Degree theory for discontinuous operators

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by  Rubén Figueroa

Degree theory for discontinuous operators
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2 Rub´en Figueroa†, Rodrigo Lo´pez Pouso† and Jorge Rodr´ıguez Lo´pez

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†Departamento de Ana´lise Matema´tica, Estat´ıstica e Optimizacio´n, Universidadede Santiago de Compostela,
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15782, Facultade deMatema´ticas, Campus Vida, Santiago, Spain.
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C Abstract
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h Weintroducea new definition of topological degree for a meaningful class of operators which need
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a notbecontinuous. Subsequently,wederiveanumberoffixedpointtheoremsforsuchoperators. Asan
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application, we deducea new existence result for first–order ODEs with discontinuousnonlinearities.
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1 2010 MSC: 26B20, 34A12.
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0 Keywords and phrases: Degree theory;Leray–Schauder degree; Discontinuous differential equations.
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0 1 Introduction
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Degreetheoryisafundamentaltoolinnonlinearanalysis,especiallyinthestudyofexistenceofsolutions
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1 to many types of problems; see [8, 10, 13]. Readers interested in the history and the development of
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v degree theory are referred to the expository paperby Mawhin [12].
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X Asawell–known fact, continuity isa basicassumption in degree theoryand theclearest limitation
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ofitsapplicability. Asanimportantparticularcase,wepointouttheusualdegree-theory-basedproofs
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ofexistenceofsolutionstoboundaryvalueproblems,whichconsistonturningtheformerproblemsinto
fixedpointproblemsofintegraloperatorsforwhichdegreetheoryapplies. However,mostdiscontinuous
differential equations, see [3, 9], fall outside that scope simply because the corresponding fixed point
operators are not continuous.
Ontheotherhand,theanalysisofdiscontinuousdifferentialequationsusuallyleansonfixedpoints
resultsformonotoneoperators, andthereforethecorrespondingexistenceresultslean,tosomeextent,
on monotonicity conditions imposed on the nonlinear parts of the considered problems.
In this paper we introduce a new definition of topological degree, which coincides with the usual
degree in the continuous case, and it is also suitable for a wide class of operators which need not be
continuous. As a consequence, this new degree proves useful in the study of discontinuous differential
equationsand, moreover, it yields new existence results which do not require monotonicity at all.
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This paper is organized as follows: in Section 2 we introduce our new definition of degree, which
is based on the degree for multivalued uppersemicontinuous operators; in Section 3 we show that the
degree satisfies the usual and desirable properties; in Section 4 we deduce some fixed point theorems,
very much along the line of the classical degree theory but replacing continuity by the less stringent
condition introduced in Section 2; finally, in Section 5, we illustrate the applicability of theory by
proving the existence of solutions to an initial value problem for a discontinuous first–order ordinary
differential equation.
2 A topological degree for discontinuous operators
Hereand henceforth,Ω denotesanonemptyopen subset of aBanach space (X,k·k) and T :Ω−→X
is an operator, not necessarily continuous.
Definition 2.1 The closed–convex envelope of an operator T : Ω −→ X is the multivalued mapping
T:Ω−→2X given by
Tx= coT B (x)∩Ω for every x∈Ω, (2.1)
ε
ε>0
\ (cid:0) (cid:1)
where B (x) denotes the closed ball centered at x and radius ε, and co means closed convex hull.
ε
Inother words, wesaythat y∈Txifforevery ε>0and every ρ>0there exist m∈N andafinite
family of vectors x ∈B (x)∩Ω and coefficients λ ∈[0,1] (i=1,2,...,m) such that λ =1 and
i ε i i
m P
y− λ Tx <ρ.
i i
(cid:13) (cid:13)
(cid:13) Xi=1 (cid:13)
(cid:13) (cid:13)
The following properties are straightf(cid:13)orward consequ(cid:13)ences of the previousdefinition.
(cid:13) (cid:13)
Proposition 2.2 In the conditions of Definition 2.1 the following statements are true:
1. Tx is closed and convex, and Tx∈Tx for all x∈Ω;
2. If TΩ⊂K for some closed and convex set K ⊂X, then TΩ⊂K.
Closed–convexenvelopes(cc–envelopes,forshort)neednotbeuppersemicontinuous(usc,forshort),
see [4, Example 1.2], unless some additional assumptions are imposed on T.
Proposition 2.3 Let T be the cc–envelope of an operator T :Ω −→ X. The following properties are
satisfied:
1. If T maps bounded sets into relatively compact sets, then T assumes compact values and it isusc;
2. If TΩ is relatively compact, then TΩ is relatively compact.
Proof. Let x ∈ Ω be fixed and let us prove that Tx is compact. We know that Tx is closed, so it
suffices to show that it is contained in a compact set. To doso, we note that
Tx= coT B (x)∩Ω ⊂coT B (x)∩Ω ⊂coT B (x)∩Ω ,
ε 1 1
ε>0
\ (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
2
and coT B (x)∩Ω is compact because it is the closed convex hull of a compact subset of a Banach
1
space; see(cid:0) [1, Theore(cid:1)m 5.35]. Hence Tx is compact for every x ∈ Ω, and this property allows us to
check that T is usc by means of sequences, see [1, Theorem 17.20]: let x →x in Ω and let y ∈Tx
n n n
for all n∈N be such that y →y; we have to prove that y ∈Tx. Let ε>0 be fixed and take N ∈N
n
such that B (x ) ⊂ B (x) for all n ≥ N. Then we have y ∈ coT(B (x )∩Ω) ⊂ coT(B (x)∩Ω)
ε n 2ε n ε n 2ε
for all n ≥ N, which implies that y ∈coT(B (x)∩Ω). Since ε > 0 was arbitrary, we conclude that
2ε
y∈Tx.
Argumentsare similar for thesecond part of theproposition. For every x∈Ω and ε>0 we have
coT(B (x)∩Ω)⊂coTΩ,
ε
and therefore Tx ⊂ coTΩ for all x ∈ Ω. Hence, TΩ is compact because it is a closed subset of the
compact set coTΩ. ⊓⊔
OurnextpropositionshowsthatTisthesmallestclosedandconvex–valueduscoperatorwhichhas
T as a selection.
Proposition 2.4 Let T be the cc–envelope of an operator T :Ω−→X.
If T˜ : Ω −→ 2X is an usc operator which assumes closed and convex values and Tx ∈ T˜x for all
x∈Ω, then Tx⊂T˜x for all x∈Ω.
Proof. Let T˜ : Ω −→ 2X be an operator in the conditions of the statement, let x ∈ Ω be fixed and
takey∈Tx; we haveto show that y∈T˜x.
First, we fix r>0 and we consider theopen set
V = B (u),
r/2
u[∈T˜x
where B (u) = {z ∈X : kz−uk < r/2} is the open ball centered at u and radius r/2. Obviously,
r/2
we haveT˜x⊂V and ρ(z,T˜x)<r/2 for all z ∈V, where ρ denotes themetric induced by the norm in
X. Furthermore, we havethat
ρ(z,T˜x)<r/2 for all z ∈coV (2.2)
because T˜x is convex.
Since T˜ is upper semicontinuous, there exists ε > 0 such that T˜(B (x)∩Ω) ⊂ V. Since T is a
0 ε0
selection of T˜, we also havethat T(B (x)∩Ω)⊂V, and then
ε0
y∈Tx= coT B (x)∩Ω ⊂coT(B (x)∩Ω)⊂coV.
ε ε0
ε>0
\ (cid:0) (cid:1)
Hencewe can findz ∈V and λ ∈[0,1], for i=1,2,...,m, such that λ =1 and
i i i
m
r P
y− λ z < .
(cid:13) i i(cid:13) 2
(cid:13) Xi=1 (cid:13)
(cid:13) (cid:13)
Since λizi ∈coV, we can use (2.2) to o(cid:13)btain that (cid:13)
(cid:13) (cid:13)
P m m
ρ(y,T˜x)≤ρ λ z ,T˜x + y− λ z <r,
i i i i
! (cid:13) (cid:13)
Xi=1 (cid:13) Xi=1 (cid:13)
(cid:13) (cid:13)
which implies that y∈T˜x because r>0 can bearbitrari(cid:13)ly small and(cid:13)T˜x is closed. ⊓⊔
(cid:13) (cid:13)
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Asa corollary of theprevious result we obtain thefollowing.
Corollary 2.5 If T :Ω−→X is continuous then Tx={Tx} for all x∈Ω.
Wearealreadyinapositiontodefineatopologicaldegreeforsomediscontinuousoperators. Inthis
case we replace continuityby condition (2.3).
Definition 2.6 Let Ω be a nonempty bounded open subset of a Banach space X and let T :Ω−→X
be such that TΩ is relatively compact, Tx6=x for every x∈∂Ω, and
{x}∩Tx⊂{Tx} for every x∈Ω∩TΩ, (2.3)
where T is the cc–envelope of T.
We define the degree of I−T on Ω with respect to 0∈X as follows:
deg(I−T,Ω,0)=deg(I−T,Ω,0), (2.4)
where the degree in the right–hand side is that of usc multivalued operators (see, e.g. [2, 13, 15]).
Let us see that deg(I−T,Ω,0) is well–defined in the conditions of Definition 2.6. First, we know
from Proposition 2.3 that TΩ is relatively compact. Second,if x∈Tx for some x∈∂Ω, then
{x}∩Tx={x} and x∈Ω∩TΩ,
which, together with condition (2.3), yields x = Tx, a contradiction with the assumptions on T.
Therefore, deg(I−T,Ω,0) is well definedand Definition 2.6 makessense.
Moreover,Definition2.6reducestotheusualLeray–SchauderdegreewhenT iscontinuous. Indeed,
when T is continuous we have Tx = {Tx} for all x ∈ Ω, so condition (2.3) is trivially satisfied, and
(2.4) is just
deg(I−T,Ω,0)=deg(I−{T},Ω,0),
wheredeg(I−{T},Ω,0)isthedegreeformultivaluedoperatorsintheparticularcaseofasingle–valued
completely continuousoperator T,which coincides with theLeray–Schauderdegree.
As usual, we shall simplify notation and write deg(I−T,Ω) instead of deg(I−T,Ω,0). The
definition of deg(I−T,Ω,p) for any p∈X reduces to thecase p=0, see [13].
3 Properties of this new degree
The degree we defined in the previous section provides, as we have seen, a generalization of Leray–
Schauderdegreeforsomekindofdiscontinuousoperators. AswesaidinIntroduction,Leray–Schauder
degreeisaverypowerfultoolthathasbeenextensivelyusedinmanycontexts,particularlyforguaran-
teing theexistence of solutions of differential equations. The utilitiy of the degree lies in thefact that
itsatisfiessometopologicalandalgebraicproperties. Nowwewillshowthatournewdegreealsofulfills
theseproperties, and this will be a consequenceof theproperties of degree for multivalued mappings.
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Proposition 3.1 Let T :Ω−→X be a mapping in the conditions of Definition 2.6. Then the degree
deg(I−T,Ω) satisfies the following properties:
1. (Homotopy invariance) Let H :Ω×[0,1]−→X a mapping such that:
(a) for each (x,t)∈Ω×[0,1] and all ε>0 there exists δ =δ(ε,x,t)>0 such that
s∈[0,1], |t−s|<δ =⇒ kH(z,t)−H(z,s)k<ε ∀z∈B (x)∩Ω;
δ
(b) H Ω×[0,1] is relatively compact;
(c) giv(cid:0)en t∈[0,(cid:1)1], if H (x):=H(x,t) and H denotes the cc-envelope of H then for all x∈Ω
t t t
we have
{x}∩H (x)⊂{H (x)}. (3.5)
t t
If x∈/ H (x) for all x∈∂Ω and all t∈[0,1] then deg(I−H ,Ω) does not depend on t.
t t
2. (Additivity) Let Ω ,Ω ⊂Ω be open and such that Ω ∪Ω =Ω and Ω ∩Ω =∅.
1 2 1 2 1 2
If 0∈/ (I−T)(Ω\(Ω ∪Ω )), then we have
1 2
deg(I−T,Ω)=deg(I−T,Ω )+deg(I−T,Ω ).
1 2
3. (Excision) Let A⊂Ω be a closed set such that 0∈/ (I−T)(∂Ω)∪(I−T)(A). Then
deg(I−T,Ω)=deg(I−T,Ω\A).
4. (Existence) If deg(I−T,Ω)6=0 then there exists x∈Ω such that Tx=x.
5. (Normalization) deg(I,Ω)=1 if and only if 0∈Ω.
Proof.
1. WedefineH as the following multivalued mapping:
H(x,t)= coH B (x)∩Ω,t .
ε
ε>0
\ (cid:0) (cid:1)
Observethat H(x,t)=H (x),where H is as in the statement.
t t
Since H Ω×[0,1] is a relatively compact set, H Ω×[0,1] is relatively compact. In addition,
themult(cid:0)ivaluedma(cid:1)ppingHis convexand closed va(cid:0)lued. Let(cid:1)usprovethat H:Ω×[0,1]→2X is
an uppersemicontinuousoperator. Tosee this,it sufficestoprovethatif x →xin Ω,t →tin
n n
[0,1]andy ∈H(x ,t )with y →y,theny∈H(x,t). Letε>0andµ>0befixed;wehaveto
n n n n
m
find x ∈B (x)∩Ω and λ ∈[0,1] (i=1,...,m) such that λ =1 and
i ε i i
i=1
X
m
y− λ H(x ,t) <µ. (3.6)
i i
(cid:13) (cid:13)
(cid:13) Xi=1 (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
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We can assume without loss of generality that ε< δ(µ/4,x,t), where δ is as in 1(a), and so we
can takeN ∈N such that
x ∈B (x)∩Ω,
N ε/2
µ
ky−y k< ,
N 2
µ
kH(z,t )−H(z,t)k< ∀z∈B (x)∩Ω.
N 4 ε
Asy ∈H(x ,t ) we knowthat thereexist x ∈B (x)∩Ωand λ ∈[0,1] (i=1,...,m) with
N N N i ε/2 i
m
λ =1 and
i
i=1
X
m
µ
y − λ H(x ,t ) < .
(cid:13) N i i N (cid:13) 4
(cid:13) Xi=1 (cid:13)
(cid:13) (cid:13)
Hence, we have (cid:13) (cid:13)
(cid:13) (cid:13)
ε ε
kx−x k≤kx−x k+kx −x k≤ + =ε,
i N N i 2 2
so x ∈B (x)∩Ω. Moreover, bytriangle inequality
i ε
m m m
y− λ H(x ,t) ≤ky−y k+ y − λ H(x ,t ) + λ (H(x ,t )−H(x ,t))
i i N N i i N i i N i
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) Xi=1 (cid:13) (cid:13) Xi=1 (cid:13) (cid:13)Xi=1 (cid:13)
(cid:13) (cid:13) µ µ µ(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13)< + + (cid:13)=µ, (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) 2 4 4 (cid:13) (cid:13) (cid:13) (cid:13)
and thus(3.6) is satisfied.
On the other hand, condition (3.5) along with x 6= H(x,t) for every (x,t) ∈ ∂Ω×[0,1], imply
that x6∈H(x,t) for all (x,t)∈∂Ω×[0,1], and so the degree deg(I−H ,Ω)=deg(I−H ,Ω) is
t t
well–defined and it is independent of t ∈ [0,1], by homotopy property of degree for multivalued
mappings, see [15].
2. Condition (2.3) and the hypothesis 0 6∈ (I−T) Ω\(Ω ∪Ω ) imply that x 6∈ Tx holds for all
1 2
x∈Ω\(Ω1∪Ω2). Then,bydirectapplicationof(cid:0)theadditivityp(cid:1)ropertyofdegreeformultivalued
mappings we conclude that
deg(I−T,Ω)=deg(I−T,Ω)=deg(I−T,Ω )+deg(I−T,Ω )
1 2
=deg(I−T,Ω )+deg(I−T,Ω ).
1 2
3. As0∈/ (I−T)(A)∪(I−T)(∂Ω),condition (2.3) implies that 0∈/ (I−T)(A)∪(I−T)(∂Ω), and
so the conclusion derives from theexcision property of thedegree for multivalued mappings.
4. Asdeg(I−T,Ω)=deg(I−T,Ω)6=0then thereexists x∈Ωsuch that x∈Tx,andso condition
(2.3) implies that x=Tx.
5. Since deg(I,Ω) = deg(I −0,Ω), and the operator 0 is continuous, our degree coincides with
Leray–Schauder’s, and thenormalization propertyis fulfilled. ⊓⊔
Remark 3.2 Note that condition (2.3) is not essential in order to define deg(I −T,Ω) in terms of
deg(I−T,Ω); in fact, to this end it suffices to require that {x}∩Tx⊂{Tx} in ∂Ω. However, we need
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this condition to be satisfied in the whole of Ω∩TΩ to guarantee the desirable existence property. As
an example, the reader can consider the mapping T :(−1,1)7−→(−1,1) defined by 1(χ −χ ).
2 (−1,0] (0,1)
Thus defined, {x}∩Tx = ∅ for x ∈ [−1/2,1/2] and deg(I −T,(−1,1)) 6= 0 (as a consequence of the
multivalued version of Borsuk’s Theorem [15]), but T has no fixed point in (−1,1).
Thehomotopy invariance propertythat we provedabovebecomes not veryuseful in practice. It is
due to the unstability of condition (3.5) requested for all t ∈ [0,1] and all x ∈ Ω, because the set of
functions satisfying this condition not very well–behaved, as we show in thefollowing example.
Example 3.3 Let T :[0,1]→[0,1] be the piecewise constant function given by
1/3 if 0 ≤x≤ 1/3,
T(x)=2/3 if 1/3 <x≤ 2/3,
 1 if 2/3 <x≤ 1.
Then it is easy to check that condition (2.3) holds for all x∈[0,1] but this is not true for the mapping
S = 1T at the point x=1/3. Indeed, in this case we have
2
1 1 1 1 1 1 1 1
S = , = 6⊂ = S .
3 3 3 6 3 3 6 3
(cid:26) (cid:27) (cid:18) (cid:19) (cid:26) (cid:27) (cid:20) (cid:21) (cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:18) (cid:19)(cid:27)
\ \
The previous example shows that even for linear homotopies condition (3.5) can fail. To overcome
thisdifficultyimproveonthepreviouspropositioninordertoavoidrequestingcondition(3.5)forallt.
Theorem 3.4 Let H :Ω×[0,1]→X be a map satisfying the following conditions:
(a) for each (x,t)∈Ω×[0,1] and all ε>0 there exists δ=δ(ε,x,t)>0 such that
s∈[0,1], |t−s|<δ =⇒ kH(z,t)−H(z,s)k<ε ∀z∈B (x)∩Ω;
δ
(b) H Ω×[0,1] is relatively compact;
(c) {x(cid:0)}∩H (x)⊂(cid:1) {H (x)} is satisfied for all x∈Ω∩H Ω when t=0 and t=1.
t t t
If x6∈H(x,t) for all (x,t)∈∂Ω×[0,1] then
deg(I−H ,Ω)=deg(I−H ,Ω).
0 1
Proof. It is possibly toprove that thedegree for multivalued mappings is well defined for
H (x)= co H B (x)∩Ω ,
t t ε
ε>0
\ (cid:0) (cid:0) (cid:1)(cid:1)
foreveryt∈[0,1],inasimilarwaythatforhomotopyinvariancepropertyabove. Therefore,homotopy
invarianceproperty of thedegree for multivalued mappings guarantees in particular that
deg(I−H ,Ω)=deg(I−H ,Ω)=deg(I−H ,Ω)=deg(I−H ,Ω),
0 0 1 1
which finishes theproof. ⊓⊔
7
We finish this section by introducing two classical results in the context of Leray–Schauder degree
that remain true when considering our new degree for discontinuous operators satisfying (2.3). The
firstoneisthewell–known factthatfordegree“onlywhathappensintheboundarymatters,”andthe
second one is the natural extension of Borsuk’s Theorem in our setting.
Proposition 3.5 Let T,S :Ω−→X be two mappings in the conditions of Definition 2.6. If Tx=Sx
for all x∈∂Ω and 0∈/ (I−T)(∂Ω)∪(I−S)(∂Ω) then deg(I−T,Ω)=deg(I−S,Ω).
Proof. The degree for themultivalued mappings T and S in Ω is well defined because T and S are in
theconditionsofDefinition2.6,soTx=Sxforallx∈∂Ωimpliesthatdeg(I−T,Ω)=deg(I−S,Ω),
see [15, Theorem 4]. Therefore, by Definition 2.6, we conclude that deg(I−T,Ω)=deg(I−S,Ω). ⊓⊔
Asthe proof of thefollowing result is similar to the previousone with the obvious changes we will
omit it.
Theorem 3.6 (Borsuk’s) Assume that 0∈Ω and that x∈Ω implies −x∈Ω, and let T :Ω−→X be
a mapping in the conditions of Definition 2.6. If 0∈/ (I−T)(∂Ω) and T(x)=−T(−x) for all x∈∂Ω
then deg(I−T,Ω) is odd.
4 Fixed point theorems
One of the most typical application of degree theory is the search for topological spaces which satisfy
the so called fixed point property, that is, topological spaces M such that any continuous mapping
T : M −→ M has a fixed point. It was proved by Dugundji [5] that in every infinite–dimensional
space there exists a fixed–point–free mapping from its closed unit ball into itself, and so some extra–
assumption regarding compacity of images is required in theinfinite–dimensional case. In this section
we will use our new degree to extend these results for operators which are not necessarily continuous
butsatisfy condition (2.3).
Theorem 4.1 LetM beabounded,closedandconvexsubsetofX containingtheorigin0initsinterior.
Let T : M → M be a mapping satisfying (2.3) with Ω = int(M) and such that T(M) is a relatively
compact subset of X. Then T has a fixed point in M.
Proof. Let Ω=int(M). The assumptions imply that Ω6=∅ and, since M is convex, we have Ω=M
and ∂M =∂Ω, see [1, Lemma 5,28].
We can assume that x 6∈ Tx for all x ∈ ∂Ω, because otherwise condition (2.3) guarantees that
x=Tx and the proof is over.
Consider the homotopy
H(x,t)=tT(x) x∈Ω, 0≤t≤1 .
(cid:0) (cid:1)
Given t ∈ [0,1), we define the set S = tx:x∈Ω , which is closed. Since 0 ∈ Ω, we deduce
from Lemma 5.28 in [1] that S ⊂ Ω. Hence(cid:8)co Ht Ω (cid:9) ⊂ S ⊂ Ωand, therefore, we have Ht Ω ⊂
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1)
8
co H Ω ⊂Ω. This implies that x6∈H (x) for all x∈∂Ω and all t∈[0,1). Moreover, since TΩ is
t t
rel(cid:0)ative(cid:0)ly(cid:1)c(cid:1)ompact, then so H Ω×[0,1] is. Now, we can apply Theorem 3.4 and we obtain that
(cid:0) (cid:1)
deg(I−T,Ω)=deg(I,Ω).
By the normalization property in Proposition 3.1, we have deg(I,Ω) = 1 and then the existence
property ensuresthat thereis x∈Ω with Tx=x. ⊓⊔
Now we relax the hypothesis that M has non-empty interior. This restriction can be remove by
using the extension of Tietze’s Theorem given by Dugundji. We omit its proof, the reader can see [5,
Theorem 4.1].
Theorem 4.2 (Dugundji) Suppose that A is a closed subset of a metric space B, and let L be a
normedlinearspace. Every continuous functionf :A→Lhasacontinuous extension F :B →Lsuch
that F(B)⊂cof(A).
The following result is a generalization of Schauder’s fixed–point Theorem for not necessarily con-
tinuous operators. It is possible to find other results in this direction in [6, 11], but here we present a
direct proof using our new degree theory.
Theorem 4.3 Let M be a non-empty, bounded, closed and convex subset of X. Let F :M →M be a
mapping such that FM is a relatively compact subset of X and
{x}∩Fx⊂{Fx} for every x∈M∩FM, (4.7)
where Fx= coF(B (x)∩M).
ε>0 ε
Then F hTas a fixed point in M.
Proof. Since M is bounded, there exists an open ball B containing the origin such that M ⊂B. By
Dugundji’sTheorem4.2,thereisacontinuousfunctionr:B→M suchthatr =I,theidentity. Let
|M
usprovenowthat theoperator T =F ◦r:B→B satisfies theconditionsin Theorem 4.1. First, note
thatTB isarelativelycompactsubsetofX becauseTB ⊂FM. Nowweprovethat{x}∩Tx⊂{Tx}
forallx∈B∩TB. SinceTB ⊂FM ⊂M,andM isaclosed andconvexsubsetofX,thenTB⊂M
and B∩TB⊂B∩M =M.
If x ∈ M, then T(x) = F(x). Moreover, as r is continuous at x and r(x) = x, then for all ε > 0
thereexists δ>0 (we can always choose δ<ε) such that
r B (x)∩B ⊂B (r(x))∩M =B (x)∩M.
δ ε ε
(cid:0) (cid:1)
Therefore, we deducethat
Tx= coT B (x)∩B = co(F ◦r) B (x)∩B ⊂ coF B (x)∩M =F(x).
δ δ ε
δ\>0 (cid:0) (cid:1) δ\>0 (cid:0) (cid:1) ε\>0 (cid:0) (cid:1)
Then {x}∩Tx ⊂ {x}∩Fx ⊂ {Fx} = {Tx} for all x ∈ M ⊃ B∩TB. By Theorem 4.1, there exists
z∈B such that T(z)=z. Since TB⊂M, we concludethat z∈M and F(z)=z. ⊓⊔
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Corollary 4.4 LetK beanon-empty, convexandcompactsubsetofX. LetT :K →K beamapping
satisfying (4.7) with M =K. Then T has a fixed point in K.
Proof. SinceTK ⊂K andK iscompact,TK isarelativelycompactsubsetofX. ApplyingTheorem
4.3, T has a fixedpoint. ⊓⊔
Theorem 4.5 (Schaefer’s) Let X be a Banach space and T : X −→ X mapping bounded sets into
relativelycompactones. AssumethatthereexistsR>0suchthatconditionx∈σTxforsomeσ∈[0,1]
implies kxk < R. If T satisfies condition (2.3) with Ω = B (0) (open ball centered at the origin and
R
radius R) then T has a fixed point in B (0).
R
Proof. For each σ ∈ [0,1] the mapping I−σT : B (0) −→ 2X satisfies that 0 ∈/ (I−σT)(∂B (0)),
R R
and so deg(I−σT,B (0)) is well–defined. Now the homotopy invariance property guarantees that
R
deg(I−T,B (0))=deg(I−T,B (0))=deg(I,B (0))=1,
R R R
and so there exists x∈B (0) such that Tx=x. ⊓⊔
R
5 Application to a first order differential problem
Inthis section we illustrate theapplicability of thedegree theory described in sections 2 and 3. Todo
so, we consider a specially simple problem, namely, theexistence of absolutely continuous solutions of
theinitial valueproblem
x′=f(t,x) for a.a. t∈I =[a,b], x(a)=x ∈R. (5.8)
a
Unliketheclassicalsituation,wedonotassumethatf :I×R−→RisaCarath´eodoryfunction. Indeed,
we shall allow f to be discontinuous over the graphs of countably many functions in the conditions of
thefollowing definition. Similar definitions can be found in [6, 11].
Definition 5.1 An admissible discontinuity curve for the differential equation x′=f(t,x) is an abso-
lutely continuous function γ :[c,d]⊂I →R satisfying one of the following conditions:
either γ′(t)=f(t,γ(t)) for a.a. t∈[c,d] (and we say that γ is viable for the differential equation),
or there exists ε>0 and ψ∈L1(c,d), ψ(t)>0 for a.a. t∈[c,d], such that
either
γ′(t)+ψ(t)<f(t,y) for a.a. t∈I and all y∈[γ(t)−ε,γ(t)+ε], (5.9)
or
γ′(t)−ψ(t)>f(t,y) for a.a. t∈I and all y∈[γ(t)−ε,γ(t)+ε]. (5.10)
We say that γ is inviable for the differential equation if it satisfies (5.9) or (5.10).
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