Sturm–Liouville Differential Inclusions with Set-Valued Reaction Term Depending on a Parameter

In this paper we study the controllability for a Cauchy problem governed by a nonlinear differential inclusion driven by a Sturm–Liouville type operator. In particular, the considered second order differential inclusion involves a set-valued reaction term depending on a parameter. The key tool in the proof of the controllability result we provide is a multivalued version of the theorem recently proved by Haddad–Yarou, here established for an initial conditions problem monitored by a nonlinear second order differential inclusion presenting the sum of two multimaps on the right-hand side. We thereby deduce the existence of a local admissible pair for the considered control problem, that is the existence of a couple of functions consisting of a control, which is a measurable function, and the correspondent trajectory, which is an absolutely continuous function with absolutely continuous derivative. Secondly, under appropriate assumptions on the involved multimaps, we obtain an increased regularity for the solutions produced by our existence result. This regularity is the same of that recently tested by Bonanno, Iannizzotto and Marras for a different type of problem, which however involves the Sturm–Liouville operator.


Introduction
In this paper, we investigate about the controllability of the following Cauchy problem governed by a nonlinear differential inclusion driven by a type Sturm-Liouville operator involving a set-valued reaction term depending on a parameter u(x(t), x (t)) ∈ U (x(t), x (t)) where Ω 1 , Ω 2 are bounded subsets of R, Ω = Ω 1 × Ω 2 , (x 0 , y 0 ) ∈ Ω, p, q, H, V are suitable maps and U is the multimap which allows the control of the present problem. To be specific, we prove the existence of a local admissible pair for problem (SLP ), that is the existence of a pair {x, u} satisfying the initial conditions in (SLP ) and such that x : [0, T ] → R is an absolutely continuous function with absolutely continuos derivative x and u : Ω → R is measurable function with respect to the Borel σ-algebra on Ω. Initially, in a series of papers, Sturm and Liouville write any linear second order differential equation in the self adjoint form one denotes the considered differential equation/inclusion as a type Sturm-Liouville differential equation/inclusion. Although the subject of Sturm-Liouville is over 180 years old, a surprising number of reasults referring to Sturm-Louville differential equations have recent origin ( [1,2,[6][7][8]). In these works, the authors achieve the existence of solutions for different typologies of problems governed by differential equations or inclusions involving Sturm-Liouville operators. The investigations in this direction is stimulated by problems related to mathematical physics, physics, engineering and others. For instance, the development of quantum mechanics in the 1920s and 1930s (the one-dimensional time-indipendent Schrödinger equation is a special case of a Sturm-Liouville equation) and the proof of the general spectral theorem due to von Neumann and Stone provide reasons for deepening the theory of Sturm-Lioville differential equations/inclusions. The aim of this paper is twofold. On one hand, we first obtain the existence of local absolutely continuous solutions with absolutely continuous derivative for a Cauchy problem governed by the following nonlinear second order differential inclusion presenting the sum of two multimaps on the right-hand side By doing so, we give a multivalued version of the Haddad-Yarou result proved in [10] (see Theorem 2.1). In the case in which the considered Hilbert space is R n , our Theorem 2.1 strictly contains the existence theorem of [10]. Moreover, we provide a result of the Lupulescu type (see [13], Corollary 2.2 and Remark 3.1). On the other hand, we reach the target of finding a local admissible pair for problem (SLP ) by employing our Theorem 2.1 related to problem (P ) and by relying on the classical Michael Selection Theorem. We also obtain a result, again concerning a Cauchy problem monitored by a Sturm-Liouville differential inclusion, which let us find solutions having a better regularity than the one recently established in [6][7][8], but having the same regularity recently tackled in [1] and [2] by Bonanno, Iannizzotto and Marras for another kind of problem, however guided by a type Sturm-Liouville operator. Finally, Appendix contains some background material with the intent of making the paper self-standing.

Existence of solutions for second order differential inclusions
In this section we present a result of existence of solutions for the following Cauchy problem governed by a nonlinear second order differential inclusion presenting the sum of two multimaps on the right-hand side there exist a linear application γ : R n → R n with the property that x, γ(x) > 0, for every x ∈ R n \{0}, and a locally Lipschitz function f : R n → R β-uniformly regular over Ω 2 , with β ≥ 0, such that where ∂ C f (y) is the Clarke subdifferential of f at y; is a Michael map which satisfies the lower Scorza-Dragoni property and such that for any bounded set B ⊆ Ω 1 ×Ω 2 there exists a compact set K B in R n such that By local (A.C.)-solution of (P ) we mean an absolutely continuous function Remark 2.1. Let us note that, denoted with Γ(R n ) the set of all linear applications γ : R n → R n satisfying x, γ(x) > 0, for every x ∈ R n \{0}, in the case in which the considered separable Hilbert space H is R n , Γ(R n ) coincides with the set Γ(H) defined in [10] and consisting of all linear applications γ : Indeed, in the case in which the considered Hilbert space is R n , a linear application γ : R n → R n immediately satisfies (γ) * (see [4], Proposition 11.2). In addition, let us underline the fact that the set Γ(R n ) also coincides with the set of all automorphisms of R n associated to positive definite matrices.
Firstly, we prove the following result Proof. First of all we observe that the multifunction G : [0, b] × Ω 1 × Ω 2 → P(R n ) satisfies the assumptions of Theorem 4.2 in Appendix. Therefore, we can say that G has a Carathéodory selection. It means that there exists a function g : Then, if we denote with N the null measure subset of [0, b] given by we can write (2) in this form Vol. 78 (2023) Now, fixed a pointz ∈ R n , we can construct the functionĝ : Since g is a Carathéodory function, it immediately follows thatĝ too is a Carathéodory function. Furthermore, fixed a bounded set B ⊆ Ω 1 × Ω 2 , from assumption (G) we have that there exists a compact set K B such that we can note that the function γ of assumption (F 2 ) belongs to the set Γ(R n ) (see Remark 2.1). Therefore, combining the fact that F assumes compact values and has properties (F 1 ), (F 2 ) with what have been just proved for the functionĝ, we can claim that F andĝ satisfy all hypothesis of Theorem 3.1 in [10]. Hence, there exist T ∈]0, b] and a local ( Consequently, by using the fact thatĝ(t, ·, ·) = g(t, ·, ·), for a.e. t ∈ [0, b], and (2), we can claim that x : [0, T ] → R n is a local (A.C.)-solution for problem (P ).

Remark 2.2.
Since the property (M) is difficult to check, we want to remember a class of functions that surely satisfies it. It is enough to consider multimaps G : [0, b] × Ω 1 × Ω 2 → P(R n ) that assume closed and convex values. For such multimaps the property (M) is obviously satisfied. Indeed, if G is a multimap assuming values in P fc (R n ), then for every close set Z ⊆ [0, b] × Ω 1 × Ω 2 such that G restricted to Z is lower semicontinuous, from Michael Theorem we can say that G has a continuous selection on Z. Remark 2.3. Let us note that, in the case in which the considered Hilbert space is R n , our Theorem 2.1 strictly contains the result established in [10] by Haddad-Yarou. Indeed, given any single valued function g : R + 0 ×Ω 1 ×Ω 2 → R n having the property (H 3 ) required in Theorem 3.1 by the authors of [10], we can consider the multimap G : Obviously G is a Michael map and, furthermore, G has the lower Scorza-Dragoni property (see [12], Proposition 2.7.16). Thus, G satisfies assumption (G) of Theorem 2.1.
where ∂f (y) is the subdifferential of f at y. Proof. First of all, we prove that the multimap F satisfies assumptions of Theorem 2.1. Of course, property (F 1 ) holds. So, all that remains is to check the validity of hypothesis (F 2 ). Since the function f of hypothesis (F 2 ) is proper, convex and lower semicontinuous, we can say that f is β-uniformly regular over Ω 2 with β ≡ 0 (see [10], Example 2.2). Therefore, being in addition the function f lower semicontinuous and locally Lipschitz, by using Proposition 4.1 of Appendix, we can write where ∂ P f (y) is the proximal subdifferential of f at y (see Appendix, Definition 4.3). Furthermore, being f a convex function too, one has that Indeed, from Definitions 4.1 and 4.3, it is obviously true that Conversely, let ξ ∈ ∂ P f (x). Then we can write that there exist σ, η > 0 for which Let y ∈ R n . Then there exists ρ ∈]0, 1[ such that ty + (1 − t)x ∈ B R n (x, η), for every t ∈]0, ρ]. By using (6), we have that, for every t ∈]0, ρ], Then, by using the convexity of f and by dividing for t ∈]0, ρ], (7) becomes Letting t ↓ 0 in (8), we have that that is ξ ∈ ∂f (x). Then also Hence, by using (4) and (5), we can say that Now, setting γ = 1 R n ∈ Γ(R n ), where 1 R n the identical function on R n , (3) becomes (see (9)) So assumption (F 2 ) of Theorem 2.1 holds. Then, by applying Theorem 2.1, we can deduce the existence of a local (A.C.)-solution for problem (P ).

Remark 2.4. Let us note that Corollary 2.2 is a multivalued version of the
Lupulescu Theorem proved in [13]. Indeed, using again Theorem 4.2 of Appendix, we have the existence of a Carathéodory selection g of the multimap G, that is g is a function of the type of [13].

Controllability for Sturm-Liouville differential inclusions
Now we present a controllability result concerning the following Cauchy problem governed by a Sturm-Liouville differential inclusion where Ω 1 and Ω 2 are two open and bounded subsets of R, Ω = Ω 1 × Ω 2 , (x 0 , y 0 ) ∈ Ω, λ, b ∈ R + and p : [0, b] → R, q : [0, b] → R, H : Ω → P fc (R), V : Ω → P k (R), U : Ω → P fc (R) satisfy the following properties: there exists γ ∈ Γ(R) and a locally Lipschitz, β-uniformly regular, with where ∂ C f (y) is the Clarke subdifferential of f at y; (HU ) H and U are bounded and lower semicontinuous. We recall that local admissible pair for problem (SLP ) is a couple {x, u} satisfying (SLP ) such that x : [0, T ] → R is an absolutely continuous function with absolutely continuous derivative x and u : Ω → R is a B(Ω)-measurable function. Proof. First of all we note that, thanks to assumption (p), in order to prove our existence result it will be enough to find a local adsmissible pair for problem Now, it is immediate to note that the multimap U satisfies assumptions of classical Michael selection Theorem (see [14], Theorem 3.2 ). So there exists a continuous selectionū : Ω → R of U . The functionū is also B(Ω)-measurable from Corollary 3.11.5 in [11]. Therefore, let us focus on the Cauchy problem Next, our goal will be to establish the existence of a local (A.C.)-solution for (SLC). To this aim, let G : [0, b] × Ω → P(R) be the so defined multimap We want to show that G satisfies hypothesis (G) of Theorem 2.1. Since the multimap H takes values in P fc (R), we can immediately say that G assumes convex and closed values. Then, from Remark 2.2, one has that G is a Michael map. Moreover, the multifunction G is lower semicontinuous in [0, b] × Ω. Indeed, by using assumptions (p), (q) and the continuity of the functionū, the so defined function On the other hand, the multimap defined by setting is lower semicontinuous in [0, b] × Ω. To see this, fix (t,x,ȳ) ∈ [0, b] × Ω. Let ((t n , x n , y n )) n be a sequence in [0, b] × Ω with the property that (t n , x n , y n ) → (t,x,ȳ) in [0, b] × Ω. We want to prove that To show that (11) holds, fixz ∈ K(t,x,ȳ). Then we can write (see (10)) (12) Then, sincew ∈ H(x,ȳ), by using (12), there exists a sequence (w n ) n , w n ∈ H(x n , y n ) for every n ∈ N, such that w n →w. In addition, since (t n ) n converges tot, the continuity of p given by (p) implies that p(t n ) → p(t). At this point, the sequence (u n ) n where is such that u n ∈ K(t n , x n , y n ), for every n ∈ N and u n → − λ p(t)w =z in R. Hence, we can conclude thatz ∈ limK(t n , x n , y n ) and, as a consequence, that (11) holds. This means that K is lower semicontinuous at the point (t,x,ȳ) ∈ [0, b]×Ω (see Proposition 1.2.6 in [12]). Thus, from the arbitrariness of (t,x,ȳ), we can deduce the lower semicontinuity of the multimap K in [0, b] × Ω. By applying Proposition 1.2.59 in [12], we can say that G is lower semicontinuous in [0, b]×Ω. Therefore, G satisfies the lower Scorza-Dragoni property (see, here Definition 4.8). Finally, since the set Ω is bounded, it is sufficient to prove the existence of a compact subset K of R such that By taking into account of (p), (q), (HU ) and by recalling that Ω 1 , Ω 2 are bounded, it is easy to chech that there exists M > 0 such that Hence ( Hence, multiplying by λ in (14), we have that which means the H-upper semicontinuity of F at (x,ȳ). From the arbitrariness of (x,ȳ), we can claim that assumption (F 1 ) of Theorem 2.1 holds. Now, we know from hypothesis (V 2 ) that So, if we defineγ : R → R given byγ = λγ, we have that whereγ is a linear application belonging to the space Γ(R). So (F 2 ) is true. We are in position to apply our Theorem 2.1 which guarantees the existence of a local (A.C.)-solution x : [0, T ] → R for problem (SLC). Now, this solution x satisfies Furthermore, if the multimap V is also bounded, every local (A.C.)-solution of (SLP ) belongs to the Sobolev space W 2,∞ ([0, T ], R).
Proof. Trivially, Theorem 3.1 guarantees the existence of a local (A.C.)-solution x : [0, T ] → R for the Cauchy problem (SLP ). We claim that x ∈ W 2,∞ ([0, T ], R). Indeed, being x, x absolutely continuous in the compact interval [0, T ], from Weierstrass Theorem we can say that x, x ∈ L ∞ ([0, T ], R). That is x ∈ W 1,∞ ([0, T ], R). On the other hand, we know that x is measurable. Therefore, by using the boundness of H and V , we can conclude that Remark 3.1. We have seen that, by requiring the further hypothesis of boundness on the multimap V , Theorem 3.2 guarantees the existence of a solution x : [0, T ] → R in W 2,∞ ([0, T ], R) for the Cauchy problem (SLP ) governed by a Sturm-Liouville differential inclusion. The study about solutions of this regularity has been recently tackled in the case V ≡ {0} in [1], [2]. It goes without saying that the authors of the aforementioned works require assumptions on the involved maps that are different from those in Theorem 3.2. Now, let X be a topological space and Y be a linear topological space. In the sequel we will use the following notations:    [12]) The Kuratowski limit inferior of a sequence (A n ) n ⊆ P f (Y ) has also this expression limA n = {y ∈ Y : y = lim n→+∞ a n , a n ∈ A n ∀n ∈ N}.
Definition 4.8 (see [5]). If (T, τ, μ) is a finite measure space and X, Y are topological spaces, a multimap F : T × X → P(Y ) satisfies the lower Scorza-Dragoni property if (SD) for every > 0 there exists a compact K ⊆ T such that μ(T \K ) < and F restricted to K × X is lower semicontinuous. Definition 4.9 (see [5]). If (T, τ, μ) is a finite measure space, a multifunction and X, Y are topological spaces, a multimap F : T ×X → P(Y ) is said to be a Michael map if it satisfies the property (M ) , i.e. for every closed set Z ⊆ T ×X such that F restricted to Z is lower semicontinuous, there exists a continuous selection of F on Z (i.e. there exists a continuous function f : Z → Y such that f (t, x) ∈ F (t, x), ∀(t, x) ∈ Z).