Recent advances in the Lefschetz fixed point theory for multivalued mappings

In 1923 S. Lefschetz proved the famous xed point theorem known as the Lefschetz xed point theorem (comp. [5], [9], [20], [21]. The multivalued case was considered for the rst time in 1946 by S. Eilenberg and D. Montgomery ([10]). They proved the Lefschetz xed point theorem for acyclic mappings of compact ANR-spaces (absolute neighbourhood retracts (see [4] or [13]) using Vietoris mapping theorem (see [4], [13], [16]) as a main tool. In 1970 Eilenberg, Montgomery's result was generalized for acyclic mappings of complete ANR-s (see [17]). Next, a class of admissible multivalued mappings was introduced ([13] or [16]). Note that the class of admissible mappings is quite large and contains as a special case not only acyclic mappings but also nite compositions of acyclic mappings. For this class of multivalued mappings several versions of the Lefschetz xed point theorem was proved (comp. [11], [13] [15], [18], [19], [27]). In 1982 G. Skordev and W. Siegberg ([26]) introduced the class of multivalued mappings so-called now (1 − n)-acyclic mappings. Note that the class (1 − n)-acyclic mappings contains as a special case n-valued mappings considered in [6], [12], [28]. We recommend [8] for the most important results connected with (1 − n)-acyclic mappings. Finally, the Lefschetz xed point theorem was considered for spheric mappings (comp. [3], [2], [7], [23]) and for random multivalued mappings (comp. [1], [2], [13]). Let us remark that the main classes of spaces for which the Lefschetz xed point theorem was formulated are the class of ANR-spaces ([4]) and MANR-spaces (multi absolute neighbourhood retracts (see [27]). The aim of this paper is to recall the most important results concerning the Lefschetz xed point theorem for multivalued mappings and to prove new versions of this theorem, mainly for AANR-spaces (approximative absolute neighborhood retracts (see [4] or [13]) and for MANR-s. We believe that this article will be useful for analysts applying topological xed point theory for multivalued mappings in nonlinear analysis, especially in di erential inclusions. Email address: gorn@mat.umk.pl (Lech Górniewicz) Received March 17, 2021, 2021, Accepted May 14, 2021, Online May 22, 2021. L. Górniewicz, Results in Nonlinear Anal. 4 (2021), 116 126 117


Auxiliary notions
In this paper all topological spaces are assumed to be metric. We shall consider the ech homology functor with compact carriers and coecients in the eld of rational numers Q (for details see [13] or [16]). For a space X by H(X) = {H n (X); n ≥ 0} we shall denote the ech homology with compact carriers and coecients in Q of the space X. Then H n (X) is a linear vector space over Q.
For any continuous map f : X → Y by f * = {f * n ; n ≥ 0} : H(X) = {H n (X); n ≥ 0} → H(Y ) = {H n (X); n ≥ 0} we denote the induced linear map, i.e. f * n : H n (X) → H n (Y ) is a linear map, n ≥ 0. Denition 1.1. A space X is called of nite type provided: (a) the dimension dimH n (X) < +∞ for every n ≥ 0, (b) H n (X) = 0 for almost all n. Denition 1.2. A space X is called acyclic provided H 0 (X) = Q and H n (X) = 0, for every n ≥ 1.
Evidently, any contractible space is acyclic; an acyclic space is of nite type. Denition 1.3. A continuous map p : Y → X is called Vietoris map provvvided: (a) p is onto, (b) for every compact set K ⊂ X the counter image p −1 (K) is compact, (c) p −1 (x) is acyclic for every x ∈ X. Theorem 1.4 (Vietoris Mapping Theorem, [4], [13], [16]). If p : Y → X is a Vietoris map, then p * n : H n (Y ) → H n (X) is a linear isomorphism for every n ≥ 0.
In what follows if p : Y → X is a Vietoris map then shall use the following symbol: p : Y =⇒ X. Let us consider the following diagram: Then, for every n ≥ 0, we have the following diagram of linear mappings: in which p * n is an isomorphism. So we can dene the following linear map: which is called by us the induced by the pair (p, q) linear map, for every n ≥ 0. Now we are going to dene the Lefschetz number. First let assume that E is a linear vector space over Q and l : E → E a linear endomorphism.
(1) Assume further that dimE < +∞, then by tr(l) we shall denote the trace of l.

We put
Ker(l (n) ) and E = E| N (l) .
Since N (l) is an invariant subspace of E we can consider the induced endomorphism l : E → E. We shall say that l is a Leray endomorphism provided dimE < +∞. Consequently we are able to dene the generalized trace Tr(l) of l by the formula: Consider a continuous mapping f : X → X. We shall say that the Lefschetz number λ(f ) of f is well dened provided f * n is a Leray endomorphism for every n ≥ 0 and Tr(f * n ) = 0 for almost all n. Then we let: Evidently, if X is a space of nite type, then is well dened. Observe that for given pair (p, q) the above consideration is exactly the same and we have provided it is well dened.
The following proposition is very usefull in what follows: Proposition 1.5. Assume that the following diagram of linear mappings @ @ @ @ @ @ @ is commutative. Then l is a Leray endomorphism if and only if l 1 is a Leray endomorphism and in that case Tr(l) = Tr(l 1 ).
Finally assume that l : E → E is a weakly nilpotent map, i.e. for all x ∈ E there exists n x > 0 suth that l nx (x) = 0. We have (see [13]): Proposition 1.6. If l : E → E is a weakly nilpotent endomorphism, then Tr(l) = 0.

Multivalued mappings
Let X and Y be two metric spaces. A multivalued map dened on X with the values in Y we shall denote by the symbol ϕ : X Y . In what follows we shall denote multivalued mappings by greek letters. We shall assume also that, for every x ∈ X the value ϕ(x) of x under ϕ is a non empty subset of Y .
Let ϕ : X Y be a multivalued mapping. We associate with ϕ the graph Γ ϕ by putting: Then we have the following diagram: in which p ϕ (x, y) = x and q ϕ (x, y) = y for every (x, y) ∈ Γ ϕ . A multivalued map ϕ : X Y is called upper semicontinuous (u.s.c.) provied for every open set U ⊂ Y the set is open in X; ϕ is called lower semicontinuous (l.s.c.) provided for every open set U ⊂ Y the following set: x ∈ X.
The following two remarks are selfevident.
for every x ∈ X; the pair p ϕ , q ϕ will be called a selective pair of ϕ (written p ϕ , q ϕ ⊂ ϕ.
For given two maps: X ϕ Y ψ Z we dene the composition ψ • ϕ of ϕ and ψ by the formula: for every x ∈ X. Observe that the composition of two acyclic maps is not an acyclic map in general. Now we are going to dene the notion of admissible multivalued maps (see [13], [16]).

Denition 2.5 ([13], [16]). A multivalued map
in which p is a Vietoris map and q is continuous such that Evidently, (comp. Remark 2.4) any acyclic map is admissible. Moreover, the following property holds true. Property 2.6. Compositions of two admissible mappings is admissible too.
Note that the class of acyclic maps was introduced by S. Eilenberg and D. Montgomery in 1946 (see [10]); the class of admissible mappings was introduced in 1976 (see [16]).

Appropriate spaces
In this section we recall types of metric spaces essential from the point of view of Lefschetz xed point theorem. Let A be a closed subset of the space X. We shall say that A is a retract of X provided there exists a continuous map r : X → A (called retraction map) such that r(x) = x, for every x ∈ X; the map Denition 3.1. We shall say that a space X is an absolute retract (X ∈ AR) provided, for every space Y and for every embedding h : Note that any metric space X can be embedded as a closed subset into a normed space. Moreover, let us add that any absolute retract is a contractible space (see [4], [13], [20]). Denition 3.2. A map r : X → Y is called multi-retraction provided there exists an admissible map ϕ : Y X such that r(ϕ(y)) = y, for every y ∈ Y .

Denition 3.3 ([27]).
A space X is called an absolute multiretract (X ∈ AMR) provided there exists a normed space E and a multi-retraction r : E → X; X is called an absolute neighbourhood multi-retrac (X ∈ ANMR) provided there exists an open subset of some normed space E and a multi-retraction r : U → X.
The following diagram illustrates the relationship between the introduced spaces: Note that all of the above inclusions are proper. Moreover, let us add that every X ∈ AMR is an acyclic space.
Finally, we shall recall notions of approximative retracts.

Denition 3.4 ([4]
, [13], [16]). A space X is called an absolute approximative retract provided for any embedding h : X → Y of X into a space Y and, for every ε > 0, there exists an ε-retraction r ε : Y → h(X) (written X ∈ AAR); if there exists an open subset U of X such that, for every ε > 0 tehre is an ε-retraction r ε : U → h(X), then X is called an absolute approximative neighbourhood retract (X ∈ AANR).
It s easy to see that AR ⊂ AAR ∩ ∪ ANR ⊂ AANR and all inclusions are proper (comp. [13]).

The Lefschetz xed point theorem for admissible mappings
First we remind denitions of some types of admissible mappings. We let K(X) = ϕ : X X; ϕ is admissible and compact .
(a) ϕ is called compact absorbing contraction (ϕ ∈ CAC(X)), if there exists an open subset U of X such that ϕ(X) is a compact subset of U and, for every x ∈ X there exists a natural number n X > 0 such that ϕ n X (x) ⊂ U ; (b) ϕ os called eventually compact if ϕ n ∈ K(X), for some n > 0; (c) ϕ is called asymptotically compact provided, for every x ∈ X, the orbit O(x) = ∞ n=1 ϕ n (x) is relatively compact and the core C(ϕ) = n≥1 ϕ n (X) is nonempty relatively compact (ϕ ∈ ASC(X)); (d) ϕ is called compact attraction if there exists a compact subset K ⊂ X such that, for every open neighbourhood V of K in X such that , for every x ∈ X, there exists n X ≥ 0 for which ϕ n X (x) ⊂ V (ϕ ∈ CA(X)).
Note that the notion of CAC-mappings was introduced in [11]. We have: Let us add that all the above inclusions are proper (see [13], [19], [25]). Let ϕ : X X be an addmissible map and (p, q) ⊂ ϕ be a selective pair of ϕ. Then we have the induced linear maps: q * n • p −1 * n : H n (X) → H n (X), for every n. So we shall say that (p, q) is a Lefschetz pair provided q * n • p −1 * n is a Leray endomorphism for every n ≥ 0 and Tr q * n • p −1 * n = 0 for almost all n ≥ 0. Consequently in this case we have well dened the Lefschetz number λ((p, q)) od the pair (p, q) by the formula (1.4).
We shall say that ϕ is a Lefschetz map provided every selected pair (p, q) ⊂ ϕ of ϕ is a Lefschetz pair. In this case we dene the Lefschetz set Λ(ϕ) of ϕ by putting: Note that, if X is of nite type, then for any admissible map ϕ the Lefschetz set Λ(ϕ) of ϕ is well dened. Observe that, if ϕ is acyclic map and X is of nite type, then the Lefschetz set Λ(ϕ) of ϕ is a singleton, i.e. Λ(ϕ) = {λ(p ϕ , q ϕ )}. Finally, it is easy to see that, if X is an acyclic space, then for every admissible map ϕ : X X is a Lefschetz map and Λ(ϕ) = {1}, i.e. for every (p, q) ⊂ ϕ, we have λ(p, q) = 1. We recommend for the details [13], [16], [19].
The following theorem is provided in [27].
Note that, for X ∈ ANR, Theorem 4.2 was proved in [11] and for X to be compact ANR-space and ϕ : X X to be acyclic map it was proved in [10]. Now we shall present essential generalization of Theorem 4.2. Theorem 4.3. If X ∈ MANR and ϕ ∈ CAC(X), then ϕ is Lefschetz map and Λ(ϕ) = {0} implies that ϕ has a xed point.
Proof. Since ϕ ∈ CAC there exists an open subset U ⊂ X such that ϕ(X) is a compact subset of U .
To show that ϕ is a Lefschetz map it is enough to prove that Λ(p, q) = Λ(p , q ). In view of Theorem 4.2 we obtain that the Lefschetz number Λ((p , q )) is well dened. In order to show that Λ((p, q)) = Λ((p , q )) we consider pairs (X, U ) and (p, q), where p, q : (p −1 (X), p −1 (U )) → (X, U ) are dened by p(u) = p(u), q(u) = q(u), for every u ∈ p −1 (X). Now we are able to dene a linear map l * n : H n (X, U ) → H n (X, U ) dened by the formula l * n = q * n • p −1 * n for every n ≥ 0.
Finally, if we assume that Λ((p, q)) = {0}, then Λ((p , q )) = 0 and from Theorem 4.2 we obtain that ϕ has a xed point. But every xed point of ϕ is also a xed point of ϕ and our theorem is proved.
Note that theorem was formulated in [18] but witout the proof. For X ∈ ANR Theorem 4.2 was proved in [11] (see also [13]). Corollary 4.4. Assume that X satises one of the following three conditions: (a) X is an acyclic MANR-space, If ϕ ∈ CAC(X), then ϕ has a xed point.
In the end of this section we shall deal with approximative retracts. First we remind the following fact: Proposition 4.5 ([16]). If X is a compact AANR-space, then it is of nite type.
The key in the proof the Lefschetz xed point theorem is the following: Proposition 4.6 ([16]). Assume that y is a compact space of nite type. Then there exists ε > 0 such that for arbitrary space Y and two conditions maps f, g : Y → Y if d(f (y), g(y)) < ε, for every y ∈ Y , then f * n = g * n , for every n ≥ 0. Theorem 4.7 ([13], [16]). Let X be a compact AANR-space and ϕ : X X an admissible map. Then Λ(ϕ) = 0 implies that ϕ has a xed point. Corollary 4.8. If X is an acyclic AANR or X ∈ AAR, then for every admissible map ϕ : X X there exists a xed point.
Open Problem 4.9. Is it possible to prove Theorem 4.7 for arbitrary X ∈ AANR and ϕ ∈ K(X) or ϕ ∈ CAC(X)?

Random admissible operators
A systematic study of random operation was iniciated in the 1950 by Czech mathematicians. We are interested in the xed point theory for multivalued random operators. For more details and references sii [1] or [13] and references there in.
By a measurable space we shall mean the pair (Ω, Σ), where the set Ω is equipped in a σ-algebra Σ of subsets. We shall use B(X) to denote the Borel σ-algebra on X. The symbol Σ ⊗ B(X) denotes the smallest σ-algebra on Ω × X which contains all the sets A × B, where A ∈ Σ and B ∈ B(X). ξ(ω) ∈ ϕ(ω, ξ(ω)), for every ω ∈ Ω.
The following proposition is crucial in what follows.
Proposition 5.4 ([1], [13]). Let X be a separable space and ϕ : Ω × X X a random operator such that the map ϕ(ω, · ) : X X has a xed point for every ω ∈ Ω, then ϕ has a random xed point. Now we are able to prove the main result of this section, i.e. the Lefschetz-type xed point theorem for random maps.
Proof. In view of Theorem 4.3 ϕ(ω, · ) is a Lefschetz map for every ω ∈ Ω. Consequently our assumption that Λ(ϕ(ω, · )) = {0} implies that ϕ(ω, · ) has a xed point. Hence our theorem follows from Proposition 5.4 and the proof is complete. Corollary 5.6. Assume that X is a separable and one of the following conditions is satised: (a) X is acyclic multi absolute neighbourhood retract, (b) X ∈ AR, (c) X ∈ ANR and it is acyclic, then an random admissible map ϕ : Ω × X X such that ϕ(ω, · ) ∈ CAC(X), for every ω ∈ Omega has a random xed point.
The formulation of the respective Lefschetz-type theorem for X to be compact AANR-space is left to the reader.

Spheric maps
In 1947 B. O'Neill [24] constructed an example of a xed point free map ϕ : K 2 K 2 which is continuous with values homotopically equivalent to S 1 (even homeomorc to S n ), where K 2 denotes the unit ball in euclidean space R 2 and S 1 denotes the unit sphere. The above example was inspiration to introduce the notion of spheric mappings. For spheric mappings see [3], [7], [13], [23]. In this section all considered topological spaces are subsets of the euclidean space R n , n ≥ 1.
Let X be a compact subsets of R n , n ≥ 2. Then R n \ X consists of two components, namely: (1) an unbounded component DX of R n \ X, i.e. x ∈ DX if and only if for every r > 0 there exists a continuous function f : [0, 1] → R n \ X such that f (0) = x and f (1) > r, since X is compact DX is the unique unbounded component of R n \ X.
(2) a union of all bounded component BX of (R n \ DX) \ X. Moreover, we let Observe that if A is a compact subset of X, then A ⊂ X; if X is an acyclic set then X = X.
Denition 6.1. Let X be a compact subset of R n and let ϕ : X X be an u.s.c. map with connected values. We say that ϕ is a spheric map provided: } is an open subset of X.
Of course any acyclic map is spheric. Evidently, if X is an acycylic set or R n \ X is connected, then ϕ is well dened.
Let ϕ : X X be a spheric map. Then ϕ : X X is acyclic map and consequently the linear map ϕ * n : H n (X) → H n (X), given by the formula ϕ * n (q ϕ ) * n (p ϕ ) −1 * n , n ≥ 0.
is well dened for every q ≥ 0. We dene the Lefschetz number λ(ϕ) of ϕ by putting: λ(ϕ) = λ ϕ , provided λ ϕ is well dened. The following proposition is crucial. Proposition 6.2. If ϕ : X X is a spheric map and ϕ : X X has a xed point, then ϕ has a xed point.
Proof. There are possibilities: (i) the boundary ∂X of X in R n is equal to X, (ii) ∂X = X.
In the case (i) we have ϕ = ϕ so our claim is evident. For the proof of (ii) assume to the contrary that ϕ has no xed points. Then the set Fix ϕ = x ∈ X; x ∈ ϕ(x) is closed subset of X and in view of Denition 6.1 we deduce that Fix ϕ is also an open subset. To obtain a contradiction it is sucent to observe that X \ {x ∈ X; x ∈ Bϕ(x)} = ∅. Indeed, we have that ∂X = ∅ (since X is compact) if x ∈ ∂X then x ∈ Bϕ(x) and hence X \ {x ∈ X; x ∈ Bϕ(x)} = ∅ and the proof is completed. From Proposition 6.2 and Theorems 4.2, 4.7 we obtain Theorem 6.3. If X is a compact subset of R n and X ∈ MANR or X ∈ AANR and ϕ : X X is a spheric map, then λ(ϕ) = 0 implies that ϕ has a xed point. Corollary 6.4. In particular, if X ∈ MAR or X ∈ AR or X ∈ AAR, then any spheric map ϕ : X X has a xed point. Remark 6.5. Evidently the respective results for random spheric maps are also possible (comp. Section 5).