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On the Cauchy problem for lower semicontinuous differential inclusions
Advances in Difference Equations volume 2016, Article number: 214 (2016)
Abstract
We provide a new proof of a classical result by Bressan on the Cauchy problem for firstorder differential inclusions with null initial condition. Our approach allows us to prove the result directly for kth order differential inclusions, under weaker regularity assumptions on the involved multifunction. Our result is the following: let a, b, M be positive real numbers, with \(M\cdot\max\{a,a^{k}\}\le b\), and let B and X be the closed balls in \(\mathbf{R}^{n}\), centered at the origin with radius b and M, respectively. Let \(F:[0,a]\times B^{k}\to2^{X}\) be a multifunction with nonempty closed values, such that F is \(\mathcal{L}([0,a])\otimes\mathcal{B}(B^{k})\)measurable, and for all \(t\in[0,a]\) the multifunction \(F(t,\cdot)\) is lower semicontinuous. Then there exists \(u\in W^{k,\infty}([0,a],\mathbf{R}^{n})\) such that \(u^{(k)}(t)\in F(t,u(t), u^{\prime}(t),\ldots,u^{(k1)}(t))\) a.e. in \([0,a]\), and \(u^{(i)}(0)=0_{\mathbf{R}^{n}}\) for all \(i=0,\ldots, k1\).
Introduction
The importance of the theory of differential inclusions is well documented in the literature. Indeed, they play a crucial role in the study of many dynamical problems coming from economics, social sciences, biology. They also provide a fundamental tool in control theory. Moreover, differential inclusions are very useful when studying differential equations with discontinuous righthand side. For a detailed introduction to differential inclusions and their applications, we refer the reader to [1, 2].
In the paper [3], A. Bressan proved his classical and celebrated result on the Cauchy problem for firstorder differential inclusions, which we now state. In what follows, the space \(\mathbf{R}^{n}\) is considered with its Euclidean norm.
Theorem 1.1
(Theorem 2 of [3])
Let \(a>0\), and let B and X be the closed balls in \(\mathbf{R}^{n}\), centered at the origin with radius b and M, respectively, with \(a\le b/M\). Let \(F: [0,a]\times B\to2^{X}\) be a lower semicontinuous multifunction with nonempty closed values.
Then there exists an absolutely continuous \(u:[0,a]\to\mathbf{R}^{n}\) such that \(u(0)=0_{\mathbf{R}^{n}}\) and \(u^{\prime}(t)\in F(t,u(t))\) for a.e. \(t\in[0,a]\).
The aim of this note is simply to propose an alternative proof of Theorem 1.1, completely independent from the original one. Our approach also allows one to state Theorem 1.1 directly for kthorder differential inclusions, under slightly weaker regularity assumptions on F. That is, the joint lower semicontinuity of the multifunction F is replaced by joint measurability and by the lower semicontinuity of \(F(t,\cdot )\). In this connection, we observe that the original proof of Theorem 1.1 does not work if F is not jointly lower semicontinuous (see the Remark at p.96 of [3]).
Before stating our result, we fix some notations. If \(a>0\), \(k,n\in \mathbf{N} \) and \(p\in[1,+\infty]\), we denote by \(W^{k,p}([0,a],\mathbf{R}^{n})\) the set of all \(u\in C^{k1}([0,a],\mathbf{R}^{n})\) such that \(u^{(k1)}\) is absolutely continuous in \([0,a]\) and \(u^{(k)}\in L^{p}([0,a],\mathbf{R}^{n})\). Moreover, we denote by \(\mathcal{L}([0,a])\) and \(\mathcal{B}(S)\) the family of all measurable subsets of \([0,a]\) and the Borel family of the topological space S, respectively. The following is our result.
Theorem 1.2
Let a, b, M be three positive real numbers, with \(M\cdot\max\{a,a^{k}\}\le b\). Let X and B be as in Theorem 1.1, and let \(F:[0,a]\times B^{k}\to2^{X}\) be a multifunction with nonempty closed values. Assume that F is \(\mathcal{L}([0,a])\otimes\mathcal {B}(B^{k})\)measurable, and that for all \(t\in[0,a]\) the multifunction \(F(t,\cdot )\) is lower semicontinuous. Then there exists a function \(u\in W^{k,\infty}([0,a],\mathbf{R}^{n})\) such that
The proof of Theorem 1.2, which will be given in Section 3, will follow from a more general result, Theorem 3.1 below. It is mainly based on a recent existence result for discontinuous selections and on an existence result for operator inclusions, stated below as Theorems 2.1 and 2.3, respectively. In Section 2, conversely, we shall give some notations and recall the results that we shall use in our proofs.
Preliminaries
Let \(n\in\mathbf{N}\). For all \(i\in\{1,\ldots,n\}\), we shall denote by \(P_{n,i}:\mathbf{R}^{n}\to\mathbf{R}\) the projection over the ith axis. The space \(\mathbf{R}^{n}\) is considered with its Euclidean norm \(\ \cdot\_{n}\). If \(A\subseteq \mathbf{R}^{n}\), we denote by \(\overline{\mathrm{conv}}(A)\) the closed convex hull of A. Moreover, we denote by \(m_{n}\) the ndimensional Lebesgue measure in \(\mathbf{R}^{n}\). If \([a,b]\) is a compact interval, we shall denote by \(AC([a,b],\mathbf{R}^{n})\) the space of all absolutely continuos functions from \([a,b]\) into \(\mathbf{R}^{n}\).
Let \(p\in[1,+\infty]\). The space \(L^{p}([a,b],\mathbf{R}^{n})\) is considered with the usual norm
As usual, we put \(L^{p}([a,b]):=L^{p}([a,b],\mathbf{R})\). Moreover, if \(x\in\mathbf{R}^{n}\) and \(r>0\), we put
Let S, V be topological spaces, let \(F:S\to2^{V}\) be a multifunction, and let \(x_{0}\in S\). We recall that the multifunction F is said to be lower semicontinuous at \(x_{0}\) if for any open set \(A\subseteq V\), with \(F(x_{0})\cap A\ne \emptyset \), there exists an open set \(B\subseteq S\) such that
The multifunction F is said to be lower semicontinuous in S if it is lower semicontinuous at each point \(x\in S\). When the multifunction F is single valued, the above notion of lower semicontinuity reduces to the usual notion of continuity (for singlevalued maps).
If \((T,\mathcal{A})\) is a measurable space, a multifuncion \(F:T\to 2^{V}\) is said to be measurable (resp., weakly measurable) if for any closed (resp., open) set \(W\subseteq V\) one has \(F^{}(W)\in \mathcal{A}\). For the basic definitions and facts about multifunctions, we refer the reader to [4] and [5]. For what concerns measurable multifunctions, we also refer to [6]. In the following, we shall make the obvious identification \((\mathbf{R}^{n})^{k}=\mathbf{R}^{nk}\) whenever it is convenient for simpler exposition.
For the sake of completeness, we now recall the results that we use in our proofs. We start with the following very recent selection theorem.
Theorem 2.1
(Theorem 2.2 of [7])
Let T and \(X_{1},X_{2},\ldots, X_{k}\) be Polish spaces, with \(k\in\mathbf{N}\), and let \(X:=\prod_{j=1}^{k}X_{j}\) (endowed with the product topology). Let \(\mu, \psi_{1},\ldots,\psi_{k}\) be positive regular Borel measures over \(T,X_{1},X_{2},\ldots, X_{k}\), respectively, with μ finite and \(\psi_{1},\ldots,\psi_{k}\) σfinite.
Let S be a separable metric space, and let \(F:T\times X\to2^{S}\) be a multifunction with nonempty complete values. Finally, let \(E\subseteq X\) be a given set. For all \(t\in T\), let \(F_{t}:X\to2^{S}\) be the multifunction defined by putting
and let \(P_{*,i}:X\to X_{i}\) be the projection over \(X_{i}\). Assume that:

(i)
the multifunction F is \(\mathcal{T}_{\mu}\otimes\mathcal{B}(X)\)measurable (\(\mathcal {T}_{\mu}\) denoting the completion of the Borel σalgebra \(\mathcal{B}(T)\) of T with respect to the measure μ);

(ii)
for a.e. \(t\in T\), one has
$$\bigl\{ x:=(x_{1},\ldots,x_{k})\in X: F_{t} \textit{ is not lower semicontinuous at } x \bigr\} \subseteq E. $$
Then there exist sets \(Q_{1},\ldots,Q_{k}\), with
and a function \(\phi:T\times X\to S\) such that:

(a)
\(\phi(t,x)\in F(t,x) \) for all \((t,x)\in T\times X\);

(b)
for all \(x:=(x_{1},x_{2},\ldots,x_{k})\in X\setminus [ (\bigcup_{i=1}^{k}P_{*,i}^{1}(Q_{i}) )\cup E ]\), the function \(\phi( \cdot ,x)\) is \(\mathcal{T}_{\mu}\)measurable over T;

(c)
for a.e. \(t\in T\), one has
$$\begin{aligned}& \bigl\{ x:=(x_{1},x_{2},\ldots,x_{k})\in X: \phi( t,\cdot ) \textit{ is discontinuous at } x \bigr\} \\& \quad \subseteq E\cup \Biggl[ \Biggl(\bigcup _{i=1}^{k} P_{*,i}^{1}(Q_{i}) \Biggr) \Biggr]. \end{aligned}$$
We also recall the following proposition.
Proposition 2.2
(Proposition 2.6 of [8])
Let \(\psi:[a,b]\times\mathbf{R}^{n}\to\mathbf{R}^{k}\) be a given function, \(E\subseteq \mathbf{R}^{n}\) be a Lebesgue measurable set, with \(m_{n}(E)=0\), and let D be a countable dense subset of \(\mathbf{R}^{n}\), with \(D\cap E=\emptyset\). Assume that:

(i)
for all \(t\in[a,b]\), the function \(\psi(t,\cdot )\) is bounded;

(ii)
for all \(x\in D\), the function \(\psi( \cdot ,x)\) is \(\mathcal{L}([a,b])\)measurable.
Let \(G:[a,b]\times\mathbf{R}^{n}\to2^{\mathbf{R}^{k}}\) be the multifunction defined by setting, for each \((t,x)\in[a,b]\times\mathbf{R}^{n}\),
Then one has:

(a)
G has nonempty closed convex values;

(b)
for all \(x\in\mathbf{R}^{n}\), the multifunction \(G(\cdot,x)\) is \(\mathcal{L}([a,b])\)measurable;

(c)
for all \(t\in[a,b]\), the multifunction \(G(t,\cdot)\) has closed graph;

(d)
if \(t\in[a,b]\), and \(\psi(t,\cdot )_{\mathbf{R} ^{n}\setminus E}\) is continuous at \(x\in\mathbf{R}^{n}\setminus E\), then one has
$$G(t,x)=\bigl\{ \psi(t,x)\bigr\} . $$
The following existence result for operator inclusions is due to Naselli Ricceri and Ricceri.
Theorem 2.3
(Theorem 1 of [9])
Let \((T,\mathcal{F},\mu)\) be a finite nonatomic complete measure space; V a nonempty set; \((X,\\cdot\_{X})\), \((Y,\\cdot\_{Y})\) two separable real Banach spaces, with Y finitedimensional; \(p,q,s\in [1,+\infty]\), with \(q<+\infty\) and \(q\le p\le s\); \(\Psi:V\to L^{s}(T,Y)\) a surjective and onetoone operator; \(\Phi :V\to L^{1}(T,X)\) an operator such that, for every \(v\in L^{s}(T,Y)\) and every sequence \(\{v_{n}\}\) in \(L^{s}(T,Y)\) weakly converging to v in \(L^{q}(T,Y)\), the sequence \(\{\Phi (\Psi ^{1}(v_{n}))\}\) converges strongly to \(\Phi (\Psi ^{1}(v))\) in \(L^{1}(T,X)\); \(\varphi:[0,+\infty[ \, \to[0,+\infty]\) a nondecreasing function such that
for all \(u\in V\).
Further, let \(F:T\times X\to2^{Y}\) be a multifunction, with nonempty closed convex values, satisfying the following conditions:

(i)
for μalmost every \(t\in T\), the multifunction \(F(t,\cdot)\) has closed graph;

(ii)
the set
$$\bigl\{ x\in X: \textit{ the multifunction } F(\cdot,x) \textit{ is weakly measurable}\bigr\} $$is dense in X;

(iii)
there exists a number \(r>0\) such that the function
$$t\to\sup_{\x\_{X}\le\varphi(r)} d\bigl(0_{Y},F(t,x)\bigr) $$belongs to \(L^{s}(T)\) and its norm in \(L^{p}(T)\) is less or equal to r.
Under such hypotheses, there exists \(\tilde{u}\in V\) such that
$$\begin{aligned}& \Psi (\tilde{u}) (t)\in F\bigl(t,\Phi (\tilde{u}) (t)\bigr)\quad\mu\textit{a.e.}, \\ & \bigl\Vert \Psi (\tilde{u}) (t)\bigr\Vert _{Y}\le\sup _{\x\_{X}\le\varphi(r)} d\bigl(0_{Y},F(t,x)\bigr)\quad\mu\textit{a.e. in } T. \end{aligned}$$
The two following results, concerning absolutely continuous functions, will be fundamental in the sequel. We recall that a function \(f:[a,b]\to[\alpha ,\beta ]\) is called an Nfunction if it maps null sets into null sets.
Theorem 2.4
(Theorem 2 of [10])
Let \(f:[a,b]\to\mathbf{R}\) be continuous and strictly monotonic. Then \(f^{1}\) is absolutely continuous if and only if \(f^{\prime}\ne0\) almost everywhere on \([a,b]\).
Theorem 2.5
(Theorem 18.25 of [11])
Let f be a continuous function of finite variation with domain \([a,b]\subseteq \mathbf{R}\) and range \([\alpha ,\beta ]\subseteq \mathbf{R}\). Then f is an Nfunction if and only if f is absolutely continuous.
Finally, we recall the following lemma.
Lemma 2.6
(Lemma 3.2 of [12])
Let \((T,\mathcal{D},\mu)\) be a complete finite measure space, X be a Polish space, Y, Z be two topological spaces, \(F:T\times X\to2^{Z}\) and \(H:T\times Y\to2^{X}\) be two multifunctions. Assume that:

(a)
F is \(\mathcal{D}\otimes\mathcal{B}(X)\)measurable;

(b)
H is \(\mathcal{D}\otimes\mathcal{B}(Y)\)measurable and has closed values.
Then the multifunction G defined by \(G(t,y)=F(t,H(t,y))\) for all \((t,y)\in T\times Y\) is \(\mathcal{D}\otimes\mathcal{B}(Y)\)measurable.
The result
Theorem 1.2 above follows from the following more general result.
Theorem 3.1
Let \(n,k\in\mathbf{N}\), \(a>0\), \(F:[0,a]\times(\mathbf{R}^{n})^{k}\to2^{\mathbf{R}^{n}}\) a bounded multifunction with nonempty closed values. Assume that F is \(\mathcal{L}([0,a])\otimes\mathcal{B}((\mathbf{R} ^{n})^{k})\)measurable, and that for all \(t\in[0,a]\) the multifunction \(F(t,\cdot)\) is lower semicontinuous. Then there exists a solution \(u\in W^{k,\infty}([0,a],\mathbf{R}^{n})\) of problem (1).
Proof
Let \(H:=\overline{\mathrm{conv}}(F([0,a]\times(\mathbf{R}^{n})^{k}))\). We now divide the proof into two steps.
Step 1. We first of all assume, in addition, that \(H\subseteq \mathbf{R}^{n}_{+}\) (the positive open orthant of \(\mathbf{R}^{n}\)). Let \(M>0\) be such that \(H\subseteq \overline{B}_{n}(0_{\mathbf{R}^{n}},M)\). By Theorem 2.1, there exist sets \(C\in\mathcal{B}(\mathbf{R})\) and \(K_{0}\in\mathcal {L}([0,a])\), with \(m_{1}(C)=m_{1}(K_{0})=0\), and a function \(\psi:[0,a]\times(\mathbf{R} ^{n})^{k}\to 2^{H}\), such that:

(a)
\(\psi(t,\xi )\in F(t,\xi )\) for all \((t,\xi )\in [0,a]\times (\mathbf{R}^{n})^{k}\);

(b)
for all \(\xi \in(\mathbf{R}^{n})^{k}\setminus[\bigcup_{i=1}^{nk} P_{nk,i}^{1}(C)]\), the function \(\psi( \cdot ,\xi )\) is \(\mathcal{L}([0,a])\)measurable;

(c)
for all \(t\in[0,a]\setminus K_{0}\), one has
$$\bigl\{ \xi \in\bigl(\mathbf{R}^{n}\bigr)^{k}:\psi(t,\cdot ) \mbox{ is discontinuous at } \xi \bigr\} \subseteq \Biggl[\bigcup _{i=1}^{nk}P_{nk,i}^{1}(C) \Biggr]. $$Since \(m_{nk}(\bigcup_{i=1}^{nk}P_{nk,i}^{1}( C))=0\), there exists a countable set \(D\subseteq (\mathbf{R}^{n})^{k}\setminus[\bigcup_{i=1}^{nk}P_{nk,i}^{1}(C)]\), which is dense in \(\mathbf{R}^{nk}\).
Let \(G:[0,a]\times(\mathbf{R}^{n})^{k}\to2^{\mathbf{R}^{n}}\) be the multifunction defined by setting, for all \((t,\xi )\in[0,a]\times(\mathbf{R}^{n})^{k}\),
By Proposition 2.2 we get:
 (a)′:

G has nonempty closed convex values;
 (b)′:

for all \(\xi \in(\mathbf{R}^{n})^{k}\), the multifunction \(G( \cdot ,\xi )\) is \(\mathcal{L}([0,a])\)measurable;
 (c)′:

for all \(t\in[0,a]\), the multifunction \(G( t, \cdot )\) has closed graph;
 (d)′:

if \(t\in[0,a]\), and the function \(\psi(t,\cdot )\) is continuous at \(\xi \in(\mathbf{R}^{n})^{k}\), then one has \(G(t,\xi )=\{\psi(t,\xi )\}\).
Moreover, we have \(G([0,a]\times(\mathbf{R}^{n})^{k})\subseteq H\subseteq \overline{B}_{n}(0,M)\).
Now, let \(\phi_{1}:L^{\infty}([0,a],\mathbf{R}^{n})\to AC([0,a],\mathbf {R}^{n})\) be defined by putting, for all \(v\in L^{\infty}([0,a], \mathbf{R}^{n})\),
For each \(j=2,\ldots,k\), let \(\phi_{j}:L^{\infty}([0,a],\mathbf{R}^{n})\to C^{j1}([0,a],\mathbf {R}^{n})\) be defined by putting, for all \(v\in L^{\infty}([0,a],\mathbf{R}^{n})\),
Let \(\Phi :L^{\infty}([0,a],\mathbf{R}^{n})\to AC([0,a],\mathbf {R}^{nk})\) be defined by putting, for all \(v\in L^{\infty}([0,a],\mathbf{R}^{n})\),
for all \(t\in[0,a]\). Now we want to apply Theorem 2.3, with \(T=[0,a]\), \(s=p=+\infty\), \(q=1\), \(X=\mathbf{R}^{nk}\), \(Y=\mathbf{R}^{n}\), \(V=L^{\infty}([0,a],\mathbf{R}^{n})\), \(\Psi (v)=v\), \(\varphi\equiv+\infty\), \(r=M\), \(F=G\) and Φ defined as above. To this aim, observe what follows.
 (a)^{′′} :

For every \(v\in L^{\infty}([0,a],\mathbf{R}^{n})\), and every sequence \(\{v_{m}\}\) in \(L^{\infty}([0,a],\mathbf{R}^{n})\), weakly converging to v in \(L^{1}([0,a],\mathbf{R}^{n})\), the sequence \(\{\Phi (v_{m})\}\) converges strongly to \(\Phi (v)\) in \(L^{1}([0,a],\mathbf{R}^{nk})\). To see this, let \(\{v_{m}\}\) and v in \(L^{\infty}([0,a],\mathbf{R}^{n})\) be fixed, with \(\{v_{m}\}\) weakly convergent to v in \(L^{1}([0,a],\mathbf{R}^{n})\). It is quite easy to check that the sequence \(\{\phi_{1}(v_{m})\}\) converges pointwise in \([0,a]\) to \(\phi_{1}(v)\). That is, for every \(t\in[a,b]\) one has \(\lim_{m\to\infty} \\phi_{1}(v_{m})(t)\phi_{1}(v)(t)\_{n}=0\). Since \(\{v_{m}\}\) is weakly convergent in \(L^{1}([0,a],\mathbf{R}^{n})\), \(\{v_{m}\}\) is bounded in \(L^{1}([0,a],\mathbf{R}^{n})\). Consequently, for all \(m\in\mathbf{N}\) and \(t\in[0,a]\) we get
$$\begin{aligned} \bigl\Vert \phi_{1}(v_{m}) (t)\phi_{1}(v) (t) \bigr\Vert _{n} \le& \biggl\Vert \int_{0}^{t}v_{m}(s) \,ds\biggr\Vert _{n}+ \bigl\Vert \phi_{1}(v) (t)\bigr\Vert _{n} \\ \le & \bigl\Vert \phi_{1}(v) (t)\bigr\Vert _{n} +\sup _{j\in\mathbf{N}}\v_{j}\_{L^{1}([0,a],\mathbf{R}^{n})}. \end{aligned}$$By the dominated convergence theorem we get
$$\lim_{m\to+\infty} \int_{0}^{a}\bigl\Vert \phi_{1}(v_{m}) (t)\phi_{1}(v) (t)\bigr\Vert _{n} \,dt=0, $$hence \(\{\phi_{1}(v_{m})\}\) converges strongly to \(\phi_{1}(v)\) in \(L^{1}([0,a],\mathbf{R}^{n})\). Of course, this implies that \(\{\phi_{1}(v_{m})\}\) also converges weakly to \(\phi_{1}(v)\) in \(L^{1}([0,a],\mathbf{R}^{n})\). Consequently, using exactly the same argument as before, it is immediately seen that \(\{\phi_{2}(v_{m})\}\) also converges strongly to \(\phi_{2}(v)\) in \(L^{1}([0,a],\mathbf{R}^{n})\). By using recursively the same argument as before, we get that for each fixed \(j=1,\ldots, k\), the sequence \(\{\phi_{j}(v_{m})\}\) converges strongly to \(\phi_{j}(v)\) in \(L^{1}([0,a],\mathbf{R}^{n})\). Of course, this implies that the whole sequence \(\{\Phi (v_{m})\}\) converges strongly to \(\Phi (v)\) in \(L^{1}([0,a],\mathbf{R}^{nk})\), as desired.
 (b)^{′′} :

Let d denote the usual pointtoset distance in \(\mathbf{R}^{n}\), with respect to the norm \(\\cdot\_{n}\). Then he function
$$\omega:t\in[a,b]\to\sup_{\xi \in\mathbf{R}^{nk}} d\bigl(0_{\mathbf {R}^{n}},G(t,\xi )\bigr) $$belongs to \(L^{\infty}([0,a])\) and \(\\omega\_{L^{\infty}([a,b])}\le M\) (as regards the measurability of ω, we refer to [9]).
Therefore, all the assumptions of Theorem 2.3 are satisfied. Consequently, there exists a function \(\hat{v}\in L^{\infty}([0,a],\mathbf{R}^{n})\) and a set \(K_{1}\in\mathcal{L}([0,a])\), with \(m_{1}(K_{1})=0\), such that
In particular, by (2) and by the above construction we get
Fix \(i\in\{1,\ldots,n\}\). Since \(H\subseteq \mathbf{R}^{n}_{+}\), by (3) we get
(where \(\hat{v}_{i}\) denotes the ith component of v̂). Then for a.e. \(t\in[0,a]\) we have
In particular, this implies that \(\phi_{1}(\hat{v})_{i}\) (which is absolutely continuous in \([0,a]\)) is strictly increasing in \([0,a]\). By Theorem 2.4, the function \((\phi_{1}(\hat{v})_{i})^{1}\) is absolutely continuous in \([0,a]\). Consequently, by Theorem 2.5, the set
has null Lebesgue measure. Now, observe that
hence, in particular,
Consequently, the function \(\phi_{2}(\hat{v})_{i}\) is strictly increasing in \([0,a]\). Therefore, by Theorem 2.4, the function
is absolutely continuous, hence by Theorem 2.5 it maps nullmeasure sets into nullmeasure sets. Therefore, if we put
we get \(m_{1}(W_{i,2})=0\). Applying recursively the same argument, if for all \(j=1,\ldots,k\) we put
we get \(m_{1}(W_{i,j})=0\). Now, put
Of course, \(m_{1}(U)=0\). Let \(t^{*}\in[0,a]\setminus U\) be fixed. Since \(t^{*}\notin\bigcup_{{{i=1,\ldots,n}\atop {j=1,\ldots,k}}} W_{i,j}\), we immediately get
hence, by (c), the function \(\psi(t^{*},\cdot)\) is continuous at \(\Phi (\hat{v})(t^{*})\). Therefore, by (2) and (d)′ we get
Hence, we have proved that
Now, take \(u:=\phi_{k}(\hat{v})\in C^{k1}([a,b],\mathbf{R}^{n})\). By the definition of the functions \(\phi_{i}\) we get
and
Therefore,
and the function u satisfies the conclusion.
Step 2. We now prove the result in its full generality. Fix \(y_{0}\in\mathbf{R}^{n}\) such that \(y_{0}+H\subseteq \mathbf{R}^{n}_{+}\). Let \(\hat{F}:[0,a]\times(\mathbf{R}^{n})^{k}\to2^{\mathbf{R}^{n}}\) be defined by putting, for all \(t\in[0,a]\) and all \(\xi =(\xi _{0},\xi _{1},\ldots,\xi _{k1})\in (\mathbf{R}^{n})^{k}\),
Of course, for every fixed \(t\in[0,a]\) the multifunction \(\hat{F}(t,\cdot )\) is lower semicontinuous in \((\mathbf{R}^{n})^{k}\). Moreover, by Lemma 2.6, F̂ is \(\mathcal{L}([0,a])\otimes\mathcal {B}((\mathbf{R}^{n})^{k})\)measurable.
By the first part of the proof, there exists \(u\in W^{k,\infty }([0,a],\mathbf{R}^{n})\) such that
It is immediate to check that the function \(v(t)=u(t)\frac{y_{0}}{k!}t^{k}\) satisfies the conclusion. □
Proof of Theorem 1.2
Let \(F^{*}:[0,a]\times(\mathbf{R}^{n})^{k}\to2^{X}\) be defined by setting
Of course, \(F^{*}\) is \(\mathcal{L}([0,a])\otimes\mathcal{B}((\mathbf{R}^{n})^{k})\)measurable, and for all \(t\in[0,a]\) the multifunction \(F^{*}(t,\cdot )\) is lower semicontinuous in \((\mathbf{R}^{n})^{k}\). By Theorem 2.1, there exists \(u\in W^{k,\infty}([0,a],\mathbf{R}^{n})\) such that
It is immediate to check that \((u(t), u^{\prime}(t),\ldots,u^{(k1)}(t))\in B^{k}\) for all \(t\in[0,a]\). Consequently, by the definition of \(F^{*}\), the function u solves problem (1). □
Remark 3.2
It is quite easy to construct examples of applications of Theorem 1.2, where the multifunction F is not jointly lower semicontinuous. For instance, one can take \(a=1/8\), \(b=1\), \(M=8\), \(n=k=1\), \(F:[0,a]\times[1,1] \to2^{\mathbf{R}}\) defined by \(F(t,x)=h(t)\cdot G(x)\), where
and \(G(x)=[x+2,x+3]\). It is routine matter to check that all the assumptions of Theorem 1.2 are satisfied (see [5, 6]), and that the multifunction F is not jointly lower semicontinuous. Therefore, Theorem 1.2 applies, while Theorem 1.1 does not.
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The second author was partially supported by the Grant MOST 1032923E039001MY3.
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Keywords
 differential inclusions
 Cauchy problem
 generalized solutions
 discontinuous selections