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Coincidence theory for compact morphisms

Fixed Point Theory and Applications20172017:19

https://doi.org/10.1186/s13663-017-0613-0

  • Received: 7 July 2017
  • Accepted: 6 September 2017
  • Published:

Abstract

In this paper we present several coincidence type results for morphisms (fractions) in the sense of Gorniewicz and Granas.

Keywords

  • morphisms
  • admissible and dominated spaces
  • coincidence
  • fixed points

MSC

  • 47H10
  • 54H25
  • 55M20

1 Introduction

Morphisms (Vietoris fractions) in the sense of Gorniewicz and Granas were introduced in 1981 and coincidence spaces were discussed. In this paper, using compact morphisms, we present a variety of coincidence (and fixed point) results on particular Hausdorff topological spaces. These spaces include \(\operatorname{ES}(\mathrm{compact})\), \(\operatorname{AES}(\mathrm{compact})\), general admissible and general dominated spaces. Our theory is motivated partly by ideas in [14].

Now we present some ideas needed in Section 2. 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 \(H(X)=\{H_{q}(X)\} \) (here X is a Hausdorff topological space) is a graded vector space, \(H_{q}(X) \) being the q-dimensional C̆ech homology group with compact carriers of X. For a continuous map \(f:X \to X\), \(H(f) \) is the induced linear map \(f_{\star}=\{ f_{\star q} \} \) where \(f_{\star q}:H_{q}(X) \to H_{q}(X)\). A space X is acyclic if X is nonempty, \(H_{q}(X)=0\) for every \(q \geq1\), and \(H_{0}(X)\approx K\).

Let X, Y and Γ be Hausdorff topological spaces. A continuous single valued map \(p:\Gamma\to X \) is called a Vietoris map (written \(p: \Gamma\Rightarrow X\)) if the following two conditions are satisfied:
  1. (i)

    for each \(x\in X\), the set \(p^{-1}(x) \) is acyclic,

     
  2. (ii)

    p is a perfect map, i.e., p is closed and for every \(x\in X\) the set \(p^{-1}(x) \) is nonempty and compact.

     
Let \(D(X,Y) \) be the set of all pairs \(X \stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} Y \) 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 \(X \stackrel{p'}{\Leftarrow} \Gamma' \stackrel{q'}{\rightarrow} Y\), we write \((p,q) \sim (p',q') \) if there are continuous maps \(f:\Gamma\to\Gamma' \) and \(g:\Gamma' \to\Gamma \) such that \(q' \circ f=q\), \(p'\circ f=p\), \(q \circ g=q' \) and \(p \circ g=p'\). The equivalence class of a diagram \((p,q) \in D(X,Y) \) with respect to is denoted by
$$\phi= \{ X \stackrel{p}{\Leftarrow} \Gamma \stackrel{q}{\rightarrow} Y\}:X \to Y $$
or \(\phi=[(p,q)] \) and is called a morphism from X to Y. We let \(M(X,Y) \) be the set of all such morphisms. Note that if \((p,q), (p_{1},q_{1})\in D(X,Y) \) (where \(X \stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} Y \) and \(X \stackrel{p_{1}}{\Leftarrow} \Gamma' \stackrel{q_{1}}{\rightarrow} Y\)) and \((p,q) \sim(p_{1},q_{1}) \), then it is easy to see (use \(q \circ g=q_{1} \) and \(p \circ g=p_{1}\) where \(g:\Gamma' \to\Gamma\)) that for \(x\in X\) we have \(q_{1} (p_{1}^{-1}(x))=q (p^{-1}(x))\). For any \(\phi\in M(X,Y) \) a set \(\phi(x)=q p^{-1} (x) \) where \(\phi=[(p,q)] \) is called an image of x under the morphism ϕ. Let \(\phi\in M(X,Y) \) and let \((p,q)\) be a representative of ϕ. We define \(\phi(X) \subseteq Y\) by \(\phi(X)=q (p^{-1}(X))\). Note \(\phi(X)\) does not depend on the representative of ϕ. Now \(\phi\in M(X,Y) \) is called compact, provided the set \(\phi(X) \) is relatively compact in Y. Note we will identify a map \(f:X \to Y\) with the morphism \(f= \{ X \stackrel{\mathit{Id}_{X}}{\Leftarrow} X \stackrel{f}{\rightarrow} Y\}:X \to Y\). Let \(X \subseteq Y\). A point \(x\in X\) is called a fixed point of a morphism \(\phi\in M(X,Y) \) if \(x\in\phi(x)\).
Let \(\phi= \{ X \stackrel{p}{\Leftarrow} \Gamma \stackrel{q}{\rightarrow} Y\}:X \to Y\) be a morphism. We define the coincidence set
$$\operatorname{Coin} (p,q)=\bigl\{ y\in\Gamma: p(y)=q(y)\bigr\} . $$
We say ϕ has a coincidence provided the set \(C(\phi)=p (\operatorname{Coin} (p,q))\) is nonempty (i.e., there exists \(x\in p (\operatorname{Coin} (p,q))\), i.e., there exists \(y\in\Gamma\) with \(x=p(y)=q(y)\)). Let \((p',q')\) be another representation of ϕ, say \(\phi= \{ X \stackrel{p'}{\Leftarrow} \Gamma' \stackrel{q'}{\rightarrow} Y\}\). Note \(p (\operatorname{Coin} (p,q))=p' (\operatorname{Coin} (p',q'))\); to see this, note that if \(x\in p (\operatorname{Coin} (p,q))\), then \(x=p(y)=q(y)\) for some \(y\in\Gamma\). Now since \((p,q) \sim(p',q') \), with \(f:\Gamma\to\Gamma'\) we have \(x=q(y)=q'(f(y))\) and \(x=p(y)=p'(f(y))\) so \(f(y) \in\Gamma'\) and \(x=q'(f(y))=p'(f(y))\), i.e., \(x \in p' (\operatorname{Coin} (p',q'))\). Thus the above definition does not depend on the choice of a representation \((p,q)\). Also \(C(\phi) \neq\emptyset\) iff \(\operatorname{Coin} (p,q) \neq\emptyset\) for any representation \((p,q)\) of ϕ.
Suppose \(\phi \in M(X,X)\) (here \(\phi= \{ X \stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} X\}\)) has a coincidence point for \((p,q)\), i.e., suppose there exists \(y\in \Gamma\) with \(p(y)=q(y)\). Now since p is surjective, there exists \(w\in X\) with \(y\in p^{-1}(w)\) (note \(p(w)=y\)) and so \(w \in q (p^{-1}(w))=\phi(w)\) (note \(p p^{-1}(w)=w\) and the set \(q (p^{-1}(w)) \) is the image of w under ϕ), i.e., ϕ has a fixed point. As a result
$$p(y)=q(y),\quad y\in\Gamma\ \bigl(\text{and let } w=p(y)\bigr)\quad \Leftrightarrow\quad w \in q \bigl(p^{-1}(w)\bigr). $$
Note that if \(w\in q (p^{-1}(w))\), then there exists \(y\in p^{-1}(w)\) with \(w=q(y)\) so \(p(y) \in p p^{-1}(w)=w\) (i.e., \(p(y)=w\)) and so \(p(y)=q(y)\). In particular if the morphism \(\phi \in M(X,X)\) (here \((p,q)\) is a representation of ϕ) has a fixed point (say w, i.e., \(w\in q (p^{-1}(w))\)), then there exists \(y\in p^{-1}(w)\) with \(q(y)=p(y)\), so ϕ has a coincidence point for \((p,q)\). We can apply this argument for any representation \((p,q)\) of ϕ (recall that if \((p_{1},q_{1})\) is another representation of ϕ, then \((p,q) \sim(p_{1},q_{1}) \) and as above \(q (p^{-1}(w))=q_{1} (p_{1}^{-1}(w))\), so \(w \in q_{1} (p_{1}^{-1}(w))\), so there exists \(y_{1} \in p_{1}^{-1}(w)\) with \(q_{1}(y_{1})=p_{1}(y_{1})\)), thus \(\operatorname{Coin} (p,q) \neq\emptyset\) for any representation \((p,q)\) of ϕ, i.e., ϕ has a coincidence.

For a subset K of a topological space X, we denote by \(\operatorname{Cov}_{X} (K) \) the set of all coverings of K by open sets of X (usually we write \(\operatorname{Cov} (K)=\operatorname{Cov}_{X} (K)\)). Given a morphism \(\phi\in M(X,X)\) and \(\alpha\in \operatorname{Cov} (X)\), a point \(x\in X \) is said to be an α-fixed point of ϕ if there exists a member \(U\in\alpha \) such that \(x\in U \) and \(\phi(x) \cap U \neq\emptyset\). Given a morphism \(\phi\in M(X,X)\) (here \(\phi= \{ X \stackrel{p}{\Leftarrow} \Gamma \stackrel{q}{\rightarrow} X\}\)) and \(\alpha\in \operatorname{Cov} (X)\), a point \(y\in\Gamma\) is said to be an α-coincidence point for \((p,q)\) if there exists a member \(U\in\alpha \) with \(p(y) \in U\) and \(q(y) \in U\). We say ϕ has an α-coincidence if ϕ has an α-coincidence point for each representation \((p,q)\) of ϕ.

Let X and Y be topological spaces. Given two morphisms \(\phi\in M(X,Y)\) and \(\psi\in M(X,Y)\) and \(\alpha\in \operatorname{Cov} (Y)\), ϕ and ψ are said to be α-close if for any \(x\in X \), there exists \(U_{x} \in\alpha\) with \(\phi(x) \cap U_{x} \neq \emptyset\) and \(\psi(x) \cap U_{x} \neq\emptyset\). Recall that, given two single valued maps \(f, g:X \to Y \) and \(\alpha\in \operatorname{Cov} (Y)\), f and g are said to be α-close if for any \(x\in X \) there exists \(U_{x} \in\alpha\) containing both \(f(x) \) and \(g(x)\). Given a morphism \(\phi\in M(X,Y)\) and a single valued map \(g:X \to Y\) and \(\alpha\in \operatorname{Cov} (Y)\), ϕ and g are said to be strongly α-close if for any \(x\in X \) there exists \(U_{x} \in\alpha\) with \(\phi(x) \subseteq U_{x}\) and \(g(x) \in U_{x}\).

Let T be the Tychonoff cube (i.e., Cartesian product of copies of the unit interval). Finally we recall the following result from the literature; see [3] (see Theorem 7.6 and the proof of Theorem 5.5) or alternatively see [5] (see Corollary 6.5 and if we take the Hausdorff locally convex topological vector space E containing T we just need to note that T is a retract of E [6]).

Theorem 1.1

Let \(\phi\in M(T,T)\) be compact. Then ϕ has a coincidence.

2 Coincidence theory

By a space we mean a Hausdorff topological space. Let Q be a class of topological spaces. A space Y is an extension space for Q (written \(Y \in\operatorname{ES}(Q)\)) if, \(\forall X \in Q\) and \(\forall K \subseteq X \) closed in X, any continuous function \(f_{0}:K \to Y \) extends to a continuous function \(f:X \to Y\).

Theorem 2.1

Let \(X\in \operatorname{ES}(\mathrm{compact}) \) and \(\phi\in M(X,X) \) is compact. Then ϕ has a coincidence.

Proof

Let \(\phi= \{ X \stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} X\}: X \to X\). We know [4] that every compact space is homeomorphic to a closed subset of the Tychonoff cube T, so as a result \(K=\overline{\phi(X)}\) can be embedded as a closed subset \(K^{\star} \) of T; let \(s:K \to K^{\star} \) be a homeomorphism. Also let \(i:K \hookrightarrow X \) and \(j:K^{\star} \hookrightarrow T \) be inclusions. Now since \(X\in \operatorname{ES}(\mathrm{compact}) \) and \(i s^{-1}:K^{\star} \to X\), \(i s^{-1}\) extends to a continuous function \(h:T \to X\). Let \(\psi=j s \phi h\) and note (see [3], see (4.2)) \(\psi\in M(T,T)\) is compact. Now Theorem 1.1 guarantees that \(j s \phi h\) has a coincidence and therefore (see [3] (Lemma 6.3)) \(h j s \phi\) has a coincidence. Thus there exists a \(y\in \Gamma\) with \(h j s q(y)=p(y)\); note \(h j s \phi= \{ X \stackrel{\overline{p}}{\Leftarrow} \Gamma\boxtimes K \stackrel{\overline{q}}{\rightarrow} X\}\) where \(\Gamma\boxtimes K=\{(z_{1},z_{2}) \in\Gamma\times K: q(z_{1})=z_{2}\}\), \(\overline{p}(z_{1},z_{2})=p f_{1}(z_{1},z_{2})\), \(\overline{q}(z_{1},z_{2})=h j s f_{2}(z_{1},z_{2})\), \(f_{1}(z_{1},z_{2})=z_{1}\) and \(f_{2}(z_{1},z_{2})=z_{2}\) so \(\overline{p}(z_{1},z_{2})=p(z_{1})\) and \(\overline{q}(z_{1},z_{2})=h j s (z_{2})=h j s q(z_{1})\). Also note \(h j(z)=i s^{-1}(z)\) for \(z\in K^{\star} \) so \(h j s (w)=i(w)=w\) for \(w\in K\). Consequently \(q(y)=p(y)\), so ϕ has a coincidence point (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ, so \(C(\phi) \neq \emptyset\). □

A space Y is an approximate extension space for Q (written \(Y \in\operatorname{AES}(Q)\)) if, \(\forall \alpha\in \operatorname{Cov} (Y)\), \(\forall X \in Q\), \(\forall K \subseteq X \) closed in X, and any continuous function \(f_{0}:K \to Y\), there exists a continuous function \(f:X \to Y \) such that \(f|_{K} \) is α-close to \(f_{0}\).

Theorem 2.2

Let \(X\in \operatorname{AES}(\mathrm{compact}) \) and \(\phi\in M(X,X) \) is compact. Then for any \(\alpha\in \operatorname{Cov}_{X} (\overline{\phi(X)})\), ϕ has an α-coincidence.

Proof

Let \(\phi= \{ X \stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} X\}: X \to X\) and let \(\alpha\in \operatorname{Cov}_{X} (K) \), where \(K=\overline{\phi(X)}\). Now K can be embedded as a closed subset \(K^{\star} \) of T; let \(s:K \to K^{\star} \) be a homeomorphism. Also let \(i:K \hookrightarrow X \) and \(j:K^{\star} \hookrightarrow T \) be inclusions. Now let \(\alpha'=\alpha\cup\{X \backslash K\} \) and note \(\alpha' \) is an open covering of X. Let the continuous map \(h:T \to X \) be such that \(h|_{K^{\star}} \) and \(s^{-1} \) are \(\alpha'\)-close (guaranteed since \(X\in \operatorname{AES}(\mathrm{compact})\)). Thus (note \(\alpha'=\alpha\cup \{X \backslash K\}\)) \(h s:K \to X \) and \(i:K \to X \) are α-close. Let \(\psi= j s \phi h \) and note \(\psi \in M(T,T)\) is compact so Theorem 1.1 guarantees that \(j s \phi h\) has a coincidence. Then from [3] (Lemma 6.3), \(h j s \phi\) has a coincidence, i.e., there exists a \(y\in \Gamma\) with \(h j s q(y)=p(y)\). Now since \(h s:K \to X \) and \(i:K \to X \) are α-close there exists \(U \in \alpha \) with
$$h s\bigl(q(y)\bigr) \in U\quad\text{and}\quad i\bigl(q(y)\bigr) \in U\quad \textit{i.e.} \quad p(y)= h s\bigl(q(y)\bigr)\in U\quad \text{and}\quad q(y) \in U. $$
Thus ϕ has an α-coincidence (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ. □

Remark 2.3

One can put conditions on the space X and the morphism ϕ so that ϕ has an α-coincidence for each \(\alpha\in \operatorname{Cov}_{X} (\overline{\phi(X)})\) would guarantee that ϕ has a coincidence; for examples we refer the reader to [7] (Lemma 1.2), [3] (Lemma 6.1), [8] (Theorem 1.4 and Remark 1.6). We say \((X,\phi)\) has the α-coincidence property if ϕ, having an α-coincidence for each \(\alpha\in \operatorname{Cov}_{X} (\overline{\phi(X)})\), guarantees that ϕ has a coincidence. Thus we have: Suppose \(X\in \operatorname{AES}(\mathrm{compact})\), \(\phi\in M(X,X) \) is compact, and \((X,\phi)\) has the α-coincidence property. Then ϕ has a coincidence.

Next we generalise the above results motivated in part from Schauder projections [4]. Let W be a space. We say W is admissible if, for all compact subsets K of W, all \(\alpha\in \operatorname{Cov}_{W}(K)\), there exists a single valued continuous map \(g_{\alpha}:K \to W \) such that:
  1. (i)

    \(g_{\alpha} \) and \(i:K \hookrightarrow W \) are α-close,

     
  2. (ii)

    \(g_{\alpha}(K) \) is contained in a subset \(C_{\alpha} \subseteq W \) and \(C_{\alpha} \) has the coincidence property (i.e., any compact \(\theta\in M(C_{\alpha},C_{\alpha}) \) has a coincidence).

     

Theorem 2.4

Let W be admissible and \(\phi\in M(W,W) \) is compact.
  1. (i)

    Then, for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-coincidence.

     
  2. (ii)

    If \((W,\phi)\) has the α-coincidence property, then ϕ has a coincidence.

     

Proof

(i) Let \(\phi= \{W\stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} W\}: W \to W\) and let \(\alpha\in \operatorname{Cov}_{W} (K) \) where \(K=\overline{\phi(W)}\). Now there exists a single valued continuous map \(g_{\alpha}:K \to W \) and \(C_{\alpha}\) as described in the definition of admissible. Let \(j_{\alpha}:C_{\alpha} \hookrightarrow W \) be the inclusion and note \(g_{\alpha} \phi j_{\alpha} \in M(C_{\alpha}, C_{\alpha})\) is compact. Since \(C_{\alpha}\) has the coincidence property, there exists \(y\in p^{-1}(C_{\alpha}) \subseteq\Gamma\) with \(g_{\alpha} q(y)=p(y)\); note \(g_{\alpha} \phi j_{\alpha}= \{ C_{\alpha} \stackrel{\overline{p}}{\Leftarrow} p^{-1}(C_{\alpha}) \boxtimes K \stackrel{\overline{q}}{\rightarrow} C_{\alpha}\}\), where \(p^{-1}(C_{\alpha}) \boxtimes K=\{(z_{1},z_{2}) \in p^{-1}(C_{\alpha}) \times K: q(z_{1})=z_{2}\}\), \(\overline{p}(z_{1},z_{2})=p f_{1}(z_{1},z_{2})\), \(\overline{q}(z_{1},z_{2})=g_{\alpha} f_{2}(z_{1},z_{2})\), \(f_{1}(z_{1},z_{2})=z_{1}\) and \(f_{2}(z_{1},z_{2})=z_{2}\) so \(\overline{p}(z_{1},z_{2})=p(z_{1})\) and \(\overline{q}(z_{1},z_{2})=g_{\alpha} (z_{2})=g_{\alpha} q(z_{1})\). Since \(g_{\alpha} \) and \(i:K \hookrightarrow W \) are α-close, there exists \(U \in\alpha \) with
$$g_{\alpha} \bigl(q(y)\bigr) \in U\quad\text{and}\quad i\bigl(q(y)\bigr) \in U\quad\textit{i.e.}\quad p(y)= g_{\alpha}\bigl(q(y)\bigr)\in U \quad \text{and}\quad q(y) \in U. $$
Thus ϕ has an α-coincidence (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ.

(ii). Immediate from the definition and part (i). □

Let W be a space. We say W is general admissible if for all compact subsets K of W and all \(\alpha\in \operatorname{Cov}_{W}(K)\), there exists a \(\psi_{\alpha} \in M(K, W) \) such that:
  1. (i)

    \(\psi_{\alpha} \) and \(i:K \hookrightarrow W \) are strongly α-close,

     
  2. (ii)

    \(\psi_{\alpha}(K) \) is contained in a subset \(C_{\alpha} \subseteq W \) and \(C_{\alpha} \) has the coincidence property.

     

In our first result we will phrase it as a fixed point result but immediately after the proof we will rephrase it as a coincidence result (see Theorem 2.6).

Theorem 2.5

Let W be general admissible and \(\phi\in M(W,W) \) is compact.
  1. (i)

    Then for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-fixed point.

     
  2. (ii)

    If \((W,\phi)\) has the α-fixed point property (i.e., ϕ, having an α-fixed point for each \(\alpha\in \operatorname{Cov}_{X} (\overline{\phi(X)})\), guarantees that ϕ has a fixed point), then ϕ has a coincidence.

     

Proof

(i) Let \(\phi= \{W\stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} W\}: W \to W\) and let \(\alpha\in \operatorname{Cov}_{W} (K) \) where \(K=\overline{\phi(W)}\). Since W is general admissible, there exists a \(\psi_{\alpha} \in M(K, W) \) (here \(\psi_{\alpha}= \{K\stackrel{p_{\alpha}}{\Leftarrow} \Gamma' \stackrel{q_{\alpha}}{\rightarrow} W\}\), i.e., \((p_{\alpha},q_{\alpha})\) is a representation of \(\psi_{\alpha}\)) and \(C_{\alpha}\) as described in the definition of general admissible. Let \(j_{\alpha}:C_{\alpha} \hookrightarrow W \) be the inclusion and note \(\psi_{\alpha} \phi j_{\alpha} \in M(C_{\alpha}, C_{\alpha})\) is compact (note ϕ is compact and the map \(x \mapsto q_{\alpha}(p_{\alpha}^{-1}(x))\) is upper semicontinuous with nonempty compact values [2]). Since \(C_{\alpha}\) has the coincidence property, \(\psi_{\alpha} \phi\) has a coincidence so (see Section 1) \(\psi_{\alpha} \phi\) has a fixed point, i.e., there exists a \(w\in C_{\alpha}\) with \(w\in \psi_{\alpha}(\phi(w))\). Now \(\psi_{\alpha} \phi= \{ C_{\alpha} \stackrel{\overline{p}}{\Leftarrow} p^{-1}(C_{\alpha}) \boxtimes \Gamma' \stackrel{\overline{q}}{\rightarrow} C_{\alpha}\}\) where \(p^{-1}(C_{\alpha}) \boxtimes\Gamma'=\{(z_{1},z_{2}) \in p^{-1}(C_{\alpha}) \times\Gamma': q(z_{1})=p_{\alpha}(z_{2})\}\), \(\overline{p}(z_{1},z_{2})=p f_{1}(z_{1},z_{2})\), \(\overline{q}(z_{1},z_{2})=q_{\alpha} f_{2}(z_{1},z_{2})\), \(f_{1}(z_{1},z_{2})=z_{1}\) and \(f_{2}(z_{1},z_{2})=z_{2}\). Now [9] (2.1), [2] (Section 40) guarantee that
$$ w\in\overline{q} \bigl( (\overline{p})^{-1}(w)\bigr)= q_{\alpha} \bigl(p_{\alpha}^{-1} \bigl(q \bigl(p^{-1}(w)\bigr) \bigr)\bigr). $$
(1)
Thus there exists a \(y \in q (p^{-1}(w))=\phi(w)\) with \(w \in q_{\alpha} (p_{\alpha}^{-1}(y))=\psi_{\alpha}(y)\). Now since \(\psi_{\alpha} \) and \(i:K \hookrightarrow W \) are strongly α-close, there exists \(U \in\alpha \) with
$$\psi_{\alpha}(y) \subseteq U \quad\text{and} \quad i(y) \in U, $$
and since \(w\in\psi_{\alpha}(y)\), we have \(w\in U\) and \(y\in U\), i.e.,
$$w\in U\quad\text{and}\quad\phi(w) \cap U \neq\emptyset\quad\bigl(\text{since }y \in\phi(w)\text{ and }y\in U\bigr). $$
Thus ϕ has an α-fixed point (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ.

(ii) From part (i) we know that ϕ has an α-fixed point for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\). Now since \((W,\phi)\) has the α-fixed point property, ϕ has a fixed point and, as in Section 1, ϕ has a coincidence. □

Theorem 2.6

Let W be general admissible and \(\phi\in M(W,W) \) is compact.
  1. (i)

    Then for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-coincidence.

     
  2. (ii)

    If \((W,\phi)\) has the α-coincidence property, then ϕ has a coincidence.

     

Proof

(i) Follow the proof in Theorem 2.5 (i) to obtain (1). Then there exists a \(y \in q (p^{-1}(w))\) with \(w \in q_{\alpha} (p_{\alpha}^{-1}(y))=\psi_{\alpha}(y)\) and since \(\psi_{\alpha} \) and \(i:K \hookrightarrow W \) are strongly α-close, there exists \(U \in\alpha \) with \(\psi_{\alpha}(y) \in U\) and \(i(y) \in U\). Thus \(w\in U\) and \(y\in U\). Also, there exists \(a\in p^{-1}(w)\) with \(y=q(a)\) and note \(p(a) =w\), i.e., \(y=q(a)\) and \(p(a)=w\). As a result
$$p(a) \in U\quad\text{and}\quad q(a) \in U, $$
so ϕ has an α-coincidence (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ.

(ii) Immediate from the definition and part (i). □

Let W be a space and C a space with the coincidence property, (i.e., any compact \(\theta\in M(C,C)\) has a coincidence). We say C dominates W if, for every compact subset K of W and every \(\alpha\in \operatorname{Cov}_{W} (K) \), there exist single valued continuous maps \(s_{\alpha}:W \to C\), \(r_{\alpha}:C \to W\) with \(r_{\alpha} s_{\alpha}: K \to W \) and \(i:K \hookrightarrow W \) α-close.

Theorem 2.7

Let W be a space and C a space with the coincidence property. Suppose \(\phi \in M(W,W) \) is compact and C dominates W. Then, for any \(\alpha \in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-coincidence.

Proof

Let \(\phi= \{W\stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} W\}: W \to W\) and let \(\alpha\in \operatorname{Cov}_{W} (K) \), where \(K=\overline{\phi(W)}\). Since C dominates W there exist single valued continuous maps \(s_{\alpha}:W \to C\), \(r_{\alpha}:C \to W\) as described in the definition. Note \(s_{\alpha} \phi r_{\alpha} \in M(C,C)\) is compact. Since C has the coincidence property, \(s_{\alpha} \phi r_{\alpha}\) has a coincidence and therefore (see [3] (Lemma 6.3)) \(r_{\alpha} s_{\alpha} \phi\) has a coincidence. Thus there exists \(y\in\Gamma\) with \(r_{\alpha} s_{\alpha} q(y)=p(y)\); note \(r_{\alpha} s_{\alpha} \phi= \{ W \stackrel{\overline{p}}{\Leftarrow} \Gamma\boxtimes K \stackrel{\overline{q}}{\rightarrow} W\}\) where \(\Gamma\boxtimes K=\{(z_{1},z_{2}) \in\Gamma\times K: q(z_{1})=z_{2}\}\), \(\overline{p}(z_{1},z_{2})=p f_{1}(z_{1},z_{2})\), \(\overline{q}(z_{1},z_{2})=r_{\alpha} s_{\alpha} f_{2}(z_{1},z_{2})\), \(f_{1}(z_{1},z_{2})=z_{1}\) and \(f_{2}(z_{1},z_{2})=z_{2}\), so \(\overline{p}(z_{1},z_{2})=p(z_{1})\) and \(\overline{q}(z_{1},z_{2})=r_{\alpha} s_{\alpha} (z_{2})=r_{\alpha} s_{\alpha} q(z_{1})\). Since \(r_{\alpha} s_{\alpha}: K \to W \) and \(i:K \hookrightarrow W \) are α-close, there exists \(U \in \alpha \) with
$$r_{\alpha} s_{\alpha} q(y)\in U\quad \text{and}\quad i(q) \in U \quad\textit{i.e.} \quad p(y)=r_{\alpha} s_{\alpha} q(y)\in U \quad\text{and}\quad q(y) \in U. $$
Thus ϕ has an α-coincidence (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ. □

Let W be a space and C a space with the coincidence property. We say C generally dominates W if, for every compact subset K of W and every \(\alpha\in \operatorname{Cov}_{W} (K) \), there exist \(S_{\alpha} \in M(W,C)\) and \(R_{\alpha} \in M(C,W)\) with \(R_{\alpha} S_{\alpha} \in M(K,W)\) and \(i:K \hookrightarrow W \) strongly α-close.

In our first result we will phrase it as a fixed point result but immediately after the proof we will rephrase it as a coincidence result (see Theorem 2.9).

Theorem 2.8

Let W be a space and C a space with the coincidence property. Suppose \(\phi \in M(W,W) \) is compact and C generally dominates W. Then for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-fixed point.

Proof

Let \(\phi= \{W\stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} W\}: W \to W\) and let \(\alpha\in \operatorname{Cov}_{W} (K) \), where \(K=\overline{\phi(W)}\). Since C generally dominates W, there exist \(S_{\alpha} \in M(W,C)\) and \(R_{\alpha}\in M(C,W)\) as described in the definition. Note with \(S_{\alpha}= \{W\stackrel{p_{1}}{\Leftarrow} \Gamma_{1} \stackrel{q_{1}}{\rightarrow} C\}\), \(R_{\alpha}= \{C\stackrel{p_{2}}{\Leftarrow} \Gamma_{2} \stackrel{q_{2}}{\rightarrow} W\}\) then [3], (4.2), guarantee we have \(R_{\alpha}S_{\alpha}= \{ W\stackrel{p_{\alpha}}{\Leftarrow} \Gamma_{1} \boxtimes\Gamma_{2} \equiv\Gamma_{3} \stackrel{q_{\alpha}}{\rightarrow} W\}\), where \(\Gamma_{1} \boxtimes \Gamma_{2}=\{(z_{1},z_{2}) \in\Gamma_{1} \times \Gamma_{2}: q_{1}(z_{1})=p_{2}(z_{2})\}\), \(p_{\alpha}(z_{1},z_{2})=p_{1} f_{1}(z_{1},z_{2})\), \(q_{\alpha}(z_{1},z_{2})=q_{2} f_{2}(z_{1},z_{2})\), \(f_{1}(z_{1},z_{2})=z_{1}\) and \(f_{2}(z_{1},z_{2})=z_{2}\). Note \(S_{\alpha} \phi R_{\alpha} \in M(C,C)\) is compact (note ϕ is compact and the map \(x \mapsto q_{1}(p_{1}^{-1}(x))\) is upper semicontinuous with nonempty compact values [2]). Since C has the coincidence property, \(S_{\alpha} \phi R_{\alpha}\) has a coincidence and therefore (see [3] (Lemma 6.3)) \(R_{\alpha} S_{\alpha} \phi\) has a coincidence. From Section 1 note \(R_{\alpha}S_{\alpha}\phi\) has a fixed point, i.e., there exists a \(w\in W\) with \(w\in R_{\alpha}S_{\alpha}\phi(w)\). Next note \(R_{\alpha}S_{\alpha}\phi= \{ W\stackrel{\overline{p}}{\Leftarrow} \Gamma \boxtimes\Gamma_{3} \stackrel{\overline{q}}{\rightarrow} W\}\), where \(\Gamma\boxtimes \Gamma_{3}=\{(z_{1},z_{2}) \in\Gamma\times \Gamma_{3}: q(z_{1})=p_{\alpha}(z_{2})\}\), \(\overline{p}(z_{1},z_{2})=p f_{1}(z_{1},z_{2})\), \(\overline{q}(z_{1},z_{2})=q_{\alpha} f_{2}(z_{1},z_{2})\), \(f_{1}(z_{1},z_{2})=z_{1}\) and \(f_{2}(z_{1},z_{2})=z_{2}\). Now it is easy to note (see [9] (2.1), [2] (Section 40)) that
$$ w\in\overline{q} \bigl( (\overline{p})^{-1}(w)\bigr)= q_{\alpha} \bigl(p_{\alpha}^{-1} \bigl(q \bigl(p^{-1}(w)\bigr) \bigr)\bigr). $$
(2)
Thus there exists a \(y \in q (p^{-1}(w))=\phi(w)\) with \(w \in q_{\alpha} (p_{\alpha}^{-1}(y))=R_{\alpha}S_{\alpha}(y)\). Now since \(R_{\alpha}S_{\alpha}\) and \(i:K \hookrightarrow W \) are strongly α-close, there exists \(U \in\alpha \) with
$$R_{\alpha}S_{\alpha}(y) \subseteq U\quad\text{and}\quad i(y) \in U, $$
and since \(w\in R_{\alpha}S_{\alpha}(y)\), we have \(w\in U\) and \(y\in U\), i.e.,
$$w\in U\quad\text{and}\quad\phi(w) \cap U \neq\emptyset\quad\bigl(\text{since }y \in\phi(w)\text{ and }y\in U\bigr). $$
Then ϕ has an α-fixed point (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ. □

Theorem 2.9

Let W be a space and C a space with the coincidence property. Suppose \(\phi \in M(W,W) \) is compact and C generally dominates W. Then for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-coincidence.

Proof

Follow the proof in Theorem 2.8 to obtain (2). Then there exists a \(y \in q (p^{-1}(w))\) with \(w \in q_{\alpha} (p_{\alpha}^{-1}(y))=R_{\alpha}S_{\alpha}(y)\) and, since \(R_{\alpha}S_{\alpha} \) and \(i:K \hookrightarrow W \) are strongly α-close, there exists \(U \in\alpha \) with \(R_{\alpha}S_{\alpha}(y) \in U\) and \(i(y) \in U\). Thus \(w\in U\) and \(y\in U\). Also there exists \(a\in p^{-1}(w)\) with \(y=q(a)\) and note \(p(a) =w\), i.e., \(y=q(a)\) and \(p(a)=w\). As a result \(p(a) \in U \) and \(q(a) \in U\), so ϕ has an α-coincidence (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ. □

3 Conclusions

In this paper, using new ideas, we present a number of coincidence and α-coincidence results for compact morphisms (Vietoris fractions) defined on a variety of admissible and dominating type spaces.

Notes

Declarations

Authors’ contributions

The author read and approved the final manuscript.

Competing interests

The author declares that he has no competing interests.

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Authors’ Affiliations

(1)
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

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