Báo cáo hóa học: " FIXED-POINT-LIKE THEOREMS ON SUBSPACES" docx

PHILIPPE BICH AND BERNARD CORNETReceived 8 June 2004 We prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and co

PHILIPPE BICH AND BERNARD CORNET

Received 8 June 2004

We prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets Our result generalizes two different kinds of theorems: the fixed-point-like theorem by Hirsch et al (1990) or Husseini et al (1990) and the fixed-point theorem by Gale and Mas-Colell (1975) (which generalizes Kakutani’s theorem (1941))

1 Introduction

In this paper, we prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets Letk be an

in-teger and letV be a Euclidean space such that 0 ≤ k ≤dimV, then the k-Grassmannian

manifold ofV, denoted G k(V), is the set of all the k-dimensional subspaces of V The

setG k(V) is a smooth compact manifold but, in general, it does not satisfy properties

such as convexity or acyclicity and its Euler characteristic may be null This prevents the use of classical fixed-point theorems as Brouwer’s [2], Kakutani’s [14], or Eilenberg-Montgomery’s theorem [7]

Our main result generalizes two different kinds of theorems: the fixed-point-like the-orem by Hirsch et al [11] or Husseini et al [13] and the fixed-point thethe-orem by Gale and Mas-Colell [8] (which generalizes Kakutani’s theorem [14]) As in [11,13], we will mainly use techniques from degree theory As a consequence of our main result, we first deduce the standard fixed-point theorems when the variable is in a convex domain (such

as Brouwer and Kakutani’s theorem) and second Borsuk-Ulam’s theorem

The main result of this paper is directly motivated by the existence problem of equilib-ria in economic models with incomplete markets; in [1], it is used to extend the classical existence result by Duffie and Shafer [6] to the nontransitive case

The paper is organized as follows The main result is stated inSection 2together with some direct consequences of it, namely, the results by Hirsch et al [11], Gale and Mas-Colell [8] and Borsuk-Ulam’s theorem The proof of the main result is given inSection 3

Copyright©2004 Hindawi Publishing Corporation

Fixed Point Theory and Applications 2004:3 (2004) 159–171

2000 Mathematics Subject Classification: 47H04, 47H10, 47H11

URL: http://dx.doi.org/10.1155/S1687182004406056

and the appendix recalls the main properties of the Grassmannian manifold, used in this paper

2 Statement of the results

2.1 Preliminaries A correspondenceΦ from a set X to a set Y is a map from X to the set

of all the subsets ofY, and the graph of Φ, denoted G(Φ), is defined by G(Φ) = {(x, y) ∈

X × Y | y ∈ Φ(x) } A mappingϕ : X → Y is said to be a selection of Φ if ϕ(x) ∈ Φ(x) for

allx ∈ X If A is a subset of X, we let Φ(A) =
x∈A Φ(x), and the restriction of Φ to A,

denotedΦ| A, is the correspondence from A to Y defined by Φ | A( x) = Φ(x) if x ∈ A If X

andY are topological spaces, the correspondence Φ is said to be lower semicontinuous

(l.s.c.) (resp., upper semicontinuous (u.s.c.)) if for every open setU ⊂ Y, the set { x ∈ X |

Φ(x) ∩ U = ∅}is open inX (resp., the set { x ∈ X | Φ(x) ⊂ U }is open inX and, for every

x ∈ X, Φ(x) is compact).

Ifx =(x1, ,x n) and y =(y1, , y n) belong toRn, we denote byx · y =n i=1x i y ithe scalar product of Rn,  x  = √ x · x the Euclidian norm If x ∈Rn and r ∈R+, we let

B(x,r) = { y ∈Rn |  x − y  < r }and B(x,r) = { y ∈Rn |  x − y  ≤ r } IfE is a vector

subspace ofRn, we denote byE ⊥ = { u ∈Rn | ∀ x ∈ E, x · u =0}the orthogonal space

toE If u1, ,u k belong toE, a vector space, we denote by span { u1, ,u k }the vector subspace ofE spanned by u1, ,u k

LetV be a Euclidean space and let k be an integer such that 0 ≤ k ≤dimV; we denote

byG k(V) the set consisting of all the linear subspaces of V of dimension k, called the

(k-)Grassmannian manifold of V Then it is known that G k(V) is a smooth manifold

of dimensionk(dimV − k) and we refer to the appendix for the properties we will use

hereafter, together with the precise definition of the manifold structure onG k(V).

2.2 The main result and some consequences The aim of this paper is to prove the

following result

Theorem 2.1 Let I, J be two finite disjoint sets For every i ∈ I, let k i be an integer and let

V i be a Euclidean space such that 0 ≤ k i ≤dimV i For every j ∈ J, let C j be a nonempty, convex, compact subset of a Euclidean space V j , and let M =i∈I G k i(V i) ×j∈J C j For i ∈ I and k =1, ,k i , let F k

i be a correspondence from M to V i with convex values, for j ∈ J, let F j be a correspondence from M to C j with convex values, and suppose that, for every i ∈ I and k =1, ,k i (resp., j ∈ J), the correspondence F k

i (resp., F j ) is either l.s.c or u.s.c.

Then, there exists ¯ x =(( ¯x i)i∈I, ( ¯x j)j∈J)∈ M such that

(i) either F k

i( ¯x) ∩ x¯i = ∅ or F k

i( ¯x) = ∅ for every i ∈ I and k =1, ,k i ; (ii) either F j( ¯ x) ∩ { x¯j } = ∅ or F j( ¯x) = ∅ for every j ∈ J.

The proof ofTheorem 2.1is given inSection 3 A first consequence ofTheorem 2.1is the following theorem by Hirsch et al [11]

Corollary 2.2 Let V1be a Euclidean space, let k1be an integer such that 0 ≤ k1≤dimV1, and for every k =1, ,k1, let f k:G k1(V1)→ V1be a continuous mapping Then, there exists

¯

x ∈ G k1(V ) such that for every k =1, ,k , f k( ¯x) ∈ x.¯

Proof Take I = {1},J = ∅, andF1k(x) = { f k(x) }for everyx ∈ G k1(V1) and for everyk =

1, ,k1 FromTheorem 2.1, there exists ¯x ∈ M = G k1(V1) such that for everyk =1, ,k1,

A second consequence ofTheorem 2.1is the following generalization of Gale and Mas-Colell’s theorem [8], which is also a generalization of Kakutani’s theorem Hereafter, we use the formulation by Gourdel [9] allowing each correspondence to be either u.s.c or l.s.c

Corollary 2.3 Let J be a finite set, for j ∈ J, let C j be a nonempty, convex, compact subset

of a Euclidean space, and let F j be a correspondence from M : =j∈J C j to C j with convex values, such that the correspondence F j is either l.s.c or u.s.c Then, there exists ¯ x =( ¯x j)j∈J ∈

M such that for every j ∈ J, either ¯x j ∈ F j( ¯ x) or F j( ¯ x) = ∅

Remark 2.4 According to our definition, an u.s.c correspondence has compact values

and without this requirement,Theorem 2.1may not be true, as we can see in the fol-lowing counterexample LetM : = G1(R2) Each elementD of G1(R2) can be written as

D t = { λ(cost,sint) | λ ∈R}, for somet ∈[0,π[ We define the correspondence F from

M toR2 byF(D0)=R× {1}andF(D t)= D t ∩(R× {1}) +{(1, 0)}ift ∈]0,π[ We let

the reader check that for every open setU ⊂R2, the set{ x ∈ M | F(x) ⊂ U }is open in

M and that F has nonempty, convex (and closed) values Yet, it is straightforward that F(x) ∩ x = ∅for everyx ∈ G1(R2)

Another consequence of our main result is the following multivalued version of Borsuk and Ulam’s theorem We denote byS nthe unit sphere ofRn+1

Corollary 2.5 For k =1, ,n, let F k be a correspondence from S n toRwith nonempty and convex values such that for every k =1, ,n, F k is either l.s.c or u.s.c Then, there exists

¯

x ∈ S n such that

∀ k ∈ {1, ,n }, F k( ¯x) ∩ F k(− x)¯ = ∅ (2.1)

Proof For every k =1, ,n, let ˆF kbe the correspondence fromS ntoRdefined by

ˆ

F k(x) =u − v | u ∈ F k(x), v ∈ F k(− x). (2.2)

We let the reader check that for everyk =1, ,n, the correspondence ˆF k has non-empty, convex values and that it is u.s.c (resp., l.s.c.) ifF k is u.s.c (resp., l.s.c.) So, to prove Corollary 2.5, it suffices to show the existence of ¯x∈ S nsuch that 0∈ Fˆk( ¯x) for

everyk =1, ,n.

We define, for everyk =1, ,n, the correspondence H k fromG n(Rn+1) to Rn+1 as follows: for everyE ∈ G n(Rn+1), we letH k(E) = Fˆk(x)x, where x is an arbitrary element of

E ⊥ ∩ S n The correspondenceH kis well defined sinceE ⊥ ∩ S n = { x, − x }for some element

x ∈ S nand since ˆF k(x)x = Fˆk(− x)( − x).

TakeI = {1},V1=Rn+1,k1= n, J = ∅, and applyTheorem 2.1to the correspondences

H k, which clearly satisfy the assumptions ofTheorem 2.1 So there exists ¯E ∈ G n(Rn+1) such that ¯E ∩ H k( ¯E) = ∅for everyk =1, ,n.

Now, if ¯x is an arbitrary point of ¯E ⊥ ∩ S n, then we have ¯E ∩ Fˆk( ¯x)¯x = ∅; from ¯x ∈ E¯⊥

and ¯x =0, we finally obtain 0∈ Fˆk( ¯x) for every k =1, ,n, which ends the proof of

3 Proof of Theorem 2.1

The proof is given in three steps, corresponding to the following three subsections The first step gives the proof under the additional assumptions thatJ = ∅and the correspon-dencesF k

i are single-valued The second step only assumes in addition thatJ = ∅ Finally, the third step gives the proof under the assumptions ofTheorem 2.1

3.1 Proof whenJ = ∅andF k

i are single-valued We first proveTheorem 2.1under the additional assumptions thatJ = ∅and theF k

i are single-valued This is exactly the state-ment below

Theorem 3.1 Let I be a finite set and for i ∈ I, let k i be an integer and let V i be a Euclidean space such that 0 ≤ k i ≤dimV i Let M : =i∈I G k i(V i) and for i ∈ I, let f i:M →(V i) k i be a continuous mapping Then, there exists ¯ x =( ¯x i)i∈I ∈ M such that

∀ i ∈ I, f i( ¯ x) ∈x¯ik i (3.1) The proof ofTheorem 3.1is given in two steps In the first step, we additionally assume that the mappings are smooth, and the second step gives the proof in the general case

3.1.1 Proof of Theorem 3.1 when the f i are smooth Let M : =i∈I G k i(V i) and define f :

M →i∈I V k i

i by

f (x) =proj(x ki

i )⊥ f i( x)i∈I forx =x i

and the subsetsZ, Z1, andZ2ofM ×i∈I V k i

i by

Z =



(x, y) ∈ M ×

i∈I

V k i

i | ∀ i ∈ I, y i ∈x k i

i ⊥

,

Z1=



(x, y) ∈ M ×

i∈I

V k i

i | y = f (x) ,

Z2=



(x, y) ∈ M ×

i∈I

V k i

i | y =0 .

(3.3)

ProvingTheorem 3.1amounts to showing the existence of ¯x ∈ M such that f (¯x) =0

or, equivalently, such thatZ1∩ Z2= ∅ For this, we will use the followingIntersection Theorem 3.2, which is a direct consequence of mod 2 intersection theory (see, e.g., [10, page 79] and [5, page 127])

Intersection theorem 3.2 Let Z be a smooth boundaryless manifold of dimension 2m and let Z1, Z2 be two compact boundaryless submanifolds of Z of dimension m If ¯Z1 is a compact boundaryless m-submanifold of Z homotopic to Z1 and if the manifolds ¯ Z1 and

Z2intersect transversally in a unique point ¯ z (which means that T z¯Z¯1+T z¯Z2= T z¯Z), then

Z1∩ Z2= ∅

The proof ofTheorem 3.1consists of checking that the above-defined setsZ, Z1, and

Z2 (together with the set ¯Z1 defined below) satisfy the assumptions of Intersection Theorem 3.2

The sets Z, Z1, and Z2 satisfy the assumptions of Intersection Theorem 3.2 We recall

that for everyi ∈ I, G k i(V i) is a smooth, boundaryless, compact manifold of dimension

k i(dim V i − k i) (seeLemma A.1in the appendix) ThusM : =i∈I G k i(V i) is a

bound-aryless, smooth, compact manifold of dimensionm =i∈I k i(dim V i − k i) Clearly Z is

a fiber bundle whose base space isM and whose fiber at x =(x i)i∈I ∈ M is the vector

space

i∈I(x k i

i )⊥which has the dimension ofM Hence, Z is a smooth manifold of

di-mension 2m.

The mapping f : M →i∈I V k i

i is a smooth mapping from Parts (c), (d), and (e) of Lemma A.1in the appendix Consequently,Z1 is a smooth compact boundaryless sub-manifold of Z of dimension m Finally, Z2 is clearly a smooth boundaryless compact submanifold ofZ of dimension m.

The manifold Z1 is homotopic to the manifold ¯ Z1that we now define For every i ∈ I,

let ¯x i ∈ G k i(V i) and let { e¯1

i, , ¯e k i

i } be an orthonormal basis of ¯x i For every i ∈ I, let

g i:G k i(V i) → V i k i andg : M →i∈I V i k ibe the mappings defined as follows:

∀ x i ∈ G k i

V i

, g i

x i

=projx i ⊥



¯

e1

i

, ,proj x i ⊥



¯

e k i

i 

∈x ⊥

i k i

=x k i

i ⊥

,

∀ x =x i

i∈I ∈ M, g(x) =g i

x i

We let

¯

Z1:=(x, y) ∈ M ×

i∈I

V k i

To show that the manifoldZ1 is homotopic to ¯Z1, we letH : [0,1] × Z1→ Z be the

continuous mapping defined byH(t,(x, f (x))) =(x,(1 − t) f (x) + tg(x)) Then H(0, ·) is the canonical inclusion fromZ1toZ, and H(1, ·)(Z1)= Z¯1

The manifolds ¯ Z1 and Z2 intersect transversally in a unique point First, notice that

¯

Z1∩ Z2= {(x,0) ∈ M ×i∈I V k i

i | g(x) =0}is the singleton ( ¯x,0) =(( ¯x i)i∈I, 0) But that

¯

Z1andZ2intersect each other transversally inZ means that T( ¯x,0) Z¯1+T( ¯x,0) Z2= T( ¯x,0) Z.

Recalling that dimT( ¯x,0) Z¯1+ dimT( ¯x,0) Z2=dimT( ¯x,0) Z =2m, we only have to show that

T( ¯x,0) Z¯1∩ T( ¯x,0) Z2= {0} Finally, noticing thatT( ¯x,0) Z¯1= {(u,Dg(¯x)(u)) | u ∈ T x¯M }and

T( ¯x,0) Z2= {(u,0) | u ∈ T x¯M }, we only have to prove that Dg(¯x) is injective, which is

proved in the following lemma

Lemma 3.3 Dg(¯x) is injective.

Proof Recalling that for every x =(x i)i∈I ∈ M, g(x) =(g i( x i))i∈I, the mappingDg(¯x) is

injective if and only if for everyi ∈ I, Dg i( ¯ x i) is injective.

So, leti ∈ I, let (ϕ,U) be a local chart of G k i(V i) at ¯ x i, and let ψ : (¯x ⊥

i )k i → G k i(V i)

be the inverse mapping ofϕ : U →( ¯x ⊥

i )k i From the appendix, if { e¯1, , ¯e k i

i }is a given orthonormal basis of ¯x i, ψ can be defined by

ψu1, ,u k i

=span

¯

e1

i +u1, , ¯e k i

i +u k i

for every

u1, ,u k i

∈x¯⊥

i k i (3.6)

Since the mappingg i ◦ ψ is the local representation g iin the chart (ϕ,U), proving that

Dg i( ¯ x i) is injective amounts to proving that D(g i ◦ ψ)(0) is injective This is a consequence

of the following claim

Claim 3.4 For all ( h1, ,h k i)∈( ¯x ⊥

i )k i , D(g i ◦ ψ)(0)(h1, ,h k i)= −(h1, ,h k i ).

Proof of Claim 3.4 Let p : V i ×( ¯x ⊥

i )k i → V ibe defined by

If we prove that for everyy ∈ V i, the derivative of the mapping p y:u → p(y,u) is the

linear mappingDp y(0) : ( ¯ x ⊥

i )k i → V idefined by

Dp y(0)( h) =

k i

k=1



y · e¯k

i

h k, ∀ h =h1, ,h k i



∈x¯⊥

i k i

thenClaim 3.4 will be proved Indeed, taking y = e¯k

i for every k =1, ,k i, we would

obtainD e¯k

i p(0)(h1, ,h k i)= h k Thus, sinceg i ◦ ψ(u) =( ¯e1

i, , ¯e k i

i )−(p e¯ 1

i(u), , p e¯ki

i (u)),

it would entailClaim 3.4

Now, for everyu =(u1, ,u k i)∈( ¯x ⊥

i )k i, there existsλ(y,u) =(λ k( y,u)) k=1, ,k i ∈Rk i

such that

p(y,u) =projψ(u) y =

k i

k=1

λ k( y,u)e¯k

i+u k

with (λ k( y,u)) satisfying

− y +k i

k=1

λ k( y,u)e¯k

i +u k

·e¯i j+u j

=0 for everyj =1, ,k i (3.10)

This can be equivalently rewritten as follows:



I k i+G(u)λ(y,u) =y ·e¯1

i+u1 

, , y ·e¯k i

i +u k i

where I k i is the k i × k i identity matrix and G(u) is the k i × k i matrix G(u) =

(u j · u k)j,k=1, ,k i Besides, foru in a neighborhood ᏺ of 0 small enough, the matrix (I k i+

G(u)) is invertible Consequently, the mapping λ( ·,·) is smooth onV ×ᏺ, which implies

that the mappingp( ·,·) is smooth onV × ᏺ Differentiating, with respect to u, the above

equality atu =0, we obtain, for everyh =(h1, ,h k i)∈( ¯x ⊥

i )k i,

DG(0)(h)λ(y,0) + D u λ(y,0)(h) =0. (3.12)

But it is clear thatDG(0) =0 Consequently,D u λ(y,0) =0

Finally, differentiating the equality p(y,u)=k i

k=1λ k(y,u)(¯e k

i +u k) at (y,0), one

ob-tains, for everyh =(h k)k i

k=1∈( ¯x ⊥

i )k,

D u p(y,0)(h) =

k i

k=1

λ k( y,0)h k =

k i

k=1



y · e¯k

i

3.1.2 Proof of Theorem 3.1 in the general case Since M is a compact manifold and V k i

i is

a Euclidean space, for everyi ∈ I, each continuous mapping f i:M → V k i

i can be approx-imated by a sequence of smooth mappings f n

i :M → V k i

i converging to f i, in the sense

that limn→∞ f n

i − f i  ∞ =0 (see, e.g., Hirsch [12]) Applying the first step to the smooth mappings f n

i , we deduce the existence of (x n

i)i∈I∈ M such that

∀ i ∈ I, f n

i 

x n

i

∈x n

ik i

(3.14)

or, equivalently,

proj(x n ⊥

i )ki f i

x n

i

From the compactness ofM, without any loss of generality, one can suppose that the

sequence (x n

i)i∈Iconverges to some element ( ¯x i)i∈I ∈ M We have

( ¯x i ⊥)ki f i

¯

x i

−proj( ¯x n ⊥

i )ki f n

i 

x n

i

≤proj

( ¯x i ⊥)ki f i

¯

x i

−proj( ¯x n i ⊥)ki f i

x n

i+f n

i − f i

Consequently, from the convergence of f n

i to f i and the continuity of the mapping (u,v) →proj(u ⊥)ki v (seeLemma A.1in the appendix), we obtain

proj( ¯x i ⊥)ki f i

¯

x i

or, equivalently,

∀ i ∈ I, f i

¯

x i

∈

¯

x ⊥

i k i⊥

=x¯ik i

which ends the proof ofTheorem 3.1

3.2 Proof of Theorem 2.1 whenJ = ∅ We now proveTheorem 2.1whenJ = ∅ The proof rests on the following claim

Claim 3.5 For every i ∈ I and every k ∈ {1, ,k i } , there exists an u.s.c correspondence ˆ F k

i from M to V i , with nonempty convex values, such that

i(x) = ∅=⇒∀ y ∈ Fˆk

i(x), ∃ λ ∈R,λy ∈ F k

i(x). (3.19)

Proof of Claim 3.5 Let i ∈ I and k ∈ {1, ,k i } We distinguish two cases

Assume first thatF k

i is l.s.c LetU k

i = { x ∈ M | F k

i(x) = ∅} ThenU k

i is an open subset

ofM and F k

i | U k

i is a l.s.c correspondence with nonempty convex values By Michael [15], there exists a continuous selectionf k

i ofF k

i | U k

i, that is, f k

i :U k

i → V iis a continuous map-ping such thatf k

i (x) ∈ F k

i(x) for every x ∈ U k

i LetB ibe the closed unit ball ofV i, and we

define the correspondence ˆF k

i fromM to B iby ˆF k

i(x) = { f k

i (x)/  f k

i (x) }ifx ∈ U k

i and

f k

i (x) =0 and ˆF k

i(x) = B iotherwise We let the reader check that the correspondence ˆF k

i

satisfies the conclusion ofClaim 3.5

We now consider the case whereF k

i is u.s.c LetU k

i = { x ∈ M | F k

i(x) = ∅}.ThenU k

i

is a closed subset ofM By Cellina [4], one can extendF k

i | U i as follows: there exists a correspondence ˆF k

i fromM to V iwhich is u.s.c., with nonempty, convex, and compact values, such that for everyx ∈ U k

i,F k

i(x) = Fˆk

We now come back to the proof of Theorem 2.1when J = ∅ For every i ∈ I and

k =1, ,k i, let ˆ F k

i be the u.s.c correspondence fromM to V i with nonempty convex (compact) values defined inClaim 3.5 By Cellina [3], for every integern, there exists a

continuous mapping f i k,n:M → V isuch that

Gf i k,n⊂ GFˆk

i

+B0,1n



Now, fori ∈ I, let f n

i :M →(V i) k ibe defined as follows:

i (x) =f1,n

i (x), , f k i,n

ApplyingTheorem 3.1to the mappings f n

i , we deduce the existence of ( ¯x n

i)i∈I∈ M

such that for everyi ∈ I, f n

i ( ¯x n)∈( ¯x n

i)k i, hence

y i k,n:= f i k,n

¯

x n

∈ x¯n

Since the correspondence ˆF k

i is bounded (M is compact and ˆF k

i is u.s.c.), the sequence (y k,n i ) is bounded Thus, without any loss of generality, one can suppose that the sequence (y k,n i ) converges to somey k

i ∈ V iwhenn tends to + ∞ Besides, from the compactness ofM, without any loss of generality, one can suppose

that ( ¯x n

i)i∈Iconverges to ¯x =( ¯x i)i∈I ∈ M when n tends to + ∞

Moreover, fromLemma A.1(d) in the appendix and fromy k,n

i ∈ x¯k,n

i , at the limit we have that

∀ i ∈ I, ∀ k =1, ,k i, y k

Since the graph of ˆF k

i is closed (it is u.s.c with compact values) and fromG( f i k,n)⊂

G( ˆF k

i) +B(0,1/n), one obtains

y k

i ∈ Fˆk

To end the proof, we assume thatF k

i( ¯x) = ∅ Since y k

i ∈ Fˆk

i( ¯x), byClaim 3.5, there exists λ ∈Rsuch thatλy k

i ∈ F k

i( ¯x) Hence λy k

i ∈ F k

i( ¯x) ∩ x¯i = ∅ (since y k

i ∈ x¯i) This

ends the proof ofTheorem 2.1

3.3 Proof of Theorem 2.1 in the general case We first prove the following lemma.

Lemma 3.6 Let C be a nonempty, convex, compact subset of a Euclidean space V Then there exists a continuous mapping ρ : G1(V ×R)→ C such that

∀ x ∈ G1(V ×R), x ∩C × {1}= ∅ =⇒ x ∩C × {1}=ρ(x),1. (3.25)

Proof Since C is compact, it is included in a closed ball B(0,k) of V We let r : V →

B(0,k + 1) be defined by r(u) = α(  u )u, where α :R+→R+is defined by

α(t) =1 ift ∈[0,k], α(t) = k + 1 − t if t ∈[k,k + 1], α(t) =0 ift ≥ k + 1.

(3.26)

Letπ1:V ×R→ V and ρ : G1(V ×R)→ C be defined by π1(x,t) = x and

ρ(x) =







projC ◦ r ◦ π1 x ∩V × {1} ifx ∩V × {1}= ∅, projC(0) ifx ∩V × {1}= ∅, (3.27) where projC:V → C denotes the projection from V to C Then, one easily sees that ρ

Proof of Theorem 2.1 UsingLemma 3.6, we first modify the correspondencesF jfor ev-ery j ∈ J and replace each nonempty compact convex set C j ⊂ V j by the Grassmannian manifoldG1(V j ×R) For everyj ∈ J, let ρ j:G1(V j ×R)→ C jbe the mapping associated

toC j ⊂ V jbyLemma 3.6 Let

ρ : ˜ M : =

i∈I

G k iV i

j∈J

G1 V j ×R−→ M : =

i∈I

G k iV i

j∈J

be defined by

ρ(x) =

x i

i∈I,ρ j

x j

j∈J



, forx =

x i

i∈I,

x j

j∈J



Fori ∈ I and k =1, ,k i, let ˜ F k

i be the correspondence from ˜M to V idefined by

˜

F k

i(x) = F k

i

For j ∈ J, let ˜F jbe the correspondence from ˜M to V j ×Rdefined by

˜

F j( x) = F j

Now, applying the result proved inSection 3.2 (i.e.,Theorem 2.1 when J = ∅) to the correspondences ˜F k

i and ˜F j, there exists x =((x i)i∈I, ( x j)j∈J)∈ M such that˜ (i) either ˜F k

i(x) ∩ x i = ∅or ˜F k

i(x) = ∅for everyi ∈ I and i =1, ,k i,

(ii) either ˜F j( x) ∩ x j = ∅or ˜F j(x) = ∅for everyj ∈ J.

Let ¯x = ρ(x) ∈ M; we end the proof by showing that it satisfies the conclusion ofTheorem 2.1 From the above, it is clearly the case fori ∈ I and k =1, ,k i, that is, we have that

(i) eitherF k

i(x) ∩ x i = ∅orF k

i(x) = ∅for everyi ∈ I and i =1, ,k i.

Now, let j ∈ J We first notice that ˜F j( x) = ∅if and only ifF j(x) = ∅ Assume now that

˜

F j(x) ∩ x j = ∅and recall thatx j ∩ F˜j( x) = x j ∩(F j( ¯x) × {1}) andF j( ¯ x) ⊂ C j

Conse-quently,x j ∩(C j × {1})= ∅and fromLemma 3.6we get

∅ = x j ∩F j( ¯ x) × {1}⊂ x j ∩C j × {1}=ρ j

x j

, 1

Hence, the equality holds and ¯x j = ρ j( x j) ∈ F j( ¯ x) This ends the proof ofTheorem 2.1

Appendix

The Grassmannian manifoldG k(V)

LetV be a Euclidean space and let k be an integer such that 0 ≤ k ≤dimV In this section,

we recall the properties ofG k(V) which are used in this paper.

First, we recall thatG k(V) is a smooth boundaryless manifold of dimension k(dimV −

k) (see, e.g., Hirsch [12] andLemma A.1) The local charts can be defined as follows Let ¯E ∈ G k(V) and let { e¯1, , ¯e k }be some given orthonormal basis of ¯E; we define the

mappingψ E¯: ( ¯E ⊥)k → G k(V) by

ψ E¯(u) =span

¯

e1+u1, , ¯e k+u k

, foru =u1, ,u k

∈E¯⊥k (A.1) Then it is easy to check that the mapping ψ E¯ is injective (see Claim A.2); so ψ E¯ is a bijection from ( ¯E ⊥)k ontoU E¯= ψ E¯(( ¯E ⊥)k) We can now consider the inverse mapping

ϕ E¯:U E¯→( ¯E ⊥)kdefined byϕ E(¯ E) = ψ −1

¯

E (E), which is clearly a bijection.

Lemma A.1 (a) G k(V) is a smooth boundaryless (i.e., C ∞ ) manifold of dimension k(dimV − k) without boundary and (U E,¯ ϕ E)¯ E¯∈ G k(V) defines a C ∞ atlas of G k(V).

(b) The set G k(V) is compact.

(c) The mapping E → E ⊥ from G k(V) to G (V) ( =dimV − k) is a smooth diffeomor-phism.

(d) The mapping p : V × G k(V) → V defined by p(x,E) =projE(x) is smooth Hence, the set {(x,E) ∈ V × G k(V) | x ∈ E } is a closed subset of V × G k(V).

(e) The mapping x → x p from G k(V) to (G k(V)) p is smooth.

We prepare the proof of the lemma with a claim

For j ∈ J, let ˜F jbe the correspondence from ˜M to... i, y k

Since the graph of ˆF k

i...

which ends the proof ofTheorem 3.1

3.2 Proof of Theorem 2.1 whenJ = ∅