Introduction
Classical courses on ordinary differential equations (ODEs) typically begin with either the Peano existence theorem or the Picard–Lindelöf existence and uniqueness theorem, both of which address Cauchy (initial value) problems defined by the equation x˙=f(t, x) with the initial condition x(0)=x₀ In this context, the function f is continuous over the domain [0, τ] × Rⁿ and satisfies a Lipschitz condition, ensuring that the difference |f(t, x) - f(t, y)| is bounded by a constant multiplied by the distance |x - y| for all t in [0, τ] and x, y in Rⁿ.
The Lipschitz condition (1.2) ensures that for boundary value problems with linear conditions close to x(0) = x₀, the principle of "uniqueness implies existence" holds true, as noted in [53] Additionally, uniqueness generally leads to continuous dependence of solutions on initial values, as demonstrated in [54, Theorem 4.1 in Chapter 4.2] This concept is further illustrated by the Poincaré translation operator Tτ: Rⁿ → Rⁿ, which operates at time τ > 0 along the trajectories defined by x˙ = f(t, x).
, (1.3) is a homeomorphism (cf [54, Theorem 4.4 in Chapter 4.2]).
Hence, besides the existence, uniqueness is also a very important problem W Orlicz
In 1932, it was demonstrated that the set of continuous functions \( f: U \to \mathbb{R}^n \), where \( U \) is an open subset of \([0, \tau] \times \mathbb{R}^n\), for which the Cauchy problem (1.1) with \((0, x_0) \in U\) lacks a unique solution, is meager, indicating that such instances are rare Consequently, most continuous Cauchy problems (1.1) are uniquely solvable The first known example of nonuniqueness was identified by M.A Lavrentev in 1925 This principle also applies to Carathéodory ordinary differential equations (ODEs), where classical solutions can be substituted with Carathéodory solutions—absolutely continuous functions that satisfy (1.1) almost everywhere—by utilizing the Lebesgue integral rather than the Riemann integral.
In 1923, H Kneser demonstrated that the solution sets for continuous Cauchy problems are continua, meaning they are compact and connected at every time This finding was further enhanced by M Hukuhara, who established that the solution set is a continuum within C([0, τ], R^n) Later, in 1942, N Aronszajn provided additional specifications regarding these continua.
R δ -sets (see Definition 2.3 below), and as a subsequence, multivalued operatorsT τ in (1.3) become admissible in the sense of L Górniewicz (see Definition 2.5 below).
If the function f(t, x) is periodic with respect to time, meaning f(t, x) ≡ f(t + τ, x), then the operator Tτ has a fixed point, denoted as x̂ ∈ R^n This fixed point x̂ corresponds to a τ-periodic solution of the ordinary differential equation ẋ = f(t, x) Consequently, studying the fixed point theory for multivalued mappings is essential for finding periodic solutions to nonuniquely solvable ordinary differential equations.
Poincaré’s operator \( T_\tau \) is consistent for differential inclusions \( \dot{x} \in F(t, x) \), where \( F \) is an upper Carathéodory mapping with nonempty, convex, and compact values This consistency suggests a direct study of such differential inclusions Additionally, initial value problems for these inclusions are generally nonuniquely solvable, in contrast to ordinary differential equations (ODEs), which is why Poincaré’s operators exhibit multivalued characteristics.
In this context, an interesting phenomenon occurs with respect to the Sharkovskii cycle coexistence theorem [95] This theorem is based on a new ordering of the positive integers, namely
2 n + 1 2 n ã ã ã 2 2 21, saying that if a continuous functiong:R→Rhas a point of periodmwithmk(in the above Sharkovskii ordering), then it has also a point of periodk.
By a period, we mean the least period, i.e a pointa∈Ris a periodic point of periodm ifg m (a)=aandg j (a)=a, for 0< j < m.
Now, consider the scalar ODE ˙ x=f (t, x), f (t, x)≡f (t+τ, x), (1.4) wheref:[0, τ] ×R→Ris a continuous function.
The Poincaré translation operator \( T_\tau \) corresponds uniquely to \( m \)-periodic points of \( T_\tau \) and \( m_\tau \)-periodic solutions of Eq (1.4) under uniqueness conditions However, unlike classical Sharkovskii’s theorem, this relationship does not extend to subharmonics of Eq (1.4) Consequently, the results yield an empty conclusion, as every bounded solution of (1.4) is either \( \tau \)-periodic or asymptotically \( \tau \)-periodic, assuming uniqueness.
The handicap arises from the assumed uniqueness condition; however, in the absence of uniqueness, the multivalued operator \( T_\tau \) in (1.3) is deemed admissible, as established in Theorem 4.17 In the context of \( \mathbb{R} \), this implies that \( T_\tau \) is upper semicontinuous, meaning its sets of values consist of either single points or compact intervals Our research, detailed in papers [16, 29, 36], presents a version of the Sharkovskii cycle coexistence theorem applicable to (1.4).
THEOREM1.1 If Eq (1.4) has anmτ-periodic solution, then it also admits akτ-periodic solution, for everyk m, with at most two exceptions, wherek mmeans thatk is less
Fig 1 Braid σ thanmin the above Sharkovskii ordering of positive integers In particular, ifm=2 k , for allk∈N, then infinitely many (subharmonic) periodic solutions of (1.4) coexist.
Theorem 1.1 is applicable solely in scenarios lacking uniqueness; otherwise, it becomes void Additionally, the right-hand side of the specified multivalued ordinary differential equation (ODE) may be represented as a multivalued upper Carathéodory mapping, characterized by nonempty, convex, and compact values, as detailed in Definition 2.10 below.
A 3τ-periodic solution of equation (1.4) suggests that, with some exceptions for specific values of k (namely k=2, 4, or 6), there exists a corresponding kτ-periodic solution However, demonstrating the existence of such a solution poses significant challenges Notably, a single 3τ-periodic solution indicates the presence of at least two additional 3τ-periodic solutions for equation (1.4).
The Sharkovskii phenomenon is fundamentally one-dimensional; however, T Matsuoka's findings indicate that three harmonic τ-periodic solutions in the planar system (1.4) generally lead to the existence of infinitely many subharmonic kτ-periodic solutions, where k is a natural number In this context, "genericity" is defined through the lens of Artin braid group theory, excluding specific simple braids that correspond to the three identified harmonics.
The following theorem was presented in [8], on the basis of T Matsuoka’s results in papers [87–89].
Theorem 1.2 establishes that under a uniqueness condition for the planar system (1.4), there exist three τ-periodic harmonic solutions whose graphs do not correspond to the braid σ_m in the factor group B_3/Z, for any integer m in N Here, σ is illustrated in Fig 1, and B_3/Z represents the factor group of the Artin braid group B_3, with Z denoting its center.
[22, Chapter III.9]) Then there exist infinitely many (subharmonic)kτ-periodic solutions of (1.4),k∈N.
REMARK1.3 In the absence of uniqueness, there occur serious obstructions, but Theo- rem 1.2 still seems to hold in many situations; for more details see [8].
The application of the Nielsen theory, as discussed in Section 3.2, may yield the desired three harmonic solutions of (1.4) Specifically, it is more feasible to identify two harmonics through the corresponding Nielsen number, while the third harmonic can be determined using the related fixed point index outlined in Section 3.3.
Forn >2, statements like Theorem 1.1 or Theorem 1.2 appear only rarely Nevertheless, iff =(f 1 , f 2 , , f n )has a special triangular structure, i.e. f i (x)=f i (x 1 , , x n )=f i (x 1 , , x i ), i=1, , n, (1.5) then Theorem 1.1 can be extended to hold inR n (see [35]).
THEOREM 1.3 Under assumption (1.5), the conclusion of Theorem 1.1 remains valid inR n
Theorem 1.3, like Theorem 1.1, is applicable only in the absence of uniqueness In the absence of the specific triangular structure outlined in (1.5), it is highly unlikely to find a comparable result to Theorem 1.1, particularly for n2.
There is also another motivation for the investigation of multivalued ODEs, i.e differ- ential inclusions, because of the strict connection with
(i) optimal control problems for ODEs,
(ii) Filippov solutions of discontinuous ODEs,
In the context of control problems represented by the equation ˙x = f(t, x, u), where u is a control parameter within a specified set U, we define a multivalued map F(t, x) = {f(t, x, u)} for u ∈ U The solutions to the control problem can be characterized by the condition ˙x ∈ F(t, x) However, when the function f is discontinuous in x, traditional Carathéodory theory becomes inapplicable for solving these equations To address this issue, one can utilize the Filippov regularization method to handle the discontinuities in f, thereby enabling a more robust analysis of the control problem.
, (1.8) whereμ(r)denotes the Lebesgue measure of the setr⊂R n and
Preliminaries
Elements of ANR-spaces
All topological spaces discussed in this article are metric, specifically focusing on topological vector spaces that qualify as Fréchet spaces A Fréchet space is defined as a complete, metrizable, locally convex space with a topology generated by a countable family of seminorms If a Fréchet space is normable, it is classified as a Banach space.
DEFINITION2.1 A (metrizable) spaceXis an absolute neighbourhood retract (ANR) if, for each (metrizable)Y and every closedA⊂Y, each continuous mappingf:A→Xis extendable over some neighbourhood ofA.
(i) IfX is an ANR, then any open subset ofX is an ANR and any neighbourhood retract ofXis an ANR.
(ii) Xis an ANR if and only if it is a neighbourhood retract of every (metrizable) space in which it is embedded as a closed subset.
A space X is classified as an ANR if it serves as a neighborhood retract of a normed linear space, meaning it can be considered a retract of an open subset within such a space.
(iv) IfXis a retract of an open subset of a convex set in a Fréchet space, then it is an
(v) IfX 1 ,X 2 are closed ANRs such thatX 1 ∩X 2 is an ANR, thenX 1 ∪X 2 is an ANR. (vi) Any finite union of closed convex sets in a Fréchet space is an ANR.
(vii) If eachx∈Xadmits a neighbourhood that is an ANR, thenXis an ANR.
DEFINITION2.2 A (metrizable) spaceXis an absolute retract (AR) if, for each (metriz- able)Y and every closedA⊂Y, each continuous mappingf:A→Xis extendable overY.
(i) Xis an AR if and only if it is a contractible (i.e homotopically equivalent to a one point space) ANR.
(ii) X is an AR if and only if it is a retract of every (metrizable) space in which it is embedded as a closed subset.
(iii) IfXis an AR andAis a retract ofX, thenAis an AR.
(iv) IfXis homeomorphic toY andXis an AR, then so isY.
(v) Xis an AR if and only if it is a retract of some normed space.
(vi) IfXis a retract of a convex subset of a Fréchet space, then it is an AR.
(vii) IfX 1,X 2 are closed ARs such thatX 1∩X 2 is an AR, thenX 1∪X 2 is an AR.
Every ANRX is locally contractible, meaning that for any point x in X and its neighborhood U, there exists a neighborhood V around x that is contractible within U Additionally, Proposition 2.2(i) establishes that every ARX is contractible, indicating that the identity map id X: X → X is homotopic to a constant map.
DEFINITION 2.3 X is called an R δ -set if, there exists a decreasing sequence {X n }of compact, contractible setsX n such thatX {X n |n=1,2, }.
In the context of metric spaces, a set X qualifies as an R δ -set if it can be represented as the intersection of a decreasing sequence of compact ARs, even though contractible spaces are not necessarily ARs Additionally, every R δ -set is acyclic concerning any continuous homology theory, such as ˇCech homology, meaning it is homologically equivalent to a single-point space Consequently, R δ -sets are inherently nonempty, compact, and connected This establishes a hierarchy where contractible spaces are a subset of acyclic spaces.
∪ convex⊂AR⊂ANR, compact+convex⊂compact AR⊂compact+contractible⊂R δ ⊂compact+acyclic, and all the above inclusions are proper.
For more details, see [47] (cf also [22,67,69]).
Elements of multivalued maps
In what follows, by a multivalued mapϕ:XY, i.e.ϕ:X→2 Y \{0}, we mean the one with at least nonempty, closed values.
DEFINITION2.4 A mapϕ:XYis said to be upper semicontinuous (u.s.c.) if, for every openU⊂Y, the set{x∈X|ϕ(x)⊂U}is open inX It is said to be lower semicontinuous
(l.s.c.) if, for every openU⊂Y, the set{x∈X|ϕ(x)∩U= ∅}is open inX If it is both u.s.c and l.s.c., then it is called continuous.
In the context of single-valued functions, a function \( f: X \to Y \) is continuous if it is either upper semi-continuous (u.s.c.) or lower semi-continuous (l.s.c.) Additionally, a compact-valued map \( \phi: X \to Y \) is continuous if and only if it satisfies the condition of Hausdorff continuity This means that the map must be continuous with respect to the metric \( d \) in \( X \) and the Hausdorff metric \( d_H \) in the set of non-empty, bounded subsets of \( Y \) The Hausdorff metric \( d_H(A, B) \) is defined as the infimum of all \( \epsilon > 0 \) such that one set is contained within the \( \epsilon \)-neighborhood of the other.
O ε (B)andB⊂O ε (A)}andO ε (B):= {x∈X| ∃y∈B: d(x, y) < ε} Every u.s.c map ϕ:XY has a closed graph ϕ , but not vice versa Nevertheless, if the graph ϕ of a compact mapϕ:XY is closed, thenϕ is u.s.c.
The important role will be played by the following class of admissible maps in the sense of L Górniewicz.
DEFINITION2.5 Assume that we have a diagramX⇐ p −→ q Y ( is a metric space), wherep: ⇒Xis a continuous Vietoris map, namely
(ii) pis proper, i.e.p − 1 (K)is compact, for every compactK⊂X,
In the context of Cech homology with compact carriers and rational coefficients, the preimage p − 1(x) is acyclic for every x in the set X Given a continuous map q: → Y, an admissible map ϕ: X → Y is defined as being induced by the function ϕ(x) = q(p − 1(x)) for all x in X Consequently, we can identify the admissible map ϕ with the pair (p, q), referred to as an admissible (selected) pair.
In the context of admissible maps, we define two mappings, \( \phi_0 = q_0 \circ p_{0} \) and \( \phi_1 = q_1 \circ p_{1} \) We say that \( \phi_0 \) is admissibly homotopic to \( \phi_1 \) (denoted as \( \phi_0 \sim \phi_1 \) or \( (p_0, q_0) \sim (p_1, q_1) \)) if there exists an admissible map from \( X \times [0,1] \) to \( Y \) that maintains the commutativity of a specific diagram.
X× [0,1] p q fork i (x)=(x, i),i=0,1, andf i : i → is a homeomorphism ontop − 1 (X×i),i=0,1,i.e.k 0 p 0 =pf 0 ,q 0 =qf 0 ,k 1 p 1 =pf 1 andq 1 =qf 1
Admissible maps are always upper semicontinuous (u.s.c.) and have nonempty, compact, and connected values Additionally, the class of admissible maps is closed under finite compositions, meaning that the composition of two or more admissible maps remains admissible A map qualifies as admissible if it can be expressed as a finite composition of acyclic maps that possess compact values, specifically u.s.c maps with acyclic and compact values.
The class of admissible maps so contains u.s.c maps with convex and compact val- ues, u.s.c maps with contractible and compact values, R δ -maps (i.e u.s.c maps with
R δ -values), acyclic maps with compact values and their compositions.
The class of compact admissible maps, denoted as K(X, Y) or K(X) for self-maps, includes maps ϕ: XY where ϕ(X) is compact If the admissible homotopy defined in Definition 2.6 remains compact, then we consider the maps ϕ₀ and ϕ₁ to be compactly admissibly homotopic within K(X, Y).
Another important class of admissible maps are condensing admissible maps denoted by
C(X, Y ) For this, we need to recall the notion of a measure of noncompactness (MNC).
LetEbe a Fréchet space endowed with a countable family of seminorms s ,s∈S
(Sis the index set), generating the locally convex topology Denoting byB=B(E)the set of nonempty, bounded subsets ofE, we can give
DEFINITION 2.7 The family of functions α= {α s } s ∈ S :B→ [0,∞) S , whereα s (B): inf{δ >0|B∈B admits a finite covering by the sets of diam s δ},s∈S, for B∈B, is called the Kuratowski measure of noncompactness and the family of functions γ {γ s } s ∈ S :B→ [0,∞) S , where γ s (B):=inf{δ >0|B∈Bhas a finite ε s -net},s∈S, for
B∈B, is called the Hausdorff measure of noncompactness.
These MNC are related as follows: γ (B)α(B)2γ (B), i.e γ s (B)α s (B)2γ s (B), for eachs∈S.
Moreover, they satisfy the following properties:
PROPOSITION2.3 Assume thatB, B 1 , B 2∈B Then we have (component-wise):
(μ 6 ) (Kuratowski condition) decreasing sequence of closed setsB n ∈Bwith n lim→∞ μ(B n )=0 ⇒
(μ 10 ) (seminorm) μ(λB)= |λ|μ(B), for every λ∈R, and μ(B 1 ∪B 2 )μ(B 1 )+ μ(B 2 ), whereμdenotes eitherαorγ.
DEFINITION2.8 A bounded mappingϕ:E⊃UE, i.e.ϕ(B)∈B, forBB⊂U, is said to beμ-condensing (shortly, condensing) ifμ(ϕ(B)) < μ(B), wheneverBB⊂U andμ(B) >0, or equivalently, ifμ(ϕ(B))μ(B)impliesμ(B)=0, wheneverBB⊂
U, whereμ= {μ s } s ∈ S :B→ [0,∞) S is a family of functions satisfying at least conditions
(μ 1 )–(μ 5 ) Analogously, a bounded mappingϕ:E⊃UEis said to be ak-set contrac- tion w.r.t.μ= {μ s } s ∈ S :B→ [0,∞) S satisfying at least conditions(μ 1 )–(μ 5 )(shortly, a k-contraction or a set-contraction) ifμ(ϕ(B))kμ(B), for some k∈ [0,1), whenever
Obviously, any set-contraction is condensing and bothα-condensing andγ-condensing maps areμ-condensing Furthermore, compact maps or contractions with compact values
(in vector spaces, also their sum) are well known to be (α, γ)-set-contractions, and so (α, γ)-condensing.
In addition to semicontinuous maps, measurable and semi-Carathéodory maps play a significant role in this context Let Y represent a separable metric space, and consider (X, U, ν) as a measurable space, where U is a σ-algebra of subsets and ν is a countably additive measure on U A common example of this setup occurs when X is a bounded domain in R^n, utilizing the Lebesgue measure.
DEFINITION 2.9 A mapϕ:Y is called strongly measurable if there exists a se- quence of step multivalued maps ϕ n :Y such thatd H (ϕ n (ω), ϕ(ω))→0, for a.a. ω∈ , as n→ ∞ In the single-valued case, one can simply replace multivalued step maps by single-valued step maps andd H (ϕ n (ω), ϕ(ω))byϕ n (ω)−ϕ(ω).
A mapϕ:Yis called measurable if{ω∈ |ϕ(ω)⊂V} ∈U, for each openV ⊂Y.
A map ϕ:Y is called weakly measurable if {ω∈ |ϕ(ω)⊂V} ∈U, for each closedV ⊂Y.
Strong measurability implies measurability, and measurability implies weak measurability When a function has compact values, measurability and weak measurability are equivalent In separable Banach spaces, strong measurability and measurability are the same for both multivalued and single-valued maps In any Banach space, a strongly measurable map with compact values can be associated with a single-valued strongly measurable selection Additionally, in separable complete spaces, every measurable function has a single-valued measurable selection, as established by the Kuratowski–Ryll-Nardzewski theorem.
Now, let = [0, a]be equipped with the Lebesgue measure andX,Y be Banach.
DEFINITION2.10 A mapϕ:[0, a] ×XY with nonempty, compact and convex values is called u-Carathéodory (resp l-Carathéodory, resp Carathéodory) if it satisfies
(i) tϕ(t, x)is strongly measurable, for everyx∈X,
(ii) xϕ(t, x)is u.s.c (resp l.s.c., resp continuous), for almost allt∈ [0, a],
ForX=R m andY =R n , one can state
(i) Carathéodory maps are product-measurable (i.e measurable as the whole(t, x) ϕ(t, x)), and
(ii) they possess a single-valued Carathéodory selection.
It need not be so for u-Carathéodory or l-Carathéodory maps Nevertheless, for u-Carathéodory maps, we have at least (againX=R m andY =R n ).
PROPOSITION 2.5 u-Carathéodory maps (in the sense of Definition 2.10) are weakly superpositionally measurable, i.e the composition ϕ(t, q(t )) admits, for every q ∈
C([0, a],R m ), a single-valued measurable selection If they are still product-measurable, then they are also superpositionally measurable, i.e the compositionϕ(t, q(t ))is measur- able, for everyq∈C([0, a],R m ).
REMARK 2.1 If X, Y are separable Banach spaces and ϕ:XY is a Carathéodory mapping, then ϕ is also superpositionally measurable, i.e.ϕ(t, q(t )) is measurable, for everyq∈C([0, a], X)(see [78, Theorem 1.3.4 on p 56]) Under the same assumptions, Proposition 2.4 can be appropriately generalized (see [73, Proposition 7.9 on p 229 and Proposition 7.23 on pp 234–235]).
If ϕ: XY is a u-Carathéodory function and X, Y are Banach spaces, then ϕ is weakly superpositionally measurable This means that for every continuous function q∈C([0, a], X), the expression ϕ(t, q(t)) has a measurable selection For further details, refer to sources such as [58, Proposition 3.5] or [78, Theorem 1.3.5].
Some further preliminaries
Assume we have again a diagram (see Definition 2.5)
X⇐ p −→ q Y wherep: ⇒Xis a Vietoris map andq: −→Y is continuous.
Takingϕ(x)=q(p − 1 (x)), for everyx∈X, and denoting as
, C(p, q): z∈ |p(z)=q(z) the sets of fixed points and coincidence points of the admissible pair(p, q), it is clear that p(C(p, q))=Fix(p, q), and so
The following Aronszajn–Browder–Gupta-type result (see [21, Theorem 3.15]; cf [22, Theorem 1.4]) is very important in order to say something about the topological structure of Fix(ϕ).
In a metric space \(X\) with a Fréchet space \(E\), consider a sequence of open convex symmetric neighborhoods \(\{U_k\}\) around the origin in \(E\) Let \(\phi: X \to E\) be a lower semi-continuous (u.s.c.) proper map that takes compact values Furthermore, assume there exists a convex symmetric subset \(C\) within \(E\) and a sequence of compact, convex-valued u.s.c proper maps \(\phi_k: X \to E\).
(ii) for everyk1, there is a convex, symmetric setV k ⊂U k ∩Csuch thatV k is closed inEand 0∈ϕ(x)impliesϕ k (x)∩V k = ∅,
(iii) for everyk1 and every u∈V k , the inclusionu∈ϕ k (x)has an acyclic set of solutions.
Then the setS= {x∈X|ϕ(x)∩ {0} = ∅}is compact and acyclic.
Now, let us assume thatEis a Fréchet space,C is a convex subset ofE,Uis an open subset ofC,μ:B→ [0,∞) S is a measure of noncompactness satisfying at least conditions
(μ 1 )–(μ 5 )in Proposition 2.3 (see Definitions 2.7 and 2.8).
Ifϕ∈C(U, C), then Fix(ϕ)can be proved relatively compact We can say more about Fix(ϕ).
DEFINITION 2.11 Let (p, q)∈C(U, C) A nonempty, compact, convex set S⊂C is called a fundamental set if:
For a homotopyχ∈C(U× [0,1], C),S⊂Cis called fundamental if it is fundamental to χ (., λ), for eachλ∈ [0,1].
(i) IfSis a fundamental set for(p, q), then Fix(p, q)⊂S.
(ii) Intersection of fundamental sets, for(p, q), is also fundamental, for(p, q).
(iii) The family of all fundamental sets for(p, q)is nonempty.
(iv) If S is a fundamental set for χ ∈C(U × [0,1], C) and P ⊂S, then the set conv(χ ((U∩S)× [0,1])∪P )is also fundamental.
For more details, see [22,67], and the references therein.
Applied fixed point principles
Lefschetz fixed point theorems
The Lefschetz theory serves as a foundational element for our exploration, particularly through the generalized Lefschetz number, which aids in defining essential classes within Nielsen theory and highlights the normalization property of the fixed point index This article focuses solely on presenting the essential facts related to these concepts.
Consider a multivalued mapϕ:XXand assume that
(i) X is a (metric) ANR-space, e.g., a retract of an open subset of a convex set in a Fréchet space,
(ii) ϕ is a compact (i.e.ϕ(X)is compact) composition of anR δ -mapp − 1 :Xand a continuous (single-valued) map q: →X, namely ϕ=q ◦p − 1 , where is a metric space.
Then an integer(ϕ)=(p, q), called the generalized Lefschetz number forϕ∈K(X), is well-defined (see, e.g., [12; 22, Chapter I.6; 67]) and(ϕ)=0 implies that
Moreover, is a homotopy invariant, namely ifϕ is compactly homotopic (in the same class of maps) withϕ:XX, then(ϕ)=(ϕ).
To understand the generalized Lefschetz number, a basic knowledge of algebraic topology, especially homology theory, is essential This article provides a concise overview of its definition, omitting proofs for brevity For those seeking in-depth information, we recommend consulting references [51,68] for the single-valued scenario and [12,22,67] for the multivalued context.
At first, we recall some algebraic preliminaries In what follows, all vector spaces are taken overQ Letf:E→Ebe an endomorphism of a finite-dimensional vector spaceE.
Ifv 1 , , v n is a basis forE, then we can write f (v i ) n j = 1 a ij v j , for alli=1, , n.
The matrix \([a_{ij}]\) is referred to as the matrix of the linear transformation \(f\) with respect to the basis \(v_1, \ldots, v_n\) For an \(n \times n\) matrix \(A = [a_{ij}]\), the trace of \(A\) is defined as \(\text{tr}(A) = \sum_{i=1}^{n} a_{ii}\) In the context of a finite-dimensional vector space \(E\), if \(f: E \to E\) is an endomorphism, the trace of \(f\), denoted \(\text{tr}(f)\), corresponds to the trace of its matrix representation with respect to a chosen basis for \(E\) In cases where \(E\) is a trivial vector space, the trace is defined to be zero: \(\text{tr}(f) = 0\) Importantly, the definition of the trace of an endomorphism is invariant and does not depend on the specific choice of basis for \(E\).
Hence, letE= {E q }be a graded vector space of a finite type.
Iff = {f q }is an endomorphism of degree zero of such a graded vector space, then the (ordinary) Lefschetz numberλ(f )off is defined by λ(f ) q
Letf:E→Ebe an endomorphism of an arbitrary vector spaceE Denote byf n :E→
Ethenth iterate off and observe that the kernels kerf ⊂kerf 2 ⊂ ã ã ã ⊂kerf n ⊂ ã ã ã form an increasing sequence of subspaces ofE Let us now put
Clearly,f mapsN (f )into itself and, therefore, induces the endomorphismf:E→Eon the factor spaceE=E/N (f ).
Letf:E→Ebe an endomorphism of a vector spaceE Assume that dimE < ∞ In this case, we define the generalized trace Tr(f )off by putting Tr(f )=tr(f ).
LEMMA3.1 Letf:E→Ebe an endomorphism If dimE