Coincidence theory for compact morphisms

Abstract

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

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 [1–4].

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:

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

• (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:

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

• (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.

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

• (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:

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

• (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.

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

• (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.

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

• (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.