Abstract
The paper deals with semilinear differential inclusions with state-dependent impulses in Banach spaces. Defining a suitable Banach space in which all the solutions can be embedded we prove the first existence result for at least one global mild solution of the problem considered. Then we characterize this result by means of a new definition of Lyapunov pairs. To this aim we give sufficient conditions for the existence of Lyapunov pairs in terms of a new concept of contingent derivative.
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Acknowledgements
The research carried out within the national group GNAMPA of INdAM.
The first and fourth author have been supported by the INdAM-GNAMPA Project 2016 “Metodi topologici, sistemi dinamici e applicazioni”.
The second and fourth author have been supported by the project Fondi Ricerca di Base 2016 “Problemi Differenziali Soggetti ad Impulsi Fissi o Variabili”, Department of Mathematics and Computer Science, University of Perugia.
The first author has been supported by the project Fondi Ricerca di Base 2016 “Metodi topologici per equazioni differenziali in spazi astratti”, Department of Mathematics and Computer Science, University of Perugia.
The third author has been supported by the Polish NCN grant no. 2013/09/B/ST1/01963.
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Appendix
Appendix
For the reader’s convenience in this section we collect some of definitions and results the proofs are based on.
In order to consider in Theorems 3.2 and 3.4 a lower semicontinuous map p we need the following approximation result contained in [15].
Lemma A.1
([15], Lemma 13.2.3) LetX be a Banach space and letg : X → (−∞, + ∞] a lower semicontinuous function with non-empty effective domain satisfying the following condition: there existtwo constantsC > 0 andr ≥ 1
Then there exists a sequence {gn}nof functions\(g_{n}:X \to \mathbb {R}\)such that every gnis Lipschitz continuous on bounded subsets of Xand gn ↑ gpointwise on Xas ngoes to infinity.
To prove the characterization of the equi-Lyapunov pair for the inclusion (2) (see Definition 1.3) we need to give the following tangency concept introduced by Cârjǎ and Postolache in [14]. Let K be a subset in E and ξ ∈ K. If A : D(A) ⊆ E → E is the infinitesimal generator of a C0-semigroup {U(t)}t≥ 0, a function \(f \in L_{loc}^{1}(\mathbb {R}^{+};E)\) is A-tangent to the setKat the pointξ ∈ K if
The class of all A-tangent functions to K at ξ ∈ K is denoted by \(\mathscr{F}_{K}^{A}(\xi )\).
Remark A.2
(14, Remark 3) It is possible to characterize the tangency by means of sequences as follows: \(f \in \mathscr{F}_{K}^{A}(\xi )\) if and only if there exist sequences \(\{h_{n}\}_{n} \subset \mathbb {R}^{+}\) and {wn}n ⊂ X with hn↓ 0 and wn → 0 such that
Moreover, we recall that a uniqueness function\(\omega :\mathbb {R}^{+}_{0} \to \mathbb {R}^{+}_{0}\) is a continuous, nondecreasing function, such that the only C1-solution of the Cauchy problem
is x ≡ 0 (see [14, Definition 4]). Given the Cauchy problem
where A : D(A) ⊆ E → E is a linear operator satisfying (A), \(F: E \multimap E\) is a multimap, a subset K ⊂ E is said to be invariant with respect to A + F if for every ξ ∈ K each mild solution u : [0, T] → E of the problem (24) is in K, that is u(t) ∈ K, for every t ∈ [0, T]. In Section 3 we use the following result.
Theorem A.3
(14, Theorem 2) LetE be a separable Banach space,A : D(A) ⊆ E → Ebe a linear operator satisfying(A),K a nonempty and closed subset ofE and \(F: E \multimap E\)anonempty, closed and bounded valued multimap. Assume that
-
(a)
there exists an open neighborhoodW ⊆ EofK, such that for everyξ ∈ Kand Ω ⊆ W, a bounded open set containingξ, there exists an uniquenessfunction \(\omega _{{\Omega }}:\mathbb {R}^{+}_{0} \to \mathbb {R}^{+}_{0}\)suchthat
$$F(x) \subset F(y) + \omega_{{\Omega}}(\|x-y\|) {B_{E}^{1}}(0), $$for every x ∈ Ω∖ K, y ∈ K ∩ Ω;
-
(b)
for every ξ ∈ K,
$$F(\xi)_{L^{1}} \subset \mathscr{F}_{K}^{A}(\xi). $$
Then Kis invariant with respect to A + F.
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Benedetti, I., Cardinali, T., Gabor, G. et al. Lyapunov Pairs in Semilinear Differential Problems with State-Dependent Impulses. Set-Valued Var. Anal 27, 585–604 (2019). https://doi.org/10.1007/s11228-018-0490-7
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DOI: https://doi.org/10.1007/s11228-018-0490-7