Articulo 13.dvi CUBO A Mathematical Journal Vol.12, No¯ 02, (199–215). June 2010 Fixed point theory for compact absorbing contractions in extension type spaces Donal O’Regan Department Of Mathematics, National University of Ireland, Galway, Ireland email: donal.oregan@nuigalway.ie ABSTRACT Several new fixed point results for self maps in extension type spaces are presented in this paper. In particular we discuss compact absorbing contractions. RESUMEN Son presentados en este art́ıculo varios resultados nuevos de punto fijo para autoaplica- ciones en espacios de tipo extensión. En particular discutimos contracciones compactas absorbentes. Key words and phrases: Extension spaces, fixed point theory, compact absorbing contractions. AMS (MOS) Subj. Class.: 47H10 1 Introduction In Sections 2, 3 and 4 we present new results on fixed point theory in extension type spaces. Section 2 discusses compact self-maps on NES, ANES and SANES spaces whereas Section 3 discusses compact absorbing contractions. In Section 4 we provide an alternative approach using projective 200 Donal O’Regan CUBO 12, 2 (2010) limits. These results improve those in the literature; see [1-3, 5, 8-11, 14-15] and the references therein. Our results were motivated in part from ideas in [1, 2, 9, 12, 15]. For the remainder of this section we present some definitions and known results which will be needed throughout this paper. Suppose X and Y are topological spaces. Given a class X of maps, X(X,Y ) denotes the set of maps F : X → 2Y (nonempty subsets of Y ) belonging to X , and Xc the set of finite compositions of maps in X . We let F(X) = {Z : FixF 6= ∅ for all F ∈X(Z,Z)} where FixF denotes the set of fixed points of F . The class A of maps is defined by the following properties: (i). A contains the class C of single valued continuous functions; (ii). each F ∈Ac is upper semicontinuous and closed valued; and (iii). Bn ∈F(Ac) for all n ∈{1, 2, ....}; here B n = {x ∈ Rn : ‖x‖≤ 1}. Remark 1.1. The class A is essentially due to Ben-El-Mechaiekh and Deguire [6]. A includes the class of maps U of Park (U is the class of maps defined by (i), (iii) and (iv). each F ∈Uc is upper semicontinuous and compact valued). Thus if each F ∈ Ac is compact valued the class A and U coincide and this is what occurs in Section 2 since our maps will be compact. The following result can be found in [6, Proposition 2.2] (see also [9 pp. 286] for a special case). Theorem 1.1. The Hilbert cube I∞ (subset of l2 consisting of points (x1,x2, ...) with |xi| ≤ 1 2i for all i) and the Tychonoff cube T (cartesian product of copies of the unit interval) are in F(Ac). We next consider the class Uκc (X,Y ) (respectively A κ c (X,Y )) of maps F : X → 2 Y such that for each F and each nonempty compact subset K of X there exists a map G ∈Uc(K,Y ) (respectively G ∈Ac(K,Y )) such that G(x) ⊆ F(x) for all x ∈ K. Theorem 1.2. I∞ and T are in F(Aκc ) (respectively F(U κ c )). Proof: Let F ∈ Aκc (I ∞,I∞) and we must show FixF 6= ∅. Now by definition there exists G ∈ Ac(I ∞,I∞) with G(x) ⊆ F(x) for all x ∈ I∞, so Theorem 1.1 guarantees that there exists x ∈ I∞ with x ∈ Gx. In particular x ∈ F x so FixF 6= ∅. Thus I∞ ∈F(Aκc ). 2 Notice [14] that Uκc is closed under compositions. The class U κ c include (see [3]) the Kakutani maps, the acyclic maps, the O’Neill maps, the approximable maps and the maps admissible with respect to Gorniewicz. For a subset K of a topological space X, we denote by CovX (K) the set of all coverings of K by open sets of X (usually we write Cov (K) = CovX (K)). Given a map F : X → 2 X and α ∈ Cov (X), a point x ∈ X is said to be an α–fixed point of F if there exists a member U ∈ α such that x ∈ U and F(x) ∩U 6= ∅. Given two maps single valued f, g : X → Y and α ∈ Cov (Y ), f and g are said to be α–close if for any x ∈ X there exists Ux ∈ α containing both f(x) and g(x). CUBO 12, 2 (2010) Fixed point theorems 201 We say f and g are α-homotopic if there is a homotopy hh : X → Y (0 ≤ t ≤ 1) joining f and g such that for each x ∈ X the values ht(x) belong to a common Ux ∈ α for all t ∈ [0, 1]. The following results can be found in [4, Lemma 1.2 and 4.7]. Theorem 1.3. Let X be a regular topological space and F : X → 2X an upper semicontinuous map with closed values. Suppose there exists a cofinal family of coverings θ ⊆ CovX (F(X)) such that F has an α–fixed point for every α ∈ θ. Then F has a fixed point. Remark 1.2. From Theorem 1.3 in proving the existence of fixed points in uniform spaces for upper semicontinuous compact maps with closed values it suffices [5 pp. 298] to prove the existence of approximate fixed points (since open covers of a compact set A admit refinements of the form {U[x] : x ∈ A} where U is a member of the uniformity [13 pp. 199] so such refinements form a cofinal family of open covers). Note also uniform spaces are regular (in fact completely regular) [7 pp. 431] (see also [7 pp. 434]). Note in Theorem 1.3 if F is compact valued then the assumption that X is regular can be removed. For convenience in this paper we will apply Theorem 1.3 only when the space is uniform. Let X, Y and Γ be Hausdorff topological spaces. A continuous single valued map p : Γ → X is called a Vietoris map (written p : Γ ⇒ X) if the following two conditions are satisfied: (i). for each x ∈ X, the set p−1(x) is acyclic (ii). p is a proper map i.e. for every compact A ⊆ X we have that p−1(A) is compact. Let D(X,Y ) be the set of all pairs X p ⇐ Γ q → Y where p is a Vietoris map and q is continuous. We will denote every such diagram by (p,q). Given two diagrams (p,q) and (p′,q′), where X p ′ ⇐ Γ′ q ′ → Y , we write (p,q) ∼ (p′,q′) if there are maps f : Γ → Γ′ and g : Γ′ → Γ such that q′ ◦f = q, p′◦f = p, q◦g = q′ and p◦g = p′. The equivalence class of a diagram (p,q) ∈ D(X,Y ) with respect to ∼ is denoted by φ = {X p ⇐ Γ q → Y} : X → Y or φ = [(p,q)] and is called a morphism from X to Y . We let M(X,Y ) be the set of all such morphisms. For any φ ∈ M(X,Y ) a set φ(x) = q p−1 (x) where φ = [(p,q)] is called an image of x under a morphism φ. Consider vector spaces over a field K. Let E be a vector space and f : E → E an endomorphism. Now let N(f) = {x ∈ E : f(n)(x) = 0 for some n} where f(n) is the nth iterate of f, and let Ẽ = E\N(f). Since f(N(f)) ⊆ N(f) we have the induced endomorphism f̃ : Ẽ → Ẽ. We call f admissible if dimẼ < ∞; for such f we define the generalized trace Tr(f) of f by putting Tr(f) = tr(f̃) where tr stands for the ordinary trace. Let f = {fq} : E → E be an endomorphism of degree zero of a graded vector space E = {Eq}. We call f a Leray endomorphism if (i). all fq are admissible and (ii). almost all Ẽq are trivial. For such f we define the generalized Lefschetz number Λ(f) by Λ(f) = ∑ q (−1)q Tr (fq). 202 Donal O’Regan CUBO 12, 2 (2010) A linear map f : E → E of a vector space E into itself is called weakly nilpotent provided for every x ∈ E there exists nx such that f nx (x) = 0. Assume that E = {Eq} is a graded vector space and f = {fq} : E → E is an endomorphism. We say that f is weakly nilpotent iff fq is weakly nilpotent for every q. It is well known [9, pp 53] that any weakly nilpotent endomorphism f : E → E is a Leray endomorphism and Λ(f) = 0. Let H be the C̆ech homology functor with compact carriers and coefficients in the field of rational numbers K 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 → X, H(f) is the induced linear map f⋆ = {f⋆ q} where f⋆ q : Hq(X) → Hq(X). With C̆ech homology functor extended to a category of morphisms (see [10 pp. 364]) we have the following well known result (note the homology functor H extends over this category i.e. for a morphism φ = {X p ⇐ Γ q → Y} : X → Y we define the induced map H (φ) = φ⋆ : H(X) → H(Y ) by putting φ⋆ = q⋆ ◦p −1 ⋆ ). Recall the following result [8 pp. 227]. Theorem 1.4. If φ : X → Y and ψ : Y → Z are two morphisms (here X, Y and Z are Hausdorff topological spaces) then (ψ ◦ φ)⋆ = ψ⋆ ◦ φ⋆. Two morphisms φ, ψ ∈ M(X,Y ) are homotopic (written φ ∼ ψ) provided there is a morphism χ ∈ M(X × [0, 1],Y ) such that χ(x, 0) = φ(x), χ(x, 1) = ψ(x) for every x ∈ X (i.e. φ = χ ◦ i0 and ψ = χ ◦ i1, where i0, i1 : X → X × [0, 1] are defined by i0(x) = (x, 0), i1(x) = (x, 1)). Recall the following result [9, pp. 231]: If φ ∼ ψ then φ⋆ = ψ⋆. Let φ : X → Y be a multivalued map (note for each x ∈ X we assume φ(x) is a nonempty subset of Y ). A pair (p,q) of single valued continuous maps of the form X p ← Γ q → Y is called a selected pair of φ (written (p,q) ⊂ φ) if the following two conditions hold: (i). p is a Vietoris map and (ii). q (p−1(x)) ⊂ φ(x) for any x ∈ X. Definition 1.1. A upper semicontinuous map φ : X → Y is said to be strongly admissible [9, 10] (and we write φ ∈ Ads(X,Y )) provided there exists a selected pair (p,q) of φ with φ(x) = q (p−1(x)) for x ∈ X. CUBO 12, 2 (2010) Fixed point theorems 203 Definition 1.2. A map φ ∈ Ads(X,X) is said to be a Lefschetz map if for each selected pair (p,q) ⊂ φ with φ(x) = q (p−1(x)) for x ∈ X the linear map q⋆ p −1 ⋆ : H(X) → H(X) (the existence of p−1⋆ follows from the Vietoris Theorem) is a Leray endomorphism. When we talk about φ ∈ Ads it is assumed that we are also considering a specified selected pair (p,q) of φ with φ(x) = q (p−1(x)). Remark 1.3. In fact since we specify the pair (p,q) of φ it is enough to say φ is a Lefschetz map if φ⋆ = q⋆ p −1 ⋆ : H(X) → H(X) is a Leray endomorphism. However for the examples of φ, X known in the literature [9] the more restrictive condition in Definition 1.2 works. We note [9, pp 227] that φ⋆ does not depend on the choice of diagram from [(p,q)], so in fact we could specify the morphism. If φ : X → X is a Lefschetz map as described above then we define the Lefschetz number (see [9, 10]) Λ (φ) (or ΛX (φ)) by Λ (φ) = Λ(q⋆ p −1 ⋆ ). If we do not wish to specify the selected pair (p,q) of φ then we would consider the Lefschetz set Λ (φ) = {Λ(q⋆ p −1 ⋆ ) : φ = q (p −1)}. Definition 1.3. A Hausdorff topological space X is said to be a Lefschetz space provided every compact φ ∈ Ads(X,X) is a Lefschetz map and Λ(φ) 6= 0 implies φ has a fixed point. Definition 1.4. A upper semicontinuous map φ : X → Y with closed values is said to be admissible (and we write φ ∈ Ad(X,Y )) provided there exists a selected pair (p,q) of φ. Definition 1.5. A map φ ∈ Ad(X,X) is said to be a Lefschetz map if for each selected pair (p,q) ⊂ φ the linear map q⋆ p −1 ⋆ : H(X) → H(X) (the existence of p −1 ⋆ follows from the Vietoris Theorem) is a Leray endomorphism. If φ : X → X is a Lefschetz map, we define the Lefschetz set Λ (φ) (or ΛX (φ)) by Λ (φ) = { Λ(q⋆ p −1 ⋆ ) : (p,q) ⊂ φ } . Definition 1.6. A Hausdorff topological space X is said to be a Lefschetz space provided every compact φ ∈ Ad(X,X) is a Lefschetz map and Λ(φ) 6= {0} implies φ has a fixed point. Recall the following result [8]. Theorem 1.5. Every open subset of the Tychonoff cube is a Lefschetz space. The following concepts will be needed in Section 4. Let (X,d) be a metric space and S a nonempty subset of X. For x ∈ X let d(x,S) = infy∈S d(x,y). Also diamS = sup{d(x,y) : x,y ∈ S}. We let B(x,r) denote the open ball in X centered at x of radius r and by B(S,r) we denote ∪x∈S B(x,r). For two nonempty subsets S1 and S2 of X we define the generalized Hausdorff distance H to be H(S1,S2) = inf{ǫ > 0 : S1 ⊆ B(S2,ǫ), S2 ⊆ B(S1,ǫ)}. 204 Donal O’Regan CUBO 12, 2 (2010) Now suppose G : S → 2X . Then G is said to be hemicompact if each sequence {xn}n∈N in S has a convergent subsequence whenever d(xn,G (xn)) → 0 as n →∞. Now let I be a directed set with order ≤ and let {Eα}α∈I be a family of locally convex spaces. For each α ∈ I, β ∈ I for which α ≤ β let πα,β : Eβ → Eα be a continuous map. Then the set { x = (xα) ∈ ∏ α∈I Eα : xα = πα,β (xβ ) ∀α, β ∈ I, α ≤ β } is a closed subset of ∏ α∈I Eα and is called the projective limit of {Eα}α∈I and is denoted by lim← Eα (or lim←{Eα,πα,β} or the generalized intersection [1, 2] ∩α∈I Eα.) 2 Preliminary Fixed Point Theory The fixed point theory presented in this section can partly be found in [9, 14]. However for the convenience of the reader we present the following elementary approach. By a space we mean a Hausdorff topological space. Let X and Y be spaces. A space Y is an neighborhood extension space for Q (written Y ∈ NES(Q)) if ∀X ∈ Q, ∀K ⊆ X closed in X, and for any continuous function f0 : K → Y , there exists a continuous extension f : U → Y of f0 over a neighbourhood U of K in X. Let X ∈ NES(compact) and F ∈ Uκc (X,X) a compact map. Now let K = F(X). We know [12] that K can be embedded as a closed subset K⋆ of T ; let s : K → K⋆ be a homeomorphism. Also let i : K →֒ X be an inclusion. Let U be an open neighbourhood of K⋆ in T and hU : U → X be a continuous extension of is−1 : K⋆ → X on U (guaranteed since X ∈ NES(compact)). Let jU : K ⋆ →֒ U be the natural embedding so hU jU = is −1. Finally let G = jU sF hU . Notice G ∈Uκc (U,U). We now assume (2.1) G ∈Uκc (U,U) has a fixed point. Then there exists x ∈ U with x ∈ Gx. Let y = hU (x), so y ∈ hU jU sF (y) i.e. y = hU jU s (q) for some q ∈ F (y). Since hU jU (z) = is −1(z) for z ∈ K⋆, we have hU jU s (q) = i (q), so y ∈ F(y). Theorem 2.1. Let X ∈ NES(compact) and F ∈ Uκc (X,X) a compact map. Also assume (2.1) holds with K, K⋆, U, s, i, jU and hU as described above. Then F has a fixed point. We discuss Theorem 2.1 for the class Ad(X,X). Let X ∈ NES(compact) and F ∈ Ad(X,X) a compact map. Also let K, K⋆, U, s, i, jU and hU as described above. Let (p,q) be a selected pair for F . Now since F hU ∈ Ad(U,X) then [9, Section 40] guarantees that there exists a selected pair (p′,q′) of F hU with (q ′)⋆ (p ′)−1⋆ = q⋆ p −1 ⋆ (hU )⋆. Notice (q′)⋆ (p ′)−1⋆ (jU )⋆ s⋆ = q⋆ p −1 ⋆ (hU )⋆ (jU )⋆ s⋆ = q⋆ p −1 ⋆ since hU jU s = is −1 s. Next note G = jU sF hU ∈ Ad(U,U) has a selected pair (p ′,jU sq ′) (since jU sq ′ (p′)−1(x) ⊆ jU sF hU (x) = G(x) for x ∈ U) and from Theorem 1.5 we know U is a Lefschetz space so (jU sq ′)⋆ (p ′)−1⋆ is a Leray endomorphism. Notice (jU )⋆ s⋆ (q ′)⋆ (p ′)−1⋆ = (jU sq ′)⋆ (p ′)−1⋆ CUBO 12, 2 (2010) Fixed point theorems 205 and from above (q′)⋆ (p ′)−1⋆ (jU )⋆ s⋆ = q⋆ p −1 ⋆ so [8, page 314, see (1.3)] (here E ′ = U′, E′′ = U′′, u = (q′)⋆ (p ′)−1⋆ , v = (jU )⋆ s⋆, f ′ = (jU sq ′)⋆ (p ′)−1⋆ and f ′′ = q⋆ p −1 ⋆ ) guarantees that q⋆ p −1 ⋆ is a Leray endomorphism and Λ (q⋆ p −1 ⋆ ) = Λ ((jU sq ′)⋆ (p ′)−1⋆ ). Thus Λ (F) is well defined. Next suppose Λ (F) 6= {0}. Then there exists a selected pair (p,q) as described above with Λ (q⋆ p −1 ⋆ ) 6= 0. Let p ′ and q′ be as described above with Λ ((jU sq ′)⋆ (p ′)−1⋆ ) = Λ (q⋆ p −1 ⋆ ) 6= 0. Now since U is a Lefschetz space there exists x ∈ U with x ∈ jU sq ′ (p′)−1(x) i.e. x ∈ G(x) so (2.1) is satisfied. Combining with Theorem 2.1 we have the following result. Theorem 2.2. Let X ∈ NES(compact) and F ∈ Ad(X,X) a compact map. Then Λ (F) is well defined and if Λ (F) 6= {0} then F has a fixed point. Remark 2.1. Theorem 2.2 says that NES(compact) spaces are Lefschetz spaces (for the class Ad). Remark 2.2. Essentially the same reasoning as in Theorem 2.2 establishes: Let X ∈ NES(compact) and F ∈ Ads(X,X) a compact map. Then Λ (F) is well defined and if Λ (F) 6= 0 then F has a fixed point i.e. NES(compact) spaces are Lefschetz spaces (for the class Ads). A space Y is a approximate neighborhood extension space for Q (written Y ∈ ANES(Q)) if ∀α ∈ Cov (Y ), ∀X ∈ Q, ∀K ⊆ X closed in X, and any continuous function f0 : K → Y , there exists a neighborhood Uα of K in X and a continuous function fα : Uα → Y such that fα|K and f0 are α-close. Let X ∈ ANES(compact) be a uniform space and F ∈Uκc (X,X) a compact upper semicontin- uous map with closed values. Also let α ∈ CovX (K) where K = F(X). To show F has a fixed point it suffices (Theorem 1.3 with Remark 1.2) to show F has an α–fixed point. Let α′ = α∪{X\K} and let K⋆, s and i be as above. Since X ∈ ANES(compact) there exists an open neighborhood Uα of K⋆ in T and fα : Uα → X a continuous function such that fα|K⋆ and s −1 are α′–close and as a result fα jUα s : K → X and i : K → X are α–close; here jUα : K ⋆ →֒ Uα is the natural imbedding. Finally let Gα = jUα sF fα. Notice Gα ∈U κ c (Uα,Uα) is a compact upper semicontinuous map with closed values. We now assume (2.2) Gα ∈U κ c (Uα,Uα) has a fixed point for each α ∈ CovX (F(X)). We still have α ∈ CovX (K) fixed and we let x be a fixed point of Gα. Now let y = fα(x) so y ∈ fα jUα sF(y) i.e. y = fα jUα s(q) for some q ∈ F(y). Now since fα jUα s and i are α–close there exists U ∈ α with fα jUα s(q) ∈ U and i(q) ∈ U i.e. q ∈ U and y = fα jUα s(q) ∈ U. Thus q ∈ U and y ∈ U so y ∈ U and F(y) ∩U 6= ∅ since q ∈ F (y). Theorem 2.3. Let X ∈ ANES(compact) be a uniform space and F ∈ Uκc (X,X) a compact upper semicontinuous map with closed values. Also assume (2.2) holds with K, K⋆, Uα, s, jUα , i and fα as described above. Then F has a fixed point. We discuss Theorem 2.3 for the class Ad(X,X). First however we need the following definition. A space Y is a strongly approximate neighborhood extension space for Q (written Y ∈ SANES(Q)) if ∀α ∈ Cov (Y ), ∀X ∈ Q, ∀K ⊆ X closed in X, and any continuous function f0 : K → Y , there 206 Donal O’Regan CUBO 12, 2 (2010) exists a neighborhood Uα of K in X and a continuous function fα : Uα → Y such that fα|K and f0 are α close and α-homotopic. Let X ∈ SANES(compact) be a uniform space and F ∈ Ad(X,X) a compact map. Also let K, K⋆, Uα, s, jUα , i and fα as described above. Let (p,q) be a selected pair for F . Now since F fα ∈ Ad(Uα,X) then [9, Section 40] guarantees that there exists a selected pair (p ′ α,q ′ α) of F fα with (q′α)⋆ (p ′ α) −1 ⋆ = q⋆ p −1 ⋆ (fα)⋆. As a result (q′α)⋆ (p ′ α) −1 ⋆ (jUα )⋆ s⋆ = q⋆ p −1 ⋆ (fα)⋆ (jUα )⋆ s⋆ = q⋆ p −1 ⋆ since fα jUα s is α homotopic to i (note fα|K⋆ and s −1 are α′-homotopic by definition). Next note Gα = jUα sF fα ∈ Ad(Uα,Uα) has a selected pair (p ′ α,jUα sq ′ α) and from Theorem 1.5 we have that (jUα sq ′ α)⋆ (p ′ α) −1 ⋆ is a Leray endomorphism. Now since (jUα )⋆ s⋆ (q ′ α)⋆ (p ′ α) −1 ⋆ = (jUα sq ′ α)⋆ (p ′ α) −1 ⋆ and from above (q′α)⋆ (p ′ α) −1 ⋆ (jUα )⋆ s⋆ = q⋆ p −1 ⋆ then [8, page 314, see (1.3)] guarantees that q⋆ p −1 ⋆ is a Leray endomorphism and we have Λ (q⋆ p −1 ⋆ ) = Λ ((jUα sq ′ α)⋆ (p ′ α) −1 ⋆ ). Thus Λ (F) is well defined. Next suppose Λ (F) 6= {0}. Then there exists a selected pair (p,q) as described above with Λ (q⋆ p −1 ⋆ ) 6= 0. Let p ′ α and q ′ α be as described above with Λ ((jUα sq ′ α)⋆ (p ′ α) −1 ⋆ ) = Λ (q⋆ p −1 ⋆ ) 6= 0. Now since Uα is a Lefschetz space there exists x ∈ Uα with x ∈ jUα sq ′ α (p ′ α) −1(x) i.e. x ∈ Gα(x) so (2.2) is satisfied. Combining with Theorem 2.3 we have the following result. Theorem 2.4. Let X ∈ SANES(compact) be a uniform space and F ∈ Ad(X,X) a compact map. Then Λ (F) is well defined and if Λ (F) 6= {0} then F has a fixed point. Remark 2.3. Theorem 2.4 says that SANES(compact) uniform spaces are Lefschetz spaces (for the class Ad). Remark 2.4. Essentially the same reasoning as in Theorem 2.4 establishes: Let X ∈ SANES(compact) be a uniform space and F ∈ Ads(X,X) a compact map. Then Λ (F) is well defined and if Λ (F) 6= 0 then F has a fixed point i.e. SANES(compact) uniform spaces are Lefschetz spaces (for the class Ads). One could in fact generalize Theorem 2.2 and Theorem 2.4 by using some results in [1]. Let X be a subset of a Hausdorff topological space and let X be a uniform space. Then X is said to be Schauder admissible if for every compact subset K of X and every open covering α ∈ CovX (K) there exists a continuous function πα : K → E such that (i). πα and i : K →֒ X are α–close; (ii). πα(K) is contained in a subset C of X with C a Lefschetz space; and (iii). πα and i : K →֒ X are homotopic. Remark 2.5. For example we could take C ∈ NES(compact) or C ∈ SANES(compact) in (ii) above (for both Ad and Ads maps). The following result can be found in [1]. CUBO 12, 2 (2010) Fixed point theorems 207 Theorem 2.5. Let X be a subset of a Hausdorff topological space and let X be a uniform space. Also suppose X is Schauder admissible. If F ∈ Ad(X,X) is a compact map then Λ (F) is well defined and if Λ (F) 6= {0} then F has a fixed point (i.e. Schauder admissible uniform spaces are Lefschetz spaces (for the class Ad)). Let X be a Hausdorff topological space and let α ∈ Cov(X). X is said to be Schauder admissible α-dominated if there exists a Schauder admissible space Xα and two continuous functions rα : Xα → X, sα : X → Xα such that rα sα : X → X and i : X → X are α–close and also that rα sα ∼ IdX . X is said to be almost Schauder admissible dominated if X is Schauder admissible α-dominated for every α ∈ Cov(X). The following result can be found in [1]. Theorem 2.6. Let X be a subset of a Hausdorff topological space and let X be a uniform space. Also suppose X is almost Schauder admissible dominated. If F ∈ Ad(X,X) is a compact map then Λ (F) is well defined and if Λ (F) 6= {0} then F has a fixed point (i.e. almost Schauder admissible dominated uniform spaces are Lefschetz spaces (for the class Ad)). Remark 2.6. A similar result holds if F ∈ Ad(X,X) is replaced by F ∈ Ads(X,X) in Theorem 2.5 and 2.6. 3 Asymptotic Fixed Point Theory Let X be a Hausdorff topological space. A map F ∈ Ad(X,X) is said to be a compact absorbing contraction (written F ∈ CAC(X,X) or F ∈ CAC(X)) if there exists Y ⊆ X such that (i). F(Y ) ⊆ Y ; (ii). F |Y ∈ Ad(Y,Y ) (automatically satisfied) is a compact map with Y a Lefschetz space; (iii). for every compact K ⊆ X there is an integer n = n(K) such that F n(K) ⊆ Y . Remark 3.1. Examples of Lefschetz spaces Y can be found in Section 2. For example Y could be NES(compact) or a SANES(compact) uniform space. Remark 3.2. If Y = U is an open subset of X then (iii) could be changed to (iii)’. for every x ∈ X there exists an integer n = n(x) such that F n(x)(x) ⊆ Y = U. To see this we show (iii)’ implies (iii). For each x ∈ X there exists n(x) such that F n(x)(x) ⊆ Y = U so by upper semicontinuity there exists an open neighborhood Ux of x in X such that F n(x)(y) ⊆ Y = U for y ∈ Ux. Let K be a compact subset of X. Then there exists an open covering {Ux1, ....,Uxn} of K. Let n = max{n(x1), ...,n(xn)} and so for x ∈ K we have F n(x) ⊆ U = Y , so (iii) is true. Remark 3.3. For a discussion on compact absorbing contractions see [9, Section 42] and [12, Section 15.5]. 208 Donal O’Regan CUBO 12, 2 (2010) Theorem 3.1. Let X be a Hausdorff topological space and F ∈ CAC(X,X). Then Λ (F) is well defined and if Λ (F) 6= {0} then F has a fixed point. Proof: Let Y be as described above. Let (p,q) be a selected pair for F so in particular q p−1(Y ) ⊆ F(Y ). Consider F |Y and let q ′, p′ : p−1(Y ) → Y be given by p′(u) = p(u) and q′(u) = q(u). Notice (p′,q′) is a selected pair for F |Y . Now since Y is a Lefschetz space then q ′ ⋆ (p ′)−1⋆ is a Leray endomorphism. Now [9, Proposition 42.2, pp 208] guarantees (see (iii)) that the homeomorphism q′′⋆ (p ′′)−1⋆ : H(X,Y ) → H(X,Y ) is weakly nilpotent (here p′′, q′′ : (Γ,p−1(Y )) → (X,Y ) are given by p′′(u) = p(u) and q′′(u) = q(u)). Then [9, pp 53] guarantees that q′′⋆ (p ′′)−1⋆ is a Leray endomorphism and Λ (q ′′ ⋆ (p ′′)−1⋆ ) = 0. Also [9, Property 11.5, pp 52] guarantees that q⋆ p −1 ⋆ is a Leray endomorphism (with Λ (q⋆ p −1 ⋆ ) = Λ (q′⋆ (p ′)−1⋆ )) so Λ (F) is well defined. Next suppose Λ (F) 6= {0}. Then there exists a selected pair (p,q) of F with Λ (q⋆ p −1 ⋆ ) 6= 0. Let (p′,q′) be as described above with Λ (q⋆ p −1 ⋆ ) = Λ (q ′ ⋆ (p ′)−1⋆ ). Then Λ (q ′ ⋆ (p ′)−1⋆ ) 6= 0 so since Y is a Lefschetz space then there exists x ∈ Y with x ∈ F |Y (x) i.e. x ∈ F x. 2 Remark 3.4. A map F ∈ Ads(X,X) is said to be a compact absorbing contraction (written F ∈ CACs(X,X) or F ∈ CACs(X)) if there exists Y ⊆ X such that (i). F(Y ) ⊆ Y ; (ii). F |Y ∈ Ads(Y,Y ) (automatically satisfied) is a compact map with Y a Lefschetz space; (iii). for every compact K ⊆ X there is an integer n = n(K) such that F n(K) ⊆ Y . Essentially the same reasoning as in Theorem 3.1 establishes the following: Let X be a Hausdorff topological space and F ∈ CACs(X,X). Then Λ (F) is well defined and if Λ (F) 6= 0 then F has a fixed point. 4 Fixed point theory in Fréchet spaces We now present another approach based on projective limits. Let E = (E,{| · |n}n∈N ) be a Fréchet space with the topology generated by a family of seminorms {|· |n : n ∈ N}; here N = {1, 2, ....}. We assume that the family of seminorms satisfies (4.1) |x|1 ≤ |x|2 ≤ |x|3 ≤ ....... for every x ∈ E. A subset X of E is bounded if for every n ∈ N there exists rn > 0 such that |x|n ≤ rn for all x ∈ X. For r > 0 and x ∈ E we denote B(x,r) = {y ∈ E : |x−y|n ≤ r∀n ∈ N}. To E we associate a sequence of Banach spaces {(En, | · |n)} described as follows. For every n ∈ N we consider the equivalence relation ∼n defined by (4.2) x ∼n y iff |x−y|n = 0. We denote by En = (E /∼n, | · |n) the quotient space, and by (En, | · |n) the completion of E n with respect to | · |n (the norm on E n induced by | · |n and its extension to En are still denoted CUBO 12, 2 (2010) Fixed point theorems 209 by | · |n). This construction defines a continuous map µn : E → En. Now since (4.1) is satisfied the seminorm | · |n induces a seminorm on Em for every m ≥ n (again this seminorm is denoted by | · |n). Also (4.2) defines an equivalence relation on Em from which we obtain a continuous map µn,m : Em → En since Em /∼n can be regarded as a subset of En. Now µn,m µm,k = µn,k if n ≤ m ≤ k and µn = µn,m µm if n ≤ m. We now assume the following condition holds: (4.3) { for each n ∈ N, there exists a Banach space (En, | · |n) and an isomorphism (between normed spaces) jn : En → En. Remark 4.1. (i). For convenience the norm on En is denoted by | · |n. (ii). In many applications En = E n for each n ∈ N. (iii). Note if x ∈ En (or E n) then x ∈ E. However if x ∈ En then x is not necessaily in E and in fact En is easier to use in applications (even though En is isomorphic to En). For example if E = C[0,∞), then En consists of the class of functions in E which coincide on the interval [0,n] and En = C[0,n]. Finally we assume (4.4) { E1 ⊇ E2 ⊇ ........ and for each n ∈ N, |jn µn,n+1 j −1 n+1 x|n ≤ |x|n+1 ∀ x ∈ En+1 (here we use the notation from [1, 2] i.e. decreasing in the generalized sense). Let lim← En (or ∩∞1 En where ∩ ∞ 1 is the generalized intersection [1, 2]) denote the projective limit of {En}n∈N (note πn,m = jn µn,m j −1 m : Em → En for m ≥ n) and note lim← En ∼= E, so for convenience we write E = lim← En. For each X ⊆ E and each n ∈ N we set Xn = jn µn(X), and we let Xn, intXn and ∂Xn denote respectively the closure, the interior and the boundary of Xn with respect to | · |n in En. For r > 0 and x ∈ En we denote Bn(x,r) = {y ∈ En : |x−y|n ≤ r}. Let M ⊆ E and consider the map F : M → 2E . Assume for each n ∈ N and x ∈ M that jn µn F (x) is closed. Let n ∈ N and Mn = jn µn(M). Since we first consider Volterra type operators we assume (note this assumption is only needed in Theorems 4.1 and 4.2) (4.5) if x, y ∈ E with |x−y|n = 0 then Hn(F x,F y) = 0; here Hn denotes the appropriate generalized Hausdorff distance (alternatively we could assume ∀n ∈ N,∀x, y ∈ M if jn µn x = jn µn y then jn µn F x = jn µn F y and of course here we do not need to assume that jn µn F (x) is closed for each n ∈ N and x ∈ M). Now (4.5) guarantees that we can define (a well defined) Fn on Mn as follows: For y ∈ Mn there exists a x ∈ M with y = jn µn(x) and we let Fn y = jn µn F x (we could of course call it F y since it is clear in the situation we use it); note Fn : Mn → C(En) and note if there exists a z ∈ M with y = jn µn(z) then jn µn F x = jn µn F z from (4.5) (here C(En) 210 Donal O’Regan CUBO 12, 2 (2010) denotes the family of nonempty closed subsets of En). In this paper we assume Fn will be defined on Mn i.e. we assume the Fn described above admits an extension (again we call it Fn) Fn : Mn → 2 En (we will assume certain properties on the extension). Now we present some Lefschetz type theorems in Fréchet spaces. Our first two results are moti- vated by Fredholm type operators. Theorem 4.1. Let E and En be as described above, C ⊆ E and F : C → 2 E. Also assume for each n ∈ N and x ∈ C that jn µn F (x) is closed and also for each n ∈ N that Fn : Cn → 2 En as described above is a closed map with x /∈ Fn(x) in En for x ∈ ∂ Cn. Suppose the following conditions are satisfied: (4.6) { for each n ∈ N, Fn ∈ CAC(Cn,Cn) and Fn : Cn → 2 En is hemicompact, (4.7) for each n ∈ N, ΛCn (Fn) 6= {0} and (4.8) { for each n ∈{2, 3, ....} if y ∈ Cn solves y ∈ Fn y in En then jk µk,n j −1 n (y) ∈ Ck for k ∈{1, ...,n− 1}. Then F has a fixed point in E. Proof: For each n ∈ N there exists yn ∈ Cn with yn ∈ Fn yn in En. Lets look at {yn}n∈N . Notice y1 ∈ C1 and j1 µ1,k j −1 k (yk) ∈ C1 for k ∈ N\{1} from (4.8). Note j1 µ1,n j −1 n (yn) ∈ F1 (j1 µ1,n j −1 n (yn)) in E1; to see note for n ∈ N fixed there exists a x ∈ E with yn = jn µn (x) so jn µn (x) ∈ Fn (yn) = jn µn F(x) on En so on E1 we have j1 µ1,n j −1 n (yn) = j1 µ1,n j −1 n jn µn (x) ∈ j1 µ1,n j −1 n jn µn F(x) = j1 µ1,n µn F(x) = j1 µ1 F(x) = F1(j1 µ1 (x)) = F1(j1 µ1,n j −1 n jn µn (x)) = F1 (j1 µ1,n j −1 n (yn)). Now (4.6) guarantees that there exists is a subsequence N⋆1 of N and a z1 ∈ C1 with j1 µ1,n j −1 n (yn) → z1 in E1 as n →∞ in N ⋆ 1 and z1 ∈ F1 z1 since F1 is a closed map. Note z1 ∈ C1 since x /∈ F1(x) in E1 for x ∈ ∂ C1. Let N1 = N ⋆ 1 \{1}. Now j2 µ2,n j −1 n (yn) ∈ C2 for n ∈ N1 together with (4.6) guarantees that there exists a subsequence N⋆2 of N1 and a z2 ∈ C2 with j2 µ2,n j −1 n (yn) → z2 in E2 as n → ∞ in N ⋆ 2 and z2 ∈ F2 z2. Also z2 ∈ C2. Note from (4.4) and the uniqueness of limits that j1 µ1,2 j −1 2 z2 = z1 in E1 since N ⋆ 2 ⊆ N1 (note j1 µ1,n j −1 n (yn) = j1 µ1,2 j −1 2 j2 µ2,n j −1 n (yn) for n ∈ N⋆2 ). Let N2 = N ⋆ 2 \{2}. Proceed inductively to obtain subsequences of integers N⋆1 ⊇ N ⋆ 2 ⊇ ......, N ⋆ k ⊆{k,k + 1, ....} and zk ∈ Ck with jk µk,n j −1 n (yn) → zk in Ek as n → ∞ in N ⋆ k and zk ∈ Fk zk. Also zk ∈ Ck. Note jk µk,k+1 j −1 k+1 zk+1 = zk in Ek for k ∈{1, 2, ...}. Also let Nk = N ⋆ k \{k}. Fix k ∈ N. Now zk ∈ Fk zk in Ek. Note as well that zk = jk µk,k+1 j −1 k+1 zk+1 = jk µk,k+1 j −1 k+1 jk+1 µk+1,k+2 j −1 k+2 zk+2 = jk µk,k+2 j −1 k+2 zk+2 = ..... = jk µk,m j −1 m zm = πk,m zm CUBO 12, 2 (2010) Fixed point theorems 211 for every m ≥ k. We can do this for each k ∈ N. As a result y = (zk) ∈ lim←En = E and also note zk ∈ Ck for each k ∈ N. Thus for each k ∈ N we have jk µk (y) = zk ∈ Fk zk = jk µk F y in Ek so y ∈ F y in E. 2 Remark 4.2. Of course one could remove x /∈ Fn(x) in En for x ∈ ∂ Cn for each n ∈ N if C is a closed subset of E. The proof follows as in Theorem 4.1 except in this case zk ∈ Ck (but not necessarily in Ck). Also from y = (zk) ∈ lim←En = E and πk,m (ym) → zk in Ek as m → ∞ we can conclude that y ∈ C = C (note q ∈ C iff for every k ∈ N there exists (xk,m) ∈ C, xk,m = πk,n (xn,m) for n ≥ k with xk,m → jk µk (q) in Ek as m → ∞). Thus zk = jk µk (y) ∈ Ck and so jk µk (y) ∈ jk µk F (y) in Ek as before. Note in fact we can remove the assumption that C is a closed subset of E if we assume F : Y → 2E with C ⊆ Y and Cn ⊆ Yn for each n ∈ N. Remark 4.3. If we replace Fn : Cn → 2 En is hemicompact in (4.6) with Fn : Cn → 2 En is hemicompact then one can remove x /∈ Fn(x) in En for x ∈ ∂ Cn and Fn : Cn → 2 En is a closed map for each n ∈ N in the statement of Theorem 4.1 since if for each n ∈ N, Fn : Cn → 2 En is hemicompact then we automatically have that zk ∈ Ck. Essentially the same reasoning as in Theorem 4.1 (with Remark 4.2) yields the following result. Theorem 4.2. Let E and En be as described above, C ⊆ E and F : C → 2 E. Also assume C is a closed subset of E, for each n ∈ N and x ∈ C that jn µn F (x) is closed and also for each n ∈ N that Fn : Cn → 2 En is as described above. Suppose the following conditions are satisfied: (4.9) for each n ∈ N, Fn ∈ CAC(Cn,Cn) is hemicompact, (4.10) for each n ∈ N, Λ Cn (Fn) 6= {0} and (4.11) { for each n ∈{2, 3, ....} if y ∈ Cn solves y ∈ Fn y in En then jk µk,n j −1 n (y) ∈ Ck for k ∈{1, ...,n− 1}. Then F has a fixed point in E. Remark 4.4. Note we can remove the assumption in Theorem 4.2 that C is a closed subset of E if we assume F : Y → 2E with C ⊆ Y and Cn ⊆ Yn for each n ∈ N. Our result two results are motivated by Urysohn type operators. In this case the map Fn will be related to F by the closure property (4.16). Theorem 4.3. Let E and En be as described above, C ⊆ E and F : C → 2 E. Also for each n ∈ N assume there exists Fn : Cn → 2 En and suppose the following conditions are satisfied: (4.12) for each n ∈ N, Fn ∈ CAC(Cn,Cn) (4.13) for each n ∈ N, ΛCn (Fn) 6= {0} 212 Donal O’Regan CUBO 12, 2 (2010) (4.14) { for each n ∈{2, 3, ....} if y ∈ Cn solves y ∈ Fn y in En then jk µk,n j −1 n (y) ∈ Ck for k ∈{1, ...,n− 1} (4.15)                      for any sequence {yn}n∈N with yn ∈ Cn and yn ∈ Fn yn in En for n ∈ N and for every k ∈ N there exists a subsequence Nk ⊆{k + 1,k + 2, .....}, Nk ⊆ Nk−1 for k ∈{1, 2, ....}, N0 = N, and a zk ∈ Ck with jk µk,n j −1 n (yn) → zk in Ek as n →∞ in Nk and (4.16)                  if there exists a w ∈ C and a sequence {yn}n∈N with yn ∈ Cn and yn ∈ Fn yn in En such that for every k ∈ N there exists a subsequence S ⊆{k + 1,k + 2, .....} of N with jk µk,n j −1 n (yn) → w in Ek as n →∞ in S, then w ∈ F w in E. Then F has a fixed point in E. Remark 4.5. Notice to check (4.15) we need to show that for each k ∈ N the sequence {jk µk,n j −1 n (yn)}n∈Nk−1 ⊆ Ck is sequentially compact. Proof: Fix n ∈ N. Now there exists yn ∈ Cn with yn ∈ Fn yn in En. Lets look at {yn}n∈N . Notice y1 ∈ C1 and j1 µ1,k j −1 k (yk) ∈ C1 for k ∈ {2, 3, ...}. Now (4.15) with k = 1 guarantees that there exists a subsequence N1 ⊆{2, 3, ....} and a z1 ∈ C1 with j1 µ1,n j −1 n (yn) → z1 in E1 as n →∞ in N1. Look at {yn}n∈N1. Now j2 µ2,n j −1 n (yn) ∈ C2 for k ∈ N1. Now (4.15) with k = 2 guarantees that there exists a subsequence N2 ⊆ {3, 4, ...} of N1 and a z2 ∈ C2 with j2 µ2,n j −1 n (yn) → z2 in E2 as n → ∞ in N2. Note from (4.4) and the uniqueness of limits that j1 µ1,2 j −1 2 z2 = z1 in E1 since N2 ⊆ N1 (note j1 µ1,n j −1 n (yn) = j1 µ1,2 j −1 2 j2 µ2,n j −1 n (yn) for n ∈ N2). Proceed inductively to obtain subsequences of integers N1 ⊇ N2 ⊇ ......, Nk ⊆{k + 1,k + 2, ....} and zk ∈ Ck with jk µk,n j −1 n (yn) → zk in Ek as n →∞ in Nk. Note jk µk,k+1 j −1 k+1 zk+1 = zk in Ek for k ∈{1, 2, ...}. Fix k ∈ N. Note zk = jk µk,k+1 j −1 k+1 zk+1 = jk µk,k+1 j −1 k+1 jk+1 µk+1,k+2 j −1 k+2 zk+2 = jk µk,k+2 j −1 k+2 zk+2 = ..... = jk µk,m j −1 m zm = πk,m zm for every m ≥ k. We can do this for each k ∈ N. As a result y = (zk) ∈ lim←En = E and also note y ∈ C since zk ∈ Ck for each k ∈ N. Also since yn ∈ Fn yn in En for n ∈ Nk and jk µk,n j −1 n (yn) → zk = y in Ek as n →∞ in Nk we have from (4.16) that y ∈ F y in E. 2 CUBO 12, 2 (2010) Fixed point theorems 213 Remark 4.6. From the proof we see that condition (4.14) can be removed from the statement of Theorem 4.3. We include it only to explain condition (4.15) (see Remark 4.5). Remark 4.7. Suppose in Theorem 4.3 we have (4.15)⋆                      for any sequence {yn}n∈N with yn ∈ Cn and yn ∈ Fn yn in En for n ∈ N and for every k ∈ N there exists a subsequence Nk ⊆{k + 1,k + 2, .....}, Nk ⊆ Nk−1 for k ∈{1, 2, ....}, N0 = N, and a zk ∈ Ck with jk µk,n j −1 n (yn) → zk in Ek as n →∞ in Nk instead of (4.15) and F : C → 2E is replaced by F : Y → 2E with C ⊆ Y and Cn ⊆ Yn for each n ∈ N and suppose (4.16) is true with w ∈ C replaced by w ∈ Y . Then the result in Theorem 4.3 is again true. The proof follows the reasoning in Theorem 4.3 except in this case zk ∈ Ck (but not necessarily in Ck) and y ∈ Y . In fact we could replace Cn ⊆ Yn above with Cn a subset of the closure of Yn in En if Y is a closed subset of E (so in this case we can take Y = C if C is a closed subset of E). To see this note zk ∈ Ck, y = (zk) ∈ lim←En = E and πk,m (ym) → zk in Ek as m →∞ and we can conclude that y ∈ Y = Y . In fact in this remark we could replace (in fact we can remove it as mentioned in Remark 4.6) (4.14) with (4.14)⋆ { for each n ∈{2, 3, ....} if y ∈ Cn solves y ∈ Fn y in En then jk µk,n j −1 n (y) ∈ Ck for k ∈{1, ...,n− 1} and the result above is again true. Essentially the same reasoning as in Theorem 4.3 (with Remark 4.7) yields the following result. Theorem 4.4. Let E and En be as described above, C ⊆ E and F : C → 2 E. Also assume C is a closed subset of E and for each n ∈ N that Fn : Cn → 2 En and suppose the following conditions are satisfied: (4.17) { for each n ∈{2, 3, ....} if y ∈ Cn solves y ∈ Fn y in En then jk µk,n j −1 n (y) ∈ Ck for k ∈{1, ...,n− 1} (4.18) for each n ∈ N, Fn ∈ CAC(Cn,Cn) (4.19) for each n ∈ N, Λ Cn (Fn) 6= {0} 214 Donal O’Regan CUBO 12, 2 (2010) (4.20)                      for any sequence {yn}n∈N with yn ∈ Cn and yn ∈ Fn yn in En for n ∈ N and for every k ∈ N there exists a subsequence Nk ⊆{k + 1,k + 2, .....}, Nk ⊆ Nk−1 for k ∈{1, 2, ....}, N0 = N, and a zk ∈ Ck with jk µk,n j −1 n (yn) → zk in Ek as n →∞ in Nk and (4.21)                  if there exists a w ∈ C and a sequence {yn}n∈N with yn ∈ Cn and yn ∈ Fn yn in En such that for every k ∈ N there exists a subsequence S ⊆{k + 1,k + 2, .....} of N with jk µk,n j −1 n (yn) → w in Ek as n →∞ in S, then w ∈ F w in E. Then F has a fixed point in E. Remark 4.8. 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