BenkafadarAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 6, No. 1, 2005

pp. 87-100

A generalized coincidence point index

N. M. Benkafadar and M. C. Benkara-Mostefa
1

Abstract. The paper is devoted to build for some pairs of contin-
uous single-valued maps a coincidence point index. The class of pairs
(f, g) satisfies the condition that f induces an epimorphism of the
∨

Cech homology groups with compact supports and coefficients in the
field of rational numbers Q. Using this concept one defines for a class of
multi-valued mappings a fixed point degree. The main theorem states
that if the general coincidence point index is different from {0}, then
the pair (f, g) admits at least a coincidence point. The results may be
considered as a generalization of the above Eilenberg-Montgomery the-
orems [12], they include also, known fixed-point and coincidence-point
theorems for single-valued maps and multi-valued transformations.

2000 AMS Classification: 54C60, 54H25, 55M20. 58C06.

Keywords: Fixed point, Concidence point, Index, Degree, Multi-valued
mapping.

1. Introduction

Let f, g : X −→ Y be two continuous single valued maps of Hausdorff topo-
logical spaces. The coincidence problem, which is a generalization of the fixed
point problem, is concerned with conditions which guarantees the existence of
a solution for the equation f(x) = g(x). A such point x ∈ X is called a coin-
cidence point of the pair of maps (f,g). The study of this problem has been
treated first in 1946 by Eilenberg-Montgomery [12]. Note that the Eilenberg-
Montgomery theorem is a natural generalization of the Lefschetz fixed point
theorem, it implies also, the fixed point theorems of Kakutani [21] and Wallace
[30]. Topological invariants for different classes of pairs of maps have been stud-
ied by many authors [9], [14], [15], [20], [22], [23], [27] and others. The purpose
of this note is to describe a generalized coincidence point index for a new class

1The authors acknowledge the support of A.N.D.R.U., (Contract No 03/06 Code CU
19905) and M.E.R.S., (Project No B*2501/04/04), Laboratory M.M.E.R.E.



88 N. M. Benkafadar and M. C. Benkara-Mostefa

of pairs of continuous maps (f,g) which satisfy the condition that f induces
a r-homomorphism [3], [4] for homology with compact carries. Moreover, one
gives several applications of the general coincidence point index in fixed point
theory for multi-valued mappings.

One uses the Dold’s fundamental class around a compact of a finite euclidean

space En [10], H denotes the
∨

Cech homology functor with compact carries and
coefficients in the field of rational numbers Q, from the category Top(2) of
Hausdorff topological pairs and continuous maps to the category Lg of graded
vector spaces over the set of rational numbers Q and linear maps of degree zero
[13], [18], [29].

2. Maps n-decomposing.

Let G1 and G2 be two additive abelian groups, τ : G1 −→ G2 be a homo-
morphism.

Definition 2.1 ([3]). A homomorphism τ is a called a r-homomorphism if τ
admits a right-inverse homomorphism.

The definition signifies, since τ : G1 −→ G2 is a r-homomorphism then there
exists a homomorphism σ : G2 −→ G1 such that τ ◦σ = IdG2, where IdG2 is
the automorphism identity on G2.

The following properties are satisfied.

Proposition 2.2. A homomorphism τ : G1 −→ G2 is a r-homomorphism if
and only if the following conditions are satisfied :

(1) τ is an epimorphism;
(2) G1 = Kerτ ⊕G , where G is a subgroup of G1.

Proposition 2.3. If G1 and G2 are two modules over a field K and if τ :
G1 −→ G2 is an epimorphism then τ is a r-homomorphism.

Proposition 2.4 ([3]). Let τ1 : G1 −→ G2 and τ2 : G2 −→ G3 be two r-
homomorphisms then their composition τ = τ2 ◦ τ1 : G1 −→ G3 is also a
r-homomorphism.

The notion of r-homomorphisms has been introduced by Borsuk and Kosinsk
[3], [4].

Let (X,A) and (Y,B) be two objects of the category Top(2) of Hausdorff

topological pairs and continuous maps and f : (X,A) −→ (Y,B) be a morphism
from the Hausdorff pair (X,A) into an other Hausdorff pair (Y,B).

Let H be the
∨

Cech homology functor with compact carries and coefficients
in the field of rational numbers Q, from the category Top(2) of Hausdorff topo-
logical pairs and continuous maps to the category Lg of graded vector spaces
over the set of rational numbers Q and linear maps of degree zero [13], [18],
[29].



A generalized coincidence point index 89

Definition 2.5. A continuous single-valued map f : (X,A) −→ (Y,B) is said
to be n-decomposing in the rank n ≥ 0 on the Hausdorff pair (Y,B) if the ho-
momorphism f∗ : Hn(X,A) → Hn(Y,B) induced by f, is a r-homomorphism.

The set of the right-inverse homomorphisms of f∗ on (Y,B) will be denoted
by Ω(f∗;Y,B).

The following propositions and corollaries, prove that the class of n-decomposing
maps is vast.

Definition 2.6 ([3]). A continuous single-valued map f : (X,A) −→ (Y,B) is
called a r-map if f admits a continuous right inverse.

Proposition 2.7. Let f : (X,A) −→ (Y,B) be a single-valued map which is a
r-map, then f is n-decomposing on (Y,B) for every rank n ≥ 0.

Corollary 2.8. A retraction r of a pair (X,A) onto (X′,A′) is n-decomposing
on the retract (X′,A′) of (X,A).

Definition 2.9 ([3]). A continuous single-valued map f : (X,A) −→ (Y,B)
is said to be a h-map if there exists a continuous single-valued g : (Y,B) −→
(X,A) such that their composition f◦g and the identity map Id(Y,B) : (Y,B) −→
(Y,B) are homotopic.

Proposition 2.10. If f : (X,A) −→ (Y,B) is a h-map, then f is n-decomposing
on (Y,B) for every n ≥ 0.

Corollary 2.11. A lower retraction r : (X,A) −→ (X′,A′) is n-decomposing
on each lower retract (X′,A′) of (X,A).

Proposition 2.12. Let f : (X,A) −→ (Y,B) be a continuous single-valued
map. If there exists a continuous single-valued map g : (Z,C) −→ (X,A)
such that their composition f ◦ g is n-decomposing on (Y,B), then f is also
n-decomposing on (Y,B).

Corollary 2.13. Let f : (X,A) −→ (Y,B) be a continuous single-valued map
and (Z,C) ⊆ (X,A). If the restriction of f on (Z,C) is n-decomposing on
(Y,B), then f is also n-decomposing on (Y,B).

Proposition 2.14. Let f : (X,A) −→ (Y,B) be a n-decomposing on (Y,B)
and g : (Y,B) −→ (Z,C) be a n-decomposing on (Z,C), then their composition
g ◦f is n-decomposing on (Z,C).

Definition 2.15 ([5]). A space X is Q-acyclic provided: (i) X is non-empty,
(ii) Hq(X) = 0 for all q > 1 and (iii) H0(X) ≈ Q.

Proposition 2.16. Let f : (X,A) −→ (Y,B) be a continuous single-valued
map such that:

(1) f is proper and surjective;
(2) f−1(B) = A;
(3) f−1(y) is Q-acyclic for every y ∈ Y.

Then the map f is n-decomposing on (Y,B) for every n ≥ 0.



90 N. M. Benkafadar and M. C. Benkara-Mostefa

Proposition 2.17. Let U be an open subset of an Euclidean space En and K
be a compact subset of U, then the injection i : (U,U\K) −→ (En,En\K) is
n-decomposing on (En,En\K).

3. Generalized coincidence point index

Let U be an open subset of an euclidean vector space En which has a fixed
orientation.

Let (f,g) be a pair of continuous single-valued maps defining as follows:

(3.1) U
f
←− X

g
−→ En

where X is an arbitrary Hausdorff topological space.

Definition 3.1. An element x ∈ X is said to be a coincidence point of the
pair (f,g) if f(x) = g(x).

Let S(f,g) be the set of all coincidence points of the pair (f,g) and F(f,g)
be the subset of U defined as follows:

F(f,g) = {u ∈ U | u ∈ g(f−1(u)}.

Lemma 3.2. One has the equality f(S(f,g)) = F(f,g).

Proof. The proof is obvious. �

Let K be a compact subset of U which contains F(f,g). Thus, one obtains
the following diagram:

(3.2) (U,U\K)
f
←− (X,X\f−1(K))

f−g
−→ (En,En\{θ})

Definition 3.3. A pair of continuous single-valued maps (f,g) as above de-
fined, is called n-admissible on (U,U\K) if f is n-decomposing on (U,U\K).

The set of all n-admissible pairs on (U,U\K) is denoted PD(U,U\K).

Let (f,g) ∈ PD(U,U \K), then if σ ∈ Ω(f∗;U,U\K) the diagram (3.2)
induces the following diagram:

(3.3)

Hn(U,U\K)
f∗
←− Hn(X,X\f

−1(K))
(f−g)∗
−→ Hn(E

n,En\{θ})

σ ց m

Hn(X,X\f
−1(K))

Let OK ∈ Hn(U,U\K) be the image of 1 under the composite map:

Z = Hn(S
n) −→ Hn(S

n,Sn\K) ≅ Hn(U,U\K)

and O{θ} ∈ Hn(E
n,En\{θ}) be the image of 1 under the composition map:

Z = Hn(S
n) −→ Hn(S

n,Sn\{θ}) ≅ Hn(E
n,En\{θ})

where Sn = En ∪{∞}.



A generalized coincidence point index 91

The elements OK and O{θ} are called the fundamental classes around the
compacts K and {θ} respectively [9], [10].

Definition 3.4. Let (f,g) be a n-admissible pair on (U,U\K). The generalized
coincidence point index of (f,g) relatively σ ∈ Ω(f∗;U,U\K) is defined as being
the rational number Iσ(f,g) which verifies the equality (f − g)∗ ◦ σ(OK) =
Iσ(f,g) ·O{θ}.

Definition 3.5. Let (f,g) be a n-admissible pair on (U,U\K). The generalized
coincidence point index of (f,g) is defined as being the set of rational numbers
I(f,g) = {Iσ(f,g) | σ ∈ Ω(f∗;U,U\K)}.

Proposition 3.6. If the single-valued map f : (X,X\f−1(K)) −→ (U,U\K)
verifies the conditions of the proposition (2.16), then the pair (f,g) is n-
admissible on (U,U\K) and I(f,g) = {I(f∗)−1(f,g)}.

Proof. The single-valued map f : (X,X\f−1(K)) −→ (U,U\K) induces an iso-
morphism f∗ : Hn(X,X\f

−1(K)) −→ Hn(U,U\K) therefore Ω(f∗;U,U\K) =
{(f∗)

−1}. �

Proposition 3.7. If F(f,g) = ∅, then I(f,g) = {0}.

Proof. Suppose that F(f,g) = ∅ then using lemma (3.2) one deduces that
S(f,g) = ∅. This equality means that f(x) 6= g(x) for each x ∈ X. Therefore
for every σ ∈ Ω(f∗;U,U\K) we have the following commutative diagram:

Hn(U,U\K)
σ
−→ Hn(X,X\f

−1(K))
(f−g)∗
−→ Hn(E

n,En\{θ})

(f −g)∗ ց m
Hn(E

n\{θ},En\{θ})

where (f −g) = f −g. One concludes the proof remarking that (f −g)∗ is
the trivial homomorphism. �

Corollary 3.8. If I(f,g) 6= {0}, then the pair (f,g) admits at least a coinci-
dence point.

Proof. This is a consequence of lemma (3.2). �

Let g : U −→ En be a continuous single-valued map defined from an open
subset U of an Euclidean vector space En and K be a compact subset of U
which contains Fix(g) = {x ∈ U | x = g(x)}. The fixed point index of g defined
in [9] is the rational Ig which verifies the equality:

(i−g)n∗ (OK) = Ig ·O{θ},

where i : U −→ En is the natural injection.

Proposition 3.9. The generalized coincidence point index of the pair (i,g) is
defined and equal to the fixed point index of g.



92 N. M. Benkafadar and M. C. Benkara-Mostefa

Proof. First note that F(i,g) = Fix(g) = {x ∈ U | x = g(x)}. Let K be
a compact subset of En which contains F(i,g) = Fix(g). So, one has the
diagram:

Hn(E
n,En\K)

i∗

←− Hn (U,U\K)
(i−g)∗
−→ Hn(E

n,En\{θ}).

Therefore, I(i,g) ·O{θ} = (i−g)∗ ◦ i
−1
∗ (OK) = (i−g)∗(OK) = Ig ·O{θ}.

�

Corollary 3.10. If I(i,g) 6= {0} then g admits at least a fixed point.

Let (f,g) and (f1,g1) be two pairs of continuous single-valued maps defining
as follows:

(3.4) U
f
←− X

g
−→ En

and

(3.5) V
f1
←− X1

g1
−→ En,

where U and V are two open subsets of En and X and X1 are two Hausdorff
topological spaces.

Let K and K1 be two compact subsets of E
n which contain F(f,g) and

F(f1,g1) respectively and such that K ⊂ K1 ⊂ V ⊂ V ⊂ U.
For instance, one obtains the following diagrams:

(3.6) (U,U\K)
f
←− (X,X\f−1(K))

f−g
−→ (En,En\{θ})

and

(3.7) (V,V\K1)
f1
←− (X1,X1\f

−1
1 (K1))

f1−g1
−→ (En,En\{θ}).

Proposition 3.11. Under the above hypotheses, assume that h : (X1,X1\f
−1
1 (K1)) −→

(X,X\f−1(K)) is a continuous single-valued map such that the following dia-
gram is commutative:

(V,V\K1)
f1
←− (X1,X1\f

−1
1 (K1))

f1−g1
−→ (En,En\{θ})

i ↓ ↓ h m

(U,U\K)
f
←− (X,X\f−1(K))

f−g
−→ (En,En\{θ})

where i is the natural injection. Then if the pair (f1,g1) ∈ PD(V,V\K1) one
can infer that (f,g) ∈PD(U,U \K) and I(f1,g1) ⊂ I(f,g).

Proof. Of course, i induces an isomorphism i∗ : Hn(V,V\K1) −→ Hn(U,U\K)
which takes OK1 in OK. Moreover, if σ ∈ Ω(f1∗,V,V\K1), then h∗ ◦σ ◦ i

−1
∗ ∈

Ω(f∗,U,U\K). �

Let (f,g) be a pair of continuous single-valued maps such that:

U
f
←− X

g
−→ En

and h : X1 −→ X be a continuous single-valued map defined between two
Hausdorff topological spaces X1 and X.



A generalized coincidence point index 93

Proposition 3.12. If h : (X1,X1\(f ◦ h)
−1(K)) −→ (X,X\f−1(K)) is n-

decomposing on (X,X\f−1(K)) and the pair (f,g) is n-admissible on (U,U\K),
then (f ◦h,g ◦h) ∈PD(U,U \K) and I(f ◦h,g ◦h) ⊂ I(f,g).

Proof. Note that F(f ◦ h,g ◦ h) ⊆ F(f,g) ⊆ K, the composition f ◦ h is n-
decomposing on (U,U\K) (see proposition 2.14), and one has the following
diagram:

(U,U\K)
f◦h
←− (X1,X1\(f ◦h)

−1(K))
f◦h−g◦h
−→ (En,En\{θ})

Let k ∈ I(f ◦ h,g ◦ h), then there exists σ ∈ Ω((f ◦h)∗ ,U,U\K) such that
(f ◦ h − g ◦ h)∗ ◦ σ(OK) = k · Oθ, therefore (f − g)∗ ◦ h∗ ◦ σ(OK) = k · Oθ.
Because h∗ ◦σ ∈ Ω(f∗;U,U\K), one deduces that k ∈ I(f,g). �

Definition 3.13. Two pairs of continuous single-valued maps defined as fol-
lows:

U
fi
←− X

gi
−→ En, i = 0,1,

are called equivariant on a compact K ⊂ En if there exist:

(1) a Hausdorff pair (X,X\X′) such that:

(U,U\K)
fi
←− (X,X\X′)

fi−gi
−→ (En,En\{θ}), i = 0,1,

(2) a pair of continuous maps (ϕ,ψ) n-admissible on (U,U\K) such that:

(U,U\K)
ϕ
←− (X,X\ϕ−1(K))

ϕ−ψ
−→ (En,En\{θ})

(3) a single-valued map h : (X,X\ϕ−1(K)) −→ (X,X\X′) n-decomposing
on (X,X\X′) such that the following diagram is commutative:

(U,U\K)
f0
←− (X,X\X′)

f0−g0
−→ (En,En\{θ})

m ↑ h m

(U,U\K)
ϕ
←− (X,X\ϕ−1(K))

ϕ−ψ
−→ (En,En\{θ})

m ↓ h m

(U,U\K)
f1
←− (X,X\X′)

f1−g1
−→ (En,En\{θ})

Proposition 3.14. If (fi,gi), i = 0,1 are two equivariant pairs on a compact
K ⊂ En, then (fi,gi) ∈PD(U,U \K), i = 0,1, and I(f0,g0) = I(f1,g1).

Proof. Assume (f0,g0) and (f1,g1) are equivariant, then f0∗ ◦ h∗ = ϕ∗ =
f1∗ ◦h∗ therefore f0∗, f1∗ are both n-decomposing on (U,U\K) and f0∗ = f1∗.
Moreover, (f0−g0)∗◦h∗ = (ϕ−ψ)∗ = (f1−g1)∗◦h∗ so (f0−g0)∗ = (f1−g1)∗. �

Definition 3.15. Two pairs (fi,gi), i = 0,1 defined as follows:

(U,U\K)
fi
←− (X,X\X′)

fi−gi
−→ (En,En\{θ}), i = 0,1,

are called homotopic on a compact K ⊂ En if the following conditions are
verified:



94 N. M. Benkafadar and M. C. Benkara-Mostefa

(1) there exists a pair of single-valued maps (ϕ,ψ) n-admissible on (U,U\K)×
[0,1] such that:

(U,U\K)× [0,1]
ϕ
←− (X,X\ϕ−1(K × [0,1]))

ϕ−Ψ
−→ (En,En\{θ}),

(2) there exists a single valued map

h : (X,X\X′) −→ (X,X\ϕ−1(K × [0,1])),

n-decomposing on (X,X\ϕ−1(K × [0,1])),
(3) the following diagram is commutative:

(U,U\K)
f0
←− (X,X\X′)

f0−g0
−→ (En,En\{θ})

χ0 ↓ h ↓ m

(U,U\K)× [0,1]
ϕ
←− (X,X\ϕ−1(K × [0,1]))

ϕ−Ψ
−→ (En,En\{θ})

χ1 ↑ h ↑ m

(U,U\K)
f1
←− (X,X\X′)

f1−g1
−→ (En,En\{θ})

where χi(x) = (x,i), for every x ∈ U and i = 0,1.

Proposition 3.16. If (f0,g0) and (f1,g1) are homotopic on a compact K ⊂ E
n

then (fi,gi) ∈PD(U,U \K), i = 0,1 and I(f0,g0) = I(f1,g1).

Proof. Of course, χ0∗ and χ1∗ are both isomorphisms and are equal, so f0∗ =
f1∗. One deduces also that f0∗ and f1∗ are both n-decomposing on (U,U\K).
In an other hand, from the commutativity of the diagram one obtains that
(f0−g0)∗◦h∗ = (f1−g1)∗◦h∗ = (ϕ−ψ)∗ therefore (f0−g0)∗ = (f0−g0)∗. �

Let (f,g) and (f′,g′) be two pairs defined by the following way:

U
f
←− X

g
−→ En

and

U′
f
′

←− X
g
′

−→ Em

where U and U′ are two open subsets of En and Em respectively.
Let K be a compact subset of En which contains F(f,g) and K′ be a compact

subset E which contains F(f′,g′).

Proposition 3.17. If the pairs (f,g) and (f′,g′) are n-admissible on (U,U\K)
and (U′,U′\K′) respectively then the pair (f ×f′,g×g′) is (n + m)-admissible
on (U ×U′,U ×U′\K ×K′) and I(f ×f′,g ×g′) ⊃ I(f,g) ·I(f′,g′).

Proof. One has the following equalities:

F(f ×f′,g ×g′) = F(f,g) ×F(f′,g′),

OK×K′ = OK ×OK′ ∈ Hn+m [(U,U\K)× (U
′,U′\K′)] =

Hn+m(U ×U
′,U ×U′\K ×K′) and the inclusion:

K ×K′ ⊃ F(f,g) ×F(f′,g′).

Therefore, if (σ,σ′) ∈ Ω(f∗,U,K) × Ω(f
′
∗,U

′,K′) one obtains the equalities:
(f ×f′ −g ×g′)∗ ◦ (σ ×σ

′)(OK×K′) =



A generalized coincidence point index 95

[(f −g)∗ ◦σ × (f
′ −g′)∗ ◦σ

′] (OK ×OK′) =
(f −g)∗ ◦σ(OK) × (f

′ −g′)∗ ◦σ
′(OK) = [Iσ(f,g) ·Iσ′ (f

′,g′)]O{θ}. �

4. Generalized fixed point degree of multi-valued mappings

Let X and Y be two Hausdorff topological spaces. A multi-valued mapping
taking X to Y is a relation F which associates to each element x ∈ X a non
empty subset F(x) ⊂ Y. Let K(Y ) be the collection of all non empty compact
subsets of Y and F : X −→ K(Y ) be a multi-valued mapping.

The subset:

ΓX(F) = {(x,y) ∈ X ×Y | y ∈ F(x)} ,

of X ×Y is called the graph of the multi-valued mapping F on X.
In this case one could define two natural projectors:

tF : ΓX(F) −→ X

and

rF : ΓX(F) −→ Y

such tF (x,y) = x, rF (x,y) = y for every (x,y) ∈ ΓX(F).
For each element x ∈ X one has the equality F(x) = rF (t

−1
F (x)). The quin-

tuple [X,Y,ΓX(F), tF ,rF ] is called the canonical representation of the multi-
valued F : X −→ K(Y ).

Let [X1,X2,X0,f1,f2] be a quintuple constituted of Hausdorff topological
spaces Xi, i = 0,1,2 and continuous maps fj : X0 −→ Xi, j = 1,2 and such
that f1 is onto and the inverse image of each element x ∈ X1 is compact, then
the equality F(x) = g ◦ f−1(x) defines a multi-valued mapping F : X1 −→
K(X2). In this case the quintuple [X1,X2,X0,f1,f2] is called a representation
of F : X1 −→ K(X2).

Two quintuples [X1,X2,X0,f1,f2], [X1,X2,X0,g1,g2] are called equivalents
if g1 ◦f

−1
1 (x) = F(x) = g2 ◦f

−1
2 (x) for each x ∈ X1.

A multi-valued mapping F : X −→ K(Y ) is called upper semi continuous if
F−1+ (V ) = {x ∈ X | F(x) ⊂ V} is an open subset of Y for every open subset
V of X.

A multi-valued G : X −→ K(Y ) is said to be a selector of F : X −→ K(Y )
if G(x) ⊆ F(x) for every element x ∈ X.

Let H be the
∨

Cech homology functor with compact carries and coefficient
in the set of rational numbers Q. A multi-valued mapping F : X −→ K(Y ) is
called to be Q-acyclic provided the image F(x) is Q-acyclic for every element
x ∈ X, F is said to be compact provided F(X) is contained in a compact
subset of Y .

More properties on multi-valued mappings can be found in [24].

Let F : U −→ K(En) be a multi-valued mapping and K be a compact
subset of U ⊆ En. In this case F(tF ,rF ) = {x ∈ U | x ∈ rF (t

−1
F )(x)} = {x ∈

U | x ∈ F(x)} = Fix(F).



96 N. M. Benkafadar and M. C. Benkara-Mostefa

Definition 4.1. A multi-valued mapping F : U −→ K(En) is called n-admissible
on (U,U\K) if the pair (tF ,rF ) of projectors:

U
tF
←− ΓU(F)

rF
−→ En

satisfies the following conditions:

(1) K ⊃ Fix(F) = {x ∈ U | x ∈ F(x)};
(2) the pair (tF ,rF ) is n-admissible on (U,U\K).

Lemma 4.2. Let F : U −→ K(En) be a multi-valued mapping n-admissible
on (U,U\K), then one has the following diagram:

Hn(U,U\K)
(tF )

∗

←− Hn(ΓU(F),ΓU\K(F))
(tF −rF )

∗

−→ Hn(E
n,En\{θ})

Proof. The proof is obvious. �

Definition 4.3. The generalized fixed point degree of a n-admissible multi-
valued mapping F on (U,U\K) is defined as the following set of rational num-
bers:

I(F ;U,K) = I(tF ,rF ) = {Iσ(tF ,rF ) | σ ∈ Ω((tF )∗ ;U,U\K)}

Let us describe some properties of this generalized fixed point degree.

Theorem 4.4. If I(F ;U,K) 6= {0} then F admits at least a fixed point i.e. a
point x ∈ U such that x ∈ F(x).

Proof. This is a consequence of corollary (3.8). �

Definition 4.5. A representation ρ = [U,En,Z,f,g] of a multi-valued mapping
F : U −→ K(En) is called n-admissible on (U,U\K) if the pair (f,g) is n-
admissible on (U,U\K) and {x ∈ U | x ∈ F(x)}⊆ K.

Let U and V be two open subsets of En, K and K1 be two compact subsets

of En such that K ⊂ K1 ⊂ V ⊂ V ⊂ U. If the restriction
∼

F : V −→ K(En)

of F : U −→ K(En) defined by the rule
∼

F(x) = F(x) for every x ∈ V admits
a representation ρ = [V,En,Z,f,g] n-admissible on (V,V\K1), so one can
consider the following diagram:

(4.8) (V,V\K1)
f
←− (Z,Z\f−1(K1))

f−g
−→ (En,En\{θ})

Let Ω(f∗;V,V\K1) be the set of the right inverse homomorphisms of:

f∗ : Hn(Z,Z\f
−1(K1)) −→ Hn(V,V\K1).

In this case one can define:

Iρ(
∼

F ;V,K1) = {Iσ(f,g) | σ ∈ Ω(f∗;V,V\K1)}.

Proposition 4.6. If a multi-valued mapping F : U −→ K(En) has a re-

striction
∼

F : V −→ K(En) which admits a representation ρ = [V,En,Z,f,g]
n-admissible on (V,V\K1) then the multi-valued mapping F is n-admissible on

(U,U\K) and Iρ(
∼

F ;V,K1) ⊂I(F ;U,K).



A generalized coincidence point index 97

Proof. The proof is a consequence of proposition (3.11) and the following com-
mutative diagram:

Hn(V,V\K1)
f∗
←− Hn(Z,Z\f

−1(K1))
(f−g)∗
−→ Hn(E

n,En\{θ})
i∗ ↓ ↓ α∗ m

Hn(U,U\K)
(tF )∗
←− Hn(ΓU(F),ΓU\K(F))

(tF −rF )∗
−→ Hn(E

n,En\{θ})

where α(z) = (f(z),g(z)) for each z ∈ Z. �

Corollary 4.7. If a multi-valued mapping F : U −→ K(En) admits a repre-
sentation ρ = [V,En,Z,f,g] n-admissible on (V,V\K1), then F is n-admissible
on (U,U\K) and Iρ(F ;V,K1) ⊂I(F ;U,K).

Proposition 4.8. Let F : U −→ K(En) be a multi-valued mapping and
Φ : U −→ K(En) be a selector of F, then if Φ is a multi-valued mapping
n-admissible on (U,U\K) the multi-valued mapping F is also n-admissible on
(U,U\K) and I(Φ;U,K) ⊂I(F ;U,K).

Proof. The proof is a consequence of the following commutative diagram:

Hn(U,U\K)
(tΦ)∗
←− Hn(ΓU(Φ),ΓU\K(Φ))

(tΦ−rΦ)∗
−→ Hn(E

n,En\{θ})
m i∗ ↓ m
Hn(U,U\K) ←−

(tF )∗
Hn(ΓU(Φ),ΓU\K(Φ)) −→

(tF −rF )∗
Hn(E

n,En\{θ})

where i is the canonical injection. �

Definition 4.9. A continuous single-valued map λ : [0,1]×U ×En −→ En is
said to be a distortion of En if for each element x ∈ U the single-valued map
λ(0,x, .) : En −→ En is the map identity.

Definition 4.10. A multi-valued F : U −→ K(En) n-admissible on (U,U\K)
distorts into the multi-valued G : U −→ K(En) if there exists a distortion of
En such that :

(1) λ(1,x,F(x)) = G(x) for every x ∈ U;
(2) x /∈ λ(t,x,F(x)) for every t ∈ [0,1] and x ∈ (U\K) .

Proposition 4.11. If a multi-valued F : U −→ K(En) n-admissible on
(U,U\K) distorts into the multi-valued G : U −→ K(En), then G is n–
admissible on (U,U\K) and I(F ;U,K) ⊂I(G;U,K).

Proof. Consider ξ : (ΓU(F),ΓU\K(F)) −→ (ΓU(G),ΓU\K(G)) defined by the
rule ξ(x,u) = (x,λ(1,x,u)) for every (x,u) ∈ ΓU(F). Form the equality tF =
tG ◦ ξ one deduces that G is n-admissible on (U,U\K) . In an other hand, the
continuous single-valued maps (tF −rF ), (tG−rG)◦ξ : (ΓU(F),ΓU\K(F)) −→
(En,En\{θ}) are homotopic by the homotopy h(t,(x,u) = x − λ(t,x,u) for
every t ∈ [0,1] and (x,u) ∈ ΓU(F). Let σ ∈ Ω ((tF )∗) , then ξ∗ ◦σ ∈ Ω ((tG)∗)
and one has the equalities: Iσ ·O{θ} = (tF −rF )∗ ◦σ(OK) = (tG −rG)∗ ◦ ξ∗ ◦
σ(OK) = Iξ∗◦σ ·O{θ}, which means that I(F ;U,K) ⊂I(G;U,K). �



98 N. M. Benkafadar and M. C. Benkara-Mostefa

Assume that U and V are two open subsets of En, K and K1 are two
compact subsets of En such that K ⊂ K1 ⊂ V ⊂ V ⊂ U.

Proposition 4.12. Let F : U −→ K(En) be a multi-valued mapping upper
semi continuous compact and Q-acyclic. If G : U −→ K(En) is a selector of F
and n-admissible on (V,V\K1), then I(G;V,K1) = I(F ;U,K) = {k}, where
k is the rational number which verifies the equality (tF −rF )∗ ◦ (tF )

−1
∗ (OK) =

k ·O{θ}.

Proof. The proof is a consequence of the Vietoris maps theorems [12], propo-
sition (4.8) and the following commutative diagram:

Hn(V,V\K1)
tG∗
←− Hn(ΓV (G),ΓV \K1(G))

(tG−rG)∗
−→ Hn(E

n,En\{θ})
i∗ ↓ j∗ ↓ m
Hn(U,U\K) ←−

tF ∗
Hn(ΓU(F),ΓU\K(F)) −→

(tF −rF )∗
Hn(E

n,En\{θ})

where i : (V,V\K1) −→ (U,U\K) and j : (ΓV (G),ΓV \K1(G)) −→ (ΓU(F),ΓU\K(F))
are the natural injections. �

Proposition 4.13. Let K be a compact Q-acyclic subset of En and F : U −→
K(En) be a multi-valued mapping such that F(U) ⊂ K, then F is n-admissible
on (U,U\K) and I(F ;U,K) = {1}.

Proof. Consider x0 ∈ K and let f : U −→ K(E
n) be the map defined by

the rule f(x) = {x0} for each x ∈ U. The quintuple ρ = [U,E
n,U,IdU,f]

is a representation n-admissible on (U,U\K) of f. Consider the following
commutative diagram:

Hn(U,U\K)
(IdU )∗
←− Hn(U,U\K)

(IdU −f)∗
−→ Hn(E

n,En\{θ})
j∗ ↓ m
Hn(E

n,En\{x0}) −→
(IdEn−f)∗

Hn(E
n,En\{θ})

where j∗ is an isomorphism induced by the natural injection and (IdEn −f)∗ is
the isomorphism induced by the homeomorphism (IdEn −f) : (E

n,En\{x0}) −→
(En,En\{θ}) defined by the rule (IdEn − f)(x) = x − x0 for every x ∈ E

n.
For instance, one deduces that Iρ(f;U,K) = {1}. In an other hand, consider
the following commutative diagram:

Hn(U,U\K)
(IdU )

∗

←− Hn(U,U\K)
(IdU −f)

∗

−→ Hn(E
n,En\{θ})

m ↓ µ∗ m
Hn(U,U\K) ←−

(tF )
∗

Hn(ΓU(i−R),ΓU\K(i−R)) −→
(tF −rF )

∗

Hn(E
n,En\{θ})

where µ(x) = (x,f(x)), for each x ∈ U. The multi-valued mapping F is n-
admissible on (U,U\K) because (IdU)∗ is an isomorphisms. From the propo-
sitions (3.9), (4.7) and the commutativity of the above diagram one infers
Iρ(f;U,K) ⊂ I(F ;U,K). The multi-valued mapping F : U −→ K(E

n) is
a selector of the upper semi continuous, compact and Q-acyclic multi-valued
mapping G : U −→ K(En) defined by the rule G(x) = K for each x ∈ U.



A generalized coincidence point index 99

Using the proposition (4.12), one deduces I(F ;U,K) = I(G;U,K) = {k} so
k = 1. �

Proposition 4.14. Let C be a compact subset of En which is a neighborhood
retract. Let F : C −→ K(C) be an upper semi continuous and Q-acyclic
multi-valued mapping. Then F admits at least a fixed point.

Proof. Consider U an open subset of En and let ρ : U −→ C be a retraction
from U into C. The multi-valued G = F ◦ρ : U −→ K(C) ⊂ K(En) is upper
semi continuous compact with Q-acyclic values, therefore I(G;U,C) = {1}.
One deduces that G admits in U, at least, a fixed point x ∈ G(x) = F(ρ(x)).
However, x ∈ C then ρ(x) = x. �

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Received May 2004

Accepted December 2004

N. M. Benkafadar (benkafadar@caramail.com)
Department of Mathematics, Faculty of Sciences, University of Constantine,
Road of Ain El Bey 25000, Constantine, Algeria

M. C. Benkara-Mostefa (karamos@yahoo.fr)
Department of Mathematics, Faculty of Sciences, University of Constantine,
Road of Ain El Bey 25000, Constantine, Algeria