Namely we define the notion of locally admissible mappings

LOCALLY ADMISSIBLE MULTI-VALUED MAPS

Mirosław Ślosarski Technical University of Koszalin Śniadeckich 2, 75–453 Koszalin, Poland

e-mail: slomir@neostrada.pl

Abstract

In this paper we generalize the class of admissible mappings as due by L. Górniewicz in 1976. Namely we define the notion of locally admissible mappings. Some properties and applications to the fixed point problem are presented.

Keywords: Lefschetz number, fixed point, absolute neighborhood multi-retracts, admissible maps, locally admissible maps.

2010 Mathematics Subject Classification:32A12, 47H10, 55M20, 54C55.

1. Introduction

In 1946 S. Eilenberg and D. Montgomery (see [5]) presented a class of multi- valued maps of acyclic images. They proved that every mapping that belongs to this class is unequivocally represented by a pair of continuous single- valued maps where one of them is a Vietoris map. Due to this important observation, they applied Vietoris theorem to examine fixed points of acyclic multi-valued maps. In 1976, L. Górniewicz (see [7]) introduced the notion of an admissible multi-valued map where he generalized the aforementioned class of mappings. It was an enormous progress in the theory of multi-valued maps, above all because the images of admissible multi-valued maps do not even have to be connected. In the paper, we attempt to generalize admissible multi-valued maps. The definition of a locally admissible multi-valued map is introduced. Examples and a few of properties are provided and theorems about fixed points are formulated.

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 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 ˘Cech 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 [2] and [5]). A space X is acyclic if:

(i) X is non-empty,

(ii) Hq(X) = 0 for every q ≥ 1 and (iii) H0(X) ≈ Q.

A continuous mapping f : X → Y is called proper if for every compact set K ⊂ Y the set f⁻¹(K) is non-empty and compact. A proper map p : X → Y is called Vietoris provided 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 non-empty subset ϕ(x) of Y is given. In such a case we say that ϕ : X ⊸ Y is a multi-valued mapping. For a multi-valued mapping ϕ : X ⊸ Y and a subset U ⊂ Y , we let:

ϕ⁻¹(U ) = {x ∈ X; ϕ(x) ⊂ U }.

If for every open U ⊂ Y the set ϕ⁻¹(U ) is open, then ϕ is called an upper semi-continuous mapping; we shall write that ϕ is u.s.c.

Proposition 2.1 (see [2, 5]). 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) =S

x∈Aϕ(x) of the set A under ϕ is a compact set;

(2.1.2) the composition ψ ◦ ϕ : X ⊸ T , (ψ ◦ ϕ)(x) = S

y∈ϕ(x)ψ(y) 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 multi-valued map. A pair (p, q) of a single-valued, continuous map of the form 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 multi-valued mapping ϕ : X ⊸ Y is called admissible provided there exists a selected pair (p, q) of ϕ.

Theorem 2.3 (see [5]). Let ϕ : X ⊸ Y and ψ : Y ⊸ Z be two admissible maps. Then the composition ψ ◦ ϕ : X ⊸ Z is an admissible map.

Theorem 2.4 (see [5]). Let ϕ : X ⊸ Y and ψ : Z ⊸ T be admissible maps. Then the map ϕ × ψ : X × Z ⊸ Y × T is admissible.

Lemma 2.5 (see [5]). 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.

Theorem 2.6 (see [2]). If p : X → Y is a Vietoris map, then an induced mapping

p_∗: H_∗(X) → H_∗(Y ) is a linear isomorphism.

Let u : E → E be an endomorphism of an arbitrary vector space. Let us put N (u) = {x ∈ E : uⁿ(x) = 0 for some n}, where uⁿis the nth iterate of u and E = E/N (u). Since u(N (u)) ⊂ N (u), we have the induced endomorphisme e

u : eE → eE defined by eu([x]) = [u(x)]. We call u admissible provided dim eE < ∞.

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 u_q are admissible, (ii) almost all fEq are trivial.

For such a u, we define the (generalized) Lefschetz number Λ(u) of u by putting

Λ(u) =X

q

(−1)^qtr( euq),

where tr( euq) is the ordinary trace of euq(comp. [2]). 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 ∈ Eq, there exists an integer n such that uⁿ_q(x) = 0. Since for a weakly nilpotent endomorphism u : E → E we have N (u) = E, we get:

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^′ u ^-

E^′′

6u^′′

E^′′

ZZ ZZ

} v

E^′ - u^′6

u

If one of u^′, u^′′ is a Leray endomorphism, then so is the other; andΛ(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}. Let us observe that if X is an acyclic or, in partic- ular, contractible space, then for every admissible map ϕ : X ⊸ X and for any pair (p, q) ⊂ ϕ the endomorphism q_∗p⁻¹_∗ : H_∗(X) → H_∗(X) is a Leray endomorphism and Λ(q_∗p⁻¹_∗ ) = 1.

Theorem 2.9 (see [10, 12]). Let ϕX : X ⊸ X be an admissible map.

Assume that there exists a pair (p, q) ⊂ ϕX such that the Lefschetz number Λ(q_∗p⁻¹_∗ ) is well defined and let m be a prime number, then Λ((q_∗p⁻¹_∗ )^m) is well defined andΛ((q∗p⁻¹_∗ )^m) ≡ Λ(q∗p⁻¹_∗ ) mod m.

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) = ϕ(x) for all x ∈ X and ϕX0 : X0 ⊸ X0 ϕX0(x) = ϕ(x) 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 ep : (Z, p⁻¹(X₀)) → (X, X₀) ep(z) = p(z), eq : (Z, p⁻¹(X₀)) → (X, X0) eq(z) = q(z) for all z ∈ Z, p : p⁻¹(X0) → X0 p(z) = p(z) and q : p⁻¹(X₀) → X₀ q(z) = q(z) for all z ∈ p⁻¹(X₀). We observe that (ep, eq) ⊂ ϕ and (p, q) ⊂ ϕX0.

Theorem 2.10 (see [5, 6]). Let ϕ : (X, X₀) ⊸ (X, X0) be an admissi- ble map of pairs and (p, q) ⊂ ϕX. If any two of endomorphisms qe_∗ep⁻¹_∗ : H(X, X₀) → H(X, X₀), q_∗p⁻¹_∗ : H(X) → H(X), q_∗p⁻¹_∗ : H(X₀) → H(X₀) are Leray endomorphisms, then so is the third and

Λ(eq_∗pe⁻¹_∗ ) = Λ(q_∗p⁻¹_∗ ) − Λ(q_∗p⁻¹_∗ ).

Theorem 2.11 (see [5]). 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 (p, q) ⊂ ψ ◦ ϕ such that q_2∗p⁻¹_2∗ ◦ q_1∗p⁻¹_1∗ = q_∗p⁻¹_∗ .

Definition 2.12. An admissible map ϕ : X ⊸ X is called a Lefschetz map provided the generalized Lefschetz set Λ(ϕ) of ϕ is well defined and Λ(ϕ) 6= {0} implies that the set F ix(ϕ) = {x ∈ X : x ∈ ϕ(x)} is non- empty.

Let Y be a metric space and let Id_Y : Y → Y be a map given by formula IdY(y) = y for each y ∈ Y .

Definition 2.13 (see [13]). 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 2.14 (see [13, 14]). A metric space X is called an absolute multi- retract (notation: X ∈ AM R) provided there exists a normed space E and an mr-map r : E → X from E onto X.

Definition 2.15 (see [13, 14]). A metric space X is called an absolute neigh- borhood multi-retract (notation: X ∈ANMR) provided there exists an open subsetU of some normed space E and an mr-map r : U → X from U onto X.

Theorem 2.16 (see [13, 14]). A space X is an AN M R if and only if there exists a metric spaceZ and a Vietoris map p : Z → X which factors through an open subset U of some normed space E, i.e., there are two continuous mapsα and β such that the following diagram

Z p ^-

X 6β U ZZ

ZZ~ α

is commutative.

Theorem 2.17 (see [13]). Let X ∈ AN M R and let V ⊂ X be an open set.

Then V ∈ AN M R.

Theorem 2.18 (see [13]). Assume that X is AN M R. Let U be an open subset in X and ϕ : U ⊸ U be an admissible and compact map, then ϕ is a Lefschetz map.

Definition 2.19 (see [5]). An admissible map ϕ_X : X ⊸ X is called a com- pact absorbing contraction (written ϕX ∈ CAC(X)) provided there exists an open set U ⊂ X such that:

(3.6.1) ϕ_X(U ) ⊂ U and the ϕ_U : U ⊸ U, ϕU(x) = ϕ_X(x) for every x ∈ X is compact (ϕX(U ) ⊂ U ),

(3.6.2) for every x ∈ X there exists n = nx such that ϕⁿ_X(x) ⊂ U .

Theorem 2.20 (see [13]). Let X ∈ AN M R and ϕ ∈ CAC(X), then ϕ is a Lefschetz map.

3. Locally admissible multi-valued map

We denote with ϕ_XY : X ⊸ Y and ψA : A ⊸ A multi-valued maps. If a nonempty set A ⊂ X, a nonempty set B ⊂ Y and ϕXY(A) ⊂ B, then a multi-valued map ϕAB : A ⊸ B is given by ϕAB(x) = ϕXY(x) for each x ∈ X.

Definition 3.1. A multi-valued map ϕXY : X ⊸ Y is called locally admis- sible 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.

The following example shows that the class of locally admissible maps is indeed wider than the class of admissible maps. Every single-valued mapping in particular is locally admissible. Let R be a real number set and N be a natural number set.

Example 3.2. Let {an}^∞_n=1 ⊂ R be a sequence given by formula:

an=

(1, for n = 1

a_n−1+ 3 · 2ⁿ⁻¹, for n > 1.

We observe that lim_n→∞ ^a₂ⁿn = 3. Let for n = 1, 2, . . . f_n: R → R be a map given by fn(x) = ₂¹nsinx + ^a₂ⁿn and let k : R → N ∪ {0} be a map given by k(x) = int(|x|) for all x ∈ R, where by int(|x|) we denote the biggest integral number not bigger than |x|. We have f_n(R) = [^an₂⁻¹n ,^an₂⁺¹n ] for each n ∈ N.

The reader will easily verify that for every two different natural numbers n, m we have

(1) fn(R) ∩ fm(R) = ∅.

It is clear that lim_n→∞fn = f₀, where f₀ : R → R is a constant map given by f₀(x) = 3 for any x ∈ R. Let ϕR : R ⊸ R, ϕ^k_R : R ⊸ R be multi-valued maps given by the formula ϕ_R(x) = {f₀(x), f₁(x), . . . , f_n(x), . . . }, ϕ^k_R(x) = {f₀(x), f₁(x), . . . , f_k(x)(x)} for any x ∈ R. We define a map ψR : R ⊸ R given by:

(2) ψR(x) = ϕR(x)\ϕ^k_R(x) for all x ∈ R.

From the construction of ϕRand from (1) it results that the pairs (IdR, fn), n = 1, 2, . . . are its only selectors (p, q) ⊂ ϕ_R. We show that a map ψ_R is locally admissible but is not admissible. Let K ⊂ R be a compact set.

Then, there exists a natural number n₀ such that for each x ∈ K k(x) ≤ n₀ and let V = {x ∈ R : dist(x, K) < ¹₂}. Then, for n = n0+ 1, n₀ + 2, . . . (Id_R, f_n) ⊂ ψ_{V R} and ψ_{V R} : V ⊸ R is admissible. Let for some n^′ ∈ N (IdR, fn^′) ⊂ ψR and let x₀ ∈ R such that k(x₀) > n^′, then fn^′(x₀) /∈ ψR(x₀), which results in an evident contradiction. Consequently, a map ψR is not admissible.

We will now prove a couple of essential properties of locally admissible map- pings.

Theorem 3.3. Let ϕXY : X ⊸ Y and ψY Z : Y ⊸ Z be locally admissible maps. Then a map Φ_XZ = (ψ_{Y Z}◦ ϕXY) : X ⊸ Z is locally admissible.

P roof. Let K ⊂ X be a compact and nonempty set. There exists an open set V ⊂ X such that K ⊂ V and ϕV Y : V ⊸ Y is admissible. Let (p, q) ⊂ ϕV Y and let ΘV Y : V ⊸ Y be a map given by ΘV Y(x) = q(p⁻¹(x)) for each x ∈ V . Then, A = ΘV Y(K) is a compact set and A ⊂ Y . From the assumption there exists an open set U ⊂ Y such that A ⊂ U and ψU Z : U ⊸ Z is admissible. Let V1 = (ΘV Y)⁻¹(U ). We have the following diagram:

V₁ −−−−→ U^ΘV1U −−−−→ Z.^{ψU Z}

A map ψ_{U Z} ◦ ΘV1U is admissible and for all x ∈ V₁ (ψ_{U Z} ◦ ΘV1U)(x) ⊂ ΦV1Z(x).

The above theorem results in the following:

Theorem 3.4. Let A ⊂ X be a nonempty set and let ϕ_XY : X ⊸ Y be a locally admissible map. Then a map ϕAY : A ⊸ Y is locally admissible.

P roof. Let i : A → X be an inclusion. Then ϕAY = ϕXY ◦ i and the proof is complete.

Another property concerns the Cartesian product of locally admissible multi- valued maps.

Theorem 3.5. Let ϕ_XY : X ⊸ Y and ψZT : Z ⊸ T be locally admissible maps. Then, a map ΦX×Z Y ×T = ϕXY × ψZT : X × Z ⊸ Y × T is locally admissible.

P roof. Let K ⊂ X × Z be a compact and nonempty set. Then, there exist compact sets K1 ⊂ X and K2 ⊂ Z such that K ⊂ K1× K2. From the assumption there exist open sets V₁ ⊂ X, V2 ⊂ Z such that K1 ⊂ V1, K₂ ⊂ V2 and ϕ_V1Y and ψ_V2T are admissible. Hence a map Φ_V1×V2Y×T = ϕV1Y × ψV2T is admissible (see 2.4) and the proof is complete.

The notion of a map of CAC (compact absorbing contraction)(see [5, 7, 8, 15]) type is known in mathematical literature. It can be defined similarly

in the class of locally admissible maps. Let ϕX : X ⊸ X be a map.

ϕⁿ_X =







IdX, for n = 0,

ϕX, for n = 1,

ϕX ◦ ϕX ◦ . . . ◦ ϕX(n-iterates), for n > 1.

Definition 3.6 (see [5]). A locally admissible map ϕX : X ⊸ X is called a compact absorbing contraction (written ϕX ∈ CACL(X)) provided there exists an open set U ⊂ X such that:

(3.6.1) ϕX(U ) ⊂ U and the ϕU : U ⊸ U, ϕU(x) = ϕX(x) for every x ∈ X is compact (ϕX(U ) ⊂ U ),

(3.6.2) for every x ∈ X there exists n = nx such that ϕⁿ_X(x) ⊂ U .

We will need some properties of maps of CACLtype. We will now formulate a lemma that is very easy to prove. Let ϕ_X : X ⊸ X be a locally admissible map. We shall say that a nonempty set B ⊂ X has absorbing property (we write B ∈ APϕ(X)) if for each x ∈ X there exists n ≥ 1 such that ϕⁿ_X(x) ⊂ B.

Lemma 3.7. Let ϕ_X ∈ CACL(X) and let U ⊂ X be an open set from the Definition 3.6.

3.7.1 Let B be a nonempty set in X and ϕX(B) ⊂ B. Then, U ∩ B 6= ∅.

3.7.2 For any n ∈ N ϕⁿ_X ∈ CACL(X).

3.7.3 Let V ⊂ X be a nonempty and open set. Assume that ϕX(V ) ⊂ V . Then,ϕ_V ∈ CACL(V ).

P roof. 3.7.1 Let B ⊂ X be a nonempty set such that ϕ_X(B) ⊂ B and let U ⊂ X be an open set from the Definition 3.6. Then, for each x ∈ B there exists a natural number n such that ϕⁿ_X(x) ⊂ U and ϕⁿ_X(x) ⊂ B. Hence U ∩ B 6= ∅.

3.7.2 It is obvious.

3.7.3 Let V ⊂ X be a nonempty and open set such that ϕ_X(V ) ⊂ V and U ⊂ X be an open set from definition 3.6. From 3.7.1 we get U ∩ V 6= ∅. We have ϕV(U ∩ V ) = ϕX(U ∩ V ) ⊂ ϕX(U ) ∩ ϕX(V ) ⊂ ϕX(U ) ∩ ϕX(V ) ⊂ U ∩ V and we observe that U ∩V ∈ APϕ(V ). It is clear that ϕ_U∩V : U ∩V ⊸ U ∩V is compact and ϕV ∈ CACL(V ).

Let ϕX : X ⊸ X be a locally admissible map. We denote by ADLϕ(X) a family of open sets V ⊂ X such that the following conditions are satisfied:

(i) ϕX(V ) ⊂ V (not necessarily compact), (ii) ϕV : V ⊸ V is admissible,

(iv) V ∈ APϕ(X).

It is clear that the family of sets ADL_ϕ(X) can be empty. If f : X → X is a single-valued continuous map, then ADLf(X) 6= ∅ and if Λ(fU), Λ(fV) are defined, then Λ(fU) = Λ(fV), where U, V ∈ ADL_f(X) (see Theorem 4.3). It is not hard to observe that if ϕ_X : X ⊸ X is locally admissible (in particular admissible) and ADLϕ(X) 6= ∅, then for two different sets U, V ∈ ADLϕ(X), Lefschetz sets Λ(ϕU), Λ(ϕV) (if they are defined) can be different. The following examples will show the property. Let C be a complex number set and let S^r = {z ∈ C : |z| = r}, where r is a positive real number.

Example 3.8. We define a sequence of maps fk : S^vk → S^vk given by formula:

fk: S^vk → S^vk :









fk(z) = z, for k = 1, vk= 1, f_k(z) = ^zk

v_k^k−1, for k = 2, 3, . . . , n, n > 3, v_k = 1 +¹_k, fk(z) = z, for k = n + 1, n + 2, . . ., vk = 1 +¹_k. Let r_k : C\{0} → S^vk be a retraction, g_k : C\{0} → C\{0} a map given by formula g_k = f_k◦ rk, k = 1, 2, . . . and let t : C\{0} → N be a map given by

t(z) =

(1, for 0 < |z| < 1, int(|z|), for |z| ≥ 1,

where by int(|z|) we denote the biggest integral number not bigger than

|z|. We define multi-valued maps ψ_C\{0}, Φ_C\{0} : C\{0} ⊸ C\{0}, given by ψ_C\{0}(z) = {g₁(z), g₂(z), . . .}, Φ_C\{0}(z) = {g₁(z), g₂(z), . . . , g_t(z)(z)} for each z ∈ C\{0}. Finally, we denote the map ϕ_C\{0} : C\{0} ⊸ C\{0} by the formula:

ϕ_C\{0}(z) =

(ψ_C\{0}(z), for 0 < |z| < 3, ψ_C\{0}(z)\Φ_C\{0}(z), for |z| ≥ 3.

We observe that ϕ_C\{0} is locally admissible, but is not admissible. The justification of this fact is similar to that in Example 3.2. It is clear that

ϕ_C\{0} is compact and for each z ∈ K = ϕ_C\{0}(C\{0}) 1 ≤ |z| ≤ 2. From mathematical literature it is known that (see [10]):

Λ(f_k) =

(1 − k, for k = 1, 2, . . . , n, 0, for k = n + 1, n + 2, . . . .

From the construction of maps g_kit results that Λ(g_k) = Λ(f_k) k = 1, 2, . . . . Let V ⊂ C\{0} be an open set such that K ⊂ V and for any z ∈ V 0 < |z| < 3 and let Vp⊂ C\{0} p = 1, 2, . . . be a sequence of open sets given by:

V_p =







V, for p = 1

V ∪

p−1S

i=1

Li for p > 1,

where Li ⊂ C\{0} is a set such that for each z ∈ Li 2 + i ≤ |z| < 3 + i i = 1, 2, . . . . We observe that for each p ∈ N V_p∈ ADLϕ(C\{0}) and

Λ(ϕVp) =







{0, −1, . . . , 1 − n}, for p = 1,

{0, −1 − p, . . . , 1 − n}, for p = 2, . . . , n − 2,

{0} for p > n − 2.

Finally, we get T∞

p=1Λ(ϕ_Vp) = {0}. We observe that (U ∈ ADL_ϕ(C\{0})) ⇒ (U is compact).

Hence, for each U ∈ ADL_ϕ(C\{0}) there exists p ∈ N such that U ⊂ V_p, so

∅ 6= \

U∈ADLϕ(C\{0})

Λ(ϕU) = {0}.

Moreover, let us notice that F ix(ϕ_C\{0}) 6= ∅ so according to Theorem 3.11, F ix(ϕ_C\{0}) ⊂T

U∈ADLϕ(C\{0})U .

Example 3.9. We define the sequence of maps fk : S^vk → S^vk from the previous Example 3.8 in the following way:

fk: S^vk → S^vk :

(fk(z) = z, for k = 1, vk= 1, f_k(z) = ^zk

v^k−1_k , for k = 2, 3, . . . , n, . . . , n > 3, v_k= 1 + ¹_k.

While we leave the next phases of the construction of the map ϕ_C\{0} un- changed, it is easy to notice that T

U∈ADLϕ(C\{0})Λ(ϕ_U) = ∅.

A situation such as that in Example 3.9 cannot occur in the class of admissi- ble maps. We observe that if ϕX : X ⊸ X is admissible, then ADLϕ(X) 6= ∅ since X ∈ ADLϕ(X).

Theorem 3.10. Let ϕX : X ⊸ X be an admissible map. Assume that for each V ∈ ADL_ϕ(X) and (p, q) ⊂ ϕ_V q_∗p⁻¹_∗ : H_∗(V ) → H_∗(V ) is a Leray endomorphism. Then, Λ(ϕX) =T

V∈ADLϕ(X)Λ(ϕV).

P roof. Let V ∈ ADLϕ(X), (p, q) ⊂ ϕX and let Λ(q_∗p⁻¹_∗ ) = c₀. We observe that a map eq_∗pe⁻¹_∗ : H_∗(X, V ) ⊸ H∗(X, V ) ((ep, eq) ⊂ ϕ, ϕ : (X, V ) ⊸ (X, V )) is weakly nilpotent, so from 2.7 and 2.10 Λ(q_∗p⁻¹_∗ ) = Λ(q_∗p⁻¹_∗ ) = c₀, where (p, q) ⊂ ϕV and p, q denote a respective contraction of p, q. Hence c₀ ∈ Λ(ϕ_V) and Λ(ϕ_X) ⊂ T

V∈ADLϕ(X)Λ(ϕ_V). It is clear that X ∈ ADL_ϕ(X) and the proof is complete.

Theorem 3.11. Let ϕ_X : X ⊸ X be a locally admissible map and let ADLϕ(X) 6= ∅. Then, F ix(ϕX) ⊂T

V∈ADLϕ(X)V .

P roof. If F ix(ϕX) = ∅, then the proof is obvious. Assume that F ix(ϕX) 6= ∅ and let x ∈ ϕX(x). We observe that x ∈ ϕⁿ_X(x) for each n ≥ 1. Let V ∈ ADL_ϕ(X), then there exists k ≥ 1 such that ϕ^k_X(x) ⊂ V . Hence x ∈ V and the proof is complete.

4. Fixed point result

Let us formulate and prove a couple of facts concerning fixed points in the class of locally admissible maps. We observe that if ϕX : X ⊸ X is admissible and for each V ∈ ADLϕ(X) Λ(ϕX) is well defined, then T

V∈ADLϕ(X)Λ(ϕV) 6= ∅ (see 3.10). If ϕX : X ⊸ X is, however, lo- cally admissible, but is not admissible and ADLϕ(X) 6= ∅ and for any V ∈ ADLϕ(X) Λ(ϕV) is well defined (Example 3.9), it may occur that T

V∈ADLϕ(X)Λ(ϕ_V) = ∅. Hence for a locally admissible mapping such that ADLϕ(X) 6= ∅, we shall formulate the Lefschetz set definition in the follow- ing way:

Definition 4.1. Let ϕX : X ⊸ X be a locally admissible map and let ADLϕ(X) 6= ∅. Assume that for each V ∈ ADLϕ(X) and for each

(p, q) ⊂ ϕV q_∗p⁻¹_∗ is a Leray endomorphism. We define Λ^L(ϕ_X) = [

V∈ADLϕ(X)

Λ(ϕ_V).

In the class of admissible mappings, the above definition of a Lefschetz set is more general.

Example 4.2. Let f, g : C\{0} → C\{0} be single-valued continuous and compact maps. Assume that Λ(g) 6= 0, F ix(f ) = ∅ and there exists d ∈ R d > 0 such that for each z ∈ f (C\{0}) ∪ g(C\{0}) |z| < d. We define a multi-valued map ϕ_C\{0}: C\{0} ⊸ C\{0} given by formula:

ϕ_C\{0}(z) =

(f (z), for |z| ≥ d, {f (z), g(z)}, for |z| < d.

We observe that the map ϕ_C\{0} is admissible, Λ(ϕ_C\{0}) = {0} and the only selective pair is the pair (Id_C\{0}, f ) ⊂ ϕ_C\{0}, but F ix(f ) = ∅. Let V ⊂ C\{0} be an open set such that f (C\{0}) ∪ g(C\{0}) ⊂ V and for each z ∈ V |z| < d. Then, V ∈ ADLϕ(C\{0}) and Λ(ϕV) 6= {0}. Hence Λ^L(ϕX) 6= {0} and ∅ 6= F ix(ϕV) ⊂ F ix(ϕ_C\{0}).

Let us remind that if the multi-valued mapping ϕX : X ⊸ X is u.s.c.

and has compact and acyclic images, then the only selective pair is the pair (p^′, q^′) ⊂ ϕX, where p^′, q^′: Γ → X are given by formulae

(3) p^′(x, y) = x, q^′(x, y) = y for every (x, y) ∈ Γ and Γ is the graph ϕ_X i.e. Γ = {(x, y) ∈ X × X; y ∈ ϕ_X(x)}.

Theorem 4.3. Let ϕX : X ⊸ X be a multi-valued u.s.c. map such that for each x ∈ X ϕX(x) is acyclic and compact. If one of Λ^L(ϕX), Λ(ϕX) is well defined, then so is the other and

Λ^L(ϕX) = Λ(ϕX).

P roof. Assume that Λ(ϕ_X) is well defined, i.e., (q^′)_∗(p^′)⁻¹_∗ : H_∗(X) → H_∗(X) is a Leray endomorphism (see (3)). Let V ∈ ADLϕ(X), then (qg^′)_∗(pg^′)⁻¹_∗ : H∗(X, V ) → H∗(X, V ) (see 2.10) is weakly nilpotent, where (ep^′, eq^′) ⊂ ϕ and ϕ : (X, V ) ⊸ (X, V ) given by ϕ(x) = ϕX(x) for each x ∈ X. Hence and from 2.7 and 2.10 we have Λ(ϕX) = Λ(ϕV) and the proof is complete.

From Theorem 4.3 it results that if f : X → X is a continuous single-valued mapping, then Λ^L(f_X) = Λ(f_X).

Theorem 4.4. Let X ∈ AN M R and let ϕ_X ∈ CACL(X). Then, the fol- lowing conditions are satisfied:

4.4.1 ADL_ϕ(X) 6= ∅,

4.4.2 for each V ∈ ADL_ϕ(X) and for each (p, q) ⊂ ϕ_V q_∗p⁻¹_∗ is a Leray endomorphism,

4.4.3 if Λ^L(ϕX) 6= {0}, then F ix(ϕX) 6= ∅.

P roof. Let U ⊂ X be a set from Definition 3.6, then there exists an open set U^′ ⊂ X such that ϕX(U ) ⊂ U^′ ⊂ U and ϕU^′ : U^′ ⊸ U^′ is admissible.

It is obvious that U^′ ∈ ADLϕ(X). Let V ⊂ X be an open set such that ϕV(V ) ⊂ V ,

(4) ϕV : V ⊸ V is admissible

and V ∈ AP_ϕ(X). Then, from 3.7.3 and (4) we get ϕ_V ∈ CAC(V ). From 2.20 and 2.17 for each (p, q) ⊂ ϕV q_∗p⁻¹_∗ is a Leray endomorphism. It is clear that if Λ^L(ϕX) 6= {0}, then F ix(ϕX) 6= ∅ and the proof is complete.

Definition 4.5. Let ϕX : X ⊸ X be a multi-valued map and let A ⊂ X be a nonempty set. We shall say that a set O_ϕⁿ(A) is an n-orbit of A if O_ϕⁿ(A) =Sn

i=0ϕⁱ_X(A), where n ∈ N ∪ {0}.

Let ϕX : X ⊸ X. We observe that if Oϕⁿ(A) is an n-orbit of A, then ϕX(Oⁿ_ϕ(A)) ⊂ O_ϕⁿ⁺¹(A).

Theorem 4.6. Let X ∈ AN M R and let ϕ_X ∈ CACL(X). Assume that there exists a compact and acyclic set A ⊂ X such that the following condi- tions are satisfied:

4.6.1 ϕⁿ_X(X) ⊂ A for some n ≥ 1,

4.6.2 the n-orbit of A is relatively compact.

Then,F ix(ϕX) 6= ∅.

P roof. Let U ⊂ X be an open set from Definition 3.6 and let K = Sm

i=0ϕⁱ_X(A) ∪ ϕX(U ), where m is a prime number and m > n. The set K is compact, so there exists an open set V ⊃ K such that ϕV X : V ⊸ X

is admissible. We choose a selected pair (p, q) ⊂ ϕV X and define a map ψ : V ⊸ X given by ψ(x) = q(p⁻¹(x)) for each x ∈ V . We observe that ψ(K) ⊂ K. Let U₁^′ = ψ⁻¹(V ), U₂^′ = ψ⁻¹(U₁^′),. . . , U_m^′ = ψ⁻¹(U_m−1^′ ) and let U^′ =Tm

i=1U_i^′. It is clear that A ⊂ K ⊂ U^′ and for each i = 1, 2, . . . , m ψⁱ(U^′) ⊂ V . For j = 1, . . . , m we have the following commutative diagrams:

(5)

ψ^j−1(U^′) ←−−−− p^{pj −1}(ψ^j−1(U^′)) −−−−→ ψ^{qj j}(U^′) x

^ij1

x

^ij2

x

^ij3 ψ^j−1(A) ←−−−− p^{pj −1}(ψ^j−1(A)) −−−−→ ψ^{qj j}(A),

where ψ⁰(U^′) ≡ U^′, ψ⁰(A) ≡ A, ψ^m(U^′) ≡ U^′, ψ^m(A) ≡ A; (pj, qj) denote respective contractions of (p, q); (p_j, q_j) denote respective contractions of (p, q); i^j₁, i^j₂, i^j₃ are inclusions. Hence

(6) Λ(ψ_U^m′) = {1},

where ψ_U^m′ : U^′ ⊸ U^′ given by ψ_U^m′(x) = ψ^m(x) for each x ∈ U^′. Let U^′′ = U ∩ U^′, then ψ_U^m′′, ψ_U′′ : U^′′ ⊸ U^′′ are admissible and compact. We define a map Φ : (U^′, U^′′) ⊸ (U^′, U^′′) given by formula Φ(x) = ψ^m(x) for each x ∈ U^′. Let (r, s) ⊂ Φ_U′ be a selected pair such that s_∗r_∗⁻¹ = q_m∗p⁻¹_m∗ ◦ . . . ◦ q1∗p⁻¹_1∗ (see 2.11 and (5)). From 3.7.2 and 3.7.3 the map Φ ∈ CAC(U^′) and from (6), 2.10 and 2.7 we get

(7) Λ(s_∗r⁻¹_∗ ) = 1,

where (r, s) ⊂ Φ_U′′ denote respective contraction of (r, s) and (8) s∗r⁻¹_∗ = (q_∗p⁻¹_∗ )^m,

where (p, q) ⊂ ψU^′′ denote respective contraction of (p, q). The set U^′′ ∈ AN M R (see 2.17). From 2.9,(7) and (8) Λ(q_∗p⁻¹_∗ ) 6= 0 so ∅ 6= F ix(ψU^′′) ⊂ F ix(ϕX) and the proof is complete.

Let us remind that we call the map ϕ_X : X ⊸ Y locally compact if for every x ∈ X there exists an open neighbourhood V ⊂ X of x point such that ϕX(V ) is a compact set. We observe that if ϕ : X ⊸ Y is locally compact, then for each compact and nonempty set K ⊂ X there exists an open set

V ⊂ X such that K ⊂ V and ϕX(V ) is compact. If X = Y , then we observe that for every compact set K ⊂ X and for every n ∈ N ϕⁿ_X(K) is compact.

Let us also remind that a multi-valued mapping ψ : X ⊸ Y is the selector of a multi-valued mapping ϕ : X ⊸ Y if for every x ∈ X ψ(x) ⊂ ϕ(x).

Lemma 4.7. Let ϕX : X ⊸ X be a locally admissible and locally compact map. Let for some n ≥ 1 M = ϕⁿ_X(X) be a compact set and we assume that the set L = ϕX(W ) is compact for an open set W ⊂ X such that Oϕⁿ⁻¹(M ) ⊂ W . Then, for each open set V ⊂ X such that Oϕⁿ⁻¹(M ∪ L) ⊂ V and ϕV X : V ⊸ X is admissible and for each admissible u.s.c selector ψ of ϕV X there exists an open set U ⊂ V such that the following conditions are satisfied:

4.7.1 ψU : U ⊸ U is a compact map and ψ(U) ⊂ U, 4.7.2 U ∈ APϕ(X).

P roof (see Lemma 42.8 [5]). Let for some n ≥ 1 M = ϕⁿ_X(X) be a compact set and let S = O_ϕⁿ⁻¹(M ) = Sn−1

i=0 ϕⁱ_X(M ), K = S. We observe that K is compact and ϕX(S) = Sn

i=1ϕⁱ_X(M ) ⊂ S ∪ M = S. Since ϕX

is locally compact, there exists an open set W ⊂ X such that K ⊂ W and L = ϕX(W ) is a compact set. Let V ⊂ X be an open set such that (Oⁿ⁻¹ϕ (M ∪ L) = K ∪Sn−1

i=0 ϕⁱ_X(L)) ⊂ V and ϕV X : V ⊸ X is admissible.

Let ψ : V ⊸ X be a map given by ψ(x) = q(p⁻¹(x)) for each x ∈ V , where (p, q) ⊂ ϕV X. We construct family open sets {V0, . . . , Vn} in V such that (9) L ∩ ψ(Vi) ⊂ V_i−1, K ∪ ψⁿ⁻ⁱ(L) ⊂ Vi, i = 0, 1, . . . , n.

It is clear that ψ(S) ⊂ S and ψ(K) = ψ(S) ⊂ ψ(S) ⊂ S = K. Let V₀= W , then K ∪ ψⁿ(L) ⊂ K ∪ ϕⁿ_X(L) ⊂ K ∪ M = K ⊂ V₀. Assume that an open set V_i (0 < i < n) satisfies the conditions of (9). K ∪ ψⁿ⁻ⁱ(L) and L\Vi are disjoint compact sets and there exists an open set V^′⊂ V such that

(10) K ∪ ψⁿ⁻ⁱ(L) ⊂ V^′ ⊂ V^′ ⊂ Vi∪ (V \L).

We define V_i+1 = ψ⁻¹(V^′). We have ψ(K ∪ ψ^n−(i+1)(L)) = ψ(K) ∪ ψ(ψ^n−(i+1)(L) ⊂ K ∪ ψⁿ⁻ⁱ(L) ⊂ V^′. Hence K ∪ ψ^n−(i+1)(L) ⊂ V_i+1. We observe that ψ(V_i+1) ⊂ V^′ ⊂ V^′ ⊂ Vi∪ (V \L), so L ∩ ψ(V_i+1) ⊂ L ∩ Vi ⊂ Vi. Let U = V₀∩V1∩. . .∩Vn, then M ⊂ K ⊂ U and ψ(U ) ⊂ ψ(V₀)∩. . .∩ψ(Vn) ⊂ L ∩ ψ(V₁) ∩ . . . ∩ ψ(Vn). Hence ψ(U ) ⊂ (L ∩ ψ(V₁)) ∩ . . . ∩ (L ∩ ψ(Vn)) ∩ L ⊂ V0∩ V1 ∩ . . . ∩ Vn = U . It is obvious that ψ(U ) ⊂ L, so ψ(U ) is compact.

A map ψU : U ⊸ U is compact, u.s.c and admissible. Since M ⊂ U, so U ∈ AP_ϕ(X) and the proof is complete.

Let us notice that Lemma 4.7 allows for modifying Theorem 4.6 in the fol- lowing way:

Theorem 4.8.LetX ∈ AN M R and let ϕX : X ⊸ X be a locally admissible and locally compact map. Assume that there exists a compact and acyclic set A ⊂ X such that ϕⁿ_X(X) ⊂ A for some n ≥ 1. Then, F ix(ϕX) 6= ∅.

P roof. ϕX : X ⊸ X is locally compact, so the set K = Oϕ^m(A) is compact, where m is a prime number and m > n. There exists an open set W ⊂ X such that K ⊂ W and L = ϕ_X(W ) is compact. Let Z = O_ϕ^m(A ∪ L). It is clear that Z is compact, so there exists an open set W^′ ⊂ X such that Z ⊂ W^′and L^′ = ϕX(W^′) is compact. Let V ⊂ X be an open set such that (Z ∪ L^′) ⊂ V and ϕV X : V ⊸ X is admissible. Choose a selected pair (p, q) ⊂ ϕV X. We define a map ψ : V ⊸ X given by formula ψ(x) = q(p⁻¹(x)) for each x ∈ V . Let V^′ = W^′∩ V and let η : V^′ ⊸ X be a map given by η(x) = ψ(x) for each x ∈ V^′. It is obvious that η is the selector of ϕ_V′X. From 4.7 we get an open set U ⊂ V^′ ⊂ W^′ ⊂ V such that ψU = ηU : U ⊸ U is compact, ψ(U ) ⊂ U and U ∈ APϕ(X). According to the reasoning from the proof of theorem 4.6, we get the thesis.

The next theorem results directly from Theorem 4.8.

Theorem 4.9.LetE be a Banach space and let ϕE be locally admissible and locally compact. Assume that there exists n ≥ 1 such that ϕⁿ_E is compact.

Then,F ix(ϕ_E) 6= ∅.

P roof. From the assumption, there exists n ≥ 1 such that the set ϕⁿ_E(E) is compact. Let A = conv(ϕⁿ_E(E)), then for each k ≥ n ϕ^k_E(E) ⊂ A and the proof is complete.

The next theorem is a straightforward conclusion from Theorem 4.9.

Theorem 4.10. Let E be a Banach space and let ϕE : E ⊸ E be locally admissible and compact. Then, F ix(ϕE) 6= ∅.

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Received 26 May 2010