@ Appl. Gen. Topol. 20, no. 1 (2019), 135-153doi:10.4995/agt.2019.10474 c© AGT, UPV, 2019 Remarks on fixed point assertions in digital topology Laurence Boxer a and P. Christopher Staecker b a Department of Computer and Information Sciences, Niagara University, NY 14109, USA; and Department of Computer Science and Engineering, State University of New York at Bu- ffalo. (boxer@niagara.edu) b Department of Mathematics, Fairfield University, Fairfield, CT 06823-5195, USA. (cstaecker@fairfield.edu) Communicated by S. Romaguera Abstract Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. We show that some of the published assertions based on these tools are incorrect or trivial; we offer improvements on others. 2010 MSC: 54H25. Keywords: digital topology; fixed point; metric space. 1. Introduction Recent papers have attempted to apply to digital images ideas from Eu- clidean topology and real analysis concerning metrics and fixed points. While the underlying motivation of digital topology comes from Euclidean topology and real analysis, some applications recently featured in the literature seem of doubtful worth. Although papers including [30, 7] have valid and interesting results for fixed points and for “almost” or “approximate” fixed points in digital topology, other published assertions concerning fixed points in digital topology are incorrect or trivial (e.g., applicable only to singletons, or functions studied forced to be constant), as we will discuss in the current paper. Received 02 July 2018 – Accepted 19 November 2018 http://dx.doi.org/10.4995/agt.2019.10474 L. Boxer and P. C. Staecker 2. Preliminaries We let Z denote the set of integers, and R, the real line. We consider a digital image as a graph (X,κ), where X ⊂ Zn for some positive integer n and κ is an adjacency relation on X. A digital metric space is [12] a triple (X,d,κ) where (X,κ) is a digital image and d is a metric for X. In [12], d was taken to be the Euclidean metric, but we will not limit our discussion to the Euclidean metric. The diameter of a metric space (X,d) is diamX = sup{d(x,y) |x,y ∈ X}. 2.1. Adjacencies. The most commonly used adjacencies for digital images are the cu-adjacencies, defined as follows. Definition 2.1. Let p,q ∈ Zn, p = (p1, . . . ,pn), q = (q1, . . . ,qn), p 6= q. Let 1 ≤ u ≤ n. We say p and q are cu-adjacent, denoted p ↔cu q or p ↔ q when the adjacency is understood, if • for at most u distinct indices i, |pi − qi| = 1, and • for all other indices j, pj = qj. Often, a cu-adjacency is denoted by the number of points in Z n that are cu-adjacent to a given point. E.g., • in Z1, c1-adjacency is 2-adjacency; • in Z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency; • in Z3, c1-adjacency is 8-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. An adjacency often used for Cartesian products of digital images is the normal product adjacency, denoted in the following by κ∗ and defined [2] as follows. Given digital images (X,κ) and (Y,λ) and points x,x′ ∈ X, y,y′ ∈ Y , we have (x,y) ↔κ∗ (x ′,y′) in X × Y if and only if one of the following holds. • x ↔κ x ′ and y = y′, or • x = x′ and y ↔λ y ′, or • x ↔κ x ′ and y ↔λ y ′. Other adjacencies for digital images are discussed in papers such as [16, 5, 6]. A digital interval is a digital image of the form ([a,b]Z,2), where a < b and [a,b]Z = {z ∈ Z |a ≤ z ≤ b}. Digital connectedness is defined in terms of adjacency. Definition 2.2 ([30]). A digital image (X,κ) is κ-connected (or connected when κ is understood) if given distinct x,y ∈ X there is a sequence {xi} n i=0 ⊂ X such that x = x0, xn = y, and xi ↔κ xi+1 for 0 ≤ i < n. 2.2. ℓp metric. Let X ⊂ R n and let x = (x1, . . . ,xn) and y = (y1, . . . ,yn) be points of X. Let 1 ≤ p ≤ ∞. The ℓp metric d for X is defined by d(x,y) = { ( ∑n i=1 |xi − yi| p) 1/p for 1 ≤ p < ∞; max{|xi − yi|} n i=1 for p = ∞. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 136 Remarks on fixed point assertions in digital topology For p = 1, this gives us the Manhattan metric d(x,y) = ∑n i=1 |xi − yi|; for p = 2, we have the Euclidean metric d(x,y) = ( ∑n i=1 |xi − yi| 2)1/2. Notice that for any ℓp metric d, if x,y ∈ Z n and d(x,y) < 1, then x = y. We note that digital images in the literature are commonly, although not exclusively, finite; also, the cu adjacencies or adjacencies based on them (e.g., the normal product adjacency for Cartesian products with cu adjacencies) are common. Many papers simply state an assumption that every digital image is a finite subset of Zn with some cu adjacency. Digital metric spaces typically use the Euclidean or some other ℓp metric. The development of classical topol- ogy has often placed emphasis on “wild” counterexamples, and in that setting finiteness and focus on ℓp metrics is very restrictive. But digital topology is motivated primarily by “real-world” digital images, which are represented in the real world as subsets of Z2 with either the c1 or c2 adjacency. When a metric is used in a real world digital image, it’s usually ℓ1 or ℓ2. Thus if in some results we assume that our images are finite and use the ℓp metric, these should be regarded as light assumptions. 2.3. Digital continuity and homotopy. Definition 2.3 ([30, 4]). A function f : (X,κ) → (Y,λ) between digital images is (κ,λ)-continuous (or just continuous when κ and λ are understood) if for every κ-connected subset X′ of X, f(X′) is a λ-connected subset of Y . Theorem 2.4 ([4]). A function f : (X,κ) → (Y,λ) between digital images is (κ,λ)-continuous if and only if x ↔κ x ′ in X implies either f(x) = f(x′) or f(x) ↔λ f(x ′) in Y . As in topology, the digital topology notion of homotopy can be understood as one function deforming in a continuous fashion into another. Precisely, we have the following. Definition 2.5 ([4]). Let f,g : (X,κ) → (Y,λ). We say f and g are homotopic, denoted f ≃(κ,λ) g or f ≃ g when κ and λ are understood, if there is a function F : X × [0,m]Z → Y for some m ∈ N such that • F(x,0) = f(x) and F(x,m) = g(x) for all x ∈ X. • The induced function Ft : X → Y defined by Ft(x) = F(x,t) is (κ,λ)- continuous for all t ∈ [0,m]Z. • The induced function Fx : [0,m]Z → Y defined by Fx(t) = F(x,t) is (2,λ)-continuous for all x ∈ X. 2.4. Cauchy sequences and complete metric spaces. The papers [12, 18, 20, 21, 23, 24, 26, 27] apply to digital images the notions of Cauchy sequence and complete metric space. Since for common metrics such as an ℓp metric, a digital metric space is discrete, the digital versions of these notions are quite limited. Recall that a sequence of points {xn} in a metric space (X,d) is a Cauchy sequence if for all ε > 0 there exists n0 ∈ N such that m,n > n0 implies c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 137 L. Boxer and P. C. Staecker d(xm,xn) < ε. If every Cauchy sequence in X has a limit, then (X,d) is a complete metric space. It has been shown that under a mild additional assumption, a digital Cauchy sequence is eventually constant. The following is an easy generalization of Proposition 3.6 of [18], where only the Euclidean metric was considered. The proof given in [18] is easily modified to give the following. Theorem 2.6. Let a > 0. If d is a metric on a digital image (X,κ) such that for all distinct x,y ∈ X we have d(x,y) > a, then for any Cauchy sequence {xi} ∞ i=1 ⊂ X there exists n0 ∈ N such that m,n > n0 implies xm = xn. An immediate consequence of Theorem 2.6 is the following. Corollary 2.7 ([18]). Let (X,d,κ) be a digital metric space. If d is a metric on (X,κ) such that for all distinct x,y ∈ X we have d(x,y) > a for some constant a > 0, then any Cauchy sequence in X is eventually constant, and (X,d) is a complete metric space. Remark 2.8. It is easily seen that the hypotheses of Theorem 2.6 and Corol- lary 2.7 are satisfied for any finite digital metric space, or for a digital metric space (X,d,κ) for which the metric d is any ℓp metric. Thus, a Cauchy sequence that is not eventually constant can only occur in an infinite digital metric space with an unusual metric. Such an example is given below. Example 2.9. Let d be the metric on (N,c1) defined by d(i,j) = |1/i − 1/j|. Then {i}∞i=1 is a Cauchy sequence for this metric that does not have a limit. 2.5. Digital fixed points and approximate fixed points. The study of the fixed points of continuous self-maps is prominent in many areas of mathematics. We say a topological space X or a digital image (X,κ) has the fixed point property if every continuous (respectively, (κ,κ)-continuous) f : X → X has a fixed point, i.e., a point p ∈ X such that f(p) = p. A version of Theorem 2.10 below was proved by Rosenfeld in [30] for the case when X is a digital picture, that is, a digital image of the form Πni=1[ai,bi]Z ⊂ Z n with the cn-adjacency. For general digital images, Theorem 2.10 was proved in [7]. Theorem 2.10. A digital image (X,κ) has the fixed point property if and only if X is a singleton. This theorem led to the study in [7] of the approximate fixed point property, an idea suggested by results of [30]. An approximate fixed point [7], called an almost fixed point in [30], of a (κ,κ)-continuous function f : (X,κ) → (X,κ) is a point p ∈ X such that f(p) = p or f(p) ↔κ p. A digital image (X,κ) has the approximate fixed point property (AFPP) [7] if for every continuous f : X → X there is an approximate fixed point of f. We have rephrased the following to conform with terminology used in this paper. Theorem 2.11 (Theorem 4.1 of [30]). Every digital picture (Πni=1[ai,bi]Z,cn) has the AFPP. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 138 Remarks on fixed point assertions in digital topology In Remark 6.2 (2) of [19], the author incorrectly attributes to [30] the claim that “Every digital image (Y,8) has the AFPP.” The attribution is incorrect, since, as we have shown above, the citation should be about digital pictures, not the more general digital images. Further, the claim is false, as the following example shows. Example 2.12. Let n ≥ 4 and let Y = ({yi} n−1 i=0 ,8) ⊂ Z 2 be a digital simple closed curve with the points yi indexed circularly. Then (Y,8) does not have the AFPP. Proof. The function f : Y → Y defined by f(yi) = y(i+2) mod n is easily seen to be (8,8)-continuous and free of approximate fixed points. � 2.6. Contraction and expansion functions. We cite several fixed point theorems for digital topology that are modeled on analogs for the topology of Rn. In section 4 below, we explore limitations on many of the types of functions introduced in this section; in many cases, their inspirations in topology are not similarly limited. In the following definitions, we assume (X,d,κ) is a digital metric space and f : X → X is a function. Several of these definitions are unmodified from their inspirations in the topology of metric spaces. Definition 2.13 ([12]). If for some α ∈ (0,1) and all x,y ∈ X, d(f(x),f(y)) < αd(x,y), then f is a digital contraction map. We say α is the multiplier. Note such a function should not be confused with a digital contraction [3], a homotopy between an identity map and a constant function. Definition 2.14. If d(f(x),f(y)) ≤ α[d(x,f(x)) + d(y,f(y))] for all x,y ∈ X, where 0 < α < 1/2, we say f is a Kannan contraction map. Definition 2.15. If d(f(x),f(y)) ≤ α[d(x,f(y)) + d(y,f(x))] for all x,y ∈ X, where 0 < α < 1/2, we say f is a Chatterjea contraction map. Definition 2.16. If d(f(x),f(y)) ≤ ad(x,f(x)) + bd(y,f(y)) + cd(x,y) for all x,y ∈ X and all nonnegative a,b,c such that a + b + c < 1, then f is a Reich contraction map. Proposition 2.17. A Reich contraction map is a digital contraction map and is a Kannan contraction map. Proof. Let f be a Reich contraction map. That f is a digital contraction map follows from the observation of [28] that in Definition 2.16, we can take a = b = 0 and obtain the conclusion from Definition 2.13. That f is a Kannan contraction map follows from the observation that in Definition 2.16, we can c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 139 L. Boxer and P. C. Staecker take a = b ∈ (0,1/2) and c = 0 to obtain the conclusion from Definition 2.14. � Definition 2.18 ([26]). Let (X,d,κ) be a digital metric space and let f : X → X be a function. If there exists α ∈ (0,1) such that for all x,y ∈ X we have d(f(x),f(y)) ≤ α max{d(x,y), d(x,f(x)) + d(y,f(y)) 2 , d(x,f(y)) + d(y,f(x)) 2 } then f is called a Zamfirescu digital contraction. Definition 2.19 ([26]). Let (X,d,κ) be a digital metric space and let f : X → X be a function. If there exists α ∈ (0,1) such that for all x,y ∈ X we have d(f(x),f(y)) ≤ α max{d(x,y), d(x,f(x)) + d(y,f(y)) 2 ,d(x,f(x)),d(y,f(y))} then f is called a Rhoades digital contraction. Proposition 2.20. Let (X,d,κ) be a digital metric space and let f : X → X be a function. If f is a digital contraction map, then f is a Zamfirescu digital contraction and a Rhoades digital contraction. Proof. If f is a digital contraction map, then d(f(x),f(y)) ≤ αd(x,y) for all x,y ∈ X. The assertion follows from Definitions 2.18 and 2.19,. � The following is a minor generalization of a definition in [20]. Therefore, results we derive in this paper for digitally (α,κ)-uniformly locally contractive functions apply to the version in [20]. Definition 2.21. Suppose 0 ≤ α < 1. Let (X,d,κ) be a digital metric space. Let f : X → X be a function such that d(x,y) ≤ 1 implies d(f(x),f(y)) ≤ αd(x,y). Then f is called digitally (α,κ)-uniformly locally contractive. Proposition 2.22. A digital contraction map with multiplier α is a digitally (α,κ)-uniformly locally contractive map. Proof. This is obvious from Definition 2.13 and Definition 2.21. � Below, we define a set of functions Ψ that will be used in the following. Definition 2.23 ([24]). Let Ψ be a set of functions ψ : [0,∞) → [0,∞) such that for each ψ ∈ Ψ we have • ψ is nondecreasing, and • there exists k0 ∈ N, a ∈ (0,1), and a convergent series ∑ ∞ k=1 vk of non-negative terms such that k ≥ k0 implies ψ k+1(t) ≤ aψk(t) + vk for all t ∈ [0,∞), where ψk represents the k-fold composition of ψ. The following will be used later in the paper. Example 2.24. The constant function with value 0 is a member of Ψ. Definition 2.25 ([31]). Let T : X → X and α : X × X → [0,∞). We say T is α-admissible if α(x,y) ≥ 1 implies α(T(x),T(y)) ≥ 1. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 140 Remarks on fixed point assertions in digital topology Definition 2.26 ([31]). Let (X,d) be a metric space, T : X → X, α : X → X, and ψ ∈ Ψ. We say T is an α−ψ-contractive mapping if α(x,y)d(T(x),T(y)) ≤ ψ(d(x,y)) for all x,y ∈ X. Remark 2.27 ([24]). A digital contraction map f : (X,d,κ) → (X,d,κ) is an α − ψ-contractive mapping for α(x,y) = 1 and ψ(t) = λt for λ ∈ (0,1). Definition 2.28 ([24]). Let (X,d,κ) be a digital metric space, T : X → X, β : X × X → [0,∞), and ψ,φ ∈ Ψ such that ψ(d(T(x),T(y))) ≥ β(x,y)ψ(d(x,y)) + φ(d(x,y)) for all x,y ∈ X. Then T is a β − ψ − φ-expansive mapping. Depending on the choice of functions β,φ in Definition 2.28, the definition may not be very discriminating, as we see in the following. Remark 2.29. Every function T : X → X is a β − ψ − φ-expansive mapping if we take β and φ to be constant functions with value 0. Proof. The assertion follows from Example 2.24 and Definition 2.28. � Definition 2.30 ([9]). Let (X,d,κ) be a digital metric space. Let T : X → X. Then T is a weakly uniformly strict digital contraction if given ε > 0 there exists δ > 0 such that ε < d(x,y) < ε + δ implies d(T(x),T(y)) < ε for all x,y ∈ X. Definition 2.31 ([24]). Let (X,d,κ) be a complete digital metric space. Let T : X → X. If T satisfies the condition d(T(x),T(y)) ≥ kd(x,y) for all x,y ∈ X and some k > 1, then T is a digital expansive mapping. Example 2.32. The function T : N → N defined by T(n) = 2n is a digital expansive mapping, using the usual Euclidean metric. This map is not c1- continuous [30]. The literature contains the following theorems concerning fixed points for such functions. The following is a digital version of the Banach contraction principle [1]. Theorem 2.33 ([12]). Let (X,d,κ) be a complete digital metric space, where d is the Euclidean metric in Zn. Let f : X → X be a digital contraction map. Then f has a unique fixed point. The following is a digital version of the Kannan fixed point theorem [25]. Theorem 2.34 ([27]). Let f : X → X be a Kannan contraction map on a digital metric space (X,d,κ). Then f has a unique fixed point in X. The following is a digital version of the Chatterjea fixed point theorem [8]. Theorem 2.35 ([27]). If f : X → X is a Chatterjea contraction map on a digital metric space (X,d,κ), then f has a unique fixed point. The following gives digital versions of Zamfirescu [32] and Rhoades [29] fixed point theorems. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 141 L. Boxer and P. C. Staecker Theorem 2.36 ([26]). Let d be the Euclidean metric on Zn and let (X,d,κ) be a digital metric space. If f : X → X is a Zamfirescu digital contraction or a Rhoades contraction map, then f has a unique fixed point. The following is a digital version of the Reich fixed point theorem [28]. We give a simpler proof of the digital version than appeared in [27]. Theorem 2.37. Let f : (X,d,κ) → (X,d,κ) be a Reich contraction map on a digital metric space. Then f has a unique fixed point in X. Proof. This follows immediately from Proposition 2.17 and Theorem 2.33. � The following is a version of the Edelstein fixed point theorem [10] for digital images. Theorem 2.38 ([20]). A digitally (α,κ)-uniformly locally contractive function on a connected complete digital metric space has a unique fixed point. 3. Digital homotopy fixed point theory The paper [13] defines a digital fixed point property as follows. The digital image (X,κ) has the digital fixed point property with respect to the digital interval [0,m]Z if for all (κ∗,κ)-continuous functions f : (X × [0,m]Z,κ∗) → (X,κ), where κ∗ is the normal product adjacency for (X,κ)×([0,m]Z,2), there is a κ-path p : [0,m]Z → X of fixed points, i.e., p(t) is a fixed point of the induced function ft : X → X defined by ft(x) = f(x,t). Also, [13] defines a digital homotopy fixed point property and states that this is equivalent to the following: A digital image (X,κ) has the digital homotopy fixed point property if for each digital homotopy f : X × [0,m]Z → X there is a κ-path p : [0,m]Z → X such that for all t ∈ [0,m]Z, p(t) is a fixed point of the induced function ft. These imply triviality, as follows. Theorem 3.1. A digital image (X,κ) has the digital fixed point property and the digital homotopy fixed point property if and only if X is a singleton. Proof. Clearly a singleton has the digital homotopy fixed point property and the digital homotopy fixed point property. Conversely, if X is not a singleton, then by Theorem 2.10 there is a continuous function f : X → X that does not have a fixed point. Let F : X × [0,m]Z → X be defined by F(x,t) = f(x). Then F is both (κ∗,κ)-continuous (where κ∗ = κ∗(κ,2) is the normal product adjacency) and a homotopy, and fails to have a fixed point for any of the induced functions ft = f. Thus, (X,κ) does not have the digital fixed point property or the digital homotopy fixed point property. � 4. Results for various contraction and expansion maps 4.1. Digital contraction maps. In both of the papers [12, 20], arguments are given for the incorrect assertion that every digital contraction map is digitally continuous. Both papers present an error of confusing (topological) continuity c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 142 Remarks on fixed point assertions in digital topology of a map between metric spaces with (digital) continuity of a map between dig- ital images. We present a counterexample to this assertion; our example is also used to show that Kannan, Chatterjea, Zamfirescu, and Rhoades contraction maps need not be digitally continuous. We use the Manhattan metric for its ease of computation, but the Euclidean or other ℓp metrics could be used to obtain similar conclusions. Example 4.1. Let X = {p1 = (0,0,0,0,0), p2 = (2,0,0,0,0), p3 = (1,1,1,1,1)} ⊂ Z 5. Let f : X → X be defined by f(p1) = f(p2) = p1, f(p3) = p2. Then f is not (c5,c5)-continuous. However, with respect to the Manhattan metric d, f is • a digital contraction map, • a Kannan contraction map, • a Chatterjea contraction map, • a Zamfirescu contraction map, • a Rhoades contraction map, • a (0.45,c5)-uniformly locally contractive function, • an α − ψ-contractive mapping for α(x,y) = 1 and ψ(t) = λt for λ ∈ (0,1), • a β − ψ − φ-expansive mapping, where ψ and φ are constant functions with the value 0, • a weakly uniformly strict digital contraction. Proof. Note f is not (c5,c5)-continuous, since p1 ↔c5 p3 ↔c5 p2, but f(X) = {p1,p2} is not c5-connected. Observe that d(p1,p2) = 2, d(f(p1),f(p2)) = 0, d(p1,p3) = 5, d(f(p1),f(p3)) = 2, d(p2,p3) = 5, d(f(p2),f(p3)) = 2. Therefore, we have, for all x,y ∈ X such that x 6= y, d(f(x),f(y)) ≤ 2/5 d(x,y) < 0.45d(x,y). Therefore, f is a digital contraction map, a Zamfirescu contraction map, a Rhoades contraction map, and a (0.45,c5)-uniformly locally contractive function. Since d(f(x),f(y)) ≤ 2/5[d(x,f(x)) + d(y,f(y))] < 0.45[d(x,f(x)) + d(y,f(y))] for all x,y ∈ X such that x 6= y, f is a Kannan contraction map. Note d(f(p1),f(p2)) = 0 < 0.45[d(p1,f(p2)) + d(p2,f(p1))], d(f(p1,f(p3))) = 2 < 0.45(2 + 5) = 0.45[d(p1,f(p3)) + d(p3,f(p1)), ] d(f(p2),f(p3)) = 2 < 0.45(0 + 5) = 0.45(d(p2,f(p3)) + d(p3,f(p2))). Therefore, f is a Chatterjea contraction map. That f is an α − ψ-contractive mapping for α(x,y) = 1 and ψ(t) = λt for λ ∈ (0,1) follows from Remark 2.27. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 143 L. Boxer and P. C. Staecker Since Example 2.24 notes that the constant function with value 0 is in Ψ, it follows from Definition 2.28 that f is a β − ψ − φ-expansive mapping. It follows easily from Definition 2.30 that f is a weakly uniformly strict digital contraction. � The following generalizes Theorem 4.7(1) of [18]. We give a proof, essentially that of [18], so we can refer to it below. Theorem 4.2. Let (X,d,c1) be a digital metric space that is c1-connected, where d is any ℓp metric in Z n. Let f : X → X be a digital contraction map. Then f is a constant function. Proof. Let α ∈ (0,1) satisfy d(f(x),f(y)) ≤ αd(x,y) for all x,y ∈ X. If x ↔c1 y in X, then d(x,y) = 1, so d(f(x),f(y)) ≤ α, which implies f(x) = f(y), since every distinct pair of points in Zn has distance of at least 1. Given x0 ∈ X, for any x ∈ X there is a path P = {x0,x1, . . . ,xm = x} ⊂ X from x0 to x such that xi ↔c1 xi+1, 0 ≤ i < m. It follows from the above that f is the constant function with value f(x0). � Theorem 4.7(2) of [18] gives examples of c2-connected images with digital contraction maps that are continuous and not constant. However, modification of Theorem 4.2 yields the following. Theorem 4.3. Let (X,d,κ) be a digital metric space that is κ-connected, where, for some M1 ≥ M2 > 0 we have that x 6= y implies d(x,y) ≥ M2 and x ↔κ y implies d(x,y) ≤ M1 Let f : X → X be a digital contraction map with multiplier α such that α < M2/M1. Then f is a constant function. Proof. Let x,y ∈ X such that x ↔κ y. Then d(x,y) ≤ M1 and d(f(x),f(y)) < αd(x,y) < (M2/M1)M1 = M2. By choice of M2, f(x) = f(y). It follows as in the proof of Theorem 4.2 that f is constant. � Remark 4.4. Notice that Theorem 4.3 applies to all connected digital images (X,d,cu), where X ⊂ Z n, 1 ≤ u ≤ n, and d is any ℓp metric. 4.2. Kannan and Chatterjea contractions. Example 4.1 shows that nei- ther a Kannan contraction map nor a Chatterjea contraction map must be constant. However, we have the following. Theorem 4.5. Let (X,d,κ) be a digital metric space of finite diameter, where d is any ℓp metric. Let f : X → X be a function. If f is a Kannan con- traction map or a Chatterjea contraction map with α as in Definition 2.14 or Definition 2.15, respectively, satisfying 0 < α < 1 2 diamX , then f is a constant function. Proof. We have d(f(x),f(y)) < 1 for all x,y ∈ X, by Definition 2.14 in the case of a Kannan contraction map, and by Definition 2.15 in the case of a Chatterjea contraction map. Since d is an ℓp metric, it follows that f(x) = f(y) for all x,y ∈ X. � c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 144 Remarks on fixed point assertions in digital topology 4.3. Reich contractions. Theorem 4.6. Let (X,d,κ) be a digital metric space of finite diameter, where d is any ℓp metric. Let f : X → X be a function. If f is a Reich contraction map with a,b,c as in Definition 2.16 satisfying a,b,c ∈ (0, 1 3 diamX ), then f is a constant function. Proof. We have d(f(x),f(y)) < 1 for all x,y ∈ X. Since d is an ℓp metric, it follows that f(x) = f(y) for all x,y ∈ X. � Also, it follows from Proposition 2.17 that Theorem 4.2 and Theorem 4.3 apply to a Reich contraction map. 4.4. (α,κ)-uniformly locally contractive functions. Theorem 2.38 turns out to be a trivial result for connected digital metric spaces that use an ℓp metric, as we see below. Theorem 4.7. Let (X,d,κ) be a κ-connected digital metric space, where d is any ℓp metric. Let f : X → X be an (α,κ)-uniformly locally contractive function. Then f is a constant function. Proof. Let x0,x ∈ X. Since X is connected, there is a κ-path in X, {xi} m i=0, from x0 to x such that xm = x and xi ↔c1 xi+1 for 0 ≤ i < m. If d(xi,xi+1) ≤ 1, then d(f(xi),f(xi+1)) ≤ αd(xi,xi+1) < 1. Since d is an ℓp metric, f(xi) = f(xi+1). It follows that f is a constant function. � 4.5. Digital expansive mappings. We saw in Example 2.32 that a digital expansive mapping need not be digitally continuous. Theorem 3.2 and Corollary 3.3 of [23] hypothesize a digital expansive map- ping T : X → X that is onto. But in “real world” image processing, a digital image is finite, and therefore cannot support such a map, as shown by the following Theorems 4.8 and 4.9. Theorem 4.8. Let (X,d,κ) be a digital metric space. If X has points x0,y0 such that d(x0,y0) = diamX > 0, then there is no self-map T : X → X that is onto and a digital expansive mapping. Proof. Suppose there is a digital expansive mapping T : X → X. Let x0,y0 ∈ X be such that d(x0,y0) = diamX > 0. Then for some k > 1, (4.1) d(T(x0),T(y0)) ≥ kd(x0,y0) = k diamX > diamX. Since statement (4.1) is contradictory, the assertion follows. � Theorem 4.9. Let (X,d,κ) be a digital metric space of more than one point. If there exist x0,y0 ∈ X such that (4.2) d(x0,y0) = min{d(x,y) |x,y ∈ X,x 6= y} then there is no self-map T : X → X that is onto and a digital expansive mapping. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 145 L. Boxer and P. C. Staecker Proof. Suppose there exists a map T : X → X that is a digital expansive mapping. Let x0,y0 ∈ X be as in equation (4.2). Let k be the expansive constant of T . Since T is onto, there exist distinct x′,y′ ∈ X such that T(x′) = x0 and T(y ′) = y0. Then d(x0,y0) = d(T(x ′),T(y′)) ≥ kd(x′,y′), which contradicts our choice of x0,y0. The assertion follows. � Note Theorem 4.9 is applicable when X is finite or when d is any ℓp metric. Remark 4.10. Example 3.8 of [23] claims that the self-map T : X → X of some subset of Z given by T(n) = 2n − 1 is onto. It is easy to see that this claim is only true for X = {1}. 4.6. β − ψ − φ-expansive mappings. An analog of Theorem 2.1 of [31] is asserted as Theorem 3.2 of [24]: Let (X,d,κ) be a complete digital metric space, β : X × X → [0,∞), and let T : X → X be a β − ψ − φ-expansive mapping for some ψ,φ ∈ Ψ. If there exist functions and such that • T −1 is β-admissible; • there exists x0 ∈ X such that β(x0,T −1(x0)) ≥ 1; and • T is digitally continuous, then T has a fixed point. However, this assertion is false, as we see in the following. Example 4.11. Let X = [−1,1]2 Z \{(0,0)} ⊂ Z2. Let β = d be the Manhattan metric on Z2. Let φ and ψ be constant functions with value 0. Let T : X → X be the map defined by T(x,y) = (−x,−y). By Remark 2.29, T is a β − ψ − φ- expansive mapping. Clearly T is β-admissible. For every p ∈ X we have β(p,T −1(p)) = d(p,−p) > 1. Also, T is both c1-continuous and c2-continuous. However, T has no fixed point. Proof. It was observed in Example 2.24 that the constant function with value 0 is a member of Ψ. It is easy to see that the assertion follows. � The following is given as Theorem 3.3 (and, with another hypothesis, as Theorem 3.7) of [24]. Theorem 4.12. Let (X,d,κ) be a complete digital metric space and let T : X → X be a β − ψ − φ-expansive mapping such that for some sequence {xn} ∞ n=1 ∈ X we have β(xn,xn+1) ≥ 1 for all n and xn → y ∈ X as n → ∞, then there is a subsequence {xnk} of {xn} ∞ n=1 such that β(xnk,y) ≥ 1 for all k. Then T has a fixed point. However, we have the following. Example 4.13. If X is finite or d is an ℓp metric, and β = d, then Theorem 4.12 is vacuously true, as no such sequence {xn} ∞ n=1 ∈ X exists. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 146 Remarks on fixed point assertions in digital topology Proof. By Corollary 2.7, xn → y implies that for some k0, k > k0 implies xn = y and therefore β(xn,y) = d(xn,y) = 0. Thus, no sequence {xn} ∞ n=1 satisfies the hypotheses of Theorem 4.12. � Remark 4.14. The following assertion is stated as Theorem 3.6 of [24]. Let (X,d,κ) be a complete digital metric space. Let T : X → X be a β − ψ − φ-expansive mapping such that (4.3) ψ(d(T(x),T(y))) ≥ β(x,y)ψ(M(x,y)) + φ(M(x,y)) for all x,y ∈ X, where M(x,y) = max{d(x,y),d(x,T(x)),d(y,T(y)), d(x,T(y)) + d(y,T(x)) 2 }. Then T has a fixed point. But this assertion is false. Proof. Consider the choices of X,d,κ,T,β,ψ,φ of Example 4.11, where we saw that T is a β −ψ−φ-expansive mapping. Since ψ and φ are constant functions with value 0, (4.3) is satisfied. However, as noted at Example 4.11, T has no fixed point. � 4.7. Remarks on [9]. The publisher of [9] identified one of the authors of the current paper, L. Boxer, as a reviewer. In fact, errors and other shortcomings mentioned in Boxer’s review remain in the published version of [9]. The assertion stated as Theorem 3.1 of [9] is the following. Let (X,d,κ) be a complete metric space such that T : X → X satisfies d(T(x),T(y)) ≤ ψ(d(x,y)) for all x,y ∈ X, where ψ : [0,∞) → [0,∞) is monotone nondecreasing and ψn(t) → 0 as n → ∞. Then T has a unique fixed point. The argument offered in proof of this assertion confuses topological conti- nuity (the “ε − δ definition”) and digital continuity (preservation of connect- edness) in order to conclude that T is continuous. However, in Example 4.1, using ψ(t) = t/2, we have a function that satisfies the hypotheses above and is not digitally continuous. Further, if we add hypotheses that are often satisfied to Theorem 3.1 of [9], then T is forced to be a constant function, as seen in the following. Proposition 4.15. Let (X,d,cu) be a digital metric space, X ⊂ Z n, such that T : X → X is (cu,cu)-continuous and satisfies d(T(x),T(y)) ≤ ψ(d(x,y)) for all x,y ∈ X, where ψ : [0,∞) → [0,∞) is monotone nondecreasing and ψn(t) → 0 as n → ∞. If X is cu-connected, d is an ℓp metric, and ψ(t) < 1/u1/p for all t ∈ [0,∞), then T is a constant function. Proof. This follows easily from Theorem 4.3. � The argument given in proof of the assertion stated as Theorem 3.3 of [9] is similarly flawed. The assertion is the following. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 147 L. Boxer and P. C. Staecker Let (X,d,κ) be a complete digital metric space and T : X → X a weakly uniformly strict digital contraction mapping. Then T has a unique fixed point z. Moreover, for any x ∈ X, limn→∞T n(x) = z. As above, Example 4.1 provides a counterexample to the claim appearing in the “proof” of this assertion that a weakly uniformly strict digital contraction mapping is digitally continuous. Therefore, we must regard the assertions stated as Theorems 3.1 and 3.3 of [9] as unproven. Since these and assertions dependent on these make up all of the new assertions of the paper, we conclude that nothing new is correctly established in [9]. 5. Common fixed points of intimate maps The paper [21] obtains a result for common fixed points of intimate maps. We show in this section that the characterization of intimate maps given in [21] can be simplified, and that the primary result of [21] is rather limited. Definition 5.1 ([21]). Let (X,d,κ) be a digital metric space. Let f,g : X → X. Let α be either the liminf or the limsup operation. If for every {xn} ∞ n=1 ⊂ X such that (5.1) lim n→∞ f(xn) = lim n→∞ g(xn) = t for some t ∈ X we have, for n sufficiently large, (5.2) αd(g(f(xn)),g(xn)) ≤ αd(f(f(xn)),f(xn)) then we say f is g-intimate. Proposition 5.2. Let (X,d,κ) be a digital metric space, where d is any ℓp metric. Let f,g : X → X. Then f is g-intimate if and only if for every sequence {xn} ∞ n=1 ⊂ X satisfying statement (5.1) we have d(g(t), t) ≤ d(f(t), t). Proof. From Theorem 2.6, a sequence {xn} ∞ n=1 ⊂ X satisfying statement (5.1) has, for n sufficiently large, f(xn) = g(xn) = t. The assertion follows easily. � Theorem 5.3 ([21]). If (X,d,κ) is a digital metric space and A,B,S,T : X → X are such that a) S(X) ⊂ B(X) and T(X) ⊂ A(X); b) for some α ∈ (0,1) and all x,y ∈ X, (5.3) d(S(x),T(y)) ≤ αF(x,y) where F(x,y) = max{d(A(x),B(y)),d(A(x),S(x)),d(B(y),T(y)), d(S(x),B(y)),d(A(x),T(y))}; c) A(X) is complete; d) S is A-intimate and T is B-intimate, c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 148 Remarks on fixed point assertions in digital topology then A, B, S, and T have a unique common fixed point. But Theorem 5.3 is limited, as shown by the following. Proposition 5.4. Suppose we assume the hypotheses of Theorem 5.3, with d being an ℓp metric. Suppose (5.4) 0 < α < inf{1/F(x,y) |x,y ∈ X} or (5.5) diam(S(X) ∪ T(X)) = diamX < ∞. Then S and T are constant functions that have the same value, and, in case (5.5), X is a singleton. Proof. Inequality (5.4) and hypothesis b) of Theorem 5.3 imply that d(S(x),T(y)) < 1, hence S(x) = T(y) for all x,y ∈ X. Since d is an ℓp metric, we have, for some x0,y0 ∈ X, S(x0) = T(y) and S(x) = T(y0) for all x,y ∈ X. Hence S and T are constant functions with the same values. Statement (5.5) implies there exist x0,y0 ∈ X such that d(S(x0),T(y0)) = diamX. Since F(x0,y0) ≤ diamX, inequality (5.3) becomes diamX ≤ αdiamX, which implies diamX = 0. Thus, X is a singleton, so S and T are constant functions with the same values. � 6. Homotopy invariant fixed point theory It is natural in topology to consider the behavior of the fixed point set when a continuous function is changed by homotopy. In classical topology for nice spaces (for example the geometric realization of any finite simplicial complex), when fixed points of a certain function exist, then there is a standard construction to change the function by homotopy to increase the number of fixed points. The more interesting question is whether or not the number of fixed points can be decreased by homotopy. The following Proposition 6.1 was the key to the proof of Theorem 2.10. Proposition 6.1 ([7]). Let (X,κ) be a connected digital image of more than one point. Let x0 ↔κ x1 in X. Then the function g : X → X defined by g(x) = { x0 if x 6= x0; x1 if x = x0, is continuous and has no fixed points. Proposition 6.2. Let (X,κ) be a connected digital image of more than one point. Then any constant map f : X → X is homotopic to a map without fixed points. Proof. Let x0,x1 ∈ X with x0 ↔κ x1. Let f : X → X be the constant map with image {x0}. Let H : X × [0,1]Z → X be defined by H(x,0) = x0; H(x,1) = x0 for x 6= x0; H(x0,1) = x1. It is easy to see that H is a homotopy from f to a function g as in Proposition 6.1 without fixed points. � c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 149 L. Boxer and P. C. Staecker Let MF(f) be the minimal number of fixed points among all continuous functions homotopic to f. For example, if X has only 1 point then clearly MF(f) = 1. If X has more than 1 point and (X,κ) is contractible, then any continuous function f : X → X is homotopic to a constant function, and it follows from Proposition 6.2 that MF(f) = 0. Several examples are given in [15] of digital images (X,κ) for which no function on X is homotopic to the identity except for the identity itself. Such images are called rigid. For example, a wedge product of two loops, each having at least 5 points, is rigid. Clearly if X is a rigid digital image having n points and id denotes the identity function, then MF(id) = |X|. In classical topology, MF(f) can often be computed by Nielsen fixed point theory; see [22]. Each fixed point is assigned an integer-valued fixed point index, which can be computed homologically. When x is an isolated fixed point of f, the fixed point index of x is denoted ind(f,x). The general definition of the index is complicated, but if the space is a smooth manifold, then f can be smoothed by homotopy so that ind(f,x) = sign(det |I − dfx|) where dfx is the derivative map and I is the identity matrix. This index is a sort of multiplicity count for the fixed point: when ind(f,x) = 0 then the fixed point at x can be removed by a homotopy. The fixed point index is homotopy invariant in the following sense: if, during some homotopy f ≃ g, the fixed point x of f moves into a fixed point y of g, then ind(f,x) = ind(g,y). Furthermore, when the fixed point set of f is finite, then the sum of all the fixed point indices equals the Lefschetz number L(f), which is the alternating sum of traces of the induced maps of f in the homology groups: L(f) = ∞ ∑ i=1 (−1)i trace(f∗i : Hi(X,Q) → Hi(X,Q)). Since ind(f,x) sums to L(f), it is often said that ind(f,x) is localized version of the Lefschetz number. In Nielsen fixed point theory, the fixed points are grouped into Nielsen classes, and the number of such classes having nonzero index sum is the Nielsen number N(f). This number is a homotopy invariant satisfying N(f) ≤ MF(f), and in many cases (for example when X is a manifold of dimension different from 2), N(f) = MF(f). The 2012 paper [11] by Ege & Karaca attempts to develop a Lefschetz fixed point theorem for digital images, but the main result is incorrect, and was retracted in the 2016 paper [7]. The same authors attempted to develop a Nielsen theory in the 2017 paper [14] based on their faulty Lefschetz theory. The theory developed in [14] is also incorrect. The main problem in [14] is inherited from problems in [11], and concerns the definition of the fixed point index. Definition 3.2 of [14] states the following. Let (X,κ) be a digital image, A ⊂ X, and f : A → X a digital map. We define the fixed point index of f as ind(f) = deg(F) where F(x) = x − f(x) and x ∈ X. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 150 Remarks on fixed point assertions in digital topology This does not give a satisfactory definition of the function F . It seems moti- vated by the classical fact that ind(f,x) = sign(det |I − dfx|), but the domain and range of F are never specified. The subtraction x − f(x) is apparently performed in X, but x − f(x) need not be a member of X. Furthermore, the degree deg(F) used is inadequate because the appropriate homology groups, as defined earlier in [14], are not necessarily isomorphic to Z, which is a requirement for the definition of the degree of a function. Ege & Karaca claim to define an integer valued Nielsen number N(f) which is a homotopy invariant (Theorem 3.6) and a lower bound for MF(f) (Theorem 3.7). Their Example 3.4, claiming that if f is a constant then N(f) = 1, yields a contradiction for connected images X such that |X| > 1, since our Proposition 6.2 implies that MF(f) = 0 for such functions f. The errors in this work are not merely mistakes but indicate fundamental flaws in the theory. Anything resembling the standard homological definitions of L(f) and the fixed point index will require that the Lefschetz number and fixed point index of the constant map equal 1. This cannot be reconciled with the fact that, when X has more than 1 point, the constant map can be changed by homotopy to have no fixed points. The authors believe that any successful theory for computing MF(f) will involve techniques very different from classical Lefschetz and Nielsen theory. The setting of digital images also allows the study of the quantity XF(f), the maximum number of fixed points among all functions homotopic to f. In classical topological fixed point theory this number is typically infinite, but for a digital image with n points, clearly XF(f) ≤ n for any f. In fact our definition of XF(f) implies that f is homotopic to the identity if and only if XF(f) = n. When f is a constant, then 1 ≤ XF(f) ≤ n, and for many choices of the image X we will have XF(f) < n. We do not know if it is possible for XF(f) = 0 for any function on a connected digital image. Although many of the concepts discussed in this paper turn out to be trivial or otherwise uninteresting, the questions of computing MF(f) and XF(f) seem to be difficult and interesting, and present opportunities for further work. Vari- ations that count approximate fixed points would also be interesting objects of study. 7. Concluding remarks Although the study of fixed points, or approximate fixed points, is impor- tant in digital topology as in other branches of mathematics, it does not appear that the use of metric spaces yields useful knowledge in this area. We have seen that metric space functions introduced to study fixed points in digital topology - digital contraction maps, Kannan contraction maps, Chatterjea contraction maps, Zamfirescu contraction maps, Rhoades contraction maps, Reich contrac- tion maps, uniformly locally contractive functions, intimate functions - often turn out to be either discontinuous or constant - hence, arguably uninteresting - when the image considered is finite or when common metrics are used. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 151 L. Boxer and P. C. Staecker It appears to us that the most natural metric function to use for a connected digital image (X,κ) is the path length metric [17]: d(x,y) is the length of a shortest κ-path from x to y. 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