Lyapunov Pairs in Semilinear Differential Problems with State-Dependent Impulses

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.


Introduction
Impulsive differential equations or inclusions describe phenomena characterized by the fact that the model parameters are subject to short-term perturbations in time. For  the periodic treatment of some diseases, impulses may correspond to administration of a drug treatment; in environmental sciences, impulses may correspond to seasonal changes or harvesting; in economics, impulses may correspond to abrupt changes of prices. For a bibliography on the theory of impulsive differential equations one can see, for instance, the monographs [4,25] and for more recent results on impulsive differential inclusions we refer the interested reader to some papers and monographs of the last decade: [7,8,18,19]. All the cited papers deal with fixed moments of impulses, but in fact as well as fixed beforehand the moments of impulses can be chosen in various ways, for instance, randomly, or determined by the state of a system. Existence results for several kind of impulsive differential inclusions with variable times of impulses in a finite dimensional framework are contained in [5,6,16,17,20]. Some interesting results on classical solutions to single-valued impulsive problems in abstract spaces (with state-dependent impulses) can be found in, e.g., [21]. As far as we know this work is the first attempt to solve the problem of the existence of mild solutions for semilinear differential inclusions with impulses at variable times in Banach spaces.
In particular, given a > 0 we consider the following semilinear differential inclusion with state-dependent impulses:
Problems with state-dependent impulses are more suitable to describe real-life phenomena, but more complicated from a theoretical point of view. Indeed, while for problems with fixed moments of impulses (i.e., with τ j constant functions for any j = 1, . . . , m) several methods from continuous problems can be adopted, this is not the case for state-dependent impulses. One of the main difficulties is to find a suitable function space, where all solutions could be embedded. Recently, in [20], where E = R n , such a function space (which is a Banach space) is introduced and, under suitable assumptions, all solutions are interpreted as elements of this space (for non-metrizable space see, e.g., [3]).
In this paper, we extend this definition from a finite-dimensional setting to an infinite one obtaining the first existence result for at least one global mild solution of the problem (I P ) (see Theorem 2.4). Then we characterize this first result by means of a new definition of a Lyapunov pair (called equi-Lyapunov pair) based on the definition of a Lyapunov pair given in [10]. Roughly speaking we prove the existence of at least one mild solution in the case that the barrier τ j is the first component of the equi-Lyapunov pair (see Theorem 3.6).
We recall that Cârjǎ in [10] considers the following autonomous Cauchy problem where A : D(A) ⊆ E → E is an operator and G : M E is a multimap, with M a nonempty set in the Banach space E. According to [10], given V , p : M → [−∞, +∞], two maps, the couple (V , p) is said to be a Lyapunov pair for the inclusion y ∈ Ay + G(y) if, for every ξ ∈ D(V ) = {x ∈ E : V (x) < +∞} there exists T > 0 and a mild solution y : [0, T ] → M to the autonomous multivalued semilinear Cauchy problem (AP ) ξ , the map t → p(y(t)) is integrable on [0, T ] and In the case where E is an Hilbert space, G is single-valued and Lipschitz continuous, and A is a not necessarily linear operator, Kocan and Soravia in [24] characterized the Lyapunov pairs using the partial differential inequality whose solutions V are meant in the viscosity sense.
Recently, in the same setting of hypotheses as in [24], but with the operator A possibly multivalued, in [1,2], Adly et al. provide primal and dual criteria for weakly lower semicontinuous Lyapunov pairs, explicitly formulated by means of the proximal and basic subdifferentials of the involved functions.
In [12] Cârjǎ and Motreanu, considering an appropriate contingent derivative, obtain a different characterization of Lyapunov pairs on Hilbert spaces, with A a linear operator and G a locally Lipschitz mapping. Their approach is based on viability results and uses the contingent derivative associated to the operator A defined as follows.
Later the same authors in [13] extend this result to arbitrary Banach spaces and to mdissipative possibly multivalued operators A. In [11] Cârjǎ and Lazu obtain an analogous characterization assuming only continuity on G but requiring that the semigroup generated by the m-dissipative operator is compact. Assuming that A is a linear operator generating a compact semigroup, Cârjǎ in [10] extends the results obtained in the previous cited papers to the case of a multivalued map F . To this aim he introduces a new concept of contingent derivative suitable for inclusions. Precisely, given a nonempty bounded set S ⊂ E, he considers the set In this paper we introduce a new type of Lyapunov pair, not depending on a mild solution.

Definition 1.3
Given two maps V , p : E → R, we say that (V , p) is an equi-Lyapunov pair for the inclusion if, for every (τ, ξ ) ∈ [0, a[×E and every mild solution y : [τ, a] → M to the multivalued semilinear Cauchy problem Following the approach described in [10], we give sufficient conditions for the existence of equi-Lyapunov pairs in terms of a new concept of A-contingent derivative defined as follows.

Definition 1.4 Let
We point out the fact that unlike all the cited results we consider a non-autonomous differential inclusion on arbitrary Banach spaces with A a linear operator generating a non necessarily compact semigroup. Actually, to obtain existence results for mild solutions of the problem (I P ) it is sufficient to consider a weaker definition of equi-Lyapunov pair, namely: Definition 1.5 Given two maps V , p : E → R, we say that (V , p) is an equi-Lyapunov pair for problem (P ) (τ,ξ ) if, for every mild solution y : [τ, a] → M to (P ) (τ,ξ ) , the map t → p(y(t)) is integrable on [τ, a] and Using this definition it is possible to consider p : E → R depending on the initial values (τ, ξ ) of the Cauchy problem.
The paper is organized as follows: in Section 2 we prove an existence result for the impulsive problem (I P ); in Section 3, after a preliminary study of the autonomous case (see Section 3.1), we give sufficient conditions to have an equi-Lyapunov pair for the differential inclusion (2) (see Section 3.2) and then we obtain another existence result for (I P ) via equi-Lyapunov pairs defined as in Definition 1.5 (see Section 3.3); in Section 4 we give the concluding remarks; finally, for the reader's convenience in the Appendix we recall some definitions and results the proofs are based on.

Semilinear Differential Problem with State Dependent Impulses
In this Section we consider the problem (I P ) under the following assumptions.
(H1) for every j = 1, . . . , m, the map τ j : E → R is a continuous function; moreover for every x ∈ E it holds that We search for the existence of mild solutions of problem (I P ) in the space A mild solution of the problem (I P ) is a map belonging to the space C such that it meets exactly once each More precisely, we give the following definition.
To prove the existence of at least one mild solution of the problem (I P ) we assume the following hypotheses on the multimap G : [0, a] × E E: x) is convex and compact; (G2) for every x ∈ E, the multimap G(·, x) admits a strongly measurable selector, i.e. there exists a strongly measurable function q : and every x ∈ E; where χ is the Hausdorff measure of noncompactness.
Remark 2.2 Notice that, under the above assumptions on the multimap G, for every (τ, ξ ) ∈ [0, a[×E the Cauchy problem admits at least one global mild solution by Theorem 4 in [9]. Therefore Moreover, we need the following viability-type condition:

Remark 2.3
Notice that, if the functions τ j , j = 1, . . . , m, are constant, then the problem (I P ) comes down to a problem with fixed moments of impulses. Clearly in this case property (H2) is trivially satisfied. Proof The proof splits in four steps.
We consider now the map w 1 2 : and by (H1.1) 1 2 is continuous by the continuity of the maps τ 2 and y 1 . Therefore, reasoning as in Step 1, we can say that there exists t 1 2 We notice that the point (t j −1 j , y j ) is on the right or belongs to the barrier j and is on the left of the barrier j +1 (assuming m+1 = {(a, x) : x ∈ E}). Moreover, the graph of y j restricted to the interval ]t j −1 j , a] is on the right of the barrier j . Further, the map y j intersects the barrier j +1 (j = m) in is a solution of the problem (I P ).
To this aim we define the map g : [0, a] → E as It is easy to see that g ∈ L 1 ([0, a]; E) and g(s) ∈ G(s, y(s)) for a.e. s ∈ [0, a].
We will prove now that With an iterative method we obtain the claimed result.
As we could see, assumption (H2) was used to obtain (7) and, in consequence, to prove that every trajectory meets each barrier exactly once. Since we consider in (I P ) a concrete initial condition, we need slightly less than (H2). Below we describe one of possible situations where (7) and analogs for all j = 2, . . . , m are implied without (H2). In what follows, by · L(E) we denote a standard norm in the space of linear bounded operators from E to itself and by B r X (x) an open ball in a metric space X centered at x and with radius r > 0.  Proof As in the proof of Theorem 2.4, let y 0 ∈ S (0,y 0 ) . Up to the first impulse time t 0 1 we have for every t ∈ [0, a]. So, denoted y 1 = y 0 (t 0 1 ) + I 1 (y 0 (t 0 1 )), we get and Now we consider a map y 1 ∈ S (t 0 1 ,y 1 ) , with g 1 ∈ L 1 ([t 0 1 , a]; E), g 1 (s) ∈ G(s, y 1 (s)) for a.e. s ∈ [t 0 1 , a]. Defining the function , by means of analogous calculations as in (9) we get Moreover, for t ∈]t 0 1 , a] we have Hence, for every t ∈]t 0 1 , a], thanks to (13) and (10) we can use (H3); so, by (11), (12), and (k) we obtain Hence, as in the proof of Theorem 2.4, we know that the graph of y 1 is on the right of 1 , and the next impulse time t 1 2 is the time of a meeting point with 2 . Also for y 2 = y 1 (t 1 2 ) + I 2 (y 1 (t 1 2 )) we obtain so the map y 2 ∈ S (t 1 2 ,y 2 ) satisfies the following inequality τ 2 (y 2 (t)) − t − τ 2 (y 2 ) + t 1 2 < 0 for every t ∈]t 1 2 , a]. Thus (5) is satisfied by y 2 . Then, proceeding iteratively, we can build a function y as in (8). Like in the proof of Theorem 2.4, this map is a solution to problem (I P ).

Existence Results via Equi-Lyapunov Pairs
Let M be a nonempty set in a Banach space E, G : [0, a] × M E be a given multimap. Consider the non-autonomous semilinear differential inclusion y (t) ∈ Ay(t) + G(t, y(t)), for a.e. t ∈ [0, a] (14) and suppose that there exists at least one global mild solution of (14). Notice that the existence of such a solution is guaranteed for example under assumptions (G1)-(G5) (see Remark 2.2).
In order to study a characterization of equi-Lyapunov pairs of (14) (see Definition 1.3), we need at first to consider an autonomous case.

Sufficient Conditions for Equi-Lyapunov Pairs: Autonomous Case
In this subsection we consider the following autonomous semilinear differential inclusion where G : M E and we obtain sufficient conditions to have equi-Lyapunov pairs for (15) . Precisely we have the following result. (ii) for every ξ ∈ E and every f ∈ G(ξ ) L 1 the next inequality holds

Theorem 3.1 Let E be a separable Banach space, A : D(A) ⊆ E → E be a linear operator satisfying (A), G : E E be a nonempty closed bounded valued multimap, V : E → R be a lower semicontinuous function, and p : E → R be a function. If
then (V , p) is an equi-Lyapunov pair for inclusion (15).
Proof Let X = E × R be the separable Banach space endowed with the norm (ξ, s) X = max{ ξ , |s|}, for every (ξ, s) ∈ X, and let F : X X be the multimap defined as We claim that F is Lipschitz on bounded sets, i.e. for every (ξ, r) ∈ X and for every R > 0 there exists a constant L R > 0 such that  , 0). It is known that A is the infinitesimal generator of the C 0 -semigroup {U (t)} t≥0 = {(U (t), 1 R )} t≥0 . Let K = epiV , we will prove now that for every (ξ, r) ∈ K we have (see (1)) F (ξ, r) L 1 ⊂ F A K (ξ, r), (16) where F A K is the class of all A -tangent functions to K at (ξ, r) introduced in the next Appendix.
Let (ξ, r) ∈ K and f ∈ F (ξ, r) L 1 , i.e. there exists g ∈ L 1 loc (R + ; E) with g(s) ∈ G(ξ ) for a.e. s ∈ R + , such that By (ii) we have that (see Definition 1.4) lim inf That is, there exist two sequences {h n } n ⊂ R + , h n ↓ 0 and {w n } n ⊂ E, w n → 0 E such that for every n ∈ N Hence, for every n ∈ N By the definition of the C 0 -semigroup {U (t)} t≥0 and by (17) this implies that Therefore, by Remark A.2, we have proven that for every ξ ∈ E, f (·) = (g(·), −p(ξ )) ∈ F A K (ξ, V (ξ )). We can estimate the right hand side of (18) by ρ +h n −p(ξ ) + 1 n for every ρ ≥ V (ξ), From we get for every mild solution y of (15) such that y(τ ) = ξ V (y(t)) + t τ p(y(s)) ds ≤ r, for all t ∈ [τ, a] and for all r ≥ V (ξ), in particular, with r = V (ξ) we obtain the claimed result.
We can prove an analogous result with p a lower semicontinuous map.
for every x, y ∈ ; (ii) for every ξ ∈ E and every f ∈ G(ξ ) L 1 the next inequality holds then (V , p) is an equi-Lyapunov pair for inclusion (15).
Proof By Lemma A.1 there exists a sequence of functions {p n } n , p n : E → R such that p n is Lipschitz continuous on bounded sets of E for every n ∈ N and p n ↑ p pointwise in E as n goes to infinity. By condition (ii) we have that for every ξ ∈ E and every f ∈ G(ξ ) L 1 By Theorem 3.1 we have that (V , p n ) is an equi-Lyapunov pair for (15).

Sufficient Conditions for Equi-Lyapunov Pairs: Non-autonomous Case
In this section, by using the results obtained in Section 3.1, we provide analogous results in the non-autonomous case (14). (jj) for every (τ, ξ ) ∈ [0, a] × E and every f ∈ G(τ, ξ ) L 1 the next inequality holds then (V , p) is an equi-Lyapunov pair for inclusion (14).
Y be the multimap defined as It is not hard to check that G satisfies assumption (i) from Theorem 3.
for every h ∈ R + . Thus, defining P : Y → R as P(z) ≡ P(t, x) = p(x), for every z = (t, x) ∈ Y , by (jj) it follows that Hence, by the arbitrariness of ζ and f , we can apply Theorem 3.1 and so we obtain that (V, P) is an equi-Lyapunov pair for the inclusion i.e. for any ζ = (τ, ξ ) ∈ Y and for any mild solution z Therefore, recalling that z(s) = (t (s), x(t (s))) for every s ∈ [0, a − τ ], where x : [τ, a] → E is a mild solution of (P ) (τ,ξ ) , it follows that Hence, setting r + τ = σ we obtain Finally, recalling that t = s + τ we have concluding the proof.
With similar arguments we can prove a result analogous to the previous one, but involving a map p as in Theorem 3.2.  for every (t, x), (s, y) ∈ ; (jj) for every (τ, ξ ) ∈ [0, a] × E and for every f ∈ G(τ, ξ ) L 1 the next inequality holds then (V , p) is an equi-Lyapunov pair for inclusion (14).

Existence of Impulsive Mild Solutions
In this Section we give existence results for problem (I P ) via equi-Lyapunov pairs. Firstly, observe that it is possible to provide results analogous to Theorems 3.1-3.4 obtaining the existence of an equi-Lyapunov pair for problem (P ) (τ,ξ ) (see Definition 1.5). We state the analog to Theorem 3.4, since we will need it in the sequel.
The proof of the above corollary is the same as the one of Theorem 3.4, since the fact that (τ, ξ ) is fixed in the hypotheses leads us to achieve a weaker thesis in the corollary with respect to the one of the corresponding theorem, i.e. an equi-Lyapunov pair for the Cauchy problem (P ) ( (4)) the next inequality holds: Hence, since p j,(τ,ξ ) takes positive values, we deduce that: for every j = 1, . . . , m, (τ, ξ ) ∈ [0, a[×E, and every y (τ,ξ ) ∈ S (τ,ξ ) it holds that which is property (H2). Now, by the hypothesis (j)' of Theorem 3.4 the multimap G is trivially upper semicontinuous with respect to the second variable, implying assumption (G3), and measurable with respect to the first. Hence by the separability of the space E and the Kuratowski-Ryll-Nardzewski Theorem (see, e.g.
so that Let t ∈ [0, a], and z ∈ G(t, D) be fixed, hence there exists x ∈ D such that z ∈ G(t, x). By (j)' with t = s and y = x j , j = 1, . . . , q δ(ε) it holds j =1 G(t, x j ), this set is compact and the family of balls is an open cover of K. So we can extract a finite sub-cover, i.e. there exist y ε 1 , . . . , y ε m(ε) ∈ K such that Hence by (22) and (23) we have Therefore the set {y ε i : i = 1, . . . , n} is a finite 2L D δ(ε)-net of G(t, D), obtaining, by the definition of δ(ε) (see (21)) that By the arbitrariness of ε > 0 we achieve condition (G5) with β(t) = 2L D for every t ∈ [0, a].
So, we can apply Theorem 2.4 and obtain the claimed result.
We finish with the result where the analytic sufficient condition for assumption (H2)' is provided.

Concluding Remarks
The research presented in the preceding sections leads us to several open problems and motivates to a further study. At first, the case where the multivalued perturbation G (see problem (I P )) is only measurable with respect to the first variable and sufficiently regular (e.g., lipschitzian) with respect to the second one should be examined. Of course, different method of proof (other than in Theorem 3.3) has to be found. The second direction of research focuses on non-compact valued perturbations and, instead, compact semigroups or compact evolution systems. We deeply believe it is possible to obtain important analogous results in this case and we take it as a subject of the future study. This development will open an opportunity to examine properties of solutions of new kinds of evolution PDEs with state-dependent impulses.
As the third open problem we can mention the question on topological properties of the mild solution set to problem (I P ). Especially, a compactness, contractibility or R δ -property would be welcome under different types of assumptions.
We hope our paper and the above remarks will be considered interesting and inspiring.
Moreover, we recall that a uniqueness function ω : R + 0 → R + 0 is a continuous, nondecreasing function, such that the only C 1 -solution of the Cauchy problem x (t) = ω(x(t)) x(0) = 0 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 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.
, for every x ∈ \ K, y ∈ K ∩ ; (b) for every ξ ∈ K, F (ξ) L 1 ⊂ F A K (ξ ). Then K is invariant with respect to A + F .