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 Ślosarski
Department of Electronics, Technical University of Koszalin, Śniadeckich 2, 75-453 Koszalin, Poland
Correspondence should be addressed to Mirosław Ślosarski, slosmiro@gmail.com
Received 5 January 2011; Accepted 2 March 2011
Academic Editor: Marlène Frigon
Copyright q 2011 Mirosław Ślosarski. 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. Górniewicz, which leads to more
general results in fixed point theory.
1. Introduction
In 1976 L. Górniewicz 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 C̆ech
homology functor with compact carriers and coefficients in the 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 {Hq X} is a graded
vector space, Hq X being the q-dimensional C̆ech homology group with compact carriers
of X. For a continuous map f : X → Y , H∗ f is the induced linear map f∗ {fq }, where
fq : Hq X → Hq Y see 1, 2. A space X is acyclic if
i X is nonempty,
ii Hq X 0 for every q ≥ 1,
iii H0 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 ϕx 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 , we let:
ϕ−1 U x ∈ X; ϕx ⊂ 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 ϕA x∈A ϕx of the set A under ϕ is a compact
set;
2.1.2 the composition ψ ◦ ϕ : X T , ψ ◦ ϕx y∈ϕx ψy, is an u.s.c. mapping;
2.1.3 the mapping ϕp : X Z, given by the formula ϕp 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 qp−1 x ⊂ ϕx 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, Y0 ⊂ Y , and X0 ϕ−1 Y0 , then the
contraction ϕ0 : X0 Y0 of ϕ to the pair X0 , Y0 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 isomorphism.
Let u : E → E be an endomorphism of an arbitrary vector space. Let us put Nu {x ∈ E : un x 0 for some n}, where un is the nth iterate of u and E E/Nu. Since
uNu ⊂ Nu, we have the induced endomorphism u
: E → E defined by u
x ux.
We call u admissible provided that dim E < ∞.
Fixed Point Theory and Applications
3
Let u {uq } : E → E be an endomorphism of degree zero of a graded vector space
E {Eq }. We call u a Leray endomorphism if
i all uq are admissible,
q are trivial.
ii almost all E
For such a u, we define the generalized Lefschetz number Λu of u by putting
Λu q ,
−1q tr u
2.3
q
q cf. 1. The following important property of a
where tr
uq is the ordinary trace of u
Leray endomorphism is a consequence of a well-known formula tru ◦ v trv ◦ 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 ∈ Eq , there exists an integer n such that
unq x 0. Since for a weakly nilpotent endomorphism u : E → E we have Nu E, we get
the following.
Proposition 2.7. If u : E → E is a weakly nilpotent endomorphism, then Λu 0.
Proposition 2.8. Assume that in the category of graded vector spaces the following diagram
commutes
E′
u
E′′
v
u′
E′
u
u′′
2.4
E′′
If one of u
,u
is a Leray endomorphism, then so is the other and Λu
Λu
.
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 Λϕ of
ϕ by putting
Λ ϕ Λ q∗ p∗−1 ; p, q ⊂ ϕ .
2.5
Let X0 ⊂ X and let ϕ : X, X0 X, X0 be an admissible map. We define two admissible
maps ϕX : X X given by ϕX x ϕx for all x ∈ X and ϕX0 : X0 X0 ϕX0 x ϕx for all x ∈ X0 . Let p, q ⊂ ϕX , where p : Z → X is a Vietoris mapping and
q : Z → X a continuous map. We shall denote by p : Z, p−1 X0 → X, X0 pz pz,
qz for all z ∈ Z, p : p−1 X0 → X0 pz pz, and
q : Z, p−1 X0 → X, X0 qz
−1
q : p X0 → X0 qz qz for all z ∈ p−1 X0 . We observe that p, q
⊂ ϕ and p, q ⊂ ϕX0 .
4
Fixed Point Theory and Applications
Proposition 2.9 see 2. Let ϕ : X, X0 X, X0 be an admissible map of pairs and p, q ⊂
ϕX . If any two of the endomorphisms q∗ p∗−1 : HX, X0 → HX, X0 , q∗ p∗−1 : HX → HX,
q∗ p−1
∗ : HX0 → HX0 are Leray endomorphisms, then so is the third 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 p1 , q1 ⊂ ϕ and p2 , q2 ⊂ ψ there exists a pair
−1
−1
◦ q1∗ p1∗
q∗ p∗−1 .
p, q ⊂ ψ ◦ ϕ such that q2∗ p2∗
Definition 2.11. An admissible map ϕ : X X is called a Lefschetz map provided that the
Lefschetz set Λϕ of ϕ is well defined and Λϕ /
{0} implies that the set Fixϕ {x ∈ X :
x ∈ ϕx} 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 ∈ CovE K there
exists a map
πα : 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, πα x ∈ V ,
2.12.2 there exists a natural number n nK such that πα K ⊂ En , where En is an ndimensional 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 IdY : Y → Y be a map given by formula IdY 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 IdY .
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.
p
Z
X
2.8
β
α
U
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 ϕX : X X be a map. Then
⎧
⎪
IdX ,
⎪
⎪
⎨
ϕnX ϕX ,
⎪
⎪
⎪
⎩
ϕX ◦ ϕX ◦ · · · ◦ ϕX n-iterates
for n 0,
for n 1,
2.9
for n > 1.
We denote multivalued maps with ϕXY : X Y , and ψZ : Z Z. If a nonempty set
A ⊂ X, a nonempty set B ⊂ Y and ϕXY A ⊂ B then a multivalued map ϕAB : A B given
by ϕAB x ϕXY x for each x ∈ X.
Definition 2.21 see 5. A multivalued map ϕXY : 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 ϕV X : V X is admissible.
Proposition 2.22 see 5. Let ϕXY : X Y and ψY Z : Y Z be locally admissible maps. Then
the map ΦXZ ψY Z ◦ ϕXY : X Z is locally admissible.
Proposition 2.23 see 5. Let A ⊂ X be a nonempty set, and let ϕXY : X Y be a locally
admissible map. Then a map ϕAY : A Y is locally admissible.
Definition 2.24 see 2, 5. A multivalued map ϕX : X X is called a compact absorbing
contraction written ϕX ∈ CACX provided there exists an open set U ⊂ X such that
2.24.1 ϕX U ⊂ U and the ϕU : U U, ϕU x ϕX x for every x ∈ X is compact
ϕX U ⊂ U,
2.24.2 for every x ∈ X there exists n nx such that ϕnX x ⊂ U.
Proposition 2.25 see 3. Let ϕX : X X be an admissible map, X ∈ ANMR, and ϕX ∈
CACX then ϕX is a Lefschetz map.
6
Fixed Point Theory and Applications
Proposition 2.26 see 5. Let ϕX ∈ CACX, and let U ⊂ X be an open set from Definition 2.24.
2.26.1 Let B be a nonempty set in X and ϕX B ⊂ B. Then U ∩ B /
∅.
2.26.2 For any n ∈ N ϕnX ∈ CACX.
2.26.3 Let V ⊂ X be a nonempty and open set. Assume that ϕX V ⊂ V . Then ϕV ∈ CACV.
3. Main result
Let X be a metric space, ϕX : X X a multivalued map, and let
ΩAD ϕ V ⊂ X : V is open, ϕV : V V is admissible, ϕV V ⊂ V .
3.1
Obviously the above family of sets can be empty. Thus we can define the following class of
multivalued maps:
ADL ϕX : X X, ΩAD ϕ /
∅ .
3.2
All the admissible maps ϕX : X X particularly belong to the above class of maps because
X ∈ ΩAD ϕ. We shall remind that the multivalued map ϕX : X X is called acyclic
if for every x ∈ X the set ϕX 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 ∈ ϕX x}, are a selective pair r, s ⊂ ϕX .
Moreover, for an acyclic map ϕX : X X, if the homomorphism s∗ r∗−1 : H∗ X →
H∗ X is a Leray endomorphism, then Lefschetz set ΛϕX consists of only one element and
Λ ϕX Λ s∗ r∗−1 .
3.4
For a certain class of multivalued maps ϕX ∈ ADL we define a generalized Lefschetz set
ΛG ϕX of a map ϕX in such a way that the conditions of the following definition are satisfied.
Let ϕV : V → V be an admissible map. One shall say that a set ΛϕV is well defined
if for every p, q ⊂ ϕV 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 ΥAD ϕ ⊂ ΩAD ϕ such that
if for any V ∈ ΥAD ϕ ΛϕV is well defined, then the following conditions are satisfied:
3.1.1 if ϕX : X X is acyclic, then ΛG ϕX {Λs∗ r∗−1 } see 3.3,
3.1.2 if ϕX : X X is admissible, then X ∈ ΥAD ϕ and
{0} ⇒ ΛG ϕX /
{0} ,
Λ ϕX /
3.1.3 ΛG ϕX /
{0} ⇒ there exists V ∈ ΥAD ϕ such that ΛϕV /
{0}.
3.5
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 singlevalued map, continuous and Λf is well defined, then
ΛG 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 ϕX : X X be an admissible map, and let ΥAD ϕ {X}. If ΛϕX is well
defined, then we define
ΛG ϕX Λ ϕX .
3.7
The above example consists of Lefschetz set definitions common in mathematical literature
and introduced by L. Górniewicz.
Example 3.3. Let ϕX : X X be an admissible map, and let ΥAD ϕ be a family of sets
V ∈ ΩAD ϕ such that there exists p, q ⊂ ϕV and there exists p, q ⊂ ϕX such that the
following diagram
H∗ (V )
q∗ (p∗ )−1
H∗ (V )
u∗
υ∗
u∗
H∗ (X)
q∗ p∗−1
3.8
H∗ (X)
is commutative. It is obvious that X ∈ ΥAD ϕ, hence ΥAD ϕ /
∅. Assume that for any V ∈
ΥAD ϕ ΛϕV is well defined. We define
ΛG ϕX Λ ϕV .
3.9
V ∈ΥAD ϕ
Justification 1
Let us notice that if ϕX is acyclic, then from the commutativity of the above diagram it results
that for every V ∈ ΥAD ϕ ΛϕV {Λs∗ r∗−1 }, hence ΛG ϕV {Λs∗ r∗−1 }. The second
and third conditions of Definition 3.1 are obvious.
Let A ⊂ X be a nonempty set, and let
Oε A x ∈ X; there exists y ∈ A such that d x, y < ε ,
where d is metric in X.
3.10
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 × X dx, y ≤ 1, let ϕX : X X be a multivalued map and let K ϕX X. Let
ΥAD ϕ V ∈ ΩAD ϕ : V O2/n K for some n .
3.11
Assume that ΥAD ϕ / ∅ and for all V ∈ ΩAD ϕ ΛϕV is well defined. We define
ΛG ϕX Λ ϕU , where
U O2/k K, k min n ∈ N; O2/n K ∈ ΥAD ϕ .
3.12
Justification 2
The first condition of Definition 3.1 results from the commutativity of the following diagram:
H∗ (U)
q∗ (p∗ )−1
H∗ (U)
u∗
υ∗
u∗
H∗ (X)
q∗ p∗−1
3.13
H∗ (X),
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 ⊂ ϕX . Condition
3.1.2 results from the fact that X O2 K ∈ ΥAD ϕ and
ΛG ϕX Λ ϕO2 K Λ ϕX .
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 ϕX : X X be a map. One shall say that a nonempty set B ⊂ X has an
absorbing property writes B ∈ APϕ if for each x ∈ X there exists a natural number n such
that ϕnX x ⊂ B.
Let ΘAD ϕ ΩAD ϕ ∩ APϕ. We observe that if ϕX : X X is admissible then
ΘAD ϕ / ∅ since X ∈ ΘAD ϕ.
Theorem 3.6 see 2. Let ϕX : X X be an admissible map. Then for any V ∈ ΘAD ϕ and for
all p, q ⊂ ϕX the homomorphism
q∗ p∗−1 : H∗ X, V −→ H∗ X, V is weakly nilpotent (see Proposition 2.9), where p, q denote a respective contraction of p, q.
3.15
Fixed Point Theory and Applications
9
Theorem 3.7. Let ϕX : X X be an admissible map. Assume that for each V ∈ ΘAD ϕ ΛϕV is
well defined. Then
Λ ϕX Λ ϕV .
V ∈ΘAD ϕ
3.16
Proof. Let V ∈ ΘAD ϕ, p, q ⊂ ϕX , and let Λq∗ p∗−1 c0 . We observe that a map q∗ p∗−1 :
p, q
⊂ ϕ, ϕ : X, V X, V is weakly nilpotent so from
H∗ X, V H∗ X, V Propositions 2.7 and 2.9 Λq∗ p∗−1 Λq∗ p−1
⊂ ϕV and p, q denote a
∗ c0 , where p, q
respective contraction of p, q. Hence c0 ∈ ΛϕV and ΛϕX ⊂ V ∈ΘAD ϕ ΛϕV . It is clear
that X ∈ ΘAD ϕ and the proof is complete.
Example 3.8. Let ϕX : X X be a multivalued map, and let
ΥAD ϕ ΘAD ϕ .
3.17
Assume that the following conditions are satisfied:
3.8.1 ΥAD ϕ / ∅,
3.8.2 for all V ∈ ΥAD ϕ ΛϕV is well defined,
3.8.3 V ∈ΥAD ϕ ΛϕV / ∅.
We define
ΛG ϕX Λ ϕV .
3.18
V ∈ΥAD ϕ
Justification 3
Condition 3.1.1 results from Proposition 2.7 and Theorem 3.6. Let us notice that if a map
ϕX : X X is admissible, then X ∈ ΥAD ϕ and from Theorem 3.7 we get
ΛG ϕX Λ ϕX ,
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. Górniewicz. It is possible to create an example see 5 of a multivalued
map ϕX : X X that is not admissible and satisfies the conditions of Example 3.8.
Example 3.9. Let ϕX : X X be a multivalued map, and let
ΥAD ϕ ΘAD ϕ .
Assume that the following conditions are satisfied:
3.9.1 ΥAD ϕ /
∅,
3.9.2 for all V ∈ ΥAD ϕ ΛϕV is well defined.
3.20
10
Fixed Point Theory and Applications
We define
ΛG ϕX Λ ϕV .
3.21
V ∈ΥAD ϕ
Justification 4
Condition 3.1.1 results from Proposition 2.7 and Theorem 3.6. If a map ϕX : X X is
admissible, then X ∈ ΥAD 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 ϕX ∈ ADL is a general Lefschetz map provided that
the following conditions are satisfied:
3.10.1 there exists ΥAD ϕ / ∅ such that conditions 3.1.1–3.1.3 are satisfied,
3.10.2 for any V ∈ ΥAD ϕ ΛϕV 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 ϕX ∈ CACX (see Definiation 2.24),
3.11.2 there exists ΥAD ϕ / ∅ such that conditions (3.1.1)–(3.1.3) are satisfied.
Then ϕX is a general Lefschetz map, and if ΛG ϕX /
{0} then FixϕX /
∅.
Proof. From the assumption ΥAD ϕ /
∅, thus we show that for all V ∈ ΥAD ϕ ΛϕV is well
defined. Let V ∈ ΥAD ϕ, then from 2.26.3 ϕV ∈ CACV, so from Propositions 2.19 and
{0}, then from 3.1.3 there exists V ∈
2.25 ΛϕV is well defined. Assume that ΛG ϕX /
{0}. By the application of 2.26.3, Propositions 2.19, and 2.25, we
ΥAD ϕ such that ΛϕV /
get ∅ /
FixϕV ⊂ FixϕX and the proof is complete.
The following is a conclusion from Theorem 3.11.
Corollary 3.12. Let X ∈ ANMR, ϕX : X X be locally admissible (not necessarily admissible),
and let ϕX ∈ CACX. Then ϕX is a general Lefschetz map, and if ΛG ϕX / {0} then FixϕX / ∅.
Proof. Let U ⊂ X be an open set from Definition 2.24, and let K ϕU U ⊂ U. We define
ΥAD ϕ ΘAD ϕ see Examples 3.8 and 3.9. The map ϕX is locally admissible, so there
exists an open set V ⊂ X such that K ⊂ V and ϕV X : V X is admissible. We observe that
U ∩ V ∈ ΥAD ϕ since ϕU∩V : U ∩ V U ∩ V is admissible, compact and U ∩ V ∈ APϕ,
∅. If we define a generalized Lefschetz set now as in Example 3.9, then from
hence ΥAD ϕ /
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. Let C be a complex number set, and let f : C \ {0} → C \ {0} be
single-valued continuous and compact map. Assume that Fixf ∅, and choose an open set
V such that fC \ {0} ⊂ V ⊂ C\{0}. Let g : V → V be a compact gV ⊂ V and continuous
map such that Λg /
0. We define a multivalued map ϕC\{0} : C \ {0} C \ {0} given by
formula
⎧
⎨fz,
ϕC\{0} z ⎩ fz, gz
for z ∈
/ V,
for z ∈ V.
3.22
The map ϕC\{0} is admissible, so ΥAD ϕ ΘAD ϕ /
∅ see Examples 3.8 and 3.9. Let
ΛG ϕ Λ ϕU .
3.23
U∈ΥAD ϕ
see Example 3.9. We observe that
Λ ϕC\{0} {0}
3.24
since the only selective pair is the pair IdC\{0} , f ⊂ ϕC\{0} , but
Fix f ∅.
3.25
It is clear that V ∈ ΥAD ϕ and ΛϕV / {0}, since from the assumption Λg / 0. Hence
ΛG ϕX /
{0},
∅/
Fix ϕV ⊂ Fix ϕC\{0} .
3.26
References
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Publishers, Dordrecht, The Netherlands, 2003.
2 L. Górniewicz, Topological Methods in Fixed Point Theory of Multi-Valued Mappings, Springer, New York,
NY, USA, 2006.
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Applications, vol. 156, no. 4, pp. 697–709, 2009.
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Theory, vol. 10, no. 2, pp. 329–346, 2009.
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