Abstract
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.
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1 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
where \(\Omega _{1}\), \(\Omega _{2}\) are bounded subsets of \(\mathbb {R}\), \(\Omega =\Omega _{1}\times \Omega _{2}\), \((x_{0},y_{0})\in \Omega \), 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 \(x:[0,T]\rightarrow \mathbb {R}\) is an absolutely continuous function with absolutely continuos derivative \(x'\) and \(u:\Omega \rightarrow \mathbb {R}\) is measurable function with respect to the Borel \(\sigma \)-algebra on \(\Omega \).
Initially, in a series of papers, Sturm and Liouville write any linear second order differential equation in the self adjoint form
and study a problem monitored by the above equation having the boundary conditions
later named Sturm–Liouville problem in the literature.
Subsequently, in the theory of differential equations (inclusions) involving the Sturm–Liouville operator
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 \(\mathbb {R}^{n}\), 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.
2 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
where \(\Omega _{1}\) and \(\Omega _{2}\) are open subsets of \(\mathbb {R}^{n}\), \((x_{0},y_{0})\in \Omega _{1}\times \Omega _{2}\), \(F:\Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}_{k}(\mathbb {R}^{n})\) and \(G:[0,b]\times \Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}(\mathbb {R}^{n})\), \(b>0\), are two maps satisfying the following assumptions (see Appendix for the involved definitions: Definitions 4.2, 4.4, 4.6, 4.8, 4.9):
- \((F_{1})\):
-
\(F:\Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}_{k}(\mathbb {R}^{n})\) is H-upper semicontinuous;
- \((F_{2})\):
-
there exist a linear application \(\gamma :\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) with the property that \(\langle x,\gamma (x)\rangle >0\), for every \(x\in \mathbb {R}^{n}\backslash \{0\}\), and a locally Lipschitz function \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\) \(\beta \)-uniformly regular over \(\Omega _{2}\), with \(\beta \ge 0\), such that
$$\begin{aligned} F(x,y)\subseteq \gamma (\partial ^{C}f(y)),\quad \forall (x,y)\in \Omega _{1}\times \Omega _{2}, \end{aligned}$$where \(\partial ^{C}f(y)\) is the Clarke subdifferential of f at y;
- (G):
-
\(G:[0,b]\times \Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}(\mathbb {R}^{n})\) is a Michael map which satisfies the lower Scorza–Dragoni property and such that for any bounded set \(B\subseteq \Omega _{1}\times \Omega _{2}\) there exists a compact set \(K_{B}\) in \(\mathbb {R}^{n}\) such that
$$\begin{aligned} G(t,x,y)\subseteq K_{B},\quad \forall (t,x,y)\in [0,b]\times B. \end{aligned}$$(1)
By local (A.C.)-solution of (P) we mean an absolutely continuous function \(x:[0,T]\rightarrow \mathbb {R}^{n}\), \(T\in ]0,b]\), with absolutely continuous derivative \(x'\) such that \(x(0)=x_{0}\), \(x'(0)=y_{0}\) and \(x''(t)\in F(x(t),x'(t))+G(t,x(t),x'(t)), \, a.\,e. \,t\in [0,T]\).
Remark 2.1
Let us note that, denoted with \(\Gamma (\mathbb {R}^{n})\) the set of all linear applications \(\gamma :\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) satisfying \(\langle x,\gamma (x)\rangle >0\), for every \(x\in \mathbb {R}^{n}\backslash \{0\}\), in the case in which the considered separable Hilbert space H is \(\mathbb {R}^{n}\), \(\Gamma (\mathbb {R}^{n})\) coincides with the set \(\Gamma (H)\) defined in [10] and consisting of all linear applications \(\gamma :H\rightarrow H\) such that
- \((\gamma )\):
-
for every \(x\in H\backslash \{0\}\), \(\langle x,\gamma (x)\rangle >0\,;\)
- \((\gamma )^{*}\):
-
the restriction of \(\gamma \) to the closed unitary ball of H centered at 0, \(\bar{B}(0,1)\), is continuous from \((\bar{B}(0,1),\tau _{w})\) into H (where \(\tau _{w}\) is the weak topology on H).
Indeed, in the case in which the considered Hilbert space is \(\mathbb {R}^{n}\), a linear application \(\gamma :\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) immediately satisfies \((\gamma )^{*}\) (see [4], Proposition 11.2).
In addition, let us underline the fact that the set \(\Gamma (\mathbb {R}^{n})\) also coincides with the set of all automorphisms of \(\mathbb {R}^{n}\) associated to positive definite matrices.
Firstly, we prove the following result
Theorem 2.1
If \(F:\Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}_{k}(\mathbb {R}^{n})\), \(G:[0,b]\times \Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}(\mathbb {R}^{n})\) satisfy assumptions \((F_{1})\), \((F_{2})\) and (G), then for every \((x_{0},y_{0})\in \Omega _{1}\times \Omega _{2}\) there exist \(T>0\) and a local (A.C.)-solution \(x:[0,T]\rightarrow \mathbb {R}^{n}\) of problem (P).
Proof
First of all we observe that the multifunction \(G:[0,b]\times \Omega _{1}\times \Omega _{2} \rightarrow \mathcal {P}(\mathbb {R}^{n})\) 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 \(g:[0,b]\times \Omega _{1}\times \Omega _{2}\rightarrow \mathbb {R}^{n}\) such that: for every \( (x,y)\in \Omega _{1}\times \Omega _{2},\,\, t\rightarrow g(t,x,y)\) is measurable; for every \(t \in [0,b],\,\, (x,y)\rightarrow g(t,x,y)\) is continuous on \(\Omega _{1}\times \Omega _{2}\) and
Then, if we denote with N the null measure subset of [0, b] given by
we can write (2) in this form
Now, fixed a point \(\bar{z}\in \mathbb {R}^{n}\), we can construct the function \(\hat{g}:\mathbb {R}^{+}_{0}\times \mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) defined by
Since g is a Carathéodory function, it immediately follows that \(\hat{g}\) too is a Carathéodory function.
Furthermore, fixed a bounded set \(B\subseteq \Omega _{1}\times \Omega _{2}\), from assumption (G) we have that there exists a compact set \(K_{B}\) such that
Then, if we set \(K^{*}_{B}=K_{B}\cap \{\bar{z}\}\), we can say that \(K^{*}_{B}\) is a compact set in \(\mathbb {R}^{n}\) such that
Now, we can note that the function \(\gamma \) of assumption \((F_{2})\) belongs to the set \(\Gamma (\mathbb {R}^{n})\) (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 function \(\hat{g}\), we can claim that F and \(\hat{g}\) satisfy all hypothesis of Theorem 3.1 in [10].
Hence, there exist \(T\in ]0,b]\) and a local (A.C.)-solution \(x:[0,T]\rightarrow \mathbb {R}^{n}\) for the Cauchy problem
Consequently, by using the fact that \(\hat{g}(t,\cdot ,\cdot )=g(t,\cdot ,\cdot )\), for a.e. \(t\in [0,b]\), and (2), we can claim that \(x:[0,T]\rightarrow \mathbb {R}^{n}\) is a local (A.C.)-solution for problem (P). \(\square \)
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 \(G:[0,b]\times \Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}(\mathbb {R}^{n})\) that assume closed and convex values. For such multimaps the property (M) is obviously satisfied. Indeed, if G is a multimap assuming values in \(\mathcal {P}_{fc}(\mathbb {R}^{n})\), then for every close set \(Z\subseteq [0,b]\times \Omega _{1}\times \Omega _{2}\) 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 \(\mathbb {R}^{n}\), our Theorem 2.1 strictly contains the result established in [10] by Haddad–Yarou.
Indeed, given any single valued function \(g:\mathbb {R}^{+}_{0}\times \Omega _{1}\times \Omega _{2}\rightarrow \mathbb {R}^{n}\) having the property \((H_{3})\) required in Theorem 3.1 by the authors of [10], we can consider the multimap \(G:[0,b]\times \Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}_{k}(\mathbb {R}^{n})\) defined as follows
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.
Next, as a consequence of Theorem 2.1, we establish the following
Corollary 2.2
Let \(F:\Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}_{k}(\mathbb {R}^{n})\) be a multimap that satisfies properties \((F_{1})\) and
- \((\hat{F}_{2})\):
-
there exists a proper, convex, locally Lipschitz and lower semicontinuous function \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\) such that
$$\begin{aligned} F(x,y)\subseteq \partial f(y),\quad \forall (x,y)\in \Omega _{1}\times \Omega _{2}, \end{aligned}$$(3)where \(\partial f(y)\) is the subdifferential of f at y.
Let \(G:\mathbb {R}^{+}_{0}\times \Omega _{1}\times \Omega _{2}\rightarrow \mathcal {P}(\mathbb {R}^{n})\) be a multimap satisfying assumption (G). Then, for every \( (x_{0},y_{0})\in \Omega _{1}\times \Omega _{2}\), there exist \(T>0\) and a local (A.C.)-solution \(x:[0,T]\rightarrow \mathbb {R}^{n}\) of problem (P).
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 \((\hat{F}_{2})\) is proper, convex and lower semicontinuous, we can say that f is \(\beta \)-uniformly regular over \(\Omega _{2}\) with \(\beta \equiv 0\) (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 \(\partial ^{P}f(y)\) 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 \(\xi \in \partial ^{P}f(x)\). Then we can write that there exist \(\sigma ,\eta >0\) for which
Let \(y\in \mathbb {R}^{n}\). Then there exists \(\rho \in ]0,1[\) such that \(ty+(1-t)x\in B_{\mathbb {R}^{n}}(x,\eta )\), for every \(t\in ]0,\rho ]\). By using (6), we have that, for every \(t\in ]0,\rho ]\),
that is
Then, by using the convexity of f and by dividing for \(t\in ]0,\rho ]\), (7) becomes
Letting \(t\downarrow 0\) in (8), we have that
that is \(\xi \in \partial f(x)\). Then also
Hence, by using (4) and (5), we can say that
Now, setting \(\gamma =1_{\mathbb {R}^{n}}\in \Gamma (\mathbb {R}^{n})\), where \(1_{\mathbb {R}^{n}}\) the identical function on \(\mathbb {R}^{n}\), (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). \(\square \)
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].
3 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 \(\Omega _{1}\) and \(\Omega _{2}\) are two open and bounded subsets of \(\mathbb {R}\), \(\Omega =\Omega _{1}\times \Omega _{2} \), \((x_{0},y_{0})\in \Omega \), \(\lambda , b\in \mathbb {R}^{+}\) and \(p:[0,b]\rightarrow \mathbb {R}\), \(q:[0,b]\rightarrow \mathbb {R}\), \(H:\Omega \rightarrow \mathcal {P}_{fc}(\mathbb {R})\), \(V:\Omega \rightarrow \mathcal {P}_{k}(\mathbb {R})\), \(U:\Omega \rightarrow \mathcal {P}_{fc}(\mathbb {R})\) satisfy the following properties:
- (p):
-
\(p\in C^{1}([0,b])\) and \(p(t)>0\), for every \(t\in [0,b]\);
- (q):
-
\(q\in C([0,b])\);
- \((V_{1})\):
-
\(V:\Omega \rightarrow \mathcal {P}_{k}(\mathbb {R})\) is H-upper semicontinuous;
- \((V_{2})\):
-
there exists \(\gamma \in \Gamma (\mathbb {R})\) and a locally Lipschitz, \(\beta \)-uniformly regular, with \(\beta \ge 0\), function \(f:\mathbb {R}\rightarrow \mathbb {R}\) over \(\Omega _{2}\) such that
$$\begin{aligned} V(x,y)\subseteq \gamma (\partial ^{C}f(y)),\quad \forall (x,y)\in \Omega _{1}\times \Omega _{2}, \end{aligned}$$where \(\partial ^{C}f(y)\) 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 \(x:[0,T]\rightarrow \mathbb {R}\) is an absolutely continuous function with absolutely continuous derivative \(x'\) and \(u:\Omega \rightarrow \mathbb {R}\) is a \(B(\Omega )\)-measurable function.
Theorem 3.1
Under the assumptions (p), (q), \((V_{1})\), \((V_{2})\) and (HU) there exists a local admissible pair \(\{x,u\}\) for problem (SLP).
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 selection \(\bar{u}:\Omega \rightarrow \mathbb {R}\) of U. The function \(\bar{u}\) is also \(B(\Omega )\)-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 \(G:[0,b]\times \Omega \rightarrow \mathcal {P}(\mathbb {R})\) 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 \(\mathcal {P}_{fc}(\mathbb {R})\), 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 \([0,b]\times \Omega \). Indeed, by using assumptions (p), (q) and the continuity of the function \(\bar{u}\), the so defined function
is obviously continuous in \([0,b]\times \Omega \). On the other hand, the multimap defined by setting
is lower semicontinuous in \([0,b]\times \Omega \). To see this, fix \((\bar{t},\bar{x},\bar{y})\in [0,b]\times \Omega \). Let \(((t_{n},x_{n},y_{n}))_{n}\) be a sequence in \([0,b]\times \Omega \) with the property that \((t_{n},x_{n},y_{n})\rightarrow (\bar{t},\bar{x},\bar{y})\) in \([0,b]\times \Omega \). We want to prove that
To show that (11) holds, fix \(\bar{z}\in K(\bar{t},\bar{x},\bar{y})\). Then we can write (see (10))
where \(\bar{w}\in H(\bar{x},\bar{y})\).
Of course, since \((t_{n},x_{n})\rightarrow (\bar{t},\bar{x})\) in \([0,b]\times \Omega _{1}\), by using (H), we have that
Then, since \(\bar{w}\in H(\bar{x},\bar{y})\), by using (12), there exists a sequence \((w_{n})_{n}\), \(w_{n}\in H(x_{n},y_{n})\) for every \(n\in \mathbb {N}\), such that \(w_{n}\rightarrow \bar{w}\).
In addition, since \((t_{n})_{n}\) converges to \(\bar{t}\), the continuity of p given by (p) implies that \(p(t_{n})\rightarrow p(\bar{t})\).
At this point, the sequence \((u_{n})_{n}\) where
is such that \(u_{n}\in K(t_{n},x_{n},y_{n})\), for every \(n\in \mathbb {N}\) and \(u_{n}\rightarrow -\frac{\lambda }{p(\bar{t})}\bar{w}=\bar{z}\) in \(\mathbb {R}\). Hence, we can conclude that \(\bar{z}\in \underline{\lim }K(t_{n},x_{n},y_{n})\) and, as a consequence, that (11) holds. This means that K is lower semicontinuous at the point \((\bar{t},\bar{x},\bar{y})\in [0,b]\times \Omega \) (see Proposition 1.2.6 in [12]). Thus, from the arbitrariness of \((\bar{t},\bar{x},\bar{y})\), we can deduce the lower semicontinuity of the multimap K in \([0,b]\times \Omega \).
By applying Proposition 1.2.59 in [12], we can say that G is lower semicontinuous in \([0,b]\times \Omega \). Therefore, G satisfies the lower Scorza–Dragoni property (see, here Definition 4.8).
Finally, since the set \(\Omega \) is bounded, it is sufficient to prove the existence of a compact subset K of \(\mathbb {R}\) such that
By taking into account of (p), (q), (HU) and by recalling that \(\Omega _{1}\), \(\Omega _{2}\) are bounded, it is easy to chech that there exists \(M>0\) such that
Hence (13) holds with \(K=\bar{B}_{\mathbb {R}}(0,M)\).
We have completely shown that the multimap G satisfies hypothesis (G) of Theorem 2.1.
Now, turn our attention to the multimap
First of all, the multimap F is H-upper semicontinuous in \(\Omega \). Indeed, fixed \((\bar{x},\bar{y})\in \Omega \) and \(\epsilon >0\), since V is H-upper semicontinuous, in correspondence of \(\frac{\epsilon }{\lambda }>0\), there exists a neighbourhood I of \((\bar{x},\bar{y})\) such that
Hence, multiplying by \(\lambda \) in (14), we have that
which means the H-upper semicontinuity of F at \((\bar{x},\bar{y})\). From the arbitrariness of \((\bar{x},\bar{y})\), we can claim that assumption \((F_{1})\) of Theorem 2.1 holds.
Now, we know from hypothesis \((V_{2})\) that
So, if we define \(\bar{\gamma }:\mathbb {R}\rightarrow \mathbb {R}\) given by \(\bar{\gamma }=\lambda \gamma \), we have that
where \(\bar{\gamma }\) is a linear application belonging to the space \(\Gamma (\mathbb {R})\). 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 \(x:[0,T]\rightarrow \mathbb {R}\) for problem (SLC).
Now, this solution x satisfies
This concludes our proof since we have shown the existence of a pair \(\{x,\bar{u}\}\) satisfying \((SLP)^{*}\) such that \(x:[0,T]\rightarrow \mathbb {R}\) is absolutely continuous with absolutely continuous derivative and \(\bar{u}:\Omega \rightarrow \mathbb {R}\) is \(B(\Omega )\)-measurable, i.e. \(\{x,u\}\) is a local admissible pair for problem \((SLP)^{*}\) and, as a consequence, for (SLP).
\(\square \)
From Theorem 3.1 can be obviously derived the result referring to the case of lack of control which it is worth to enunciate.
Theorem 3.2
Under the assumptions (p), (q), \((V_{1})\), \((V_{2})\) and (H) there exists a local (A.C.)-solution for the Cauchy problem
Furthermore, if the multimap V is also bounded, every local (A.C.)-solution of (SLP) belongs to the Sobolev space \(W^{2,\infty }([0,T],\mathbb {R})\).
Proof
Trivially, Theorem 3.1 guarantees the existence of a local (A.C.)-solution \(x:[0,T]\rightarrow \mathbb {R}\) for the Cauchy problem (SLP).
We claim that \(x\in W^{2,\infty }([0,T],\mathbb {R})\). Indeed, being \(x,x'\) absolutely continuous in the compact interval [0, T], from Weierstrass Theorem we can say that \(x,x'\in L^{\infty }([0,T],\mathbb {R})\). That is \(x\in W^{1,\infty }([0,T],\mathbb {R})\).
On the other hand, we know that \(x''\) is measurable. Therefore, by using the boundness of H and V, we can conclude that \(x''\in L^{\infty }([0,T],\mathbb {R})\), i.e. \(x\in W^{2,\infty }([0,T],\mathbb {R})\). \(\square \)
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 \(x:[0,T]\rightarrow \mathbb {R}\) in \(W^{2,\infty }([0,T],\mathbb {R})\) 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 \(V\equiv \{0\}\) 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.
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Acknowledgements
This research is carried out within the national group GNAMPA of INdAM. The first author is partially supported by the Department of Mathematics and Computer Science of the University of Perugia (Italy) and by the projects “Fondi di funzionamento per la ricerca dipartimentale-Anno 2021”, “Metodi della Teoria dell’Approssimazione, Analisi Reale, Analisi Nonlineare e loro applicazioni” and “Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni”, funded by 2018 and 2019 basic research fund of the University of Perugia, and by a 2020 GNAMPA-INdAM Project “Processi evolutivi con memoria descrivibili tramite equazioni integro-differenziali”.
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Open access funding provided by Universitá degli Studi di Perugia within the CRUI-CARE Agreement. The authors have not disclosed any funding.
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Appendix
Appendix
In this paper we shall denote with \(\langle .,.\rangle \) the scalar product on \(\mathbb {R}^{n}\) and with \(\Vert .\Vert \) the norm on \(\mathbb {R}^{n}\). Moreover, \(B_{\mathbb {R}^{n}}(x,r)\) will refer to the open ball in \(\mathbb {R}^{n}\) centered at \(x\in \mathbb {R}^{n}\) with radius \(r>0\).
Given a function \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\cup \{+\infty \}\), the effective domain of f is the so defined set
Definition 4.1
([15, §1.3]). Let \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\cup \{+\infty \}\) be a convex function. For every \(x\in domf\), the set
is called the subdifferential of f at x.
Definition 4.2
([9, §2.1]). Let \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\cup \{+\infty \}\) be a lower semicontinuous function. The Clarke subdifferential of f at \(x\in domf\) is the subset \(\partial ^{C}f(x)\) of \(\mathbb {R}^{n}\) defined by
where \(f^{\uparrow }(x;y)\) is the generalized Rockafellar directional derivative of f at x with respect to the vector y given by
Definition 4.3
([9, §1.2]). Let \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\cup \{+\infty \}\) be a lower semicontinuous function. The proximal subdifferential of f at \(x\in domf\) is the subset \(\partial ^{P}f(x)\) of \(\mathbb {R}^{n}\) consisting of all the vectors \(\xi \in \mathbb {R}^{n}\) such that there exist \(\sigma ,\eta >0\) for which
Definition 4.4
([10, Definition 2.1]). Let \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\cup \{+\infty \}\) be a lower semicontinuous function and let \(\Omega \) be a nonempty open subset of \(\mathbb {R}^{n}\) with the property that \(\Omega \subseteq domf\). The function f is said to be \(\beta \)-uniformly regular over \(\Omega \) if there exists \(\beta \ge 0\) such that for every \(x\in \Omega \) and for all \(\xi \in \partial ^{P}f(x)\) one has
Proposition 4.1
([10, Proposition 2.4], (ii)). Let \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\) be a locally Lipschitz function and let \(\Omega \) be a nonempty open set in \(\mathbb {R}^{n}\). If f is uniformly regular over \(\Omega \), then the proximal subdifferential of f at x, \(\partial ^{P}f(x)\), coincides with the Clarke subdifferential of f at x, \(\partial ^{C}f(x)\), at any point \(x\in \mathbb {R}^{n}\).
Now, let X be a topological space and Y be a linear topological space. In the sequel we will use the following notations:
Definition 4.5
([12, §1.2]). A multimap \(F:X\rightarrow \mathcal {P}(Y)\) is said to be lower semicontinuous at \(x_{0}\in X\) if, for every \(\,\Omega =\mathring{\Omega }\subseteq Y\) with \(F(x_{0})\cap \Omega \ne \emptyset \), there exists a neighbourhood V of \(x_{0}\) such that \(F(x)\cap \Omega \ne \emptyset , \,\,\forall x\in V\).
If \(Y=(Y,d)\) is a metric space, then we can define the concept of ball centered at \(S\subseteq Y\) with radius \(\epsilon \) as
where
Definition 4.6
([12, §1.2]). A multimap \(F:X\rightarrow \mathcal {P}(Y)\) is said to be H-upper semicontinuous in \(x_{0}\in X\) if \(\forall \,\epsilon >0\) there exists a neighbourhood V of \(x_{0}\) such that \(F(x)\subseteq B(F(x_{0}),\epsilon ),\,\,\forall x\in V\).
Definition 4.7
(Definition 1.1.42 in [12]). Let \(Y=(Y,d)\) be a metric space. Let \((A_{n})_{n}\) be a sequence in \(\mathcal {P}_{f}(Y)\). The Kuratowski limit inferior of the sequence \((A_{n})_{n}\) is the set defined by
Remark 4.1
(Remark 1.1.43 in [12]) The Kuratowski limit inferior of a sequence \((A_{n})_{n}\subseteq \mathcal {P}_{f}(Y)\) has also this expression
Definition 4.8
(see [5]). If \((T,\tau ,\mu )\) is a finite measure space and X, Y are topological spaces, a multimap \(F:T\times X\rightarrow \mathcal {P}(Y)\) satisfies the lower Scorza–Dragoni property if (SD) for every \(\epsilon >0\) there exists a compact \(K_{\epsilon }\subseteq T\) such that \(\mu (T{\setminus } K_{\epsilon })<\epsilon \) and F restricted to \(K_{\epsilon }\times X\) is lower semicontinuous.
Definition 4.9
(see [5]). If \((T,\tau ,\mu )\) is a finite measure space, a multifunction and X, Y are topological spaces, a multimap \(F:T\times X\rightarrow \mathcal {P}(Y)\) is said to be a Michael map if it satisfies the property (M) , i.e. for every closed set \(Z\subseteq T\times X\) 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 \(f:Z\rightarrow Y\) such that \(f(t,x)\in F(t,x),\,\,\forall (t,x)\in Z)\).
Theorem 4.2
(Theorem 3.1 in [5]). Let T, X and Y be Hausdorff topological spaces and let \(\mu \) be a Radon measure on T such that \(\mu (T)<+\infty \). If \(F:T\times X\rightarrow \mathcal {P}(Y)\) is a multimap satisfying properties (SD) and (M), then F has a Carathéodory selection, i.e. there exists a function \(f:T\times X\rightarrow Y\) such that:
-
1.
for every \( t\in T\), \(f(t,\cdot )\) is continuous on X;
-
2.
for every \( x\in X\), \(f(\cdot ,x)\) is measurable;
-
3.
for \(\mu \)-a.e. \(t\in T\) and for every \( x\in X\), \(f(t,x)\in F(t,x)\).
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Cardinali, T., Continelli, E. Sturm–Liouville Differential Inclusions with Set-Valued Reaction Term Depending on a Parameter. Results Math 78, 80 (2023). https://doi.org/10.1007/s00025-023-01857-y
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DOI: https://doi.org/10.1007/s00025-023-01857-y
Keywords
- Sturm–Liouville differential inclusions
- nonlinear second order differential inclusions
- Clarke subdifferential
- controllability