Funkcialaj Ekvacioj, 37
(1994) 101-114
Non Convex Set-Valued Systems in Banach Spaces1
By
G. CONTI, P. NISTRI and P. ZECCA
(Universita di Firenze, Italy)
Introduction
§0.
In this paper we give sufficient conditions for the solvability of set-valued
systems of the form
(1)
$¥left¥{¥begin{array}{l}0¥in F(x,y)¥¥0=g(x,y),¥end{array}¥right.$
where
$F$
and
$g$
are defined in the following way
$F(x, y)=y-¥overline{F}(¥mathrm{x}, y)$
and
$g(x, y)=X-¥overline{g}(x, y)$
,
where ¥ ¥ ,
, with $X$ ,
Banach spaces,
is an upper semicontinuous,
compact multivalued map,
is a continuous map; both maps are defined on
the closure of a suitable open subset $U$ of the product
and take values
$X$
in
and
respectively. Our approach is a development, in various aspects,
of the one introduced in [8] in the case when
is a continuous, compact,
singlevalued map. Roughly speaking, we solve the first equation in terms of
as a function of “the parameter” considering the application
, and hence we introduce the solution set $S(x)$ in the second
one. The fixed points of the composite function
are the solutions
of system (1). In recent years upper semicontinuous multivalued maps with
non convex values have been studied by several authors, see e.g. [1], [3], [6],
[7]. The paper is organized as follows.
In Section 1 we introduce the definition and the properties of
sets
and recall a graph approximation theorem for upper semicontinuous multivalued maps with
values, given independently in [1] and [7]. Then we
state the definitions of weighted maps, introduced in [4].
In Section 2 we state two existence results for system (1).
$ chi in X$
$y¥in ¥mathrm{Y}$
$¥overline{F}$
$¥mathrm{Y}$
$¥overline{g}$
$X¥times ¥mathrm{Y}$
$¥mathrm{Y}$
$¥overline{F}$
$x$
$y$
$x$
$-¥circ S(¥mathrm{x})=$
$¥{y ¥in ¥mathrm{Y}:¥mathrm{O}¥in F(x, y)¥}$
$¥overline{g}(¥mathrm{x}, S(¥mathrm{x}))$
$UV^{¥infty}$
$UV^{¥infty}$
Research supported in part by a National Research Project
$¥mathrm{M}.¥mathrm{U}$
.R.S.T. 40%.
102
G. CONTI, P. NISTRI and P. ZECCA
In Section 3 an application to control theory of the obtained results, is
given.
Definitions, notations and preliminary results
§1.
be metric spaces. A set-valued map $M$
Definition 1.1: Let $X$ and
from $X$ into 7, with nonempty values, is said to be upper semicontinuous at
$¥in X$ if for any neighborhood
of $M(x)$ there exists a neighborhood $U$ of
such that $M(u)¥subset V$ for any $u¥in U$ . If, for every $x¥in X$ , $M$ is upper semicontinuous at , then $M$ is said to be upper semicontinuous (u.s.c.) on $X$ .
$¥mathrm{Y}$
$V$
$x$
$x$
$x$
sends bounded sets into relatively compact sets, then it is said
compact. $M$ is said proper if, for each compact set $K$ of 7, $M^{-1}(K)$ is compact.
with the symbol
We will denote a multivalued map $M$ from $X$ to
$M:X-¥circ Y$
is compact and $M(x)$ is closed for
It is well known that, if
$M$ has closed graph, i.e.
$M$
$x
¥
in
X$
,
if
if
only
.
and
is
any
, then
.
implies
,
$M:X¥
circ
¥
mathrm{Y}$
$/ower$
semicontinuous (l.s.c.) on $X$
is said
The multivalued map
of every $y¥in M(x)$, there exists
if, for every $¥in X$ and every neighborhood
a neighborhood $U$ of such that $ M(u)¥cap V¥neq¥emptyset$ for all $u¥in U$ . If $M:X-¥circ ¥mathrm{Y}$
is both u.s.c. and l.s.c. on $X$ , then $M$ is said continuous on $X$ .
If
$M$
$¥mathrm{Y}$
$¥overline{M(X)}$
$¥chi_{n}¥in X$
$¥mathrm{u}.¥mathrm{s}.¥mathrm{c}$
$y_{n}¥rightarrow ¥mathcal{Y}¥mathrm{o}’ y_{n}¥in M(x_{n})$
$X_{n}¥rightarrow ¥mathrm{x}_{0}$
$¥mathcal{Y}¥mathrm{o}¥in M(x_{0})$
$V$
$x$
$x$
Definition 1.2: Let $X$ be a metric space and let $K$ be a compact subset
set if, for any neighborhood $U$ of $K$ , there
of $X$ . We say that $K$ is a
can be
exists a neighborhood , $K¥subset V¥subset U$ , such that any two points in
$U$
such that the inclusion
joined by a path in , and there is a base point
$U$
for each
induces the trivial homomorphism
of
into
positive integer .
$UV^{¥infty}$
$V$
$V$
$¥mathcal{Y}¥mathrm{o}¥in K$
$¥pi_{n}(V, y_{0})¥rightarrow¥pi_{n}(U, y_{0})$
$V$
$n$
be metric spaces and let $M:X-¥circ ¥mathrm{Y}$ be an
Let $X$ and
mapupper semicontinuous multivalued mapping. We say that $M$ is a
$¥in X$ .
for each
is
ping if the set
Definition 1.3:
$¥mathrm{Y}$
$UV^{¥infty}-$
$M(x)¥subset ¥mathrm{Y}$
$UV^{¥infty}$
$x$
is said to be an graph approximation,
Definition 1.4: A map
Gr $M$ ,
shortly -approximation, of the multivalued map $M:X-¥circ ¥mathrm{Y}$ if Gr
where Gr $M$ denotes the graph of $M$ and Gr $M$ denotes the set of points
of distance less than from the set Gr $M$ . Moreover we will say that a map
for all
is a pointwise approximation for $M:X-¥circ ¥mathrm{Y}$ if
$¥mu:X¥rightarrow ¥mathrm{Y}$
$¥epsilon-$
$¥mu¥subset¥epsilon$
$¥mathrm{e}$
$¥epsilon$
$¥epsilon$
$¥mu:X¥rightarrow ¥mathrm{Y}$
$x$
$¥in X$
$¥mu(x)¥in¥epsilon M(¥mathrm{x})$
$¥epsilon-$
.
Lemma 1.1:
holds
([7]) If for a compact
set
$K$
, one
of
the following conditions
103
Non Convex Set-Valued Systems
is a fundamental absolute retract;
has trivial shape;
$K$ is an $R_{¥delta}-set;$
$K$ is an absolute retract;
$K$ is contractible;
set.
then $K$ is a
1)
2)
3)
4)
5)
$K$
$K$
$UV^{¥infty}$
The classes of absolute retracts and absolute neighborhood retracts are denoted
by $AR$ and $ANR$ , respectively, (see [5]).
a metric
Theorem 1.1: ([1], [7]): Let $X$ be a compact $ANR$ space,
$M:X-¥circ ¥mathrm{Y}$
continumapping
exists
a
, there
and any
space. For any
ous approximation,
of $M$ .
$¥mathrm{Y}$
$¥epsilon>0$
$UV^{¥infty}-$
$¥mu:X¥rightarrow ¥mathrm{Y}$
$¥epsilon-$
-mapping the notion of
We want to recall here (see [7]) that for an
The relative definiproperties
is
well
defined.
fixed point index with the usual
tion is given by using Theorem 1.1.
$¥mathrm{U}¥mathrm{V}^{¥infty}$
Definition 1.5: Let $U$ be an open subset of a compact $ANR$ space $X$ .
mapping.
be an
the boundary of $U$ . Let
Denote by
. Then we can define
Assume that it is fixed points free on
(see [7]). Notice that this index satisfies all the standard properties.
$UV^{¥infty}$
$¥phi:¥overline{U}¥rightarrow E$
$¥partial U$
$¥partial U$
$¥mathrm{i}¥mathrm{n}¥mathrm{d}(X, ¥phi, U)$
be topological Hausdorff spaces. A finite
Definition 1.6: Let $X$ and
¥
¥
will be called a weighted map
valued upper semicontinuous map
(shortly -map) if, to each and $y¥in M(x)$ a multiplicity or weight $¥mathrm{m}(y, M(x))¥in$
$Z$ is assigned in such a way that the following property holds:
, then
with
if $U$ is an open set in
$¥mathrm{Y}$
$M:X- circ
mathrm{Y}$
$x$
$¥mathrm{w}$
$¥partial U¥cap M(x)=¥emptyset$
$¥mathrm{Y}$
$¥sum_{y¥in M(x)¥cap U}¥mathrm{m}(y, M(x))=¥sum_{y^{¥prime}¥in M(x^{¥prime})¥cap U}¥mathrm{m}(y^{¥prime}, M(¥chi^{¥prime}))$
whenever
$x^{¥prime}$
is close enough to
Definition 1.7:
$x$
The number
, (see [4] and
[8]).
$¥mathrm{i}(M(x), U)=¥sum_{y¥in M(x)¥cap U}¥mathrm{m}(y, M(x))$
will be called
the index or multiplicity of $M(x)$ in $U$ . If $U$ is a connected set, the number
¥
does not depend on $x¥in X$ . In this case the number
¥
will be called the index of the weighted map $M$ .
$¥mathrm{i}(M)=$
$ mathrm{i}(M(x), U)$
$ mathrm{i}(M(x), U)$
Definition 1.8: Let $X$ be a Banach space
ball in $X$ of radius , centered at the origin.
¥
verifies the
semicontinuous map
¥
¥
if for all
for which
on
strictly separated by an hyperplane, i.e. for all ¥ ¥
, the dual space of $X$ , such
ous functional ¥
for all $y¥in M(-x)$ .
and
$r$
$M:¥overline{B}(0, r)$
$x in partial B(0, r)$
$¥partial B(0, r)$
$x^{*} in X^{*}$
$¥mathrm{x}^{*}(y)<0$
$- circ X$
and let
be the closed
We will say that the upper
Borsuk-Ulam (B. $U.$ ) property
,
and
are
$x in partial B(0, r)$ there exists a continuthat $x^{*}(y)>0$ for all $y¥in M(x)$
$¥mathrm{O}¥not¥in M(¥mathrm{x})$
$¥overline{B}(0, r)$
$M(¥mathrm{x})$
$M(-¥mathrm{x})$
G. CONTI, P. NISTRI and P. ZECCA
104
For sake of simplicity, in the sequel we will write
Definition 1.9:
Let
$X$
,
be metric spaces.
$¥mathrm{Y}$
$B$
instead of
Given
$B(0, r)$
.
we denote
$S¥subset X¥times ¥mathrm{Y}$
by
$S(¥mathrm{x})=¥{y ¥in ¥mathrm{Y}:(x, y)¥in S¥}$
;
$S(A)=¥{y ¥in ¥mathrm{Y}:(x, y)¥in S, x¥in A¥}$
and
$S_{X}=S¥cap(¥{¥mathrm{x}¥}¥times ¥mathrm{Y})$
$¥ovalbox{¥tt¥small REJECT}^{F}=¥{x ¥in X:S_{x}^{F}¥cap¥partial U=¥emptyset¥}$
then
If
is defined in all
from infinity of the equation
$X¥times ¥mathrm{Y}$
for
$S_{A}=S¥cap(A¥times ¥mathrm{Y})$
$S^{F}=¥{(¥mathrm{x}, y)¥in¥overline{U}:y¥in¥overline{F}(¥mathrm{x}, y)¥}$
$F$
;
$A¥subset X$
;
;
.
$¥ovalbox{¥tt¥small REJECT}^{F}=¥{x$
$¥mathrm{O}¥in F(x, y)¥}$
$¥in X:x$
is not a bifurcation point
.
An existence result
§2.
We want now to prove the existence of solutions for the system
(1)
or
$¥{_{x=¥overline{g}(x,y)}^{y¥in¥overline{F}(¥mathrm{x},y)}$
$¥left¥{¥begin{array}{l}0¥in y-¥overline{F}(x,y)=F(¥mathrm{x},y)¥¥0=¥mathrm{x}-¥overline{g}(¥mathrm{x},y)=g(¥mathrm{x},y).¥end{array}¥right.$
For this we have the following.
Theorem 2.1: Let $X$ be a Banach space and let $K$ be a compact, convex
subset of a Banach space Y. Let $U$ be a relatively open, bounded, convex set
a continuous compact
mapping and
an
in
$r>0$
,
and that
such that
map. Suppose that there exists
be the application defined by $T(x)=x-¥overline{T}(x)$ ,
, $U(0))¥neq 0$ . Let
and $S(x)=¥{y ¥in K:y¥in¥overline{F}(x, y)¥}$ . Let us suppose that for
where
any
, $T(x)$ and $T(-x)$ are strictly separated by an
such that
hyperplane. Then system (1) has a solution.
$X¥times K,¥overline{F}:¥overline{U}-¥circ K$
$UV^{¥infty}-$
$¥overline{g}:¥overline{U}¥rightarrow X$
$¥overline{B}¥subset¥ovalbox{¥tt¥small REJECT}^{F}$
$¥mathrm{i}¥mathrm{n}¥mathrm{d}(K$
$T:¥overline{B}-¥circ X$
$¥overline{F}(0, ¥cdot)$
$¥overline{T}(¥mathrm{x})=¥overline{g}(¥mathrm{x}, S(x))$
$x¥in¥partial B$
$¥mathrm{O}¥not¥in T(x)$
is upper semicontinuous.
First, we prove that the map
is
and nonempty for every
compact
assumptions,
that, under our
Observe
$¥mathrm{x}-¥circ S(x)$
$S(¥mathrm{x})$
Lemma 2.1:
The multivalued map
$x-¥circ S(x)$
is
$u.s.c$
. at every
$x$
$x$
$¥in¥ovalbox{¥tt¥small REJECT}^{F}$
$¥in¥ovalbox{¥tt¥small REJECT}^{F}$
.
.
is upper semicontinuous at
, we shall prove that
Let
is an open neighborhood of $S(x_{0})$ , then there exists an open
, that is, if
$N$
. To prove this,
, such that $S(x)¥subset V$,
neighborhood
,
of
$y
¥
in
S(x_{0})$
, with
the
form
neighborhoods
of
let
and let us consider
in
a neighborhood of in , such that
and
a neighborhood of
Proof.
$x_{0}$
$S$
$x_{0}¥in¥ovalbox{¥tt¥small REJECT}^{F}$
$V$
$¥forall ¥mathrm{x}¥in N$
$N¥subset¥ovalbox{¥tt¥small REJECT}^{F}$
$x_{0}$
$N_{x_{0}}¥times V_{y}$
$¥ovalbox{¥tt¥small REJECT}^{F}$
$N_{x_{0}}$
$x_{0}$
$V_{y}$
$y$
$¥mathrm{X}$
105
Non Convex Set-Valued Systems
and
$V_{y}¥subset U(¥mathrm{x}_{0})¥cap V$
$N_{x_{0}}¥times V_{y}¥subset U$
.
there exists a finite number, say
By the compactness of
. Let
, of neighborhoods of the previous form covering
$S_{x_{0}}=S¥cap(¥{¥mathrm{x}_{0}¥}¥times K)$
$s$
$S_{x_{0}}$
and
$N_{0}=¥bigcap_{i=1}^{s}N_{i}$
.
$V^{¥prime}=¥bigcup_{¥iota=1}^{s}V_{i}$
, we have that
Clearly for each neighborhood $N$ of , with ¥
such
. Let us prove that there exists a neighborhood $N$ of
and
$x
¥
in
N$
. Suppose not, then there exists a bounded
for all
that
is upper
and
. As
,
with
sequence
$K$
is compact, we may assume (by
semicontinuous with compact values and
, that
. Then
passing to a subsequence, if necessary) that
.
, contradicting
is
$x_{0}$
$N¥times V^{¥prime}¥subset U$
$N subset N_{0}$
$V^{¥prime}¥subset V$
$x_{0}$
$S(¥mathrm{x})¥subset V^{¥prime}$
$¥{(x_{n}, y_{n})¥}$
$x_{n}¥rightarrow x_{0}$
$y_{n}¥in¥overline{F}(¥chi_{n}, y_{n})$
$¥mathcal{Y}¥mathrm{o}¥in¥overline{F}(x_{0}, y_{0})$
$y_{n}¥rightarrow ¥mathcal{Y}¥mathrm{o}$
$S(x_{0})¥subset V^{¥prime}$
$¥mathcal{Y}¥mathrm{o}¥in S(x_{0})$
$¥square $
Lemma 2.2: Let $X=R^{n}$ , $K$ as in Theorem 2.1 and let
such that if
for each neighborhood $W$ of there exists
.
, then
approximation of
$¥epsilon>0$
$S_{B}^{F}$
$¥overline{B}¥subset¥ovalbox{¥tt¥small REJECT}^{F}$
Then
is an
.
$¥overline{f}:¥overline{U}_{B}¥rightarrow K$
$S_{B}^{f}¥subset W$
$¥overline{F}_{¥backslash ¥overline{U}_{B}}$
$¥epsilon-$
Proof.
$¥overline{F}$
$y_{n}¥not¥in V^{¥prime}$
is a closed set.
In fact, let
and
for any $n¥in N$. As
, we have that
. As
, that is
is upper semicontinuous with compact values, then
is a compact set. Let $W$
as a closed subset of
. Then
an -neighborhood of , $V¥subset W$, with
be a neighborhood of ,
. Since
is a compact set, we have that
and
. Let
$S_{B}^{F}$
$¥{(x_{n}, y_{n})¥}¥subset S_{B}^{F}$
$¥{(¥mathrm{x}_{n}, y_{n})¥}¥subset S_{B}^{F}$
$(¥mathrm{x}_{0}, y_{0})$
$(x_{n}, y_{n})¥rightarrow$
$y_{n}¥in¥overline{F}(¥mathrm{x}_{n}, y_{n})$
$¥overline{F}$
$¥mathcal{Y}¥mathrm{o}¥in¥overline{F}(¥chi_{0}, y_{0})$
$(x_{0}, y_{0})¥in S_{B}^{F}$
$¥overline{B}¥times K$
$S_{B}^{F}$
$S_{B}^{F}$
$V$
$¥partial V¥cap¥partial W=$
$S_{B}^{F}$
$¥epsilon_{1}$
$A$
$A=¥overline{U}_{B}¥backslash V$
$¥epsilon_{2}=¥mathrm{d}(¥partial V, ¥partial W)$
$¥emptyset$
$¥{|s-¥tilde{y}|, s¥in¥overline{F}(¥tilde{¥mathrm{x}},¥tilde{y})¥}=¥epsilon_{3}>0$
$¥inf_{(¥overline{x},¥tilde{y})¥in A}$
Let
$¥epsilon=¥min$
$¥{¥epsilon_{1}, ¥epsilon_{2}, ¥epsilon_{3}¥}$
$(x, y)¥in¥overline{U}_{B}¥backslash W$
and let
and let
be an -approximation of
$(x,
y)$
is in , then there exists
, so
$¥overline{f}:¥overline{U}¥rightarrow ¥mathrm{Y}$
$y=¥overline{f}(x, y)$
.
$¥mathrm{s}$
$S_{B}^{f}$
$¥overline{F}_{¥backslash ¥overline{U}_{B}}$
Let
.
$(¥overline{x},¥overline{y})¥in¥overline{U}_{B}$
such that
$|(x, y)$
As
$|(¥mathrm{x}, y)$
$-(¥overline{¥mathrm{x}},¥overline{y})|+|¥overline{f}(¥mathrm{x}, y)$
$-(¥overline{x},¥overline{y})|<¥epsilon$
$|¥overline{y}-z|<¥epsilon$
, then
and
$(x, y)¥in¥overline{U}_{B}¥backslash W$
it follows that
. Thus
$(¥overline{x},¥overline{y})¥not¥in A$
$|(x, y)$
for some
$-z|<¥epsilon$
$-(¥overline{¥mathrm{x}},¥overline{y})|<¥epsilon$
$(¥overline{x},¥overline{y})¥not¥in S_{B}^{F}$
.
$z¥in¥overline{F}(¥overline{x},¥overline{y})$
.
Since
we get that
. This is an absurd, since
$y=¥overline{f}(x, y)$
$(¥overline{x},¥overline{y})¥in V¥backslash S_{B}^{F}$
, and so
.
$(¥overline{x},¥overline{y})¥not¥in V$
Lemma 2.3: Let $X=R^{n}$ and let $K$ be a compact, convex subset
there exists
. There exists an
such that for all
and
that
is an approximation of
;
is a finite subset of $U(x)$ ,
a)
$R^{m}$
$¥epsilon_{0}>0$
$¥overline{f_{¥backslash ¥overline{U}_{B}}}$
$S_{B}^{f}$
b)
$¥epsilon-$
$¥epsilon<¥epsilon_{0}$
of
$¥mathrm{Y}=$
, such
$¥overline{f}:U¥rightarrow ¥mathrm{Y}$
$¥overline{F}_{¥backslash ¥overline{U}_{B}}$
$¥forall x¥in¥overline{B}$
.
$¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(0, ¥cdot), U(0))=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{f}(0, ¥cdot), U(0))$
Proof.
Theorem 4.5 in [7] ensures the existence of a positive number,
106
G. CONTI, P. NISTRI and P. ZECCA
say , with the property that for any
every -approximation of
. Fix
has the same index of
and let
approximabe an
. Using the same arguments of Lemma 2.4 of [8] we can prove
tion of
-pointwise approximation of
:
,
that there exists
that satisfies
is an -approximation of
property ). Then
that satisfies property
. The map
is then any continuous extension of
of , and b) follows
immediately from the choice of .
$¥epsilon<¥epsilon_{0}$
$¥epsilon_{0}$
$¥overline{F}_{¥backslash ¥overline{U}_{B}}$
$¥overline{F}_{¥backslash ¥overline{U}_{B}}$
$¥mathrm{e}$
$¥tilde{f}:¥overline{U}_{B}¥rightarrow K$
$¥epsilon<¥epsilon_{0}$
$¥epsilon/2$
$¥overline{F}_{¥backslash ¥overline{U}_{B}}$
$¥overline{U}_{B}¥rightarrow K$
$¥tilde{f}_{1}$
$¥tilde{f,}$
$¥epsilon/2$
$¥tilde{f}_{1}$
$¥mathrm{a}$
$¥overline{F}_{¥backslash ¥overline{U}_{B}}$
$¥mathrm{e}$
$¥overline{U}$
$¥overline{f}$
$¥mathrm{a})$
$¥tilde{f_{1}}$
$¥square $
$¥epsilon$
Lemma 2.4: Let $X$ be a Banach space and $K$ a convex, compact subset
of any Banach space Y. Let $U$ be an open subset in $X¥times K$ . Let us suppose
that for all
the equation
has only isolated solutions. Then
lication $x-¥circ S(x)$ is a -map and
, for any
$y¥in¥overline{F}(x, y)$
$x¥in¥ovalbox{¥tt¥small REJECT}^{F}$
$¥mathrm{i}(S)=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(x, ¥cdot), U(¥mathrm{x}))$
$w$
$¥mathrm{x}theapp¥in¥ovalbox{¥tt¥small REJECT}^{F}$
.
already seen that the map $x-¥circ S(x)$ is an upper semicontinuous map. We want to prove that it is possible to associate an integer
to any $y¥in S(x)$ with the property of Definition 1.6. If
is an
isolated solution for , then there exists a neighborhood
of
such that
. Let us define
. Using the excision property of the fixed point index,
does not depend on the
choice of .
Let $W$ be an open set in $K$ such that
. As
is upper
semicontinuous there exists a ball
such that
we get
Proof. We have
$¥mathrm{m}(y, S(x))$
$y$
$¥overline{F}$
$¥Omega$
$¥Omega¥cap S(¥mathrm{x})=¥{y¥}$
$y$
$¥mathrm{m}(y, S(x))=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(x, ¥cdot), ¥Omega)$
$¥mathrm{m}(y, S(x))$
$¥Omega$
$ S(x)¥cap¥partial W=¥emptyset$
$¥overline{B}(x, r)$
$¥partial W=¥emptyset$
$S$
$¥forall¥chi^{¥prime}¥in¥overline{B}(x, r)$
$ S(x^{¥prime})¥cap$
.
Let us consider now the following homotopy:
$H:[0,1]$ $¥times W-¥circ K$ defined by
$H(t, y)=¥overline{F}(tx+(1-t)x^{¥prime}, y)$
As
tween
and
of the index, we get
$tx+(1-t)¥mathrm{x}^{¥prime}¥in¥overline{B}(x, r)$
$¥overline{F}(x, ¥cdot)_{¥backslash W}$
for all
$¥overline{F}(x^{l_{ }},¥cdot)_{¥backslash W}$
.
.
is an admissible homotopy beFrom this fact, using the additivity property
$t¥in[0,1]$ ,
$H$
$¥sum_{y¥in S(x)¥cap W}¥mathrm{m}(y, S(x))=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(x, ¥cdot), W)=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(¥chi^{l_{ }},¥cdot), W)$
$=¥sum_{y¥in S(x^{¥prime})¥cap W}¥mathrm{m}(y, S(¥chi^{¥prime}))$
.
$¥square $
The following lemmata, whose proof can be found in [8], also hold.
Lemma 2.5:
with
such that
$¥mathrm{i}(M)¥neq 0$
.
Let
If
$¥mathrm{O}¥in M(x)$
Lemma 2.6:
$M$
be an open ball in
and let $M:B-¥circ R^{n}$ be a -map
verifies the B. U. condition on , then there exists
$B$
$R^{n}$
$w$
$x¥in¥overline{B}$
$¥partial B$
.
Let
$M=I-¥overline{M}:¥overline{B}-¥circ X$
be a compact vector
field
satisfying
107
Non Convex Set-Valued Systems
the B. U. property. Then there exists
$M$ ,
, satisfies the B. U. property.
$¥overline{¥epsilon}>0$
such that every
$¥epsilon-$
of
approximation
$0<¥epsilon<¥overline{¥epsilon}$
We can now give the proof of Theorem 2.1
Theorem 2.1. Observe that the system (1) has solutions if and
. Assume to the contrary that
for all
only if
, for some
such
that
for
all
.
In
fact
. Then there exists
,
.
,
such that
if not, then there exist sequences
and
such that
,
,
It follows that there exists
and
is an upper semicontinuous compact vector
. As
, and
such that
field, there exists a subsequence
$T(x)$ , absurd.
such that
On the other hand from Lemma 2.6 there exists
. property. Let
every -approximation
of ,
verifies the
$V
¥
subset
U$
$¥delta=¥min$
$y)
¥
in
U:(x,$
$V=$
defined by
Gr }.
and let
( , )
{ ,
is an open set being the inverse image of the open set Gr ,
Clearly
trought the continuous map $(/, g)$ , where I stands for the identity map.
We divide now the proof in three parts.
.
First part: $X=R^{n}$ ,
approximation of
Let
be that one given in Lemma 2.2, i.e. every
Proof of
$x¥in¥overline{B}$
$¥mathrm{O}¥in T(x)$
$x$
$¥in¥overline{B}$
$¥mathrm{O}¥not¥in T(x)$
$¥epsilon_{1}>0$
$¥{¥epsilon_{n}¥}$
$¥{X_{n}¥}¥subset¥overline{B}$
$¥epsilon_{n}¥rightarrow 0$
$0¥in¥epsilon_{n}T(¥epsilon_{n}x_{n})$
$¥{y_{n}¥}$
$¥{x_{n}^{¥prime}¥}¥subset¥overline{B}$
$y_{n}¥in T(¥chi_{n}^{¥prime})$
$|y_{n}|<¥epsilon_{n}$
$T$
$y_{n}¥rightarrow 0$
$|x_{n}^{¥prime}-¥chi_{n}|<¥epsilon_{n}$
$x¥in¥overline{B}$
$0¥not¥in¥epsilon_{1}T(¥epsilon_{1}x)$
$ 0¥in$
$X_{n_{k}}^{l}¥rightarrow X¥in¥overline{B}$
$¥{¥chi_{n_{k}}^{¥prime}¥}¥subset¥{x_{n}^{¥prime}¥}$
$¥epsilon_{2}$
$T^{¥prime}$
$¥epsilon_{2}$
$T^{¥prime}:¥overline{B}-¥circ X$
$T$
$¥mathrm{B}.¥mathrm{U}$
$¥{¥epsilon_{1}, ¥epsilon_{2}¥}$
$(¥mathrm{x}$
$g$
$x$
$y$
$T$
$)¥in¥delta$
$V$
$T$
$¥delta$
$K¥subset ¥mathrm{Y}=R^{m}$
$¥epsilon^{*}$
$¥epsilon^{*}$
$¥overline{f}$
.
$¥mathrm{m}¥mathrm{i}¥mathrm{n}¥overline{F}¥mathrm{h}¥mathrm{a}¥mathrm{s}¥mathrm{t}¥mathrm{h}¥mathrm{e}¥mathrm{p}¥mathrm{r}¥mathrm{o}¥mathrm{p}¥mathrm{e}¥mathrm{r}¥mathrm{t}¥mathrm{y}¥mathrm{t}¥mathrm{h}¥mathrm{a}¥mathrm{t}S_{B}^{f}¥subset V¥{¥epsilon^{*},¥epsilon_{0}, ¥delta¥}.¥mathrm{B}¥mathrm{y}¥mathrm{L}¥mathrm{e}¥mathrm{m}¥mathrm{m}¥mathrm{a}¥mathrm{t}¥mathrm{a}2.3$
$¥mathrm{a}¥mathrm{L}¥mathrm{e}¥mathrm{t}¥mathrm{n}¥mathrm{d}2¥epsilon_{0}.¥mathrm{g}¥mathrm{i}¥mathrm{v}¥mathrm{e}¥mathrm{n}4¥mathrm{t}¥mathrm{h}¥mathrm{e}¥mathrm{r}¥mathrm{e}¥mathrm{b}¥mathrm{e}¥mathrm{x}¥mathrm{i}¥mathrm{s}¥mathrm{t}¥mathrm{s}¥mathrm{a}¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{t}¥mathrm{i}¥mathrm{n}¥mathrm{u}¥mathrm{o}¥mathrm{u}¥mathrm{s}¥mathrm{y}¥mathrm{L}¥mathrm{e}¥mathrm{m}¥mathrm{m}¥mathrm{a}2.3¥mathrm{a}¥mathrm{n}¥mathrm{d}1¥mathrm{m}¥mathrm{a}¥mathrm{p}¥overline{f}¥mathrm{e}¥mathrm{t}¥epsilon^{¥prime}=.$
.
which is an ’-approximation for
on
and such that the set-valued
map
.
defined by $x-¥circ S^{¥prime}(x)=S^{f}(x)$ is a -map such that
The index of the map is given by
$¥overline{V}¥rightarrow R^{n}$
$¥overline{F}$
$¥overline{V}_{B}$
$¥mathrm{e}$
$S^{¥prime}:¥overline{B}-¥circ R^{m}$
$S_{B}^{¥prime}¥subset V$
$¥mathrm{w}$
$¥mathrm{i}(S^{¥prime})=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(0, ¥cdot), V(0))=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(0, ¥cdot), U(0))¥neq 0$
.
is a -map with index
The set valued map
verify the
have
that
Gr
Gr $T$ ; but
.
As
we
(see [4]).
. Then
, which
property, then there exists $x¥in B$ such that
, as
.
contradicts the fact that
$X=R^{n}$
Banach space.
,
Second part:
-approximation
there exists a
of
on
As
is a compact
pointwise approximation
. On the other hand there exists a
of
whose range is contained in a finite dimensional convex set
.
By Lemma 2.2 and the properties of the index, we get
$T^{¥prime}(x)=g(¥mathrm{x}, S^{¥prime}(x))$
$¥mathrm{i}(T^{¥prime})=¥mathrm{i}(S^{¥prime})¥neq 0$
$¥mathrm{w}$
$ T^{¥prime}¥subset¥delta$
$S_{B}^{¥prime}¥subset V$
$T^{¥prime}$
$¥mathrm{O}¥in T^{¥prime}(x)$
$0¥not¥in¥epsilon_{1}T(¥epsilon_{1}x)$
$¥mathrm{B}.¥mathrm{U}$
$¥mathrm{O}¥in¥delta T(¥delta x)$
$¥delta¥leq¥epsilon_{1}$
$K¥subset ¥mathrm{Y}$
$¥overline{U}_{B}$
$¥mathrm{A}¥mathrm{N}¥mathrm{R}$
$¥overline{U}_{B}$
$¥overline{F}$
$¥tilde{f}$
$¥epsilon^{¥prime}/2$
$¥overline{f}$
$¥epsilon^{¥prime}/2$
$¥tilde{f}$
$K_{1}$
$0¥neq ¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(0, ¥cdot), V(0))=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{f}(0, ¥cdot), V(0))=¥mathrm{i}¥mathrm{n}¥mathrm{d}(K_{1},¥overline{f}(0, ¥cdot)_{¥backslash r_{1}}, V(0)¥cap K_{1})$
,
. Then
, with $f=I-¥overline{f.}$ Let $g_{1}=$
and
and
. These two maps satisfy the hypotheses of Theorem 2.1,
then, for the first part of the proof, the set valued map
with
$g_{¥backslash V_{1}}$
$V_{1}=V¥cap(X¥times K_{1})$
$S_{B}^{f}¥subset V_{1}$
$ S_{B}^{f}¥neq¥emptyset$
$¥overline{f_{1}}=¥overline{f_{¥backslash V_{1}}}$
$T^{¥prime¥prime}(x)=g_{1}(x, S^{f_{1}}(¥mathrm{x}))$
108
G. CONTI, P. NISTRI and P. ZECCA
we have that Gr
Gr $T$, contradicting
has a zero in . As
.
the fact that
$X$
Banach spaces.
Third part: ,
on
be an -pointwise approximation of
with finite
:
Let
be the subspace containing the range of
and
dimensional range. Let
deand
. Let
let
$T^{
¥
prime}(x)=g_{2}(x,
S(x))$
-approximation
by
then,
.
of
Lemma
is
an
fined by
. By the
sufficiently small,
. property on
2.6, for
satisfies the
has then a zero on
, which is absurd.
second part of the proof
$ T^{¥prime¥prime}¥subset¥delta$
$S_{B^{1}}^{f}(¥mathrm{x})¥subset V$
$¥overline{B}$
$0¥not¥in¥epsilon_{1}T(¥epsilon_{1}x)$
$¥mathrm{Y}$
$¥overline{g}_{2}$
$¥overline{U}_{B}¥rightarrow X$
$¥overline{U}_{B}$
$¥overline{g}$
$¥mathrm{s}$
$X_{1}¥subset X$
$¥overline{g}_{2}$
$T^{¥prime}:¥overline{B}^{¥prime}=¥overline{B}¥cap X_{1}-¥circ X_{1}$
$¥overline{F}_{2}=¥overline{F}_{¥backslash (X_{1}¥times K)¥cap¥overline{U}_{B}}$
$g_{2}=I-¥overline{g}_{2¥backslash (X_{1}¥times K)¥cap¥overline{U}_{B}}$
$T^{¥prime}$
$¥epsilon>0$
$¥mathrm{e}$
$T_{¥backslash ¥overline{B}^{¥prime}}$
$T^{¥prime}$
$¥partial B^{¥prime}$
$¥mathrm{B}.¥mathrm{U}$
$¥overline{B}^{¥prime}¥subset¥overline{B}$
$T^{¥prime}$
$¥square $
With a similar proof to that one of Theorem 1.4 of [8] we can prove the
following
Theorem 2.2: Let $X$ and $K$ as in Theorem 2.1. Let $U$ be an open,
mapping. Consider
bounded set in $X¥times K$ and let
be an
$Q
¥
subset
X$
$x
¥
in
Q$
such that for every
on
we have
compact, convex set
we have
. Assume that for some (and hence for all)
,
$U(x))¥neq 0$ .
is any continuous map such that
for
If
$x
¥
in
Q$
any
, then there exists a solution $(x, y)¥in U$ of (1) with $x¥in Q$ and $y¥in S(x)$ .
$¥overline{F}:¥overline{U}-¥circ K$
$UV^{¥infty}-$
$a$
$y¥not¥in¥overline{F}(x, y)$
$¥chi¥in Q$
$¥partial U(x)$
$¥overline{g}(x, S(¥mathrm{x}))¥subset Q$
$¥overline{g}:S_{Q}¥rightarrow X$
§3.
$¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(¥mathrm{x}, ¥cdot)$
Application
Consider the following nonlinear boundary value control problem
(C)
$¥left¥{¥begin{array}{l}¥ddot{¥mathrm{x}}+x=h(t,¥mathrm{x},¥dot{¥mathrm{x}},u),t¥in[0,¥pi]¥¥¥mathrm{x}(0)=¥mathrm{x}(¥pi),¥dot{¥mathrm{x}}(0)=¥dot{x}(¥pi),¥¥u(t)¥in U(t,¥mathrm{x}(t)),¥mathrm{a}.¥mathrm{a}.t¥in[0,¥pi].¥end{array}¥right.$
Assume the following conditions
¥ ¥
¥
, given by $(t, p, q, r)¥rightarrow h(t, p, q, r)$ , is tthe function
-continuous;
measurable and
the multivalued map $U:[0, ¥pi]¥times R^{n}-¥circ R^{n}$ , $(t, p)$ $-¥circ U(t, p)$ is -measurable
and -continuous for
. $t¥in[0, ¥pi]$ with nonempty and compact values;
$t
¥
. in[0, ¥pi]$ and any $p¥in R^{n}$ . Moreover, the set $U(t, p)$ is
for
star-shaped with respect to the origin for any
;
. $¥in[0, ¥pi]$ and any $p¥in R^{n}$ .
there exists $¥rho>0$ such that $|U(t, p)|<¥rho$ for
In order to formulate our conditions on the control law
, $u(t)=$
$(u_{1}(t), ¥ldots, u_{n}(t))¥in U(t, x(t))$, we need some preliminaries.
First, we recall that
$v
¥
in
L^{
¥
infty}([0,
¥
pi],
R)$
every essentially bounded function
can be regarded, except
for a set of measure zero, as the uniform limit of a sequence of simple functions
where, for any fixed $n¥in N$,
$h:[0,
$¥mathrm{h}_{1})$
pi] times R^{3n} rightarrow R^{n}$
$(p, q, ¥mathrm{r})$
$¥mathrm{c}_{0})$
$¥mathrm{t}$
$¥mathrm{a}.¥mathrm{a}$
$¥mathrm{p}$
$¥mathrm{c}_{1})$
$¥mathrm{O}¥in U(t, p)$
$¥mathrm{a}.¥mathrm{a}$
$(¥mathrm{r}, p)¥in[0, ¥pi]¥times R^{n}$
$¥mathrm{c}_{2})$
$¥mathrm{a}.¥mathrm{a}$
$r$
$t¥rightarrow u(t)$
$¥{¥phi_{n}¥}_{n¥in N^{¥prime}}$
$¥phi_{n}=¥sum_{j=1}^{n¥cdot 2^{n}}a_{j}¥cdot¥chi_{E_{i}}$
,
$a_{j}=¥mathrm{e}¥mathrm{s}¥mathrm{s}¥inf_{t¥in B_{j}}u(t)$
Non Convex Set-Valued Systems
and the sets
$E_{j}$
,
$1¥leq j¥leq n¥cdot 2^{n}$
109
, are given by
,
$E_{j}=¥{t ¥in[0, ¥pi]:j¥cdot 2^{-n}¥leq v(t)¥leq(j+1)¥cdot 2^{-n}¥}$
for sufficiently large
(see [2]). Obviously if is a simple function,
is assumed).
values of . (The convention
From the covering theorem of Vitali, we have that for all
there exists a finite or countable family of nondegenerate closed intervals
having no internal point in common
such that, except for a set of
measure zero, we get
$¥phi_{n}=v$
$v$
$¥chi_{¥emptyset}=0$
$n$
$1¥leq j¥leq n¥cdot 2^{n}$
$¥{I_{j}^{k}¥}_{k=1}^{K(j)}$
$E_{j}=¥bigcup_{k=1}^{K(j)}I_{j}^{k}$
Let
$(a_{j}^{i}, ¥beta_{j}^{i})_{i=1}^{D(j)}$
.
be the connected components of the set
$¥bigcup_{k=1}^{K(j)}I_{j}^{k}$
where
$D(j)$
is
finite or infinite, and let
be the set: $D_{n}=¥{D(1), D(2), ¥ldots, D(j), ¥ldots, D(n¥cdot 2^{n})¥}$ .
Let
be any positive constant. Let us denote by
the closed ball
. For all
let us consider now the sequence
where
$D_{n}$
$R$
$U_{R}$
$¥overline{B}(0, R)¥subset L^{¥infty}([0, ¥pi], R)$
$v¥in U_{R}$
$¥{¥phi_{n}¥}_{n¥in N}$
$¥phi_{n}=¥sum_{j=1}^{n¥cdot 2^{n}}a_{j}¥cdot¥chi_{E_{j}}=¥sum_{j=1}^{n¥cdot 2^{n}}¥sum_{i¥in D(j)}a_{j}^{i}¥cdot¥chi_{(¥alpha_{i}^{¥mathrm{i}},¥beta_{j}^{¥mathrm{i}})}$
,
$a_{j}^{i}=a_{j}¥forall i¥in D(j)$
The last equality holds except for a set of measure zero. Let
and define
$d(v)=¥lim_{n¥rightarrow}¥sup_{+¥infty}d_{n}$
.
For
$u¥in L^{¥infty}([0, ¥pi], R^{n})$
we define
.
$d_{n}=¥sum_{j=1}^{n¥cdot 2^{n}}D(j)$
$d(u)=¥max_{i}d(u_{i}(t))$
.
Let ¥ ¥
a continuous, non decreasing, non negative function
¥
¥
¥
with
.
Assume the following condition on the controls
There exists $N¥in R_{+}$ such that $¥eta(d(u))¥leq N$ for all $u¥in L^{¥infty}([0, ¥pi], R^{n})$ such
that $u(t)¥in U(t, x(t))$ for any $x¥in AC([0, ¥pi], R^{n})$ .
The meaning of the function
is to indicate that we pay a cost each
time that we allow the control function to be nonconstant or jump. Therefore
we confine ourselves to consider only piecewise constant controls with a finite
number of switches.
We can write the problem
$ eta:R cup$
$¥{¥infty¥}¥rightarrow R¥cup$
$¥{¥infty¥}$
$ eta( infty)= infty$
$u$
$¥mathrm{c}_{3})$
$¥eta(d(¥cdot))$
(CI)
$¥left¥{¥begin{array}{l}¥ddot{¥mathrm{x}}+¥mathrm{x}=h(t,x,¥dot{x},u)¥¥x(0)=¥mathrm{x}(¥pi),¥dot{x}(0)=¥dot{¥mathrm{x}}(¥pi),¥end{array}¥right.$
in the form
(CV)
$¥{_{z(0)=z(¥pi)}^{¥dot{z}=Az+K(t,z,u)}$
where
$A=¥left¥{¥begin{array}{ll}0 & I¥¥-I & 0¥end{array}¥right¥}$
,
$z=¥left¥{¥begin{array}{l}X¥¥X_{1}¥end{array}¥right¥}$
and
G. CONTI, P. NISTRI and P. ZECCA
110
.
$K(t, z, u)=¥left¥{¥begin{array}{l}0¥¥h(t_{X,X_{1}},,u)¥end{array}¥right¥}$
,
We put $|z|=|x|+|x_{1}|$ , $X=L^{1}([0, ¥pi], R^{n})$,
$V=L^{1}([0,
¥
pi],
R^{2n})$
.
, and
$L:D(L)¥subset Z¥rightarrow V$ defined by
operator
linear
Consider the
$z(0)=z(¥pi)¥}$ and
$¥mathrm{Y}=L^{¥infty}([0, ¥pi], R^{n})$
$Z=AC([0, ¥pi]$ ,
$R^{2n})$
for
$(Lz)(t)=¥dot{z}(t)-Az(t)$
The operator
$L$
admits an inverse
$¥mathrm{a}.¥mathrm{a}$
$¥ovalbox{¥tt¥small REJECT}:V¥rightarrow D(L)$
.
$t¥in[0, ¥pi]$
$D(L)=¥{z$
$¥in Z$
:
.
defined by
$z=¥ovalbox{¥tt¥small REJECT} w$
where
$¥left¥{¥begin{array}{l}¥dot{z}-Az=w¥¥z(0)=z(¥pi)¥end{array}¥right.$
For a fixed
, that is
, let
$u¥in ¥mathrm{Y}$
$ff(¥cdot, u):V¥rightarrow V$
be the Nemitskii operator generated by
$¥mathrm{X}$
for
$ff(z, u)=K(t, z(t), u(t))$
$¥mathrm{a}.¥mathrm{a}$
.
$t¥in[0, ¥pi]$
Therefore problem (CV) can be rewritten, for a given
$Z=¥overline{g}(z, u)$
where
$¥overline{g}:V¥times ¥mathrm{Y}¥rightarrow V$
.
, in the operator form
$u¥in 7$
,
is given by
$¥overline{g}=¥ovalbox{¥tt¥small REJECT}$
○
$ff$
,
It is clear that is a compact operator on $V¥times K$ , for any compact set
and the continuity of the operators
via the compact imbedding of $D(L)$ into
. Let us define now the multivalued Nemitskii operator
and
as follows
$K¥subset ¥mathrm{Y}$
$¥overline{g}$
$V$
$¥overline{F}:V-¥circ ¥mathrm{Y}$
$¥ovalbox{¥tt¥small REJECT}$
$¥ovalbox{¥tt¥small REJECT}$
$¥overline{F}(z):=$
{
$u¥in ¥mathrm{Y}:u(t)¥in U(t,$ $x(t))$
for
where the vector $x=x(t)$ represents the
by
.
we can also denote the set
(C) in the form
$¥overline{F}(x)$
$¥overline{F}(z)$
(3. 1)
$¥mathrm{n}$
$¥mathrm{a}.¥mathrm{a}$
.
$t¥in[0,$ $¥pi];¥eta(d(u))¥leq N$
},
-first components of ; obviously,
Finally, we can write the problem
$z$
$¥left¥{¥begin{array}{l}u¥in¥overline{F}(z)¥¥z=¥overline{g}(z,u)¥end{array}¥right.$
In order to apply Theorem 2.1, we have to prove the following
i)
$¥mathrm{i}¥mathrm{i})$
Proposition 3.1:
for all $x¥in X$ ;
is star shaped with respect to the origin
is a compact subset of ;
is u.s.c. for all $x¥in X$ .
$0¥in¥overline{F}(¥mathrm{x})$
$¥overline{F}(x)$
$¥mathrm{I}¥mathrm{m}¥overline{F}$
$¥mathrm{i}¥mathrm{i}¥mathrm{i})$
$¥overline{F}$
$¥mathrm{i}¥mathrm{v})$
$¥mathrm{Y}$
for
all
$x¥in X$
;
111
Non Convex Set-Valued Systems
immediately from hypothesis ).
for all $¥lambda¥in[0,1]$ .
; we want to show that
Assume
. on $[0, ¥pi]$ and $¥mu(u)¥leq N$ , then from ) it follows that
Since
$¥lambda u(t)¥in U(t, x(t))$ and
. $t¥in[0, ¥pi]$ .
for every $¥lambda¥in[0,1]$ and for
for every $¥lambda¥in[0,1]$ .
Hence
, is a bounded set of ,
, i.e.
From ) it follows that the
$M>0$
. Using
say it is contained in a ball of radius
) and the arguments
of [9] we can prove that it is also compact.
; let
be a sequence such
Let
be a sequence in $X$ ,
$[0,
¥pi]$
. on
and $¥mu(u_{n})¥leq N$ .
, i.e.
and
that
. $t¥in[0, ¥pi]$ and
for
Hence, passing to a subsequence if necessary,
. on $[0, ¥pi]$ and $¥mu(u)¥leq N$ . So
so from ) it follows that
.
has closed graph and the statement is
Proof. i) follows
$¥mathrm{c}_{1}$
$u¥in¥overline{F}(x)$
$¥mathrm{i}¥mathrm{i})$
$¥in¥overline{F}(x)$
$¥mathrm{k}¥mathrm{u}$
$u(t)¥in U(t, ¥mathrm{x}(t))¥mathrm{a}.¥mathrm{e}$
$¥mathrm{c}_{1}$
$¥mu(¥lambda u)¥leq N$
$¥mathrm{a}.¥mathrm{a}$
$¥in¥overline{F}(x)$
$¥lambda u$
$¥mathrm{I}¥mathrm{m}¥overline{F}$
$¥mathrm{i}¥mathrm{i}¥mathrm{i})$
$¥overline{F}(X)$
$¥mathrm{Y}$
$¥mathrm{c}_{2}$
$¥mathrm{c}_{3}$
$¥{x_{n}¥}$
$¥mathrm{i}¥mathrm{v})$
$u_{n}(t)¥in U(t, x_{n}(t))¥mathrm{a}.¥mathrm{e}$
$u_{n}¥in¥overline{F}(x_{n})$
$u_{n}¥rightarrow u$
$¥{u_{n}¥}$
$¥mathrm{x}_{n}¥rightarrow x$
$u_{n}(t)¥rightarrow u(t)$
$¥mathrm{a}.¥mathrm{a}$
$¥overline{F}$
$u(t)¥in U(t, x(t))¥mathrm{a}.¥mathrm{e}$
$¥mathrm{c}_{0}$
$¥square $
$¥dot{¥mathrm{p}}¥mathrm{r}¥mathrm{o}¥mathrm{v}¥mathrm{e}¥mathrm{d}$
Moreover, assume that the following conditions are satisfied.
$|K(t, z, u)|¥leq a|z|+b|u|¥mathrm{a}.¥mathrm{e}$ . in $[0, ¥pi]$ and for all $u¥in R^{n}$ and
$z¥in R^{2n}$
$¥mathrm{h}_{2})$
$0<a<¥frac{1}{¥pi}$
and $b>0$ ;
let $¥mathrm{i}_{K}(t, z, u)=¥langle z, Az+K(t, z, u)¥rangle=¥langle z, K(t, z, u)¥rangle$ , where
.
the inner product of ,
Assume that
$¥mathrm{h}_{3})$
$¥alpha$
$M$
denotes
$¥langle¥alpha, ¥beta¥rangle$
$¥beta¥in R^{2n}$
for every
$¥int_{¥mathrm{o}}^{¥pi}¥lim_{|z|¥rightarrow}¥inf_{¥infty}¥frac{¥mathrm{i}_{K}(t,z,u)}{|z|^{2}}>0$
Here
, where
$u¥in R^{n}$
is the constant of Proposition 3.l-(ii).
Theorem 3.1: If assumptions
then problem (C) has solutions.
$¥mathrm{h}_{1}$
),
$¥mathrm{h}_{2}$
),
$¥mathrm{h}_{3}$
such that
$|u|¥leq M$
.
We have the following result.
),
$¥mathrm{c}_{0}$
),
$¥mathrm{c}_{1}$
),
$¥mathrm{c}_{2}$
),
$¥mathrm{c}_{3}$
) are satisfied,
Hypotheses
) and ) ensure the existence of a constant $R>0$
with
such that, if is a solution of (CV) corresponding to a control
, then $¥max_{t¥in[0,¥pi]}|z(t)|<R$ .
Proof.
$¥mathrm{h}_{2}$
$¥mathrm{h}_{3}$
$u¥in ¥mathrm{Y}$
$z$
$|u|_{¥mathrm{Y}}¥leq M$
In fact, assume the contrary. Then there exists a sequence
such that
periodic solutions of (CV), with corresponding
$u_{n}¥in¥overline{F}(z_{n})$
$¥rightarrow¥infty$
as
on
.
Consider
$¥frac{d}{dt}¥frac{|z(t)|^{2}}{2}=¥langle z_{n}(t),¥dot{z}_{n}(t)¥rangle=¥langle z_{n}(t)$
Dividing by
$[¥tau, ¥tau+1]$ , taking in account of
$u_{n}(t)¥rangle ¥mathrm{a}.¥mathrm{e}$
$ t¥in$
$ n¥rightarrow¥infty$
.
$[0, ¥pi]$
.
of
$¥pi-$
$¥max_{t¥in[0,¥pi]}|z_{n}(t)|^{2}$
,
$Az_{n}(t)+K(t,$
$z_{n}(t)$
,
,
,
and integrating over
, we obtain
) and the boundedness of
$1+|z_{n}(t)|^{2}$
$¥mathrm{h}_{2}$
$¥{z_{n}¥}$
$[¥tau, t]$
$¥tau¥in R$
$¥{u_{n}¥}$
$¥frac{1}{2}(¥ln(1+|z_{n}(t)|^{2})-¥ln(1+|z_{n}(¥tau)|^{2}))=¥int_{¥tau}^{t}¥frac{¥langle z_{n}(s),Az_{n}(s)+K(s,z_{n}(s),u_{n}(s))¥rangle}{1+|z_{n}(s)|^{2}}ds$
$¥leq C$
;
G. CONTI, P. NISTRI and P. ZECCA
112
since
$z_{n}$
is a periodic function we get
$¥ln$
so that
have
$¥mathrm{h}_{2})$
$¥max_{t¥in[0,¥pi]}(1+|z_{n}(t)|^{2})¥leq 2C+¥ln¥min_{t¥in[0,¥pi]}(1+|z_{n}(t)|^{2})$
$¥min(1+|z_{n}(t)|^{2})¥rightarrow¥infty$
for
$ n¥rightarrow¥infty$
Hence
.
$|z_{n}(t)|¥neq 0$
$¥int_{0}^{¥pi}t¥in ¥mathrm{l}0,¥frac{¥langle z_{n}¥pi ¥mathrm{J}(s),Az_{n}(s)+K(s,z_{n}(s),u_{n}(s))¥rangle}{|z_{n}(s)|^{2}}ds=¥ln¥frac{|z_{n}(¥pi)|}{|z_{n}(0)|}=0$
for large
for large
$n$
and we
.
Using
$n$
we obtain
$0=¥lim_{n¥rightarrow}¥inf_{¥infty}¥int_{0}^{¥pi}¥frac{¥langle z_{n}(s),Az_{n}(s)+K(s,z_{n}(s),u_{n}(s))¥rangle}{|z_{n}(s)|^{2}}ds¥geq¥int_{0}^{¥pi}¥frac{¥mathrm{i}_{K}(s,z_{n}(s),u_{n}(s))}{|z_{n}(s)|^{2}}ds$
,
contradicting
).
containing all the solutions of problem (CV)
an open ball in
Let
is a compact
. Since
with
corresponding to a control
is a convex and compact subset of Y. Take
subset of 7, then
,
an arbitrary compact and convex subset such that
. Let $W=B¥times K$ . Clearly all the possible solutions of (3.1) are
$W$
. Finally, consider the following homotopy: $H:[0,1]$
in
and
¥
¥
$K-¥circ K$
is given by
, where ¥
defined by
for any
for any $u¥in K$ . We have that
$
¥
lambda
¥
in[0,1]$
Therefore,
for any
, since
and any
$K$ such that
$U(0)$
open
in
any
relatively
set
is
, where
$¥mathrm{h}_{3}$
$V$
$B$
$u¥in ¥mathrm{Y}$
$|u|_{¥mathrm{Y}}¥leq M$
$¥mathrm{I}¥mathrm{m}¥overline{F}$
$¥overline{¥mathrm{c}¥mathrm{o}}¥mathrm{I}¥mathrm{m}¥overline{F}$
$K¥supset¥overline{¥mathrm{c}¥mathrm{o}}¥mathrm{I}¥mathrm{m}¥overline{F}$
$K¥subset ¥mathrm{Y}$
$¥partial K¥cap$
$¥overline{¥mathrm{c}¥mathrm{o}}¥mathrm{I}¥mathrm{m}¥overline{F}=¥emptyset$
$¥overline{B}¥subset¥ovalbox{¥tt¥small REJECT}^{F}$
$¥times$
$ overline{F}:K times V- circ K$
$H(¥lambda, u)=¥lambda¥overline{F}(u, 0)$
$u¥not¥in H(¥lambda, u)$
$¥overline{F}(u, z)=¥overline{F}(z)¥subset K$
$0¥in¥overline{F}(u, z)$
$¥partial K$
$ u¥in$
$(u, z)¥in ¥mathrm{Y}¥times V$
$¥mathrm{i}¥mathrm{n}¥mathrm{d}(K,¥overline{F}(¥cdot, 0), U(0))¥neq 0$
$U(0)¥supset¥overline{F}(0)$
.
$¥square $
We end with the following result.
Lemma 3.1:
¥
,
with
$S(z)= overline{F}(z)$
$Froo/$.
-A map
For sufficiently large $R>0$ , the map $T(z)=z-¥overline{g}(z, S(z))$,
satisfies the B. U. condition on $¥partial B=¥partial B(0, R)¥subset V$.
The proof is based on the following result stated in [8].
$T:¥ovalbox{¥tt¥small REJECT}(0, R)=-¥circ V$
multivalued map
satisfies the
$¥mathrm{B}.¥mathrm{U}$
. condition if and only if the
$¥tilde{T}(z)=Q(¥overline{¥mathrm{c}¥mathrm{o}}T(z))-¥overline{¥mathrm{c}¥mathrm{o}}(T(-z))$
Here $Q(A)=$ $¥{¥lambda x:¥lambda¥in[0,1], x¥in A¥}$ and
, for
of . Consider now
$v¥in¥overline{g}(z, S(z))$
$A$
$¥overline{¥mathrm{c}¥mathrm{o}}(A)$
has no zeros on
$¥partial¥ovalbox{¥tt¥small REJECT}$
.?
is the closure of the convex hull
. We have that
$z¥in¥partial¥ovalbox{¥tt¥small REJECT}(0, R)$
$¥left¥{¥begin{array}{l}¥dot{v}-Av=K(t,z(t),u(t))¥¥v(0)=v(¥pi)¥end{array}¥right.$
for some
$u¥in S(z)$
where
.
Therefore
$v(t)=¥int_{0}^{¥pi}G(t, s)K(s, z(s), u(s))ds$
$G(t, s)=¥left¥{¥begin{array}{l}[(e^{-A}-I)^{-1}+I]e^{A(t^{-}¥mathrm{s})}¥¥(e^{-A}-I)^{-1}e^{A(t^{-}s)}¥end{array}¥right.$
$00¥leq s¥leq t¥leq¥pi¥leq t¥leq s¥leq¥pi’$
.
113
Non Convex Set-Valued Systems
We get
norm
$|v(t)|¥leq¥int_{0}^{¥pi}|G(t, s)||K(s, z(s), u(s))|ds$
$¥mu(A)$
of the matrix
$A$
Using the fact that the logarithm
.
is zero we obtain
$|v(t)|¥leq¥int_{0}^{¥pi}|K(s, z(s), u(s))|ds¥leq a¥pi R+b¥pi M$
), for sufficiently large $R>0$ we have that
there exists $z¥in¥partial B(0, R)$ , $¥lambda¥in[0,1]$ ,
By
$|v|_{L^{1}}<R$
$¥mathrm{h}_{2}$
$y_{1}¥in¥overline{¥mathrm{c}¥mathrm{o}}(T(z))$
$¥lambda z$
.
.
Assume now that
such that
$¥mathrm{a}¥mathrm{n}¥mathrm{d}¥mathrm{y}_{2}¥in¥overline{¥mathrm{c}¥mathrm{o}}(T(-z))$
$-¥lambda y_{1}+Z+y_{2}=0$
.
Thus
$0=|(1+¥lambda)z-(¥lambda y_{1}-y_{2})|_{L^{1}}¥geq(1+¥lambda)|z|_{L^{1}}-¥lambda|y_{1}|_{L^{1}}+|y_{2}|_{L^{1}}>0$
,
since
clude that $T:B(0,
$|y_{1}|_{L^{1}}$
$|y_{2}|_{L^{1}}<R$
Remark 3.1
R)$
Let
,
Therefore, using the above result of [8], we can con. condition.
satisfies the
.
$-¥circ V$
$x¥in X$
$¥mathrm{B}.¥mathrm{U}$
and let
$¥square $
$f:R^{n}¥times R^{n}¥rightarrow R^{n}$
defined as follows
$f(x(t), u(t))=¥left¥{¥begin{array}{l}1¥mathrm{i}¥mathrm{f}u(t)¥in U(t,x(t))¥¥+¥infty ¥mathrm{i}¥mathrm{f}u(t)¥not¥in U(t,x(t)).¥end{array}¥right.$
If
, we have that
$u¥in ¥mathrm{Y}$
on
$[0, ¥pi]$
.
$¥int_{0}^{¥pi}f(¥mathrm{x}(t), u(t))dt$
if and only if
$=1$
Hence we can define the map
$¥overline{F}:X-¥circ ¥mathrm{Y}$
$¥overline{F}(x)=¥{u$ $¥in ¥mathrm{Y}:¥int_{0}^{¥pi}f(x(t), u(t))dt$
$¥leq M$
,
$u(t)¥in U(t, x(t))¥mathrm{a}.¥mathrm{e}$
.
as follows
$¥mu(u)¥leq N¥}$
.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Anichini, G., Conti, G. and Zecca, P., Approximation and Selection for Nonconvex Multifunctions in Infinite Dimensional Spaces, Boll. Un. Mat. Ital. 7, 4-B, pp. 411-422, (1990).
Bartle, R.G., The elements of integration, M. Wiley & Sons Inc., (1966).
Bressan, A., Cellina, A. and Colombo, G., Upper Semicontinuous Differential Inclusion
Without Convexity, Proc. A.M.S. 106, pp. 771-775, (1989).
Darbo, G., Teoria dell’ Omologia in una Categoria di Mappe Plurivalenti Ponderate,
Rend. Sem. Mat. Univ. Padova, 28, pp. 188-224, (1958).
Dugundji, J., Topology, Allyn and Bacon Inc., Boston (1965).
Gorniewicz, L., On the Solution Sets of Differential Inclusions, J. Math. Anal, and Appl.,
113, pp. 235-244, (1986).
Gorniewicz, L., Granas, A. and Kryszewski, W., On the Homotopy Method in the Fixed
Point Index Theory of Multivalued Mappings of Compact ANR’s, Preprint.
114
[8]
[9]
G. CONTI, P. NISTRI and P. ZECCA
Massabo, I., Nistri, P. and Pejsachowicz, J., On the Solvability of Nonlinear Equations
in Banach Spaces, in Fixed Point Theorems, E. Fadell and G. Fournier Eds., Lecture
Notes in Math. 886, Springer & Verlag, pp. 270-299, (1980).
, J. Optim. Theory and
Nistri, P. and Zecca, P., An Optimal Control Problem in
Appl., 65, pp. 289-304, (1990).
$L^{¥infty}$
nuna adreso:
G. Conti
Dipartimento di Matematica applicata
alle Scienze Economiche e Sociali
Universita di Firenze
via Montebello 7
50123 Firenze
Italy
P. Nistri, P. Zecca
Dipartimento di Sistemi e Informatica
Universita di Firenze
via S. Marta 3
50139 Firenze
Italy
(Ricevita la 8-an de novembro, 1991)