Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 216146, 11 pages
doi:10.1155/2011/216146
Research Article
Generalized Lefschetz Sets
Mirosław
´
Slosarski
Department of Electronics, Technical University of Koszalin,
´
Sniadeckich 2, 75-453 Koszalin, Poland
Correspondence should be addressed to Mirosław
´
Slosarski,
Received 5 January 2011; Accepted 2 March 2011
Academic Editor: Marl
`
ene Frigon
Copyright q 2011 Mirosław
´
Slosarski. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We generalize and modify Lefschetz sets defined in 1976 by L. G
´
orniewicz, which leads to more
general results in fixed point theory.
1. Introduction
In 1976 L. G
´

orniewicz introduced a notion of a Lefschetz set for multivalued admissible
maps. The paper attempts at showing that Lefschetz sets can be defined on a broader class of
multivalued maps than admissible maps. This definition can be presented in many ways, and
each time it is the generalization of the definition from 1976. These generalizations essentially
broaden the class of admissible maps that have a fixed point. Also, they are a homologic
tool for examining fixed points for a class of multivalued maps broader than just admissible
maps.
2. Preliminaries
Throughout this paper all topological spaces are assumed to be metric. Let H
∗
be the
˘
Cech
homology functor with compact carriers and coefficients in t he field of rational numbers Q
from the category of Hausdorff topological spaces and continuous maps to the category of
graded vector spaces and linear maps of degree zero. Thus H
∗
X{H
q
X} is a graded
vector space, H
q
X being the q-dimensional
˘
Cech homology group with compact carriers
of X. For a continuous map f : X → Y, H
∗
f is the induced linear map f
∗
 {f

q
}, where
f
q
: H
q
X → H
q
Ysee 1, 2. A space X is acyclic if
i X is nonempty,
ii H
q
X0 for every q ≥ 1,
iii H
0
X ≈ Q.
2 Fixed Point Theory and Applications
A continuous mapping f : X → Y is called proper if for every compact set K ⊂ Y the set
f
−1
K is nonempty and compact. A proper map p : X → Y is called Vietoris provided that
for every y ∈ Y the set p
−1
y is acyclic. Let X and Y be two spaces, and assume that for
every x ∈ X a nonempty subset ϕx of Y is given. In such a case we say that ϕ : X  Y is a
multivalued mapping. For a multivalued mapping ϕ : X  Y and a subset U ⊂ Y ,welet:
ϕ
−1

U




x ∈ X; ϕ

x

⊂ U

. 2.1
If for every open U ⊂ Y the set ϕ
−1
U is open, then ϕ is called an upper semicontinuous
mapping; we will write that ϕ is u.s.c.
Proposition 2.1 see 1, 2. Assume that ϕ : X  Y and ψ : Y  T are u.s.c. mappings with
compact values and p : Z → X is a Vietoris mapping. Then
2.1.1 for any compact A ⊂ X, the image ϕA

x∈A
ϕx of the set A under ϕ is a compact
set;
2.1.2 the composition ψ ◦ ϕ : X  T, ψ ◦ ϕx

y∈ϕx
ψy, is an u.s.c. mapping;
2.1.3 the mapping ϕ
p
: X  Z, given by the formula ϕ
p
xp

−1
x, is u.s.c
Let ϕ : X  Y be a multivalued map. A pair p, q of single-valued, continuous maps
is called a selected pair of ϕ written p, q ⊂ ϕ if the following two conditions are satisfied:
i p is a Vietoris map,
ii qp
−1
x ⊂ ϕx for any x ∈ X.
Definition 2.2. A multivalued mapping ϕ : X  Y is called admissible provided that there
exists a selected pair p, q of ϕ.
Proposition 2.3 see 2. Let ϕ : X  Y and ψ : Y  Z be two admissible maps. Then the
composition ψ ◦ ϕ : X  Z is an admissible map.
Proposition 2.4 see 2. Let ϕ : X  Y and ψ : Z  T be admissible maps. Then the map
ϕ × ψ : X × Z  Y × T is admissible.
Proposition 2.5 see 2. If ϕ : X 
Y is an admissible map, Y
0
⊂ Y, and X
0
 ϕ
−1
Y
0
, then the
contraction ϕ
0
: X
0
 Y
0

of ϕ to the pair X
0
,Y
0
 is an admissible map.
Proposition 2.6 see 1. If p : X → Y is a Vietoris map, then an induced mapping
p
∗
: H
∗

X

−→ H
∗

Y

2.2
is a linear i somorphism.
Let u : E → E be an endomorphism of an arbitrary vector space. Let us put Nu
{x ∈ E : u
n
x0 for some n}, where u
n
is the nth iterate of u and

E  E/Nu. Since
uNu ⊂ Nu, we have the induced endomorphism u :


E →

E defined by uxux.
We call u admissible provided that dim

E<∞.
Fixed Point Theory and Applications 3
Let u  {u
q
} : E → E be an endomorphism of degree zero of a graded vector space
E  {E
q
}. We call u a Leray endomorphism if
i all u
q
are admissible,
ii almost all

E
q
are trivial.
For such a u, we define the generalized Lefschetz number Λu of u by putting
Λ

u



q


−1

q
tr

u
q

, 2.3
where tru
q
 is the ordinary trace of u
q
cf. 1. The following important property of a
Leray endomorphism is a consequence of a well-known formula tru ◦ vtrv ◦ u for
the ordinary trace. An endomorphism u : E → E of a graded vector space E is called
weakly nilpotent if for every q ≥ 0 and for every x ∈ E
q
, there exists an integer n such that
u
n
q
x0. Since for a weakly nilpotent endomorphism u : E → E we have NuE,weget
the following.
Proposition 2.7. If u : E → E is a weakly nilpotent endomorphism, then Λu0.
Proposition 2.8. Assume that in the category of graded vector spaces the following diagram
commutes
E
′
E

′
u
′
u
u
v
u
′′
E
′′
E
′′
2.4
If one of u

,u

is a Leray endomorphism, then so is the other and Λu

Λu

.
Let ϕ : X  X, be an admissible map. Let p, q ⊂ ϕ, where p : Z → X is a Vietoris
mapping and q : Z → X a continuous map. Assume that q
∗
◦ p
−1
∗
: H
∗

X → H
∗
X is a
Leray endomorphism for all pairs p, q ⊂ ϕ. For such a ϕ, we define the Lefschetz set Λϕ of
ϕ by putting
Λ

ϕ



Λ

q
∗
p
−1
∗

;

p, q

⊂ ϕ

. 2.5
Let X
0
⊂ X and let ϕ : X, X
0

  X, X
0
 be an admissible map. We define two admissible
maps ϕ
X
: X  X given by ϕ
X
xϕx for all x ∈ X and ϕ
X
0
: X
0
 X
0
ϕ
X
0
x
ϕx for all x ∈ X
0
.Letp, q ⊂ ϕ
X
, where p : Z → X is a Vietoris mapping and
q : Z → X a continuous map. We shall denote by p : Z, p
−1
X
0
 → X, X
0
 pzpz,

q : Z, p
−1
X
0
 → X, X
0
 qzqz for all z ∈ Z, p : p
−1
X
0
 → X
0
pzpz,and
q : p
−1
X
0
 → X
0
qzqz for all z ∈ p
−1
X
0
. We observe that p, q ⊂ ϕ and p, q ⊂ ϕ
X
0
.
4 Fixed Point Theory and Applications
Proposition 2.9 see 2. Let ϕ : X, X
0

  X, X
0
 be an admissible map of pairs and p, q ⊂
ϕ
X
. If any two of the endomorphisms q
∗
p
−1
∗
: HX, X
0
 → HX, X
0
, q
∗
p
−1
∗
: HX → HX,
q
∗
p
−1
∗
: HX
0
 → HX
0
 are Leray endomorphisms, then so is the t hird and

Λ

q
∗
p
−1
∗

Λ

q
∗
p
−1
∗

− Λ

q
∗
p
−1
∗

. 2.6
Proposition 2.10 see 2. If ϕ : X  Y and ψ : Y  T are admissible, then the composition
ψ ◦ ϕ : X  T is admissible, and for every p
1
,q
1

 ⊂ ϕ and p
2
,q
2
 ⊂ ψ there exists a pair
p, q ⊂ ψ ◦ ϕ such that q
2∗
p
−1
2∗
◦ q
1∗
p
−1
1∗
 q
∗
p
−1
∗
.
Definition 2.11. An admissible map ϕ : X  X is called a Lefschetz map provided that the
Lefschetz set Λϕ of ϕ is well defined and Λϕ
/
 {0} implies that the set Fixϕ{x ∈ X :
x ∈ ϕx} is nonempty.
Definition 2.12. Let E be a topological vector space. One shall say that E is a Klee admissible
space provided that for any compact subset K ⊂ E and for any open cover α ∈ Cov
E
K there

exists a map
π
α
: K −→ E 2.7
such that the following two conditions are satisfied:
2.12.1 for each x ∈ K there exists V ∈ α such that x, π
α
x ∈ V ,
2.12.2 there exists a natural number n  n
K
such that π
α
K ⊂ E
n
, where E
n
is an n-
dimensional subspace of E.
Definition 2.13. One shall say that E is locally convex provided that for each x ∈ E and for
each open set U ⊂ E such that x ∈ U there exists an open and convex set V ⊂ E such that
x ∈ V ⊂ U.
It is clear that if E is a normed space, then E is locally convex.
Proposition 2.14 see 1, 2. Let E be locally convex. Then E is a Klee admissible space.
Let Y be a metric space, and let Id
Y
: Y → Y be a map given by formula Id
Y
yy
for each y ∈ Y .
Definition 2.15 see 3. A map r : X → Y of a space X onto a space Y is said to be an mr-map

if there is an admissible map ϕ : Y  X such that r ◦ ϕ  Id
Y
.
Definition 2.16 see 3, 4. A metric space X is called an absolute multiretract notation: X ∈
AMR provided there exists a locally convex space E and an mr-map r : E → X from E onto
X.
Definition 2.17 see 3, 4. A metric space X is called an absolute neighborhood multiretract
notation: X ∈ ANMR provided that there exists an open subset U of some locally convex
space E and an mr-map r : U → X from U onto X.
Fixed Point Theory and Applications 5
Proposition 2.18 see 3, 4. A space X is an ANMR if and only if there exists a metric space Z
and a Vietoris map p : Z → X which factors through an open subset U of some locally convex E, that
is, there are two continuous maps α and β such that the following diagram is commutative.
Z
p
X
β
α
U
2.8
Proposition 2.19 see 3. Let X ∈ ANMR, and let V ⊂ X be an open set. Then V ∈ ANMR.
Proposition 2.20 see 3. Assume that X is ANMR. Let U be an open subset in X and ϕ : U  U
an admissible and compact map, then ϕ is a Lefschetz map.
Let ϕ
X
: X  X be a map. Then
ϕ
n
X


⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Id
X
, for n  0,
ϕ
X
, for n  1,
ϕ
X
◦ ϕ
X
◦···◦ϕ
X

n-iterates

for n>1.
2.9
We denote multivalued maps with ϕ
XY
: X  Y,andψ
Z

: Z  Z. If a nonempty set
A ⊂ X, a nonempty set B ⊂ Y and ϕ
XY
A ⊂ B then a multivalued map ϕ
AB
: A  B given
by ϕ
AB
xϕ
XY
x for each x ∈ X.
Definition 2.21 see 5. A multivalued map ϕ
XY
: X  Y is called locally admissible
provided for any compact and nonempty set K ⊂ X there exists an open set V ⊂ X such
that K ⊂ V and ϕ
VX
: V  X is admissible.
Proposition 2.22 see 5. Let ϕ
XY
: X  Y and ψ
YZ
: Y  Z be locally admissible maps. Then
the map Φ
XZ
ψ
YZ
◦ ϕ
XY
 : X  Z is locally admissible.

Proposition 2.23 see 5. Let A ⊂ X be a nonempty set, and let ϕ
XY
: X  Y be a locally
admissible map. Then a map ϕ
AY
: A  Y is locally admissible.
Definition 2.24 see 2, 5. A multivalued map ϕ
X
: X  X is called a compact absorbing
contraction written ϕ
X
∈ CACX provided there exists an open set U ⊂ X such that
2.24.1 ϕ
X
U ⊂ U and the ϕ
U
: U  U, ϕ
U
xϕ
X
x for every x ∈ X is compact

ϕ
X
U ⊂ U,
2.24.2 for every x ∈ X there exists n  n
x
such that ϕ
n
X

x ⊂ U.
Proposition 2.25 see 3. Let ϕ
X
: X  X be an admissible map, X ∈ ANMR, and ϕ
X
∈
CACX then ϕ
X
is a Lefschetz map.
6 Fixed Point Theory and Applications
Proposition 2.26 see 5. Let ϕ
X
∈ CACX, and let U ⊂ X be an open set from Definition 2.24.
2.26.1 Let B be a nonempty set in X and ϕ
X
B ⊂ B.ThenU ∩ B
/
 ∅.
2.26.2 For any n ∈ N ϕ
n
X
∈ CACX.
2.26.3 Let V ⊂ X be a nonempty and open set. Assume that
ϕ
X
V  ⊂ V .Thenϕ
V
∈ CACV.
3. Main result
Let X be a metric space, ϕ

X
: X  X a multivalued map, and let
Ω
AD

ϕ



V ⊂ X : V is open,ϕ
V
: V  V is admissible, ϕ
V

V

⊂ V

. 3.1
Obviously the above family of sets can be empty. Thus we can define the following class of
multivalued maps:
ADL 

ϕ
X
: X  X, Ω
AD

ϕ


/
 ∅

. 3.2
All the admissible maps ϕ
X
: X  X particularly belong to the above class of maps because
X ∈ Ω
AD
ϕ. We shall remind that the multivalued map ϕ
X
: X  X is called acyclic
if for every x ∈ X the set ϕ
X
x is nonempty, acyclic, and compact. It is known from the
mathematical literature that an acyclic map is admissible and the maps
r, s : Γ → X given by r

x, y

 x, s

x, y

 y for every

x, y

∈ Γ, 3.3
where Γ{x, y ∈ X × Y ; y ∈ ϕ

X
x}, are a selective pair r, s ⊂ ϕ
X
.
Moreover, for an acyclic map ϕ
X
: X  X, if the homomorphism s
∗
r
−1
∗
: H
∗
X →
H
∗
X is a Leray endomorphism, then Lefschetz set Λϕ
X
 consists of only one element and
Λ

ϕ
X



Λ

s
∗

r
−1
∗

. 3.4
For a certain class of multivalued maps ϕ
X
∈ ADL we define a generalized Lefschetz set
Λ
G
ϕ
X
 of a map ϕ
X
in such a way that the conditions of the following definition are satisfied.
Let ϕ
V
: V → V be an admissible map. One shall say that a set Λϕ
V
 is well defined
if for every p, q ⊂ ϕ
V
the map q
∗
p
−1
∗
: H
∗
V  → H

∗
V  is a Leray endomorphism.
Definition 3.1. Assume that there exists a nonempty family of sets Υ
AD
ϕ ⊂ Ω
AD
ϕ such that
if for any V ∈ Υ
AD
ϕΛϕ
V
 is well defined, then the following conditions are satisfied:
3.1.1 if ϕ
X
: X  X is acyclic, then Λ
G
ϕ
X
{Λs
∗
r
−1
∗
} see 3.3,
3.1.2 if ϕ
X
: X  X is admissible, then X ∈ Υ
AD
ϕ and


Λ

ϕ
X

/

{
0
}

⇒

Λ
G

ϕ
X

/

{
0
}

, 3.5
3.1.3Λ
G
ϕ
X


/
 {0} ⇒ there exists V ∈ Υ
AD
ϕ such that Λϕ
V

/
 {0}.
Fixed Point Theory and Applications 7
From the above definition it in particular results that see 3.1.1 if f : X → X is a single-
valued map, continuous and Λf is well defined, then
Λ
G

f

Λ

f

. 3.6
We shall present a few examples proving that Lefschetz sets can be defined in many ways
while retaining the conditions contained in Definition 3.1.
Example 3.2. Let ϕ
X
: X  X be an admissible map, and let Υ
AD
ϕ{X}.IfΛϕ
X

 is well
defined, then we define
Λ
G

ϕ
X

Λ

ϕ
X

. 3.7
The above example consists of Lefschetz set definitions common in mathematical literature
and introduced by L. G
´
orniewicz.
Example 3.3. Let ϕ
X
: X  X be an admissible map, and let Υ
AD
ϕ be a family of sets
V ∈ Ω
AD
ϕ such that there exists p, q ⊂ ϕ
V
and there exists p, q ⊂ ϕ
X
such that the

following diagram
H
∗
(V )
H
∗
(V )
q
∗
(p
∗
)
−1
q
∗
p
−1
∗
H
∗
(X)
H
∗
(X)
u
∗
u
∗
υ
∗

3.8
is commutative. It is obvious that X ∈ Υ
AD
ϕ, hence Υ
AD
ϕ
/
 ∅. Assume that for any V ∈
Υ
AD
ϕΛϕ
V
 is well defined. We define
Λ
G

ϕ
X



V ∈Υ
AD

ϕ

Λ

ϕ
V


. 3.9
Justification 1
Letusnoticethatifϕ
X
is acyclic, then from the commutativity of the above diagram it results
that for every V ∈ Υ
AD
ϕΛϕ
V
{Λs
∗
r
−1
∗
}, hence Λ
G
ϕ
V
{Λs
∗
r
−1
∗
}. The second
and third conditions of Definition 3.1 are obvious.
Let A ⊂ X be a nonempty set, and let
O
ε


A



x ∈ X; there exists y ∈ A such that d

x, y

<ε

, 3.10
where d is metric in X.
8 Fixed Point Theory and Applications
Example 3.4. Let X, d be a metric space, where d is a metric such that, for each x, y ∈
X × Xdx, y ≤ 1, let ϕ
X
: X  X be a multivalued map and let K  ϕ
X
X.Let
Υ
AD

ϕ



V ∈ Ω
AD

ϕ


: V  O
2/n

K

for some n

. 3.11
Assume that Υ
AD
ϕ
/
 ∅ and for all V ∈ Ω
AD
ϕΛϕ
V
 is well defined. We define
Λ
G

ϕ
X

Λ

ϕ
U

, where

U  O
2/k

K

,k min

n ∈ N; O
2/n

K

∈ Υ
AD

ϕ

.
3.12
Justification 2
The first condition of Definition 3.1 results from the commutativity of the following diagram:
q
∗
(p
∗
)
−1
q
∗
p

−1
∗
H
∗
(X)
u
∗
u
∗
υ
∗
H
∗
(X),
H
∗
(U)
H
∗
(U)
3.13
where u
∗
 i
∗
is a homomorphism determined by the inclusions i : U → X, v
∗
 q
∗
p

−1
∗
.
The maps
p, q are the respective contractions of maps p, q, p, q ⊂ ϕ
X
. Condition
3.1.2 results from the fact that X  O
2
K ∈ Υ
AD
ϕ and
Λ
G

ϕ
X

Λ

ϕ
O
2
K

Λ

ϕ
X


. 3.14
Satisfying Condition 3.1.3 is obvious.
Before the formulation of another example, let us introduce the following definition
and provide necessary theorems.
Definition 3.5. Let ϕ
X
: X  X be a map. One shall say that a nonempty set B ⊂ X has an
absorbing property writes B ∈ APϕ if for each x ∈ X there exists a natural number n such
that ϕ
n
X
x ⊂ B.
Let Θ
AD
ϕΩ
AD
ϕ ∩ APϕ. We observe that if ϕ
X
: X  X is admissible then
Θ
AD
ϕ
/
 ∅ since X ∈ Θ
AD
ϕ.
Theorem 3.6 see 2. Let ϕ
X
: X  X be an admissible map. Then for any V ∈ Θ
AD

ϕ and for
all p, q ⊂ ϕ
X
the homomorphism

q
∗
p
−1
∗
: H
∗

X, V

−→ H
∗

X, V

3.15
is weakly nilpotent (see Proposition 2.9), where p, q denote a respective contraction of p, q.
Fixed Point Theory and Applications 9
Theorem 3.7. Let ϕ
X
: X  X be an admissible map. Assume that for each V ∈ Θ
AD
ϕΛϕ
V
 is

well defined. Then
Λ

ϕ
X



V ∈Θ
AD

ϕ

Λ

ϕ
V

.
3.16
Proof. Let V ∈ Θ
AD
ϕ, p, q ⊂ ϕ
X
,andletΛq
∗
p
−1
∗
c

0
. We observe that a map q
∗
p
−1
∗
:
H
∗
X, V   H
∗
X, V p, q ⊂ ϕ, ϕ : X, V   X, V  is weakly nilpotent so from
Propositions 2.7 and 2.9 Λq
∗
p
−1
∗
Λq
∗
p
−1
∗
c
0
, where p, q ⊂ ϕ
V
and p, q denote a
respective contraction of p, q. Hence c
0
∈ Λϕ

V
 and Λϕ
X
 ⊂

V ∈Θ
AD
ϕ
Λϕ
V
.Itisclear
that X ∈ Θ
AD
ϕ and the proof is complete.
Example 3.8. Let ϕ
X
: X  X be a multivalued map, and let
Υ
AD

ϕ

Θ
AD

ϕ

. 3.17
Assume that the following conditions are satisfied:
3.8.1Υ

AD
ϕ
/
 ∅,
3.8.2 for all V ∈ Υ
AD
ϕΛϕ
V
 is well defined,
3.8.3

V ∈Υ
AD
ϕ
Λϕ
V

/
 ∅.
We define
Λ
G

ϕ
X



V ∈Υ
AD


ϕ

Λ

ϕ
V

. 3.18
Justification 3
Condition 3.1.1 results from Proposition 2.7 and Theorem 3.6. Let us notice that if a map
ϕ
X
: X  X is admissible, then X ∈ Υ
AD
ϕ and from Theorem 3.7 we get
Λ
G

ϕ
X

Λ

ϕ
X

, 3.19
and condition 3.1.2 is satisfied. Condition 3.1.3 is obvious.
It is crucial to notice that the definition of Lefschetz set encompassed in this example

agrees in the class of admissible maps with the familiar definition of a Lefschetz set
introduced by L. G
´
orniewicz. It is possible to create an example see 5 of a multivalued
map ϕ
X
: X  X that is not admissible and satisfies the conditions of Example 3.8.
Example 3.9. Let ϕ
X
: X  X be a multivalued map, and let
Υ
AD

ϕ

Θ
AD

ϕ

. 3.20
Assume that the following conditions are satisfied:
3.9.1Υ
AD
ϕ
/
 ∅,
3.9.2 for all V ∈ Υ
AD
ϕΛϕ

V
 is well defined.
10 Fixed Point Theory and Applications
We define
Λ
G

ϕ
X



V ∈Υ
AD

ϕ

Λ

ϕ
V

. 3.21
Justification 4
Condition 3.1.1 results from Proposition 2.7 and Theorem 3.6. If a map ϕ
X
: X  X is
admissible, then X ∈ Υ
AD
ϕ and hence condition 3.1.2 is satisfied. Condition 3.1.3 is

obvious.
The definition of a Lefschetz set in Example 3.9 is much more general than the
definition in Example 3.8, and as consequence it encompasses a broader class of maps. This
definition ignores the inconvenient assumption 3.8.3.
Let us define a Lefschetz map by the application of the new Lefschetz set definition.
Definition 3.10. One shall say that a map ϕ
X
∈ ADL is a general Lefschetz map provided that
the following conditions are satisfied:
3.10.1 there exists Υ
AD
ϕ
/
 ∅ such that conditions 3.1.1–3.1.3 are satisfied,
3.10.2 for any V ∈ Υ
AD
ϕΛϕ
V
 is well defined.
We will formulate, and then prove, a very general fixed point theorem.
Theorem 3.11. Let X ∈ ANMR. Assume that the following conditions are satisfied:
3.11.1 ϕ
X
∈ CACX (see Definiation 2.24),
3.11.2 there exists Υ
AD
ϕ
/
 ∅ such that conditions (3.1.1)–(3.1.3) are satisfied.
Then ϕ

X
is a general Lefschetz map, and if Λ
G
ϕ
X

/
 {0} then Fixϕ
X

/
 ∅.
Proof. From the assumption Υ
AD
ϕ
/
 ∅, thus we show that for all V ∈ Υ
AD
ϕΛϕ
V
 is well
defined. Let V ∈ Υ
AD
ϕ, then from 2.26.3 ϕ
V
∈ CACV, so from Propositions 2.19 and
2.25 Λϕ
V
 is well defined. Assume that Λ
G

ϕ
X

/
 {0}, then from 3.1.3 there exists V

∈
Υ
AD
ϕ such that Λϕ
V


/
 {0}. By the application of 2.26.3,Propositions2.19,and2.25,we
get ∅
/
 Fixϕ
V

 ⊂ Fixϕ
X
 and the proof is complete.
The following is a conclusion from Theorem 3.11.
Corollary 3.12. Let X ∈ ANMR, ϕ
X
: X  X be locally admissible (not necessarily admissible),
and let ϕ
X
∈ CACX.Thenϕ

X
is a general Lefschetz map, and if Λ
G
ϕ
X

/
 {0} then Fixϕ
X

/
 ∅.
Proof. Let U ⊂ X be an open set from Definition 2.24,andletK 
ϕ
U
U ⊂ U. We define
Υ
AD
ϕΘ
AD
ϕsee Examples 3.8 and 3.9. The map ϕ
X
is locally admissible, so there
exists an open set V ⊂ X such that K ⊂ V and ϕ
VX
: V  X is admissible. We observe that
U ∩ V ∈ Υ
AD
ϕ since ϕ
U∩V

: U ∩ V  U ∩ V is admissible, compact and U ∩ V  ∈ APϕ,
hence Υ
AD
ϕ
/
 ∅. I f we define a generalized Lefschetz set now as in Example 3.9, then from
Theorem 3.11 we get a thesis.
Fixed Point Theory and Applications 11
Finally we shall provide an example which shows that the new Lefschetz set definition
is more general than the definition of Lefschetz set for admissible maps already familiar in
mathematical literature.
Example 3.13 see 5.LetC be a complex number set, and let f : C \{0}→C \{0} be
single-valued continuous and compact map. Assume that Fixf∅, and choose an open set
V such that
fC \{0} ⊂ V ⊂ C\{0}.Letg : V → V be a compact gV  ⊂ V  and continuous
map such that Λg
/
 0. We define a multivalued map ϕ
C\{0}
: C \{0}  C \{0} given by
formula
ϕ
C\{0}

z


⎧
⎨
⎩

f

z

, for z /∈ V,

f

z

,g

z


for z ∈ V.
3.22
The map ϕ
C\{0}
is admissible, so Υ
AD
ϕΘ
AD
ϕ
/
 ∅ see Examples 3.8 and 3.9.Let
Λ
G

ϕ




U∈Υ
AD

ϕ

Λ

ϕ
U

. 3.23
see Example 3.9. We observe that
Λ

ϕ
C\{0}


{
0
}
3.24
since the only selective pair is the pair Id
C\{0}
,f ⊂ ϕ
C\{0}
,but

Fix

f

 ∅. 3.25
It is clear that V ∈ Υ
AD
ϕ and Λϕ
V

/
 {0}, since from the assumption Λg
/
 0. Hence
Λ
G

ϕ
X

/

{
0
}
, ∅
/
 Fix

ϕ

V

⊂ Fix

ϕ
C\{0}

. 3.26
References
1 J. Andres and L. G
´
orniewicz, Topological Principles for Boundary Value Problems, Kluwer Academic
Publishers, Dordrecht, The Netherlands, 2003.
2 L. G
´
orniewicz, Topological Methods in Fixed Point Theory of Multi-Valued Mappings, Springer, New York,
NY, USA, 2006.
3 R. Skiba and M.
´
Slosarski, “On a generalization of absolute neighborhood retracts,” Topology and Its
Applications, vol. 156, no. 4, pp. 697–709, 2009.
4 M.
´
Slosarski, “On a generalization of approximative absolute neighborhood retracts,” Fixed Point
Theory, vol. 10, no. 2, pp. 329–346, 2009.
5 M.
´
Slosarski, “Locally admissible multi-valued maps,” admitted for printing in Discussiones
Mathematicae. Differential Inclusions, Control and Optimization.