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Contractibility and fixed point property: the case of decomposable sets
Nonlmem Analysis, Theory, Methods Printed m Great Britain.
& Applrcotrons,
Vol. 18. No. 7, pp, 689-695,
0362-546X/92 $5.00 + .OO C 1992 Pergamon Press plc
1992.
CONTRACTIBILITY AND FIXED POINT PROPERTY: THE CASE OF DECOMPOSABLE SETS CARLO MARICONDA SISSA-ISAS, via Beirut 2/4, 34014 Trieste, Italy (Received Key words and phrases:
4 February
1991; received for publication
Fixed point, decomposable,
2 May 1991)
increasing family, Liapunov, convex hull, retract,
dimension. INTRODUCTION SCHAUDER’S fixed point theorem [l] asserts that any compact mapping of a convex set into itself has a fixed point. Various attempts have been made to do fixed point theory in spaces more general than convex sets, but no general theory is known. A theorem by Lefschetz [2, corollary, p. 3591 leads one to believe that each compact, homologically trivial and locally connected set has the fixed point property, but Borsuk [3] gave an example to refute this conjecture. His set is neither contractible nor locally contractible, hence it leaves open the following version of Lefschetz’s corollary. CONJECTURE. Any contractible,
locally contractible
set has the fixed point property
for compact
mappings. Borsuk showed in [4] that this conjecture is true in finite dimensional metric spaces (see theorem 1). Some particular cases of the conjecture-for decomposable subsets of L’--were proved by Cellina [5] and Fryszkowski [6]. Furthermore, it has been proved that each decomposable subset of L'is a retract of L' [7]. As Smart pointed out in a recent paper [S], these authors do not mention or use the contractibility and local contractibility properties that these spaces possess: Smart proved a weaker version of their theorem-an LJ’-continuous map of a decomposable set M into an L”-compact subset of M has a fixed point-by a method related to the contraction process in decomposable sets. Notice that this requirement is quite strong: it means that no “cut and paste” of functions of T(M) is possible; as an example the L'precompact subset of L'[O,l] ]~t~,~t, 01 E [0, 11) is not L”-precompact. Here I generalize Smart’s arguments in order to prove that each LP-compact mapping (1 5 p < co) of a decomposable subset of Lp into itself has a fixed point. This can be regarded as a further step of the project aimed at the conjecture. The main tool is a consequence of Liapunov’s theorem on the range of a vector measure as it is formulated in [9]: it enables the construction of a nonconvex analogue of Schauder’s projection onto a nonsymmetrical nonconvex analogue of the convex hull of a finite number of points. Its advantage with respect to the more familiar decomposable hull is that the first is compact whereas the second is not, unless it is trivial (see for instance [lo]). As a consequence, each compact map of a topological space into a decomposable set can be approximated by a sequence of functions having a finite dimensional range. This property is in common with compact maps into a convex set [ll, Chapter VI]. 689
C. MARICONDA
690 NOTATIONS
AND
PRELIMINARY
RESULTS
1. A subset A of a topological space X is said to be contractible in the space X to a set B c X provided that the inclusion A Q X is homotopic (in the space X’) to a map with values belonging to B. If A = X and B consists of a single point, then X is said to be contractible. Definition
Definition 2. A topological space X is said to be locally contractible at a point x0 E X provided that each neighbourhood U of x,, contains a neighbourhood U’ of x,, which is contractible in U to a point. The space X is said to be locally contractible if it is locally contractible at each of its points. It is clear that any convex set (and, as a consequence, any absolute retract) is contractible and locally contractible (see [4] for details). The next theorem shows that the converse is true for compact and finite dimensional spaces. By the dimension of a compact space X we understand the covering dimension of X [12, Chapter V]. Specifically, we say dim X I n provided that each open covering of X has a refinement of order 5 n (the order of a covering being the largest integer n such that there are n + 1 members of the covering which have a nonempty intersection); if dim X I n is true and dim X I n - 1 is false, then we say dim X = n. 1 [4, V(10.5)]. A finite dimensional compact metric space is an absolute retract if and only if it is contractible and locally contractible. THEOREM
Definition 3. Let (X, d) be a compact metric space and r be a nonnegative real number. The space X is said to have Hausdorff r-measure zero if for each E > 0 there exists a finite decomposition of X X=A,u**.UAk such that (diamA,)’ + .a. + (diam A,)’ < E where diam A = sup{&,
6) : a, b E A J.
The connection between the topological concept of dimension and the metrical concept of Hausdorff r-measure, given by the following theorem, will be used later. 2 [12, theorem VII.31. Let n E N and X be a compact space of (n + I)-measure zero. Then dim X 5 n. THEOREM
In what follows, we consider a measure space (Z, nt, p) where YE is a o-algebra of subsets of Z (measurable sets), ZJis a complete positive nonatomic measure such that ,u(Z) < +a, (E,
XA .
Definition 4. A nonempty subset S of L* is said to be decomposable if whenever f and g are in S and A is in 3n then fxA + gxrlA is in S.
691
Contractibility and fixed point property
Decomposable sets have many properties in common with convex sets: see, for instance [5, 7, 131. In particular, Smart [14] pointed out that the decomposable subsets of P[O, l] are contractible and locally contractible. We prove here a more general statement. PROPOSITION
1. Each decomposable
subset S of Z!‘(Z) is contractible
and locally contractible.
Proof. Liapunov’s theorem on the range of a vector measure [9, proposition existence of a family (A,), Eto,il of measurable subsets of Z such that
SC t * AsCAt,
A,=
0,
A, = I,
P(A,)
1. l] provides the
(t E IO, 11).
= W(Z)
(1)
I claim that for any fixed f in S the map @: @, 8) - fXA, +
&%\A,
describes a contraction of S to f and of any c-neighbourhood of S to f. Clearly, the claim is proved if we show that the previous map is continuous. For this purpose, fix a positive E, and let t, E [0, 11, g, E S. Let 6 be such that
Let t and g be such that jig - g,(l, < c/3 and It - t,l < d/p(Z). Since we have: @(t>g) - @(&, &I) = =
fkA,
-
XA,,)
+
gXI\A,
f&A,
-
XA,,)
+
kXZ\A,
-
gOXZ\A,, -
gOXZ\A,)
+
kOXZ\A,
-
gOXZ\A,,)
then it turns out that l/P
Ilw, 8) - wo, go)II,5
l/P +
llg
-
golIp
+
(.\’A,AA,, If” dp > where A AB = (A\B) U (B\A). Hence, by (1) and (2)
A,M,o
kelp
@ >
Ilw g)- @(to, go)Il,< ; + ; + ; = E. The claim is proved. Definition
5. A family (A,),. Lo,11of measurable sets is called increasing if the following properties hold: A, = I; ss t * A,cA,. Ao= 0;
Let us consider a family of positive vector-valued measures P, = (,u:, . . . . &) which are absolutely continuous with respect to P. The space M” of such vector measures is endowed with the topology induced by the norm I(,u~\I,the total variation of ,u,. The proposition formulated below is [9, proposition 1.21 where p. is replaced by (,ZJ,v). PROPOSITION 2. Let X be a compact topological space and v be an absolutely continuous positive measure with respect top. Assume that the map x c-) pu, E M” is continuous. Then, for every E > 0 there exists an increasing family (At), ELo,,I of measurable sets with the following properties:
]/&I,) P(A,)
- k(Z)]
= W(Z),
< s
(t E 10, 11, x E XI;
v(A,) = tv(Z)
(t E 10, 11).
(3) (4)
C. MARICONDA
692
MAINRESULTS THEOREM
3. Let S be a decomposable
P-compact
subset K of S. Then
subset of P(Z), T has a fixed point.
T be an P-continuous
The proof is modelled on a standard proof of Schauder’s second in [I] and in Smart’s proof of theorem 3 in the case where K is an A replacement for the convex combinations used in Schauder’s f , , . . . , f,, be fixed in Lp and Q. = (A,),. tO,il be an increasing Analogous to what is described in [14], if ci 2 0 and ci + ... +
mapping
into an
theorem, given for instance L”-compact subset of S [S]. projection is studied: let family of measurable sets. c, = 1 we use the notation
Clf, @a **- @a Gfn for the function f&i,,
+ --’ +
f2X~A,,+,,\A,,)
+
“*
+
fnX(I\A,,+...+r,_,)
convex combination” which equals fi on &I+... +.I\&.,+.., +Cim,. We regard this as a “nonlinear (with respect to @) of fi , . . ., f, with weights cl, . . ., c,,. These combinations make up the “nonlinear convex hull” of fi , . . . , f,, with respect to the increasing family a; it will be denoted by nocoa(fi,
. . ..f.).
set S then noco,(fi , . . . , f,) As is evident, if fi , . . . , f, belong to a decomposable in S. The rest of the proof follows after theorems 4 and 5 and lemma 1 below.
is contained
sets is called refining Definition 6. Let $ E L’. A family a = (A,), E to, i1 of measurable respect to (I, nt, ,u)) if and only if the following equalities hold:
iA,ddru =tjj#du (tE
4 (with
VA11).
sets. 4. Let fi , . . . , f, E Lp and a = (A,), E to, i1 be an increasing family of measurable FurtherThen nocoe (fi ,. . . , f,) is (i) compact, (ii) contractible and (iii) locally contractible. more, if C? is refining (1fi ( + . . . + 1f,l)” then (iv) noco,(f, , . . . , f,) is finite dimensional.
THEOREM
Proof.
Let C be the convex
compact
subset of R”-’ defined
by
C=((C,,...,C,_I)Ell?-1:05clI~~~~Cn_1~1) and let Y: C --f noco,(f,
, . . . , f,,)
Y(c it *a., %1)
be the function =
fixA,,
+
defined
fZ&A,,\A,,)
by +
‘-*
+
fn&T\A,,_,).
Since the integral operator is absolutely continuous with respect to the measure P, then Y is noco,( f, , . . . , f,) can be continuous hence noco,(f, , . . . , f,) = Y(C) is compact. Furthermore contracted to fi by the process described in the proof of proposition 1. The proof of local contractibility is essentially the proof of [14, theorem 2(iii)] taking into account that we have Y(c) instead off’. I claim that Y In (iv) let @ = (A,),. to 1l be an increasing family refining (Ifi 1 + . . . + 1f,l)“. satisfies a Holder condition. With a view to prove the above assumption, set for any x = (x 1, . . ..X._i) E E?-’ the norm of x to be [lx]] = /x1( + . . - + Jx,_,J. Since Y(c) and Y(d)
693
Contractibility and fixed point property
tl (Ad,/Adi_,) then we have:
agree on Ui (A,&__,)
Furthermore,
(If, I + ... + jfnl)P, hence the following
Q. is refining
A M (Ihl !’ d,
equalities
(i =
+ a*.+ IfnIYdP = Idi - CiI (IfiI + **. + IfnO’@ ,Ii
As a’consequence,
if we set Q = (S, (Ifi I + .-- + (f,l)Pdp)l’P, IIY(c) - Yap, , . . . ,f,)
Now, we prove that noco,(f, Letc>OandB,,...,B,besuchthat
5
hold: 1, . . ..n
-
we have
(5)
2Q11c- d”‘.
has np-measure
zero.
(6)
(diam B,)” + .. k + (diam Bk)n < L.
2Q
Such a decomposition Then
exists since the (n nocoO (fi ) . ..,f,)
1) dimensional
= Y(C)
polytope
C has n-measure
zero.
= Y(B,) u **. u Y(Bk).
Since, by (5), diam Y(Bi) I 2Q(diam
1).
(i = 1, . . . , k)
Bi)l’P
(7)
then (6) and (7) imply diam Y(B1)“p + . .. + diam Y(Bk)“p < E and this proves the claim. Theorem 2 yields the conclusion. The fixed point property for nocao (f, , . . . , f,,) now follows
immediately
5. Let fr , . . . , f,, E Lp and A = (A,),. ,O,1l be an increasing + +.= + If,\)“. Then noco,(fr, . . . . f,,) has the fixed point property.
THEOREM
(Ifil
The proof
of theorem
3 is based on the following
from theorem family
1.
refining
lemma.
“Schauder’s projection”). Let K be a compact subset of Lp. Then family Q. = (A,), E[O,11 refining E > 0 there exist fi, . . ., f, E K, an increasing (If11+ --. + If,l)” and a continuous mapping P, of K into noco,(f, , . . . , f,) such that IlP,f - f (Ip s E for all f in K.
LEMMA 1 (a nonconvex
for
every
Proof. Let fi , . . . , f, in K be such that their ~12~‘~ neighbourhoods and fEK, ,L&, v be the absolutely continuous measures (with P#)
=
i
A If - fiI"Q
(i = 1, . . . . n);
v(A)= AClf,l + -.a+ If,l)” .i
dp.
cover K. Let, for 1 5 i I n respect to p) defined by
C. MARICONDA
694
Clearly the map increasing family
f EK y (,uj,. . . , &)
hence
is continuous, C? = (A,), E to, il with the properties: I&(A)
(lfl i ‘41
+ ... +
- M)V)I
by proposition
(te[O,l],feK,i=l,...,
< 2
if,zlY& = 1 (lfil + ..* + If,,h-b iI MA,)
2 there exists an
n);
(8)
(t E LO, 11);
(t E [O, 11).
= 040
(9) (10)
ForlIiInandfEKset
ri(f) = max O,& - Ilf -fill, ; > ( thus for each f E K some ri(f)
is nonzero.
Then set
ri(f) ci(f) = Cjrj(f) so that for each f, C i Ci(f) = 1 and the map f ++(c,(f), . . . , c,(f)) is continuous. the map P,:K+ noco,(fi, . . ..f.) defined by
I claim that
p,(f) = c1(f)fl @a*** @a c,(f)fn has the required be di(f) = cl(f)
properties. In fact, set for any f E K and i = 1, . . . , n - 1 the function + se. + Ci(f). Then we have:
II&(f) -fII;
= c /Ad,w,Ad_,,lf-fiI’+.
i
By (8), for each i = 1, . . . . n - 1 the following
di to
(11)
I I inequality
If -fiI”dP 5 ci(f)
! AdLn’Ad,-,U,
.iI
holds:
If
-fiI”dP + g*
Since Ci( f) # 0 if and only if 11 f - fill, < ~12~‘~then
IIP&(f)- f 11;I ;
+;
= EP.
The continuity of P, follows from the fact that the family Q. refines (I fi ( + . -a + Ifnl)p : in this situation, the proof of theorem 4(iv) shows that the map Y is continuous. Since, for any f in K, we have P,(f) = Y(d(f)), then the conclusion follows from the continuity of the function d = (d, , . . . , d,). Proof of theorem 3. Fix k E N. Then, by lemma 1, there exist fi , . . . , f,, E K, an increasing family a = (A,), E to, i1 refining (I fi I + . . . + 1f,l)p and a continuous function P,,k of K into such that [IP,,k(f) -f jjp I l/k for all f in K. Let i be the inclusion of nocoa(fi,. . . ,f,) nocoQ (fi , . . . , f,) into S. Let us consider the composed map Pl,k 0 T 0 i.
Contractibility
s
-
T
K
i
S/k
PI,*
0
noco,vI, . . ..fJ
695
and fixed point property
-
This function maps the set nocoe (Jr, . . . , f,) point xk . Thus
To i
I
nocou(fi, . . ..fJ
into itself,
hence,
by theorem
5, it has a fixed
By the compactness of K, T has a fixed point. “Schauder’s projection” of lemma The following consequence of the nonconvex a further property of decomposable sets which is in common with convex sets.
1 provides
THEOREM 6. Let X be a topological space, S be a decomposable subset of P(I) and T: X + S be a continuous compact mapping. Then for each E > 0 there exists a continuous map T,: X --f S whose range has a finite dimension and such that /T,(x) Proof.
Let K be a compact
subset
T(x)\l,
VXEX.
< s
of S such that T(X) C K and fix E > 0. Let
P,: K -+ noco,(f,,
. . ..f.,)
be the nonconvex “Schauder’s projection” of lemma T, = P, 0 T fulfils the requirements of the claim. Acknowledgemenr--I wish to thank the preparation of this work.
Professors
1. Then,
A. Cellina and G. Colombo
by theorem
4 (iv), the map
for the useful conversations
we had during
REFERENCES 1. SMART D.R., Fixed Point Theorems. Cambridge University Press, London (1974). 2. LEFSCHETZ S., Topology. Amer. Math. Sot., New York (1930). 3. BORSUK K., Sur un continu acyclique qui se laisse transformer topologiquement en lui-mCme sans points invariants, Fund. Math. 24, 51-58 (1935). 4. BORSUK K., Theory of Retracts. PWN-Polish Scientific Publishers (Monografie Matematiczne), Warszawa (1967). 5. CELLINA A., A fixed point theorem for subsets of Lr, Lectures Notes in Mathemafics (Multifunctions and Integrands), Vol. 1091, pp. 129-137 (1984). 6. FRYSZKOWSKI A., The generalisation of Cellina’s fixed point theorem, Stud. Math. 78, 213-215 (1984). 7. BRESSAN A. & COLOMBOG., Extensions and selections of maps with decomposable values, Stud. Math. 90, 69-86 (1988). 8. SMART D.R., Full contractibility and Cellina’s theorem (to appear). 9. FRYSZKOWSKI A., Continuous selections for a class of non-convex multivalued maps, Stud. Math. 76, 163-174 (1983). 10. CELLINA A. & MARICONDA C., Kuratowski’s index of a decomposable set, Bull. PO/. Acad. Sci. 37, 679-685 (1989). 1 I. BREZIS H., Analyse Fonctionnelle-Theorie et applications. Masson, Paris (1983). 12. HUREWICZ W. & WALLMAN H., Dimension Theory. Princeton University Press, New Jersey (1941). 13. OLECH C., Decomposability as a substitute of convexity, Lectures Notes in Mathematics (Multifunctions and Integrands), Vol. 1091, pp. 193-205 (1984). 14. SMART D.R., A nonlinear convex hull (to appear).
& Applrcotrons,
Vol. 18. No. 7, pp, 689-695,
0362-546X/92 $5.00 + .OO C 1992 Pergamon Press plc
1992.
CONTRACTIBILITY AND FIXED POINT PROPERTY: THE CASE OF DECOMPOSABLE SETS CARLO MARICONDA SISSA-ISAS, via Beirut 2/4, 34014 Trieste, Italy (Received Key words and phrases:
4 February
1991; received for publication
Fixed point, decomposable,
2 May 1991)
increasing family, Liapunov, convex hull, retract,
dimension. INTRODUCTION SCHAUDER’S fixed point theorem [l] asserts that any compact mapping of a convex set into itself has a fixed point. Various attempts have been made to do fixed point theory in spaces more general than convex sets, but no general theory is known. A theorem by Lefschetz [2, corollary, p. 3591 leads one to believe that each compact, homologically trivial and locally connected set has the fixed point property, but Borsuk [3] gave an example to refute this conjecture. His set is neither contractible nor locally contractible, hence it leaves open the following version of Lefschetz’s corollary. CONJECTURE. Any contractible,
locally contractible
set has the fixed point property
for compact
mappings. Borsuk showed in [4] that this conjecture is true in finite dimensional metric spaces (see theorem 1). Some particular cases of the conjecture-for decomposable subsets of L’--were proved by Cellina [5] and Fryszkowski [6]. Furthermore, it has been proved that each decomposable subset of L'is a retract of L' [7]. As Smart pointed out in a recent paper [S], these authors do not mention or use the contractibility and local contractibility properties that these spaces possess: Smart proved a weaker version of their theorem-an LJ’-continuous map of a decomposable set M into an L”-compact subset of M has a fixed point-by a method related to the contraction process in decomposable sets. Notice that this requirement is quite strong: it means that no “cut and paste” of functions of T(M) is possible; as an example the L'precompact subset of L'[O,l] ]~t~,~t, 01 E [0, 11) is not L”-precompact. Here I generalize Smart’s arguments in order to prove that each LP-compact mapping (1 5 p < co) of a decomposable subset of Lp into itself has a fixed point. This can be regarded as a further step of the project aimed at the conjecture. The main tool is a consequence of Liapunov’s theorem on the range of a vector measure as it is formulated in [9]: it enables the construction of a nonconvex analogue of Schauder’s projection onto a nonsymmetrical nonconvex analogue of the convex hull of a finite number of points. Its advantage with respect to the more familiar decomposable hull is that the first is compact whereas the second is not, unless it is trivial (see for instance [lo]). As a consequence, each compact map of a topological space into a decomposable set can be approximated by a sequence of functions having a finite dimensional range. This property is in common with compact maps into a convex set [ll, Chapter VI]. 689
C. MARICONDA
690 NOTATIONS
AND
PRELIMINARY
RESULTS
1. A subset A of a topological space X is said to be contractible in the space X to a set B c X provided that the inclusion A Q X is homotopic (in the space X’) to a map with values belonging to B. If A = X and B consists of a single point, then X is said to be contractible. Definition
Definition 2. A topological space X is said to be locally contractible at a point x0 E X provided that each neighbourhood U of x,, contains a neighbourhood U’ of x,, which is contractible in U to a point. The space X is said to be locally contractible if it is locally contractible at each of its points. It is clear that any convex set (and, as a consequence, any absolute retract) is contractible and locally contractible (see [4] for details). The next theorem shows that the converse is true for compact and finite dimensional spaces. By the dimension of a compact space X we understand the covering dimension of X [12, Chapter V]. Specifically, we say dim X I n provided that each open covering of X has a refinement of order 5 n (the order of a covering being the largest integer n such that there are n + 1 members of the covering which have a nonempty intersection); if dim X I n is true and dim X I n - 1 is false, then we say dim X = n. 1 [4, V(10.5)]. A finite dimensional compact metric space is an absolute retract if and only if it is contractible and locally contractible. THEOREM
Definition 3. Let (X, d) be a compact metric space and r be a nonnegative real number. The space X is said to have Hausdorff r-measure zero if for each E > 0 there exists a finite decomposition of X X=A,u**.UAk such that (diamA,)’ + .a. + (diam A,)’ < E where diam A = sup{&,
6) : a, b E A J.
The connection between the topological concept of dimension and the metrical concept of Hausdorff r-measure, given by the following theorem, will be used later. 2 [12, theorem VII.31. Let n E N and X be a compact space of (n + I)-measure zero. Then dim X 5 n. THEOREM
In what follows, we consider a measure space (Z, nt, p) where YE is a o-algebra of subsets of Z (measurable sets), ZJis a complete positive nonatomic measure such that ,u(Z) < +a, (E,
XA .
Definition 4. A nonempty subset S of L* is said to be decomposable if whenever f and g are in S and A is in 3n then fxA + gxrlA is in S.
691
Contractibility and fixed point property
Decomposable sets have many properties in common with convex sets: see, for instance [5, 7, 131. In particular, Smart [14] pointed out that the decomposable subsets of P[O, l] are contractible and locally contractible. We prove here a more general statement. PROPOSITION
1. Each decomposable
subset S of Z!‘(Z) is contractible
and locally contractible.
Proof. Liapunov’s theorem on the range of a vector measure [9, proposition existence of a family (A,), Eto,il of measurable subsets of Z such that
SC t * AsCAt,
A,=
0,
A, = I,
P(A,)
1. l] provides the
(t E IO, 11).
= W(Z)
(1)
I claim that for any fixed f in S the map @: @, 8) - fXA, +
&%\A,
describes a contraction of S to f and of any c-neighbourhood of S to f. Clearly, the claim is proved if we show that the previous map is continuous. For this purpose, fix a positive E, and let t, E [0, 11, g, E S. Let 6 be such that
Let t and g be such that jig - g,(l, < c/3 and It - t,l < d/p(Z). Since we have: @(t>g) - @(&, &I) = =
fkA,
-
XA,,)
+
gXI\A,
f&A,
-
XA,,)
+
kXZ\A,
-
gOXZ\A,, -
gOXZ\A,)
+
kOXZ\A,
-
gOXZ\A,,)
then it turns out that l/P
Ilw, 8) - wo, go)II,5
l/P +
llg
-
golIp
+
(.\’A,AA,, If” dp > where A AB = (A\B) U (B\A). Hence, by (1) and (2)
A,M,o
kelp
@ >
Ilw g)- @(to, go)Il,< ; + ; + ; = E. The claim is proved. Definition
5. A family (A,),. Lo,11of measurable sets is called increasing if the following properties hold: A, = I; ss t * A,cA,. Ao= 0;
Let us consider a family of positive vector-valued measures P, = (,u:, . . . . &) which are absolutely continuous with respect to P. The space M” of such vector measures is endowed with the topology induced by the norm I(,u~\I,the total variation of ,u,. The proposition formulated below is [9, proposition 1.21 where p. is replaced by (,ZJ,v). PROPOSITION 2. Let X be a compact topological space and v be an absolutely continuous positive measure with respect top. Assume that the map x c-) pu, E M” is continuous. Then, for every E > 0 there exists an increasing family (At), ELo,,I of measurable sets with the following properties:
]/&I,) P(A,)
- k(Z)]
= W(Z),
< s
(t E 10, 11, x E XI;
v(A,) = tv(Z)
(t E 10, 11).
(3) (4)
C. MARICONDA
692
MAINRESULTS THEOREM
3. Let S be a decomposable
P-compact
subset K of S. Then
subset of P(Z), T has a fixed point.
T be an P-continuous
The proof is modelled on a standard proof of Schauder’s second in [I] and in Smart’s proof of theorem 3 in the case where K is an A replacement for the convex combinations used in Schauder’s f , , . . . , f,, be fixed in Lp and Q. = (A,),. tO,il be an increasing Analogous to what is described in [14], if ci 2 0 and ci + ... +
mapping
into an
theorem, given for instance L”-compact subset of S [S]. projection is studied: let family of measurable sets. c, = 1 we use the notation
Clf, @a **- @a Gfn for the function f&i,,
+ --’ +
f2X~A,,+,,\A,,)
+
“*
+
fnX(I\A,,+...+r,_,)
convex combination” which equals fi on &I+... +.I\&.,+.., +Cim,. We regard this as a “nonlinear (with respect to @) of fi , . . ., f, with weights cl, . . ., c,,. These combinations make up the “nonlinear convex hull” of fi , . . . , f,, with respect to the increasing family a; it will be denoted by nocoa(fi,
. . ..f.).
set S then noco,(fi , . . . , f,) As is evident, if fi , . . . , f, belong to a decomposable in S. The rest of the proof follows after theorems 4 and 5 and lemma 1 below.
is contained
sets is called refining Definition 6. Let $ E L’. A family a = (A,), E to, i1 of measurable respect to (I, nt, ,u)) if and only if the following equalities hold:
iA,ddru =tjj#du (tE
4 (with
VA11).
sets. 4. Let fi , . . . , f, E Lp and a = (A,), E to, i1 be an increasing family of measurable FurtherThen nocoe (fi ,. . . , f,) is (i) compact, (ii) contractible and (iii) locally contractible. more, if C? is refining (1fi ( + . . . + 1f,l)” then (iv) noco,(f, , . . . , f,) is finite dimensional.
THEOREM
Proof.
Let C be the convex
compact
subset of R”-’ defined
by
C=((C,,...,C,_I)Ell?-1:05clI~~~~Cn_1~1) and let Y: C --f noco,(f,
, . . . , f,,)
Y(c it *a., %1)
be the function =
fixA,,
+
defined
fZ&A,,\A,,)
by +
‘-*
+
fn&T\A,,_,).
Since the integral operator is absolutely continuous with respect to the measure P, then Y is noco,( f, , . . . , f,) can be continuous hence noco,(f, , . . . , f,) = Y(C) is compact. Furthermore contracted to fi by the process described in the proof of proposition 1. The proof of local contractibility is essentially the proof of [14, theorem 2(iii)] taking into account that we have Y(c) instead off’. I claim that Y In (iv) let @ = (A,),. to 1l be an increasing family refining (Ifi 1 + . . . + 1f,l)“. satisfies a Holder condition. With a view to prove the above assumption, set for any x = (x 1, . . ..X._i) E E?-’ the norm of x to be [lx]] = /x1( + . . - + Jx,_,J. Since Y(c) and Y(d)
693
Contractibility and fixed point property
tl (Ad,/Adi_,) then we have:
agree on Ui (A,&__,)
Furthermore,
(If, I + ... + jfnl)P, hence the following
Q. is refining
A M (Ihl !’ d,
equalities
(i =
+ a*.+ IfnIYdP = Idi - CiI (IfiI + **. + IfnO’@ ,Ii
As a’consequence,
if we set Q = (S, (Ifi I + .-- + (f,l)Pdp)l’P, IIY(c) - Yap, , . . . ,f,)
Now, we prove that noco,(f, Letc>OandB,,...,B,besuchthat
5
hold: 1, . . ..n
-
we have
(5)
2Q11c- d”‘.
has np-measure
zero.
(6)
(diam B,)” + .. k + (diam Bk)n < L.
2Q
Such a decomposition Then
exists since the (n nocoO (fi ) . ..,f,)
1) dimensional
= Y(C)
polytope
C has n-measure
zero.
= Y(B,) u **. u Y(Bk).
Since, by (5), diam Y(Bi) I 2Q(diam
1).
(i = 1, . . . , k)
Bi)l’P
(7)
then (6) and (7) imply diam Y(B1)“p + . .. + diam Y(Bk)“p < E and this proves the claim. Theorem 2 yields the conclusion. The fixed point property for nocao (f, , . . . , f,,) now follows
immediately
5. Let fr , . . . , f,, E Lp and A = (A,),. ,O,1l be an increasing + +.= + If,\)“. Then noco,(fr, . . . . f,,) has the fixed point property.
THEOREM
(Ifil
The proof
of theorem
3 is based on the following
from theorem family
1.
refining
lemma.
“Schauder’s projection”). Let K be a compact subset of Lp. Then family Q. = (A,), E[O,11 refining E > 0 there exist fi, . . ., f, E K, an increasing (If11+ --. + If,l)” and a continuous mapping P, of K into noco,(f, , . . . , f,) such that IlP,f - f (Ip s E for all f in K.
LEMMA 1 (a nonconvex
for
every
Proof. Let fi , . . . , f, in K be such that their ~12~‘~ neighbourhoods and fEK, ,L&, v be the absolutely continuous measures (with P#)
=
i
A If - fiI"Q
(i = 1, . . . . n);
v(A)= AClf,l + -.a+ If,l)” .i
dp.
cover K. Let, for 1 5 i I n respect to p) defined by
C. MARICONDA
694
Clearly the map increasing family
f EK y (,uj,. . . , &)
hence
is continuous, C? = (A,), E to, il with the properties: I&(A)
(lfl i ‘41
+ ... +
- M)V)I
by proposition
(te[O,l],feK,i=l,...,
< 2
if,zlY& = 1 (lfil + ..* + If,,h-b iI MA,)
2 there exists an
n);
(8)
(t E LO, 11);
(t E [O, 11).
= 040
(9) (10)
ForlIiInandfEKset
ri(f) = max O,& - Ilf -fill, ; > ( thus for each f E K some ri(f)
is nonzero.
Then set
ri(f) ci(f) = Cjrj(f) so that for each f, C i Ci(f) = 1 and the map f ++(c,(f), . . . , c,(f)) is continuous. the map P,:K+ noco,(fi, . . ..f.) defined by
I claim that
p,(f) = c1(f)fl @a*** @a c,(f)fn has the required be di(f) = cl(f)
properties. In fact, set for any f E K and i = 1, . . . , n - 1 the function + se. + Ci(f). Then we have:
II&(f) -fII;
= c /Ad,w,Ad_,,lf-fiI’+.
i
By (8), for each i = 1, . . . . n - 1 the following
di to
(11)
I I inequality
If -fiI”dP 5 ci(f)
! AdLn’Ad,-,U,
.iI
holds:
If
-fiI”dP + g*
Since Ci( f) # 0 if and only if 11 f - fill, < ~12~‘~then
IIP&(f)- f 11;I ;
+;
= EP.
The continuity of P, follows from the fact that the family Q. refines (I fi ( + . -a + Ifnl)p : in this situation, the proof of theorem 4(iv) shows that the map Y is continuous. Since, for any f in K, we have P,(f) = Y(d(f)), then the conclusion follows from the continuity of the function d = (d, , . . . , d,). Proof of theorem 3. Fix k E N. Then, by lemma 1, there exist fi , . . . , f,, E K, an increasing family a = (A,), E to, i1 refining (I fi I + . . . + 1f,l)p and a continuous function P,,k of K into such that [IP,,k(f) -f jjp I l/k for all f in K. Let i be the inclusion of nocoa(fi,. . . ,f,) nocoQ (fi , . . . , f,) into S. Let us consider the composed map Pl,k 0 T 0 i.
Contractibility
s
-
T
K
i
S/k
PI,*
0
noco,vI, . . ..fJ
695
and fixed point property
-
This function maps the set nocoe (Jr, . . . , f,) point xk . Thus
To i
I
nocou(fi, . . ..fJ
into itself,
hence,
by theorem
5, it has a fixed
By the compactness of K, T has a fixed point. “Schauder’s projection” of lemma The following consequence of the nonconvex a further property of decomposable sets which is in common with convex sets.
1 provides
THEOREM 6. Let X be a topological space, S be a decomposable subset of P(I) and T: X + S be a continuous compact mapping. Then for each E > 0 there exists a continuous map T,: X --f S whose range has a finite dimension and such that /T,(x) Proof.
Let K be a compact
subset
T(x)\l,
VXEX.
< s
of S such that T(X) C K and fix E > 0. Let
P,: K -+ noco,(f,,
. . ..f.,)
be the nonconvex “Schauder’s projection” of lemma T, = P, 0 T fulfils the requirements of the claim. Acknowledgemenr--I wish to thank the preparation of this work.
Professors
1. Then,
A. Cellina and G. Colombo
by theorem
4 (iv), the map
for the useful conversations
we had during
REFERENCES 1. SMART D.R., Fixed Point Theorems. Cambridge University Press, London (1974). 2. LEFSCHETZ S., Topology. Amer. Math. Sot., New York (1930). 3. BORSUK K., Sur un continu acyclique qui se laisse transformer topologiquement en lui-mCme sans points invariants, Fund. Math. 24, 51-58 (1935). 4. BORSUK K., Theory of Retracts. PWN-Polish Scientific Publishers (Monografie Matematiczne), Warszawa (1967). 5. CELLINA A., A fixed point theorem for subsets of Lr, Lectures Notes in Mathemafics (Multifunctions and Integrands), Vol. 1091, pp. 129-137 (1984). 6. FRYSZKOWSKI A., The generalisation of Cellina’s fixed point theorem, Stud. Math. 78, 213-215 (1984). 7. BRESSAN A. & COLOMBOG., Extensions and selections of maps with decomposable values, Stud. Math. 90, 69-86 (1988). 8. SMART D.R., Full contractibility and Cellina’s theorem (to appear). 9. FRYSZKOWSKI A., Continuous selections for a class of non-convex multivalued maps, Stud. Math. 76, 163-174 (1983). 10. CELLINA A. & MARICONDA C., Kuratowski’s index of a decomposable set, Bull. PO/. Acad. Sci. 37, 679-685 (1989). 1 I. BREZIS H., Analyse Fonctionnelle-Theorie et applications. Masson, Paris (1983). 12. HUREWICZ W. & WALLMAN H., Dimension Theory. Princeton University Press, New Jersey (1941). 13. OLECH C., Decomposability as a substitute of convexity, Lectures Notes in Mathematics (Multifunctions and Integrands), Vol. 1091, pp. 193-205 (1984). 14. SMART D.R., A nonlinear convex hull (to appear).