A Topological Coincidence Theory for Multifunctions via Homotopy

Abstract

A new simple result is presented which immediately yields the topological transversality theorem for coincidences.

1. Introduction

The topological transversality theorem of Granas [1] states that if F and G are continuous compact single valued maps and $F\cong G$ then F is essential if and only if G is essential. These concepts were generalized to multimaps (compact and noncompact) and for $\mathsf{\Phi }$–essential maps in a general setting (see [2,3] and the references therein). In this paper we approach this differently and we present a very general topological transversality theorem for coincidences.

For convenience we desribe now a class of maps one could consider in this setting. Let $X$ and $Z$ be subsets of Hausdorff topological spaces. We will consider maps $F:X\to K\left(Z\right)$; here $K\left(Z\right)$ denotes the family of nonempty compact subsets of $Z$. A nonempty topological space is said to be acyclic if all its reduced Čech homology groups over the rationals are trivial. Now $F:X\to K\left(Z\right)$ is called acyclic if $F$ has acyclic values.

2. Topological Transversality Theorem

In this paper we will consider two classes $\mathbf{A}$ and $\mathbf{B}$ of maps. These are abstract classes which include many types of maps in the literature (see Remark 1). Let $E$ be a completely regular space (i.e., a Tychonoff space) and $U$ an open subset of $E$. We let $\overline{U}$ (respectively, $\partial U$) denote the closure (respectively, the boundary) of U in E.

Definition 1.

We say $F\in A(\overline{U},E)$ if $F\in \mathbf{A}(\overline{U},E)$ and $F:\overline{U}\to K\left(E\right)$ is a upper semicontinuous (u.s.c.) compact map.

Remark 1.

Examples of $F\in \mathbf{A}(\overline{U},E)$ might be that $F:\overline{U}\to K\left(E\right)$ has convex values or $F:\overline{U}\to K\left(E\right)$ has acyclic values.

In this paper we fix a $\mathsf{\Phi }\in B(\overline{U},E)$ (i.e., $\mathsf{\Phi }\in \mathbf{B}(\overline{U},E)$ and $\mathsf{\Phi }:\overline{U}\to K\left(E\right)$ is a u.s.c. map).

Definition 2.

We say $F\in A_{\partial U}(\overline{U},E)$ if $F\in A(\overline{U},E)$ and $F\left(x\right)\cap \mathsf{\Phi }\left(x\right)=\varnothing$ for $x\in \partial U$.

Next we consider homotopy for maps in $A_{\partial U}(\overline{U},E)$. We present two interpretations.

Definition 3.

Two maps $F,G\in A_{\partial U}(\overline{U},E)$ are said to be homotopic in $A_{\partial U}(\overline{U},E)$, written $F\cong G$ in $A_{\partial U}(\overline{U},E)$, if there exists a u.s.c. compact map $\mathsf{\Psi }:\overline{U}\times [0,1]\to K\left(E\right)$ with $\mathsf{\Psi }(·,\eta (·))\in \mathbf{A}(\overline{U},E)$ for any continuous function $\eta :\overline{U}\to [0,1]$ with $\eta (\partial U)=0$, $\mathsf{\Phi }\left(x\right)\cap {\mathsf{\Psi }}_t\left(x\right)=\varnothing$ for any $x\in \partial U$ and $t\in (0,1)$ (here ${\mathsf{\Psi }}_t\left(x\right)=\mathsf{\Psi }(x,t)$), ${\mathsf{\Psi }}_0=F$ and ${\mathsf{\Psi }}_1=G$.

Remark 2.

Alternatively we could use the following definition for ≅ in $A_{\partial U}(\overline{U},E)$: $F\cong G$ in $A_{\partial U}(\overline{U},E)$ if there exists a u.s.c. compact map $\mathsf{\Psi }:\overline{U}\times [0,1]\to K\left(E\right)$ with $\mathsf{\Psi }\in \mathbf{A}(\overline{U}\times [0,1],E)$, $\mathsf{\Phi }\left(x\right)\cap {\mathsf{\Psi }}_t\left(x\right)=\varnothing$ for any $x\in \partial U$ and $t\in (0,1)$ (here ${\mathsf{\Psi }}_t\left(x\right)=\mathsf{\Psi }(x,t)$), ${\mathsf{\Psi }}_0=F$ and ${\mathsf{\Psi }}_1=G$. If we use this definition then we always assume for any map $\mathsf{\Theta }\in \mathbf{A}(\overline{U}\times [0,1],E)$ and any map $f\in \mathbf{C}(\overline{U},\overline{U}\times [0,1])$ then $\mathsf{\Theta }\circ f\in \mathbf{A}(\overline{U},E)$; here $\mathbf{C}$ denotes the class of single valued continuous functions.

Definition 4.

Let $F\in A_{\partial U}(\overline{U},E)$. We say $F$ is Φ–essential in $A_{\partial U}(\overline{U},E)$ if for every map $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$ there exists a $x\in U$ with $\mathsf{\Phi }\left(x\right)\cap J\left(x\right)\ne \varnothing$.

We now present a simple result. From this result the topological transversality theorem will be immediate. In our next theorem E will be a completely regular topological space and U will be an open subset of $E$.

Theorem 1.

Let $F\in A_{\partial U}(\overline{U},E)$ and let $G\in A_{\partial U}(\overline{U},E)$ be Φ–essential in $A_{\partial U}(\overline{U},E)$. Also suppose

$$\left\{\begin{array}{c}\mathit{for}\mathit{any}\mathit{map}J\in A_{\partial U}(\overline{U},E)\mathit{with}J{{|}_{\partial U}=F|}_{\partial U}\hfill \\ \mathit{we}\mathit{have}G\cong J\mathit{in}{A}_{\partial U}(\overline{U},E).\hfill \end{array}\right.$$

(1)

Then F is Φ–essential in $A_{\partial U}(\overline{U},E)$.

Proof. 

In the proof below we assume ≅ in $A_{\partial U}(\overline{U},E)$ is as in Definition 3. Let $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$. From (1) there exists a u.s.c. compact map $H^J:\overline{U}\times [0,1]\to K\left(E\right)$ with $H^J(·,\eta (·))\in \mathbf{A}(\overline{U},E)$ for any continuous function $\eta :\overline{U}\to [0,1]$ with $\eta (\partial U)=0$, $\mathsf{\Phi }\left(x\right)\cap H_t^J\left(x\right)=\varnothing$ for any $x\in \partial U$ and $t\in (0,1)$ (here $H_t^J\left(x\right)=H^J(x,t)$), $H_0^J=G$ and $H_1^J=J$. Let

$$K=\left\{x\in \overline{U}:\mathsf{\Phi }\left(x\right)\cap H^J(x,t)\ne \varnothing \mathrm{for}\mathrm{some}t\in [0,1]\right\}$$

and

$$D=\left\{(x,t)\in \overline{U}\times [0,1]:\mathsf{\Phi }\left(x\right)\cap H^J(x,t)\ne \varnothing \right\}.$$

Now $D\ne \varnothing$ (note G is $\mathsf{\Phi }$–essential in $A_{\partial U}(\overline{U},E)$) and D is closed (note $\mathsf{\Phi }$ and $H^J$ are u.s.c.) and so D is compact (note $H^J$ is a compact map). Let $\pi :\overline{U}\times [0,1]\to \overline{U}$ be the projection. Now $K=\pi \left(D\right)$ is closed (see Kuratowski’s theorem ([4], p. 126) and so in fact compact (recall projections are continuous). Also note $K\cap \partial U=\varnothing$ (since $\mathsf{\Phi }\left(x\right)\cap H_t^J\left(x\right)=\varnothing$ for any $x\in \partial U$ and $t\in [0,1]$) so since E is Tychonoff there exists a continuous map (called the Urysohn map) $\mu :\overline{U}\to [0,1]$ with $\mu (\partial U)=0$ and $\mu \left(K\right)=1$. Let $R\left(x\right)=H^J(x,\mu \left(x\right))$. Now $R\in A_{\partial U}(\overline{U},E)$ with ${R|}_{\partial U}{=G|}_{\partial U}$ (note if $x\in \partial U$ then $R\left(x\right)=H^J(x,0)=G\left(x\right)$ and $R\left(x\right)\cap \mathsf{\Phi }\left(x\right)=G\left(x\right)\cap \mathsf{\Phi }\left(x\right)$). Now since G is $\mathsf{\Phi }$–essential in $A_{\partial U}(\overline{U},E)$ there exists a $x\in U$ with $\mathsf{\Phi }\left(x\right)\cap R\left(x\right)\ne \varnothing$ (i.e., $\mathsf{\Phi }\left(x\right)\cap H_{\mu \left(x\right)}^J\left(x\right)\ne \varnothing$). Thus $x\in K$ so $\mu \left(x\right)=1$ and $\mathsf{\Phi }\left(x\right)\cap H_1^J\left(x\right)\ne \varnothing$ that is, $\mathsf{\Phi }\left(x\right)\cap J\left(x\right)\ne \varnothing$. □

Remark 3.

(i). In the proof of Theorem 1 it is simple to adjust the proof if we use ≅ in $A_{\partial U}(\overline{U},E)$ from Remark 2 if we note $H^J(x,\mu \left(x\right))=H^J\circ g\left(x\right)$ where $g:\overline{U}\to \overline{U}\times [0,1]$ is given by $g\left(x\right)=(x,\mu (x\left)\right)$.

(ii). One could replace u.s.c. in the Definition of $A(\overline{U},E)$, $B(\overline{U},E)$, Definition 3 and Remark 2 with any condition that guarantees that K in the proof of Theorem 1 is closed; this is all that is needed if E is normal. If E is Tychonoff and not normal the one can also replace the compactness of the map in $A(\overline{U},E)$, Definition 3 and Remark 2 with any condition that guarantees that K in the proof of Theorem 1 is compact.

(iii). Theorem 1 immediately yields a general Leray–Schauder type alternative for coincidences. Let E be a completely metrizable locally convex space, U an open subset of $E$, $F\in A_{\partial U}(\overline{U},E)$, $G\in A_{\partial U}(\overline{U},E)$ is Φ–essential in $A_{\partial U}(\overline{U},E)$, $\mathsf{\Phi }\left(x\right)\cap \left[tF\right(x)+(1-t\left)G\right(x\left)\right]=\varnothing$ for $x\in \partial U$ and $t\in (0,1)$, and $\eta (·)J(·)+(1-\eta (·))G(·)\in \mathbf{A}(\overline{U},E)$ for any continuous function $\eta :\overline{U}\to [0,1]$ with $\eta (\partial U)=0$ and any map $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$. Then F is Φ–essential in $A_{\partial U}(\overline{U},E)$.

The proof is immediate from Theorem 1 since topological vector spaces are completely regular and note if $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$ then with $H^J(x,t)=tJ\left(x\right)+(1-t)G\left(x\right)$ note $H_0^J=G$, $H_1^J=J$, $H^J:\overline{U}\times [0,1]\to K\left(E\right)$ is a u.s.c. compact (see [5], Theorem 4.18) map, and $H^J(·,\eta (·))\in \mathbf{A}(\overline{U},E)$ for any continuous function $\eta :\overline{U}\to [0,1]$ and $\mathsf{\Phi }\left(x\right)\cap H_t^J\left(x\right)=\varnothing$ for $x\in \partial U$ and $t\in (0,1)$ (if $x\in \partial U$ and $t\in (0,1)$ then since ${J|}_{\partial U}{=F|}_{\partial U}$ we note that $\mathsf{\Phi }\left(x\right)\cap H_t^J\left(x\right)=\mathsf{\Phi }\left(x\right)\cap [tF\left(x\right)+(1-t)G\left(x\right)]$) so as a result $G\cong J$ in $A_{\partial U}(\overline{U},E)$ (i.e., (1) holds). (Note E being a completely metrizable locally convex space can be replaced by any (Hausdorff) topological vector space E if the space E has the property that the closed convex hull of a compact set in E is compact. In fact it is easy to see, if we argue differently, that all we need to assume is that E is a topological vector space).

With this simple result we now present the topological transversality theorem. Assume

$$\cong \mathrm{in}{A}_{\partial U}(\overline{U},E)\mathrm{is}\mathrm{an}\mathrm{equivalence}\mathrm{relation}$$

(2)

and

$$\mathrm{if}F,G\in A_{\partial U}(\overline{U},E)\mathrm{with}F{{|}_{\partial U}=G|}_{\partial U}\mathrm{then}F\cong G\mathrm{in}{A}_{\partial U}(\overline{U},E).$$

(3)

In our next theorem E will be a completely regular topological space and U will be an open subset of $E$.

Theorem 2.

Assume (2) and (3) hold. Suppose F and G are two maps in $A_{\partial U}(\overline{U},E)$ with $F\cong G$ in $A_{\partial U}(\overline{U},E)$. Now F is Φ–essential in $A_{\partial U}(\overline{U},E)$ if and only if G is Φ–essential in $A_{\partial U}(\overline{U},E)$.

Proof. 

Assume G is $\mathsf{\Phi }$–essential in $A_{\partial U}(\overline{U},E)$. Let $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$. We will show $G\cong J$ in $A_{\partial U}(\overline{U},E)$ (i.e., we will show (1)) and then Theorem 1 guarantees that F is $\mathsf{\Phi }$–essential in $A_{\partial U}(\overline{U},E)$. Note $G\cong J$ in $A_{\partial U}(\overline{U},E)$ is immediate since from (3) we have $J\cong F$ in $A_{\partial U}(\overline{U},E)$ and since $F\cong G$ in $A_{\partial U}(\overline{U},E)$ then (2) guarantees that $G\cong J$ in $A_{\partial U}(\overline{U},E)$. Similarly if F is $\mathsf{\Phi }$–essential in $A_{\partial U}(\overline{U},E)$ then G is $\mathsf{\Phi }$–essential in $A_{\partial U}(\overline{U},E)$. □

Remark 4.

Suppose E is a (Hausdorff) topological vector space, U is a open convex subset of E and $F\in \mathbf{A}(\overline{U},E)$ means $F:\overline{U}\to K\left(E\right)$ has acyclic values then immediately (2) holds (we use the definition of ≅ in $A_{\partial U}(\overline{U},E)$ from Definition 3). Suppose

$$\mathit{there}\mathit{exists}a\mathit{retraction}r:\overline{U}\to \partial U.$$

(4)

(Note (4) is satisfied if E is an infinite dimensional Banach space).

Then (3) holds (we use the definition of ≅ in $A_{\partial U}(\overline{U},E)$ from Definition 3). To see this let r be in (4), $F,G\in A_{\partial U}(\overline{U},E)$ with ${F|}_{\partial U}{=G|}_{\partial U}$. Consider $F^{\star }$ given by $F^{\star }\left(x\right)=F\left(r\left(x\right)\right)$, $x\in \overline{U}$. Note $F^{\star }\left(x\right)=G\left(r\left(x\right)\right)$, $x\in \overline{U}$ since ${F|}_{\partial U}{=G|}_{\partial U}$. Now take

$$\mathsf{\Lambda }(x,\lambda )=G(2\lambda r\left(x\right)+(1-2\lambda )x)=G\circ j(x,\lambda )\mathit{for}(x,\lambda )\in \overline{U}\times \left[0,\frac{1}{2}\right]$$

(here $j:\overline{U}\times \left[0,\frac{1}{2}\right]\to \overline{U}$ (note $\overline{U}$ is convex) is given by $j(x,\lambda )=2\lambda r\left(x\right)+(1-2\lambda )x$) it is easy to see that

$$G\cong F^{\star }\mathit{in}{A}_{\partial U}(\overline{U},E);$$

note $\mathsf{\Lambda }:\overline{U}\times \left[0,\frac{1}{2}\right]\to K\left(E\right)$ is a u.s.c. compact map and also for a fixed $x\in \overline{U}$ note $\mathsf{\Lambda }(x,\mu (x\left)\right)=G\left(j\right(x,\mu \left(x\right)\left)\right)$ has acyclic values and so $\mathsf{\Lambda }(·,\eta (·))\in \mathbf{A}(\overline{U},E)$ for any continuous function $\eta :\overline{U}\to [0,1]$ with $\eta (\partial U)=0$, and finally note $\mathsf{\Phi }\left(x\right)\cap {\mathsf{\Lambda }}_t\left(x\right)=\varnothing$ for $x\in \partial U$ and $t\in \left[0,\frac{1}{2}\right]$ (note if $x\in \partial U$ and $t\in \left[0,\frac{1}{2}\right]$ then since $r\left(x\right)=x$ we have $\mathsf{\Phi }\left(x\right)\cap {\mathsf{\Lambda }}_t\left(x\right)=\mathsf{\Phi }\left(x\right)\cap G\left(x\right)$). Similarly with

$$\mathsf{\Theta }(x,\lambda )=F((2-2\lambda )r\left(x\right)+(2\lambda -1)x)\mathit{for}(x,\lambda )\in \overline{U}\times \left[\frac{1}{2},1\right]$$

it is easy to see that

$$F^{\star }\cong F\mathit{in}{A}_{\partial U}(\overline{U},E).$$

Consequently $F\cong G$ in $A_{\partial U}(\overline{U},E)$ so (3) holds.

It is easy to present examples of $\mathsf{\Phi }$–essential maps if one uses coincidence result from the literature.

In our next theorem E will be a (Hausdorff) topological space and $U$ will be an open subset of $E$.

Theorem 3.

Let $\mathsf{\Phi }\in B(\overline{U},E)$ and $F\in A_{\partial U}(\overline{U},E)$. Assume the following conditions hold:

$$\left\{\begin{array}{c}\mathit{there}\mathit{exists}a\mathit{retraction}r:E\to \overline{U}\hfill \\ \mathit{with}r\left(w\right)\in \partial U\mathit{if}w\in E\backslash U\hfill \end{array}\right.$$

(5)

and

$$\left\{\begin{array}{c}\mathit{for}\mathit{any}\mathit{map}J\in A_{\partial U}(\overline{U},E)\mathit{with}J{{|}_{\partial U}=F|}_{\partial U}\hfill \\ \left(i\right).\mathit{there}\mathit{exists}aw\in \overline{U}\mathit{with}rJ\left(w\right)\cap \mathsf{\Phi }\left(w\right)\ne \varnothing ,\mathit{and}\hfill \\ \left(\mathit{ii}\right).\mathit{there}\mathit{is}\mathit{no}z\in E\backslash U\mathit{and}y\in \overline{U}\mathit{with}z\in J\left(y\right)\mathit{and}r\left(z\right)\in \mathsf{\Phi }\left(y\right).\hfill \end{array}\right.$$

(6)

Then $F$ is Φ–essential in $A_{\partial U}(\overline{U},E)$.

Proof. 

Let $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$. Now (6) (i) implies there exists a $w\in \overline{U}$ with $rJ\left(w\right)\cap \mathsf{\Phi }\left(w\right)\ne \varnothing$. Then there exists a $z\in J\left(w\right)$ with $r\left(z\right)\in \mathsf{\Phi }\left(w\right)$. Note $z\in E\backslash U$ or $z\in U$. If $z\in E\backslash U$ then $z\in J\left(w\right)$, $w\in \overline{U}$ and $r\left(z\right)\in \mathsf{\Phi }\left(w\right)$ which contradicts (6) (ii). Thus $z\in U$ so $r\left(z\right)=z$ and as a result $z\in J\left(w\right)$ and $z(=r(z\left)\right)\in \mathsf{\Phi }\left(w\right)$ that is, $\mathsf{\Phi }\left(w\right)\cap J\left(w\right)\ne \varnothing$. □

Remark 5.

(i). Suppose $\mathsf{\Phi }=i$ (identity) and $F\in \mathbf{A}(\overline{U},E)$ means $F:\overline{U}\to K\left(E\right)$ has acyclic values. Then (6) (i) holds (i.e., there exists a $w\in \overline{U}$ with $w\in rJ\left(w\right)$) from a theorem of Eilenberg and Montgomery [6,7] (note r is continuous and J is an acyclic u.s.c. compact map).

(ii). Now let us consider (5) and (6) (ii). Now in addition assume E is a locally convex topological vector space, $0\in U$ and U an open convex subset of E. Let

$$r\left(x\right)=\frac{x}{max\{1,\mu (x\left)\right\}}\mathit{for}x\in E,$$

where μ is the Minkowski functional on $\overline{U}$ (i.e., $\mu \left(x\right)=inf\{\alpha >0:x\in \alpha \overline{U}\}$). Now (5) holds

First let $\mathsf{\Phi }=i$. If we assume a Leray–Schauder type condition

$$x\notin \lambda F\left(x\right)\mathit{for}x\in \partial U\mathit{and}\lambda \in (0,1)$$

(7)

then (6) (ii) holds. To see this let $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$. Suppose there is a $z\in E\backslash U$ and $y\in \overline{U}$ with $z\in J\left(y\right)$ and $r\left(z\right)\in \mathsf{\Phi }\left(y\right)$ (i.e $r\left(z\right)=y$ since $\mathsf{\Phi }=i$). Now

$$y=r\left(z\right)=\frac{z}{\mu \left(z\right)}\mathit{with}\mu \left(z\right)\ge 1\mathit{since}z\in E\backslash U,$$

so $y\in \lambda J\left(y\right)$ with $0<\lambda =\frac{1}{\mu \left(y\right)}\le 1$. Note $y=r\left(z\right)\in \partial U$ since $z\in E\backslash U$ so $y\in \lambda F\left(y\right)$ since ${J|}_{\partial U}{=F|}_{\partial U}$. This contradicts (7).

Next we do not assume $\mathsf{\Phi }=i$. Assume

$$\left\{\begin{array}{c}\mathit{for}\mathit{any}\mathit{map}J\in A_{\partial U}(\overline{U},E)\mathit{with}J{{|}_{\partial U}=F|}_{\partial U}\hfill \\ \mathit{if}y\in \overline{U},z\in E\backslash U\mathit{with}z\in J\left(y\right)\hfill \\ \mathit{and}r\left(z\right)\in \mathsf{\Phi }\left(y\right)\mathit{then}y\in \partial U\hfill \end{array}\right.$$

(8)

and

$$\mathsf{\Phi }\left(x\right)\cap \lambda F\left(x\right)=\varnothing \mathit{for}x\in \partial U\mathit{and}\lambda \in (0,1).$$

(9)

Then (6) (ii) holds. To see this let $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$. Suppose there is a $z\in E\backslash U$ and $y\in \overline{U}$ with $z\in J\left(y\right)$ and $r\left(z\right)\in \mathsf{\Phi }\left(y\right)$. Now (8) guarantees that $y\in \partial U$. Also $r\left(z\right)=\frac{z}{\mu \left(z\right)}$ with $\mu \left(z\right)\ge 1$, so $r\left(z\right)\in \mathsf{\Phi }\left(y\right)$ and $r\left(z\right)\in \frac{1}{\mu \left(z\right)}J\left(y\right)$. Thus $\varnothing \ne \mathsf{\Phi }\left(y\right)\cap \lambda J\left(y\right)=\mathsf{\Phi }\left(y\right)\cap \lambda F\left(y\right)$ (since ${J|}_{\partial U}{=F|}_{\partial U}$) with $0<\lambda =\frac{1}{\mu \left(y\right)}\le 1$, and this contradicts (9).

(iii). One also has a ”dual” version of Theorem 3 if we consider $Jr$ instead of $rJ$. Let $\mathsf{\Phi }\in B(E,E)$ (i.e., $\mathsf{\Phi }\in \mathbf{B}(E,E)$ and $\mathsf{\Phi }:E\to K\left(E\right)$ is a u.s.c. map), $F\in A_{\partial U}(\overline{U},E)$ and assume (5) holds. In addition suppose

$$\left\{\begin{array}{c}\mathit{for}\mathit{any}\mathit{map}J\in A_{\partial U}(\overline{U},E)\mathit{with}J{{|}_{\partial U}=F|}_{\partial U}\hfill \\ \mathit{there}\mathit{exists}aw\in E\mathit{with}Jr\left(w\right)\cap \mathsf{\Phi }\left(w\right)\ne \varnothing \hfill \end{array}\right.$$

(10)

and

$$\left\{\begin{array}{c}\mathit{there}\mathit{is}\mathit{no}y\in E\backslash U\mathit{and}z\in \partial U\mathit{with}\hfill \\ z=r\left(y\right)\mathit{and}F\left(z\right)\cap \mathsf{\Phi }\left(y\right)\ne \varnothing .\hfill \end{array}\right.$$

(11)

Then $F$ is Φ–essential in $A_{\partial U}(\overline{U},E)$.

The proof is immediate since for any $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}{=F|}_{\partial U}$ from (10) there exists a $y\in E$ with $Jr\left(y\right)\cap \mathsf{\Phi }\left(y\right)\ne \varnothing$, so if $z=r\left(y\right)$ then $J\left(z\right)\cap \mathsf{\Phi }\left(y\right)\ne \varnothing$. If $y\in E\backslash U$ then $z\in \partial U$ and $\varnothing \ne J\left(z\right)\cap \mathsf{\Phi }\left(y\right)=F\left(z\right)\cap \mathsf{\Phi }\left(y\right)$ (since ${J|}_{\partial U}{=F|}_{\partial U}$), a contradiction. Thus $y\in U$ so $z=r\left(y\right)=y$ and $J\left(y\right)\cap \mathsf{\Phi }\left(y\right)\ne \varnothing$.

In our next theorem E will be a (Hausdorff) topological space and $U$ will be an open subset of $E$.

Theorem 4.

Let $\mathsf{\Phi }\in B(E,E)$ and assume:

$$0\in \mathbf{A}(\overline{U},E)\mathit{where}0\mathit{denotes}\mathit{the}\mathit{zero}\mathit{map}$$

(12)

$$\left\{\begin{array}{c}\mathit{for}\mathit{any}\mathit{map}J\in A_{\partial U}(\overline{U},E){\mathit{with}J|}_{\partial U}=\left\{0\right\}\mathit{and}\hfill \\ R\left(x\right)=\left\{\begin{array}{c}J\left(x\right),x\in \overline{U}\hfill \\ \left\{0\right\},x\in E\backslash \overline{U},\hfill \end{array}\right.\hfill \\ \mathit{there}\mathit{exists}ay\in E\mathit{with}\mathsf{\Phi }\left(y\right)\cap R\left(y\right)\ne \varnothing \hfill \end{array}\right.$$

(13)

and

$$\mathit{there}\mathit{is}\mathit{no}z\in E\backslash U\mathit{with}\mathsf{\Phi }\left(z\right)\cap \left\{0\right\}\ne \varnothing .$$

(14)

Then the zero map is Φ–essential in $A_{\partial U}(\overline{U},E)$.

Proof. 

Note $0\in A_{\partial U}(\overline{U},E)$ (see (12) and (14)). Let $J\in A_{\partial U}(\overline{U},E)$ with ${J|}_{\partial U}=\left\{0\right\}$. Let R be as in (13) so there exists a $y\in E$ with $\mathsf{\Phi }\left(y\right)\cap R\left(y\right)\ne \varnothing$. We have two cases, namely $y\in U$ and $y\in E\backslash U$. If $y\in E\backslash U$ then $R\left(y\right)=\left\{0\right\}$ so $\mathsf{\Phi }\left(y\right)\cap \left\{0\right\}\ne \varnothing$, and this contradicts (14). Thus $y\in U$ so $\mathsf{\Phi }\left(y\right)\cap J\left(y\right)\ne \varnothing$. □

Remark 6.

(i). Suppose $F\in \mathbf{A}(\overline{U},E)$ means $F:\overline{U}\to K\left(E\right)$ has acyclic values. If $\mathsf{\Phi }\in B(E,E)$ and (13) and (14) are satisfied then Theorem 4 guarantees that zero map is Φ–essential in $A_{\partial U}(\overline{U},E)$.

Suppose E is a completely metrizable locally convex space, U is an open convex subset of E, $0\in U$, $F\in A_{\partial U}(\overline{U},E)$, $\mathsf{\Phi }\in B(E,E)$ and assume (4), (9) (namely $\mathsf{\Phi }\left(x\right)\cap \lambda F\left(x\right)=\varnothing$ for $x\in \partial U$ and $\lambda \in (0,1)$), (13) and (14) hold. Then Theorem 2 and Remark 4 guarantees that F is Φ–essential in $A_{\partial U}(\overline{U},E)$. This is immediate since a homotopy (Definition 3) from F to $\left\{0\right\}$ is $\mathsf{\Psi }(x,t)=tF\left(x\right)$ (here $t\in [0,1]$ and $x\in \overline{U}$). To see this note $\mathsf{\Psi }:\overline{U}\times [0,1]\to K\left(E\right)$ is a upper semicontinuous compact (see [5], Theorem 4.18) map and also note for a fixed $t\in [0,1]$ and a fixed $x\in \overline{U}$ that ${\mathsf{\Psi }}_t\left(x\right)$ is acyclic valued (recall homeomorphic spaces have isomorphic homology groups) so ${\mathsf{\Psi }}_t\in A_{\partial U}(\overline{U},E)$ and this immediately implies $\mathsf{\Psi }(·,\eta (·))\in \mathbf{A}(\overline{U},E)$ for any continuous function $\eta :\overline{U}\to [0,1]$, $\eta (\partial U)=0$ since for $x\in \overline{U}$ fixed note $\mathsf{\Psi }(x,\mu \left(x\right))={\mathsf{\Psi }}_{\mu \left(x\right)}\left(x\right)={\mathsf{\Psi }}_t\left(x\right)$ with $t=\mu \left(x\right)\in [0,1]$. Note E being a completely metrizable locally convex space can be replaced by any (Hausdorff) topological vector space E if the space E has the property that the closed convex hull of a compact set in E is compact. In fact it is easy to see, if we argue differently, that all we need to assume is that E is a topological vector space.

(ii). It is very easy to extend the above ideas to the $(L,T)$ Φ–essential maps in [2].

Now we consider d–$\mathsf{\Phi }$–essential maps. Let $E$ be a completely regular topological space and $U$ an open subset of $E$. For any map $F\in A(\overline{U},E)$ write $F^{\star }=I\times F:\overline{U}\to K(\overline{U}\times E)$, with $I:\overline{U}\to \overline{U}$ given by $I\left(x\right)=x$, and let

$$d:\left\{{\left(F^{\star }\right)}^{-1}\left(B\right)\right\}\cup \{\varnothing \}\to \mathsf{\Omega }$$

(15)

be any map with values in the nonempty set $\mathsf{\Omega }$ where $B=\left\{(x,\mathsf{\Phi }\left(x\right)):x\in \overline{U}\right\}$.

Definition 5.

Let $F\in A_{\partial U}(\overline{U},E)$ and write $F^{\star }=I\times F$. We say $F^{\star }:\overline{U}\to K(\overline{U}\times E)$ is d–Φ–essential if for every map $J\in A_{\partial U}(\overline{U},E)$ (write $J^{\star }=I\times J$) with ${J|}_{\partial U}{=F|}_{\partial U}$ we have that $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$.

Remark 7.

If $F^{\star }$ is $d$–Φ–essential then

$$\varnothing \ne {\left(F^{\star }\right)}^{-1}\left(B\right)=\{x\in \overline{U}:(x,F\left(x\right))\cap (x,\mathsf{\Phi }\left(x\right))\ne \varnothing \},$$

so there exists a $x\in U$ with $(x,\mathsf{\Phi }(x\left)\right)\cap (x,F(x\left)\right)\ne \varnothing$ (i.e., $\mathsf{\Phi }\left(x\right)\cap F\left(x\right)\ne \varnothing$).

In our next theorem E will be a completely regular topological space and U will be an open subset of $E$.

Theorem 5.

Let $B=\left\{(x,\mathsf{\Phi }\left(x\right)):x\in \overline{U}\right\}$, d is defined in (15), $F\in A_{\partial U}(\overline{U},E)$ and $G\in A_{\partial U}(\overline{U},E)$ (write $F^{\star }=I\times F$ and $G^{\star }=I\times G$). Suppose $G^{\star }$ is d–Φ–essential and

$$\left\{\begin{array}{c}\mathit{for}\mathit{any}\mathit{map}J\in A_{\partial U}(\overline{U},E)\mathit{with}J{{|}_{\partial U}=F|}_{\partial U}\hfill \\ \mathit{we}\mathit{have}G\cong J\mathit{in}{A}_{\partial U}(\overline{U},E)\mathit{and}\hfill \\ d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(G^{\star }\right)}^{-1}\left(B\right)\right).\hfill \end{array}\right.$$

(16)

Then $F^{\star }$ is d–Φ–essential.

Proof. 

In the proof below we assume ≅ in $A_{\partial U}(\overline{U},E)$ is as in Definition 3. Consider any map $J\in A_{\partial U}(\overline{U},E)$ (write $J^{\star }=I\times J$) and ${J|}_{\partial U}{=F|}_{\partial U}$. From (16) there exists a u.s.c. compact map $H^J:\overline{U}\times [0,1]\to K\left(E\right)$ with $H^J(·,\eta (·))\in \mathbf{A}(\overline{U},E)$ for any continuous function $\eta :\overline{U}\to [0,1]$ with $\eta (\partial U)=0$, $\mathsf{\Phi }\left(x\right)\cap H_t^J\left(x\right)=\varnothing$ for any $x\in \partial U$ and $t\in (0,1)$ (here $H_t^J\left(x\right)=H^J(x,t)$), $H_0^J=G$, $H_1^J=J$ and $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(G^{\star }\right)}^{-1}\left(B\right)\right)$. Let ${\left(H^J\right)}^{\star }:\overline{U}\times [0,1]\to K(\overline{U}\times E)$ be given by ${\left(H^J\right)}^{\star }(x,t)=(x,H^J(x,t))$ and let

$$K=\left\{x\in \overline{U}:(x,\mathsf{\Phi }\left(x\right))\cap {\left(H^J\right)}^{\star }(x,t)\ne \varnothing \mathrm{for}\mathrm{some}t\in [0,1]\right\}.$$

Now $K\ne \varnothing$ is closed, compact and $K\cap \partial U=\varnothing$ so since E is Tychonoff there exists a Urysohn map $\mu :\overline{U}\to [0,1]$ with $\mu (\partial U)=0$ and $\mu \left(K\right)=1$. Let $R\left(x\right)=H^J(x,\mu \left(x\right))$ and write $R^{\star }=I\times R$. Now $R\in A_{\partial U}(\overline{U},E)$ (if $x\in \partial U$ then $\mu \left(x\right)=0$ so $R\left(x\right)=G\left(x\right)$) with ${R|}_{\partial U}{=G|}_{\partial U}$. Since $G^{\star }$ is d–$\mathsf{\Phi }$–essential then

$$d\left({\left(G^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(R^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing ).$$

(17)

Now since $\mu \left(K\right)=1$ we have

$$\begin{array}{ccc}\hfill {\left(R^{\star }\right)}^{-1}\left(B\right)& =& \left\{x\in \overline{U}:(x,\mathsf{\Phi }\left(x\right))\cap (x,H^J(x,\mu \left(x\right)))\ne \varnothing \right\}=\left\{x\in \overline{U}:(x,\mathsf{\Phi }\left(x\right))\cap (x,H^J(x,1))\ne \varnothing \right\}\hfill \\ & =& {\left(J^{\star }\right)}^{-1}\left(B\right),\hfill \end{array}$$

so from (17) we have $d\left({\left(G^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$. Now combine with the above and we have $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$. □

Also note one could adjust the proof in Theorem 5 if we use ≅ in $A_{\partial U}(\overline{U},E)$ in Remark 2.

In our next theorem E will be a completely regular topological space and U will be an open subset of $E$.

Theorem 6.

Let $B=\left\{(x,\mathsf{\Phi }\left(x\right)):x\in \overline{U}\right\}$, d is defined in (15) and assume (2) and (3) hold. Suppose F and G are two maps in $A_{\partial U}(\overline{U},E)$ (write $F^{\star }=I\times F$ and $G^{\star }=I\times G$) and $F\cong G$ in $A_{\partial U}(\overline{U},E)$. Then $F^{\star }$ is d–Φ–essential if and only if $G^{\star }$ is d–Φ–essential.

Proof. 

In the proof below we assume ≅ in $A_{\partial U}(\overline{U},E)$ is as in Definition 3. Assume $G^{\star }$ is d–$\mathsf{\Phi }$–essential. Let $J\in A_{\partial U}(\overline{U},E)$ (write $J^{\star }=I\times J$) and ${J|}_{\partial U}{=F|}_{\partial U}$. If we show (16) then $F^{\star }$ is d–$\mathsf{\Phi }$–essential from Theorem 5. Now (3) implies $J\cong F$ in $A_{\partial U}(\overline{U},E)$ and this together with $F\cong G$ in $A_{\partial U}(\overline{U},E)$ and (2) guarantees that $G\cong J$ in $A_{\partial U}(\overline{U},E)$. It remains to show $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(G^{\star }\right)}^{-1}\left(B\right)\right)$. Note since $G\cong F$ in $A_{\partial U}(\overline{U},E)$ let $H:\overline{U}\times [0,1]\to K\left(E\right)$ be a u.s.c. compact map with $H(·,\eta (·))\in \mathbf{A}(\overline{U},E)$ for any continuous function $\eta :\overline{U}\to [0,1]$ with $\eta (\partial U)=0$, $\mathsf{\Phi }\left(x\right)\cap H_t\left(x\right)=\varnothing$ for any $x\in \partial U$ and $t\in (0,1)$ (here $H_t\left(x\right)=H(x,t)$), $H_0=G$ and $H_1=F$. Let $H^{\star }:\overline{U}\times [0,1]\to K(\overline{U}\times E)$ be given by $H^{\star }(x,t)=(x,H(x,t))$ and let

$$K=\left\{x\in \overline{U}:(x,\mathsf{\Phi }\left(x\right))\cap H^{\star }(x,t)\ne \varnothing \mathrm{for}\mathrm{some}t\in [0,1]\right\}.$$

Now $K\ne \varnothing$ and there exists a Urysohn map $\mu :\overline{U}\to [0,1]$ with $\mu (\partial U)=0$ and $\mu \left(K\right)=1$. Let $R\left(x\right)=H(x,\mu (x\left)\right)$ and write $R^{\star }=I\times R$. Now $R\in A_{\partial U}(\overline{U},E)$ with ${R|}_{\partial U}{=G|}_{\partial U}$ so since $G^{\star }$ is d–$\mathsf{\Phi }$–essential then $d\left({\left(G^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(R^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$. Now since $\mu \left(K\right)=1$ we have

$$\begin{array}{ccc}\hfill {\left(R^{\star }\right)}^{-1}\left(B\right)& =& \left\{x\in \overline{U}:(x,\mathsf{\Phi }\left(x\right))\cap (x,H(x,\mu \left(x\right)))\ne \varnothing \right\}=\left\{x\in \overline{U}:(x,\mathsf{\Phi }\left(x\right))\cap (x,H(x,1))\ne \varnothing \right\}\hfill \\ & =& {\left(F^{\star }\right)}^{-1}\left(B\right),\hfill \end{array}$$

so $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(G^{\star }\right)}^{-1}\left(B\right)\right)$. □

Also note one could adjust the proof in Theorem 6 if we use ≅ in $A_{\partial U}(\overline{U},E)$ in Remark 2.

Remark 8.

It is very easy to extend the above ideas to the $(L,T)$ d–Φ–essential maps in [3].