Non-autonomous Second Order Differential Inclusions with a Stabilizing Effect

In this paper we prove the existence of mild solutions for a problem governed by a semilinear non-autonomous second order differential inclusion where a stabilization of the solution is expected due to the control of the reaction term. In order to obtain our existence theorem, first we study a more general problem with a differential inclusion which involves a perturbation guided by an operator N:I→C(C(I;X);X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N :I \rightarrow C(C(I;X);X)$$\end{document}, where X is a Banach space. Finally we show an illustrative example of application of our results to a problem involving a wave equation.


Introduction
The goal of this work is to consider theoretical mathematical models governed by differential inclusions which adapt themselves to study controllability problems in medicine, in life sciences or in other real phenomena. In particular, in this paper we prove the existence of mild solutions for the following nonlocal abstract problem for a semilinear non-autonomous second order differential inclusion Then, as consequence of Theorem 3.5, in Sect. 4 we are able to deduce the existence of mild solutions for the mentioned nonlocal problem (NLP ) (see Theorem 4.1).
As application of our abstract result presented in Sect. 3, in Sect. 5 we obtain the controllability of a non-autonomous Cauchy problem guided by a wave equation, where the perturbation is subject to an opportune operator N .
Finally, Appendix contains some background material intended to make the paper self-contained.

Problem Setting
We consider the following nonlocal problem in a real Banach space X

where {A(t)} t∈I is a family of bounded linear operators A(t): D(A) → X,
where D(A), independent on t ∈ I, is a subspace dense in X, generating a fundamental system {S(t, s)} (t,s)∈I×I , F is a multimap and N : I → C(C(I; X); X) is a map.
The concept of fundamental system, introduced by Kozak in [14], is recently used by Henríquez et al. [11] and by Cardinali, Gentili in [3].
In most of works, for every t ∈ I the linear operator A(t): D(A) → X is also closed (see [10,11,14]), but we leave out this property on A(t), since it is not necessary in order to obtain only the existence of mild solutions. As in [11], a map S : I × I → L(X), where L(X) denotes the space of all bounded linear operators in X with the norm . L(X) , is said to be a fundamental operator if {S(t, s)} (t,s)∈I×I is a fundamental system. Moreover, for each (t, s) ∈ I × I, we consider the operator C(t, s) = − ∂S ∂s (t, s): X → X (2.1) and the family of linear operators {C(t, s)} (t,s)∈I×I . Now we recall that by using Banach-Steinhaus Theorem there exist two constants K, K 1 > 0 such that (see [11]) Further, if we assume that the fundamental system {S(t, s)} (t,s)∈I×I has also the following property x is continuous on I × I, we can claim that the family {C(t, s)} (t,s)∈I×I satisfies For the sake of completeness we recall the fundamental Cauchy operator G S : L 1 (I; X) → C(I; X), introduced in [3] as and some its properties (see [3], Theorem 3.2 and Remark 3.3) Let N : I → C(C(I; X); X) be a map such that (N1) for every u ∈ C(I; X), N(.)(u) is strongly measurable; (N2) for every u ∈ C(I; X), N(.)(u) is bounded; (FN) there exist α ∈ L 1 + (I) and a non-decreasing map ψ : Then the multimap S 1 F,N : C(I; X) → P(L 1 (I; X)) given by Proof. We show that the multimap S 1 F,N assumes non empty values. First of all, fixed u ∈ C(I; X), we define the following function q u : and by (N 1) we say that q u is strongly measurable. Moreover, since the multimap F has compact values and it satisfies (F 1) and (F 2), by applying Theorem 1.3.5 of [13] there exists a strongly measurable selection f u : I → X of the multimap F (., q u (.)). Further by (F N) and by taking into account the boundedness of N (.)(u) (see (N 2)), we have where it is easy to see that m u ∈ L 1 + (I). Hence f u ∈ L 1 (I; X), therefore S 1 F,N (u) = ∅ and so S 1 F,N is well-defined. Let us note that in [4], sufficient conditions are given to obtain (F 1) without the separability of the Banach space X.
Then we show some properties for the multioperator S 1 F,N . Proof. Let us fix sequences (u n ) n , u n ∈ C(I; X), and (f n ) n , f n ∈ S 1 F,N (u n ), such that u n →ū in C(I; X) and f n f in L 1 (I; X).
According to Mazur's Theorem we have the existence of a double sequence of nonnegative numbers (α n,k ) n,k such that (II) ∞ k=n α n,k = 1, ∀ n ∈ N; (III) the following sequence (f n ) n , wheref n is defined bỹ converges tof in the normed space L 1 (I; X). Passing if necessary to a subsequence, we can assume that (f n ) n converges tof a.e. on I.

The Solution Multioperator
Now, in order to establish some properties of the solution multioperator, we assume the following property on the family {A(t)} t∈I :  Let F : I × X → P kc (X) be a map which satisfies hypotheses (F 1) and (F 2) of Proposition 3.1 and the following property: has nonempty closed convex values. Moreover Γ is totally bounded and upper semicontinuous.
Proof. First, for all u ∈ C(I; X), Proposition 3.1 implies that there exists is strongly measurable (see [13], Theorem 1.

3.5) and by (P 2) it is also B-integrable on [0, t].
Moreover v is also continuous on Finally Γ takes convex values thanks to the convexity of the values of F . From now on we proceed by steps.
Step 1: The solution multioperator Γ has closed graph (therefore Γ has also closed values).
Let (u n ) n be a sequence in C(I; X) such that u n → u * in C(I; X) and (v n ) n be a sequence in We will prove that there exists Fixed t ∈ I, since C(t, 0), S(t, 0) ∈ L(X) and g, h are continuous in u * , it follows that for n → ∞ the following convergences are true. Now we deduce that the set {f n } n is integrably bounded. Indeed, since where by (N 2) the constant P is not depending on t. Therefore {f n } n is integrably bounded. Furthermore, let us show that the set {f n (t)} n is relatively compact in X for a.e. t ∈ I.
Indeed, by using (F 3) and the monotonicity of the Hausdorff MNC η, for a.e. t ∈ I, being the set Since the map N (t) ∈ C(C(I; X); X), the compactness of W (see (3.8)) implies that also N (t)(W ) is compact. Hence by (3.9) we have Now, being the sequence (f n ) n semicompact, we can use Proposition 4.2.1 of [13] to conclude that the set {f n } n is weakly compact in L 1 (I; X), so w.l.o.g. there exists f * ∈ L 1 (I; X) such that f n f * in L 1 (I; X). Hence, since the fundamental Cauchy operator G S satisfies the mentioned properties (G S 1) and (G S 2), we are in a position to apply Theorem 5.1.1 of [13] and we deduce (3.10) Consequently, by passing to the limit in (3.5), properties (3.6, (3.7) and (3.10)) imply the following and, by recalling that v n → v * in C(I; X), the uniqueness of the limit algorithm guarantees that Finally, according to Proposition 3.3 we have that f * ∈ S 1 F,N (u * ). Hence we can conclude that v * ∈ Γ(u * ), therefore Γ has closed graph.
We fix a bounded subset Ω of C(I; X).
Step 2a: First we show that Γ(Ω) is equicontinuous on I.
. By using (P 4), (P 3), (gh2), (P 2), (F N) and (N 2) , let us note that, for every t 1 , t 2 where G, H, P Ω and C are respectively the positive constants so defined: Obviously this inequality also holds if t 1 > t 2 .
So the set Γ(Ω) is equilipschitzian and therefore it is also equicontinuous on I.
Step 2b: Next we prove that Γ(Ω)(t) is relatively compact in X, for every t ∈ I.
To this end, we consider the set Our first goal is to show that hypotheses of Theorem 5.23 of [12] are satisfied in order to prove the relative compactness of the set for every t ∈ I. Being X a separable Banach space, the space L 1 (I; X) is separable too.
Hence the closed set S Ω L 1 is separable. Therefore there exists a countable set (3.12) In a preliminary way, let us define the multimap G : I → P(X) in the following way Now, we obtain that G is measurable. By the completeness and the separability of the Banach space X, it is enough to prove that (see [12], Proposition 2.3) there exists a countable set E of measurable selectors of G such that (3.14) To this aim, we define the set of functions where Q is the countable set of non negative rational sequences so defined Clearly, the countability of Q implies that the set E is countable too. Moreover, let us note that every γ ∈ E is a measurable selector of G (see (3.13)). Now, fixed t ∈ I, we prove that (3.14) holds. By (3.15) and (3.13) obviously we can write To get the other inclusion it is sufficient to prove that  ..., k and m ∈ N, converging respectively to α 1 , . . . , α k ) ∈ E(t). Therefore (3.16) is true. Then, being (3.14) proved, we can conclude that the multimap G is measurable.
Next we show that G is integrably bounded. First of all, if u n ∈ Ω is such that f n ∈ S 1 F,N (u n ), for every n ∈ N, from (3.13), by recalling hypotheses (F N) and (N 2) , for every n ∈ N, we can write the following inequality and being ω Ω ∈ L 1 + (I), we can conclude that G is integrably bounded.
From (3.17) and by taking into account (A) , it is easy to say that, for every s ∈ [0, t], the set Φ t (s) is compact and convex. Moreover, by combining (3.17) with (P 2), we have that Φ t is integrably bounded.
Now we shall prove that Φ t is graph measurable. Again in virtue of the separability of X, it is enough to prove that Φ t is measurable (see [12], Proposition 1.7).
To this aim we define the following multimap: and we note that the measurability of G implies that G * t is measurable too. Next, since for every s ∈ [0, t], G * t (s, .) is obviously lower semicontinuous on the separable Banach space X and by applying Proposition 7.17 of [12], we deduce that G * t is (S.D.)-lower semicontinuous. Now, in a preliminary way, we consider the multimap Φ * t : [0, t] × X → P(X) defined by and we show that Φ * t is (S.D.)-lower semicontinuous. In fact G * t is (S.D.)-l.s.c. and hence for every > 0, there exists a closed . We can findz ∈ G * t (s,x) such thatȳ = S(t,s)z (see (3.19)). Then since G * t is l.s.c. at (s,x), from Proposition 2.6 of [12] there exists a net (z Next, since this estimate is true

by (A) and (3.20), we can conclude that
S(t, s α )z α →ȳ in X.
Then, according to Proposition 2.6 of [12], Φ * t is l.s.c. on T × X. Further by Proposition 2.38 of [12] we deduce that also the closed multimap Φ * t is (S.D.)-l.s.c. . Moreover fixed s ∈ [0, t], being Φ * t (s, .) = S(t, s)G(s) constant on X, Φ * t (s, .) is also l.s.c. on the separable Banach space X. Therefore, from Theorem 3.2 of [19] we can say that Φ * t is M(R)×B(X)-measurable. Therefore it is easy to see that the multimap Φ t , defined in (3.18), is measurable too.
Hence we can invoke Proposition 1.7 of [12] in order to say that Φ t is graph measurable.
Finally, by using the proof of Theorem 5.23 of [12] we can get t 0 Φ t (s) ds ∈ P kc (X). Now, let us note that the following inclusion holds. To this end let us fix t 0 S(t, s)f (s) ds ∈ H Ω (t) (see (3.11)). By (3.12) we deduce that there exists a subsequence (f n k ) k of (f n ) n : Obviously Riesz Theorem implies that (see (3.13)) Therefore we can conclude that (3.22) is satisfied and so, by (3.21) we have that the set H Ω (t) is relatively compact. Now we note that Then, by taking into account (gh1) and the continuity of operators C(t, 0) and S(t, 0), we can conclude that the set Γ(Ω)(t) is relatively compact in X. So, bearing in mind also the equicontinuity of Γ(Ω), we have that Γ(Ω) is relatively compact in C(I; X). Therefore the multimap Γ is totally bounded, hence Γ is also locally compact. Now, by recalling that Γ has closed graph, we can use Proposition 4.1.16 of [5] and deduce that Γ is upper semicontinuous in C(I; X).

Main Abstract Existence Result
Finally we are in a position to prove the existence at least of a mild solution for the nonlocal problem (N -NLP ).

S(t, s)f (s)ds, t ∈ I, f ∈ S 1 F,N (u) , u ∈ C(I; X).
In the setting of our hypotheses we are in a position to use Proposition 3.4 and so we can say that Γ has nonempty closed convex values, it is totally bounded and upper semicontinuous. Now, in order to prove the existence of a fixed point for the multioperator Γ, we show that the set Ω = {u ∈ C(I; X) : ∃ λ > 1 such that λu ∈ Γ(u)} is bounded. Let us fix u ∈ Ω, then there exists λ > 1 such that λu ∈ Γ(u) and f u ∈ S 1 F,N (u). At this point we observe that, by using (P 2), (F N), (N 2) s and (gh2), we obtain Hence we can conclude that the set Ω is bounded in C(I; X). Finally, we are ready to proceed to the application of Theorem 6.1 and we have the existence at least of one fixed point for the solution multioperator Γ, i.e. there exists at least one mild solution for (N -NLP ) problem.
Remark 3.6. Let us note that (N -NLP ) problem can be rewritten as the following problem [3], by considering the map N : I → C(C(I; X); X) so defined Unfortunately we note that Theorem 3.5 does not allow to prove the existence of mild solutions for (NLP ) * problem because N does not satisfy (N 2) s of Theorem 3.5.

Existence Result with a Stabilizing Effect
Now, as a consequence of Theorem 3.5, we are in a position to provide a result for the nonlocal problem (NLP ). Let F : I × X → P kc (X) be a map with following hypotheses: F (., x) has a strongly measurable selection; (F 2) for a.e. t ∈ I, F (t, .) is upper semicontinuous on X; (F 4) there exist α ∈ L 1 + (I) and a non-decreasing map ψ : Let g, h : C(I; X) → X be two functions which satisfy following properties: Then there exists at least one mild solution for the nonlocal problem (NLP ).
Proof. First, fixed R > 0, we note that we can define the map N : I → C(C(I; X); X) in the following way: for every t ∈ I, N (t): C(I; X) → X is such that Next we observe that for every u ∈ C(I; X), we have that N (.)(u) = u R (.) is continuous on I, so N (.)(u) is also strongly measurable and moreover obviously N (t)(v) X ≤ R, ∀ t ∈ I and ∀ v ∈ C(I; X). Therefore N satisfies (N 1) and (N 2) s of Theorem 3.5. Further, for every u ∈ C(I; X), since u R (t) X ≤ R, by combining (F 4) with (4.1), we have the following so (F N) holds. Therefore Theorem 3.5 establishes the existence of at least one mild solution for the nonlocal problem (NLP ).

An Application: Controllability with Stabilizing Effect
In this section we apply the theory developed in Sect. 3 to study the following controllability problem The following application shows a situation in which our Theorem 3.5 works, but Theorem 4.1 of [3] not.
First, as in [11], we will use the identification between functions defined on the quotient group T = R/2πZ with values in C, and 2π-periodic functions from R to C. In order to model the problem above in an abstract form, we consider the space X = L 2 (T; C), i.e. the space of all functions x : R → C, 2π-periodic and 2-integrable in [0, 2π], endowed with the usual norm . 2 . Moreover we denote by H 1 (T, C) and by H 2 (T; C) respectively the following subspaces of L 2 (T; C) and we assume that the operator A 0 is the infinitesimal generator of a strongly continuous cosine family {C 0 (t)} t∈R , where C 0 (t): L 2 (T; C) → L 2 (T; C), for every t ∈ R (see references in [11]). Moreover we fix the function P : I → L(H 1 (T; C); L 2 (T; C)) defined in this way where b : I → R is of class C 1 on I. Now we are in a position to define the family {A(t) : t ∈ I} where, for every t ∈ I, A(t): H 2 (T; C) → L 2 (T; C) is an operator so defined In Theorem 1.2 of [10], Henríquez has proved that this family generates a fundamental system {S(t, s)} (t,s)∈I×I , which is also compact (see [11], §4). Moreover, let us consider p : I → C a continuous map, ϕ 1 , ϕ 2 : I → C such that (ϕ) ϕ 1 , ϕ 2 ∈ L 1 (I; C); and the map f : I × C → C having the following properties: (f 1) f (t, x(.)) ∈ L 2 (T; C), for every t ∈ I and for every x ∈ L 2 (T; C); (f 2) for every x ∈ L 2 (T; C), the map t → f (t, x(.)) is weakly measurable; (f 3) there exists α ∈ L 1 + (I) such that, for every k = 1, 2, we have for a.e. t ∈ I and for every z, w ∈ C; (5.5) . In what follows we revise functions w, u : I × R → C such that w(t, .), u(t, .) ∈ L 2 (T; C), for every t ∈ I, as two maps x, v : I → L 2 (T, C) respectively so defined Now we define the functionf : I × L 2 (T; C) → L 2 (T; C) such that and by hypothesis (f 1) we have thatf is correctly defined. Next we consider the multimap U : I → P(L 2 (T; C)) so defined where, for every t ∈ I and i = 1, 2,φ i (t): R → C is given bỹ Since obviously for every t ∈ I,φ 1 (t),φ 2 (t) ∈ L 2 (T; C), the multimap U is well posed. Moreover we define the multimap F : I × L 2 (T, C) → P(L 2 (T; C)) in the following way (see (5.6, (5.7))) Finally we introduce the function N : I → C(C(I; L 2 (T; C)); L 2 (T; C)) such that, for every t ∈ I and for every h ∈ C(I; L 2 (T; C)), the map N (t)(h) is so defined (5.12) for every ξ ∈ R.
Clearly, for every t ∈ I and h ∈ C(I; L 2 (T; C)), being the map N (t)(h) constant on T, we have that N (t)(h) ∈ L 2 (T, C) and therefore N is also correctly defined.
At this point let us show that we can apply the Cauchy version of our Theorem 3.5 (see Remark 3.7) .
By recalling the continuity of the function p : I → C, after some standard calculations, we can say that, for every h ∈ C(I; L 2 (T; C)), the map N (.)(h) is even lipschitzian, so N satisfies (N 1). Moreover the continuity of p implies that N has also property (N 2) s . Now we prove that (F N) holds. Indeed, from the definition of the norm in L 2 (T; C) and by bearing in mind hypotheses (f 3), (f 4) and (ϕ), we can write (see (5.10, (5.8) and (5.12))) where the functionα : I → R + belongs to L 1 + (I), since α, f (., 0) C ∈ L 1 + (I) and P = max s∈[0,T ] p(s) C . Therefore we can conclude that (F N) condition is satisfied (with function ψ : R + → R + identically egual to 1).
Moreover, having U compact and convex values, we can say that also F takes compact and convex values.
On the other hand F satisfies hypothesis (F 1), i.e. for every x ∈ L 2 (T; C), F (., x) has a strongly measurable selection. Indeed, from (f 2) and by the separability of L 2 (T; C), we have that, for every x ∈ L 2 (T; C),f (., x) is strongly measurable (see [5], Corollary 3.10.5). Then by the strong measurability of ϕ i , we get thatφ i is also strongly measurable. Further obviously it appears that ϕ i (t) ∈ U (t), for every t ∈ I (see (5.9)). So we can conclude thatf (., x) +φ i is a strongly measurable selection of F (., x) and hence (F 1) holds. Now we prove that F satisfies hypotheses (F 2) and (F 3). To this end, let us consider a set V ⊂ I having null measure and such that inequalities in (f 3) are true.
Firstly, in order to have (F 2), fixed t ∈ I \ V , we define following multimaps Now we show thatf (t, .) is continuous for everyx ∈ L 2 (T; C). Fixed (x n ) n a sequence in L 2 (T; C) such that x n →x in L 2 (T; C), by applying (f 3) we obtain Moreover also H t is upper semicontinuous in L 2 (T; C), being a constant multimap. Since G t and H t have also compact values, we are in the hypotheses of Proposition 2.59 of [12], so we can conclude that the multimap F (t, .) = G t + H t is u.s.c. in L 2 (T; C). Therefore, for almost every t ∈ I, F (t, .) is u.s.c. in L 2 (T; C), i.e. (F 2) holds.
So we can conclude that F satisfies hypothesis (F 3) (with m(t) .
= α(t), ∀ t ∈ I). By means of the arguments above presented, we are in a position to apply the Cauchy version of our Theorem 3.5. Then we can deduce that there exists a continuous functionx : I → L 2 (T; C) that is a mild solution for (5.13) problem, i.e.x which is strongly measurable, beingq andf (., N (.)(x)) strongly measurable (see [13], Theorem 1.3.5).
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Appendix
In this paper X is a Banach space with the norm . X and, if P(X) is the family of all non-empty subsets of X, we will use the following notations: Further, let I = [0, T ] be an interval of the real line endowed with the usual Lebesgue measure μ.
A function f : I → X is said to be strongly measurable if there exists a sequence (f n ) n , f n : I → X, of simple functions which converges to f almost everywhere in I (see [5], Definition 3.10.1 (a)). A function f : I → X is called weakly measurable if for every x * ∈ X * , the R-valued function t →< x * , f(t) > is measurable (where X * is the dual topological space of X) (see [5], Definition 3. 10.1 (b)). Moreover, we denote by C(I; X), the space consisting of all continuous functions from I to X provided with the norm . ∞ of the uniform convergence, by L 1 (I; X) the space of all X-valued Bochner-integrable functions on I with the norm u 1 = T 0 u(t) X dt and by L 1 + (I) = {f ∈ L 1 (I; R) : f (t) ≥ 0, for a.e. t ∈ I}.
A countable set {f n } n ⊂ L 1 (I; X) is said to be semicompact if (i) {f n } n is integrably bounded, i.e. there exists ω ∈ L 1 + (I) such that f n (t) ≤ ω(t), for a.e. t ∈ I and for every n ∈ N; (ii) the set {f n (t)} n is relatively compact in X, for a.e. t ∈ I. A multimap F : I → P(X) is said to be graph measurable if GrF = {(t, x) ∈ I × X : x ∈ F (t)} ∈ Σ × B(X), where Σ is the σ-algebra on I and B(X) is the Borel σ-algebra on X.
A multimap F : I → P b (X) is said to have a strongly measurable selection if there exists a strongly measurable function f : I → X such that f (t) ∈ F (t), for almost everywhere t ∈ I.
Furthermore a multimap F : X → P(X) is called totally bounded if F maps bounded subsets of X into relatively compact sets of X and F is called locally compact if, for every x ∈ X, there exists a neighbourhood U of the point x such that F (U ) is relatively compact in X.
Next the multimap F is called upper semicontinuous (u.s.c.) inx ∈ X if, for every open set A of X such that F (x) ⊂ A, there exists a neighborhood U (x) : F (x) ⊂ A, for every x ∈ U (x) and F : X → P(X) is called lower semicontinuous (l.s.c.) inx ∈ X if, for every open set A of X such that A ∩ F (x) = ∅, there exists a neighborhood U (x) : F (x) ∩ A = ∅, for every x ∈ U (x).
Finally a multimap G : I × X → P(X) is said Scorza-Dragoni lower semicontinuous (S.D.) − l.s.c. if, for every > 0, there exists a closed set T ⊆ I : μ(I \ T ) < such that G| T ×X is lower semicontinuous. Now we enunciate the following well-known fixed point theorem proved by Martelli in [15]. Theorem 6.1. Let X be a separable Banach space and let F : X → P bf c (X) be a multimap such that (i) F is totally bounded and upper semicontinuous; (ii) The set Ω = {x ∈ X : ∃ λ > 1 such that λx ∈ F (x)} is bounded.
Then F has at least a fixed point.
Next, let us denote by O n the zero-element of R n and by the partial ordering given by the standard positive cone R n 0,+ . = (R + 0 ) n , i.e. x y if and only if y − x ∈ R n 0,+ ; clearly, x ≺ y means that x y and x = y. Moreover we present the concept of measure of noncompactness in X (see [13], Definition 2.1.1). Definition 6.2. A function η : P b (X) → R n 0,+ is said to be a measure of noncompactness (MNC, for short) in the Banach space X if, for every Ω ∈ P b (X), the following properties are satisfied: (η 1 ) η(Ω) = 0 if and only ifΩ is compact; (η 2 ) η(co(Ω)) = η(Ω), for all Ω ∈ P b (X).