setminus M(s)\). We want to prove that there exists \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\) with \(\tilde {\varphi }(s)=x\) such that \(\tilde {\varphi }(t)\in \tilde {\omega }(\hat {D},t)\setminus M(t)\).
As \(x\in \tilde {\omega }(\hat {D},s)\), there exist \(s_{n}\longrightarrow -\infty \), εn→0 and \(\tilde {\varphi }_{n}\in \tilde {{\mathscr{G}}}(s_{n})\) such that \(\tilde {\varphi }_{n}(s_{n})\in D(s_{n})\) and \(\tilde {\varphi }_{n}(s+\varepsilon _{n})\longrightarrow x\). We may assume, without loss of generality, that s + εn < t for all \(n\in \mathbb {N}\).
Each impulsive trajectory \(\tilde {\varphi }_{n}\), which is defined on \([s_{n},+\infty )\), has Nn ≥ 0 jump times. We consider τn the last jump time of \(\tilde {\varphi }_{n}\) on the interval [sn, s + 5ξ/4]. If there are no jump times we take τn = sn. We will split the proof into three different cases.
- Case 1.:
-
Up to a subsequence (denoted the same), there exists ε ∈ (0, ξ/2) such that τn < s − ε.
We know that there exist \(\psi _{n}\in {\mathscr{G}}(s-\varepsilon /2)\) such that \(\tilde {\varphi }_{n}(r)=\psi _{n}(~r)\) for r ∈ [s − ε/2, s + 5ξ/4]. By pullback \(\mathfrak {D}\)-asymptotical compactness, we may assume \(\tilde {\varphi }_{n}(s-\varepsilon /2)\longrightarrow y\). We may assume that there exists \(\psi \in {\mathscr{G}}(s-\varepsilon /2)\) with ψn(r)→ψ(r) for r ≥ s − ε/2, by definition of generalized process. First, \(\tilde {\varphi }_{n}(s+\varepsilon _{n})=\psi _{n}(s+\varepsilon _{n})\), which converges to ψ(s), so x = ψ(s). This implies that \(\tilde {\varphi }_{n}(s)\) converges to x.
We claim that ψ(r)∉M(r) for r ∈ [s − ε/2, s + ξ]. If ψ(r) ∈ M(r) for some r ∈ [s − ε/2, s + ξ], then ϕ(ψ, s − ε/2) ≤ r − (s − ε/2). But \(\tilde {\varphi }_{n}\) has no jump times on [s − ε/2, s + 5ξ/4], so ϕ(ψn, s − ε/2) ≥ (s + 5ξ/4) − (s − ε/2). Then Condition (NT) would imply that
a contradiction.
We take \(\tilde {\alpha }\in \tilde {{\mathscr{G}}}(s+\xi )\) with \(\tilde {\alpha }(s+\xi )=\psi (s+\xi )\) and define
Then \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\), \(\tilde {\varphi }(s)=\psi (s)=x\) and finally \(\tilde {\varphi }_{n}(t)=\psi _{n}(t)\longrightarrow \psi (t)=\tilde {\varphi }(t)\), so \(\tilde {\varphi }(t)\in \tilde {\omega }(\hat {D},t)\setminus M(t)\).
- Case 2.:
-
Up to a subsequence (denoted the same), there exists ε ∈ (0, ξ/2) such that τn > s + ε.
We know that τn is the only jump time of \(\tilde {\varphi }_{n}\) in [s − ξ/2, s + 5ξ/4] because of Condition (H), and we can assume that τn > s + ε for all \(n\in \mathbb {N}\). For each \(n\in \mathbb {N}\) there exist \(\psi _{n}\in {\mathscr{G}}(s-\xi /2)\) and \(\theta _{n}\in {\mathscr{G}}(\tau _{n})\) such that
We may assume that \(\tilde {\varphi }_{n}(s-\xi /2)\longrightarrow y\) by pullback \(\mathfrak {D}\)-asymptotical compactness, and by definition of generalized process for ψn we may assume that there exist a subsequence (denoted the same) and \(\psi \in {\mathscr{G}}(s-\xi /2)\) such that ψn(r)→ψ(r) for r ≥ s − ξ/2. Furthermore we have that \(\tilde {\varphi }_{n}(s+\varepsilon _{n})=\psi _{n}(s+\varepsilon _{n})\), which converges to ψ(s) by Proposition 1, so x = ψ(s). This implies that \(\tilde {\varphi }_{n}(s)\) also converges to x.
- Subcase 1.:
-
Up to a subsequence (denoted the same), there exists δ > 0 such that τn > t + δ.
We claim that ψ(r)∉M(u) for r ∈ [s − ξ/2, t + δ). If ψ(r) ∈ M(r) for some r ∈ [s − ξ/2, t + δ), then ϕ(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
a contradiction.
We take \(\tilde {\alpha }\in \tilde {{\mathscr{G}}}(t+\delta /2)\) with \(\tilde {\alpha }(t+\delta /2)=\psi (t+\delta /2)\) and define
Then \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\), \(\tilde {\varphi }(s)=\psi (s)=x\) and \(\tilde {\varphi }_{n}(t)=\psi _{n}(t)\), which converges to \(\psi (t)=\tilde {\varphi }(t)\), so \(\tilde {\varphi }(t)\in \tilde {\omega }(\hat {D},t)\setminus M(t)\).
- Subcase 2.:
-
Up to a subsequence (denoted the same), there exists δ > 0 such that τn < t − δ.
As τn ∈ (s + ε, t − δ), we may assume that τn converges to \(\bar {\tau }\in [s+\varepsilon ,t-\delta ]\). We have ψn(τn) ∈ M(τn), so \(\psi (\bar {\tau })\in M(\bar {\tau })\) by Proposition 1. We also have that \(\theta _{n}(\tau _{n})=\tilde {\varphi }(\tau _{n})\in I_{\tau _{n}}(\psi _{n}(\tau _{n}))\). By the collective upper semicontinuity of I, there exist a subsequence of \(\{\tilde {\varphi }_{n}(\tau _{n})\}_{n}\), still denoted the same, and \(z\in I_{\bar {\tau }}(\psi (\bar {\tau }))\) such that \(\theta _{n}(\tau _{n})=\tilde {\varphi }_{n}(\tau _{n})\) converges to z, so \(z\in \tilde {\omega }(\hat {D},\bar {\tau })\).
Once again, we claim that ψ(r)∉M(r) for \(r\in (s-\xi /2,\bar {\tau })\). If ψ(r) ∈ M(r) for some \(r\in (s-\xi /2,\bar {\tau })\), then ϕ(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
a contradiction.
We can assume by Condition (G3’) that there exists \(\theta \in {\mathscr{G}}(\bar {\tau })\) such that 𝜃n(un) converges to 𝜃(u) for any sequence {un}n with un ≥ τn and un converging to u. Furthermore, 𝜃(r)∉M(r) for r ∈ [τn, s + 5ξ/4] because \(\theta (\bar {\tau })\in I_{\bar {\tau }}(M(\bar {\tau }))\) and Condition (H) applies.
We take \(\tilde {\alpha }\in \tilde {{\mathscr{G}}}(s+\xi )\) with \(\tilde {\alpha }(s+\xi )=\theta (s+\xi )\) and define
We have that \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\), \(\tilde {\varphi }(s)=x\) and \(\tilde {\varphi }_{n}(t)=\theta _{n}(t)\), which converges to \(\theta (t)=\tilde {\varphi }(t)\), so \(\tilde {\varphi }(t)\in \tilde {\omega }(\hat {D},t)\setminus M(t)\).
- Subcase 3.:
-
τn converges to t.
We have that ψn(τn) ∈ M(τn), so ψ(t) ∈ M(t) using Proposition 1 and that \(\hat {M}\) is collectively closed. We also have that \(\theta _{n}(\tau _{n})=\tilde {\varphi }_{n}(\tau _{n})\in I_{\tau _{n}} (\psi _{n}(\tau _{n}))\). By the collective upper semicontinuity of I, there exist a subsequence of \(\{\tilde {\varphi }_{n}(\tau _{n})\}_{n}\), still denoted the same, and z ∈ It(ψ(t)) such that \(\theta _{n}(\tau _{n})=\tilde {\varphi }_{n}(\tau _{n})\) converges to z, so \(z\in \tilde {\omega }(\hat {D},t)\).
We claim that ψ(r)∉M(r) for r ∈ [s − ξ/2, t). If ψ(r) ∈ M(r) for some r ∈ [s − ξ/2, t), then ϕ(ψ, s − ξ/2) ≤ r − (s − ξ/2). But we know that τn is the only jump time of \(\tilde {\varphi }_{n}\) in [s − ξ/2, s + 5ξ/4], so ϕ(ψn, s − ξ/2) = τn − (s − ξ/2), and Condition (NT) implies that
a contradiction. We take \(\tilde {\alpha }\in \tilde {{\mathscr{G}}}(t)\) with \(\tilde {\alpha }(t)=z\) and we define
Then \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\), \(\tilde {\varphi }(s)=x\) and \(\tilde {\varphi }_{n}(\tau _{n})=\theta _{n}(\tau _{n})\) converges to \(z=\tilde {\alpha }(t)=\tilde {\varphi }(t)\), so \(\tilde {\varphi }(t)\in \tilde {\omega }(\hat {D},t)\setminus M(t)\).
- Case 3.:
-
τn converges to s
In this case τn is the only jump time of \(\tilde {\varphi }_{n}\) in [s − ξ/2, s + 5ξ/4]. Then there exist \(\psi _{n}\in {\mathscr{G}}(s-\xi /2)\) and \(\theta _{n}\in {\mathscr{G}}(\tau _{n})\) such that
By pullback \(\mathfrak {D}\)-asymptotical compactness we may assume \(\tilde {\varphi }_{n}(s-\xi /2)\) converges to y, and by definition of generalized process we may assume that there exists \(\psi \in {\mathscr{G}}(s-\xi /2)\) such that ψn(r)→ψ(r) for r ≥ s − ξ/2.
- Subcase 1.:
-
Up to a subsequence, still denoted the same, s + εn < τn
We have that \(\tilde {\varphi }_{n}(s+\varepsilon _{n})=\psi _{n}(s+\varepsilon _{n})\), which converges to ψ(s) by Proposition 1, so x = ψ(s). Furthermore, ψn(τn) ∈ M(τn), so ψ(s) ∈ M(s) by Proposition 1 and the collective closedness of \(\hat {M}\). 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), so ψ(s) ∈ M(s) by Proposition 1 and the fact that \(\hat {M}\) is collectively closed. Furthermore, \(\tilde {\varphi }_{n}(\tau _{n})=\theta _{n}(\tau _{n})\in I_{\tau _{n}}(\psi _{n}(\tau _{n}))\). Using the collective upper semicontinuity of I, there exist a subsequence \(\{\tilde {\varphi }_{n}(\tau _{n})\}_{n}\), still denoted the same, and z ∈ Is(ψ(s)) such that \(\theta _{n}(\tau _{n})=\tilde {\varphi }(\tau _{n})\) converges to z, so \(z\in \tilde {\omega }(\hat {D},s)\).
By Condition (G3’) we assume that there exists \(\theta \in {\mathscr{G}}(\bar {\tau })\) such that 𝜃n(rn) converges to 𝜃(r) for any sequence {rn}n with rn ≥ τn and rn converging to r. We have that 𝜃(u)∉M(u) for u ∈ [s, s + ξ] because of Condition (H). We take \(\tilde {\alpha }\in \tilde {{\mathscr{G}}}(s+\xi )\) with \(\tilde {\alpha }(s+\xi )=\theta (s+\xi )\) and define
Then \(\tilde {\varphi }\in \tilde {{\mathscr{G}}}(s)\), \(\tilde {\varphi }_{n}(s+\varepsilon _{n})=\theta _{n}(s+\varepsilon _{n})\) converges to 𝜃(s) = z, so \(x=z=\tilde {\varphi }(s)\). Finally \(\tilde {\varphi }_{n}(t)=\theta _{n}(t)\) converges to \(\theta (t)=\tilde {\varphi }(t)\), so \(\tilde {\varphi }(t)\in \tilde {\omega }(\hat {D},t)\setminus M(t)\). □
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Caraballo, T., Uzal, J.M. Dynamics of Nonautomous Impulsive Multivalued Processes. Set-Valued Var. Anal 31, 7 (2023). https://doi.org/10.1007/s11228-023-00667-2
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DOI: https://doi.org/10.1007/s11228-023-00667-2