Some General Theorems for Compact Acyclic Multifunctions

Abstract

We present general Leray-Schauder type theorems for compact acyclic Multifunctions, using the topological transversality theorem by the author.

1. Introduction

If F and G are continuous compact single valued maps and $F\cong G$, then F is essential [1] if and only if G is essential. This result was extended to multimaps in a variety of settings; see [2,3] and the references therein. In this paper, using the topological transversality by the author for compact acyclic maps [2,4,5,6,7], we establish a variety of Leray-Schauder type theorems which are useful from an application viewpoint. Please note that essential maps automatically generate fixed point results for these maps. As a result, our theory can be applied when considering variation methods, iteration methods, perturbation methods, degree theory methods, upper and lower solution methods, etc. (see for example [8,9,10]). Many problems which arise naturally in applications can be formulated in the form $x\in Fx$ and we can relate it to a simpler problem via the family of problems $x\in \lambda Fx$, $0\le \lambda \le 1$. If the zero map is essential then under appropriate conditions (see our Leray-Schauder type alternatives) F will be essential (so it automatically has a fixed point).

Let $H$ be the Čech 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. Now $H\left(X\right)=\left\{H_q\left(X\right)\right\}$ (where X is a Hausdorff topological space) is a graded vector space, $H_q\left(X\right)$ being the $q$-dimensional Čech homology group with compact carriers of $X$. For a continuous map $f:X\to X$, $H\left(f\right)$ is the induced linear map $f_{\star }=\left\{f_{\star q}\right\}$ where $f_{\star q}:H_q\left(X\right)\to H_q\left(X\right)$. A space X is called acyclic if X is nonempty, $H_q\left(X\right)=0$ for every $q\ge 1$, and $H_0\left(X\right)\approx K$. Let $X$ and $Z$ be subsets of Hausdorff topological spaces. Consider a map $F:X\to K\left(Z\right)$ where $K\left(Z\right)$ denotes the family of nonempty compact subsets of $Z$. Now $F:X\to K\left(Z\right)$ is called acyclic if $F$ is upper semicontinuous (u.s.c.) with acyclic values.

2. Topological Transversality Theorem

We begin this section with essential maps. Let $E$ be a completely regular topological space (i.e., a Tychonoff space) and $U$ an open subset of $E$ (now $\overline{U}$ is the closure of $U$ in $E$ and $\partial U$ the boundary of $U$ in $E$).

Definition 1.

Write$F\in AC(\overline{U},E)$if$F:\overline{U}\to K\left(E\right)$is an acyclic compact map.

Definition 2.

Write$F\in A{C}_{\partial U}(\overline{U},E)$if$F\in AC(\overline{U},E)$with$x\notin F\left(x\right)$when$x\in \partial U$.

Definition 3.

Considering two maps$F,G\in A{C}_{\partial U}(\overline{U},E)$and they are said to be homotopic in$A{C}_{\partial U}(\overline{U},E),$written$F\cong G$in$A{C}_{\partial U}(\overline{U},E)$if there exists a u.s.c. compact map$\mathsf{\Psi }:\overline{U}\times [0,1]\to K\left(E\right)$with${\mathsf{\Psi }}_t\in A{C}_{\partial U}(\overline{U},E)$for each$t\in [0,1]$,${\mathsf{\Psi }}_1=F$and${\mathsf{\Psi }}_0=G$(where${\mathsf{\Psi }}_t\left(x\right)=\mathsf{\Psi }(t,x)$).

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

Definition 4.

A map $F\in A{C}_{\partial U}(\overline{U},E)$ is essential in $A{C}_{\partial U}(\overline{U},E)$ if for any $G\in A{C}_{\partial U}(\overline{U},E)$ with ${G|}_{\partial U}{=F|}_{\partial U}$ and with $G\cong F$ in $A{C}_{\partial U}(\overline{U},E)$ there exists a $x\in U$ with $x\in G\left(x\right)$.

In [4] we established the topological transversality theorem.

Theorem 1.

Consider two maps $F$ and $G$ in $A{C}_{\partial U}(\overline{U},E)$ with $F\cong G$ in $A{C}_{\partial U}(\overline{U},E)$. Now $F$ is essential in $A{C}_{\partial U}(\overline{U},E)$ iff $G$ is essential in $A{C}_{\partial U}(\overline{U},E)$.

We now present two ways of proceeding from here.

Approach 1.

This approach is motivated from [4,7] (here we present a more general result). We consider the question: If $F$ and $G$ are two maps in $A{C}_{\partial U}(\overline{U},E)$ with ${G|}_{\partial U}{=F|}_{\partial U}$ is $F\cong G$ in $A{C}_{\partial U}(\overline{U},E)$? We will now show that this is true if E is a topological (Hausdorff) vector space, U is convex and

$$\begin{array}{c}\hfill \mathrm{there}\mathrm{exists}\mathrm{a}\mathrm{retraction}\left(\mathrm{continuous}\right)r:\overline{U}\to \partial U.\end{array}$$

(1)

[Note if E is an infinite dimensional Banach space and U is convex then [7] we know $\left(1\right)$ holds].

Let $F,G$ be two maps in $A{C}_{\partial U}(\overline{U},E)$ with ${G|}_{\partial U}{=F|}_{\partial U}$ and let r be as in $\left(1\right)$. Consider $F^{\star }$ defined 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 ${G|}_{\partial U}{=F|}_{\partial U}$. With

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

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

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

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 for any fixed $x\in \overline{U}$ and $t\in \left[0,\frac{1}{2}\right]$ note ${\mathsf{\Lambda }}_t\left(x\right)=G\left(j(x,t)\right)$ has acyclic values and finally note if $x\in \partial U$ and $\lambda \in \left[0,\frac{1}{2}\right]$ with $x\in {\mathsf{\Lambda }}_{\lambda }\left(x\right)$ then $x\in G(2\lambda x+(1-2\lambda \left)x\right)=G\left(x\right)$, a contradiction.

With

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

we have

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

Consequently $G\cong F$ in $A{C}_{\partial U}(\overline{U},E)$.

In this situation we replace Definition 4 with:

Definition 5.

A map $F\in A{C}_{\partial U}(\overline{U},E)$ is essential in $A{C}_{\partial U}(\overline{U},E)$ if any map $G\in A{C}_{\partial U}(\overline{U},E)$ with ${G|}_{\partial U}{=F|}_{\partial U}$ there exists a $x\in U$ with $x\in G\left(x\right)$.

Recall topological vector spaces are Tychonoff so from Theorem 1 we have:

Theorem 2.

Let $E$ be a topological vector space, $U$ an open convex subset of $E$ and suppose $\left(1\right)$ holds. Suppose $F,G\in A{C}_{\partial U}(\overline{U},E)$ with $F\cong G$ in $A{C}_{\partial U}(\overline{U},E)$. Now $F$ is essential (Definition 5) in $A{C}_{\partial U}(\overline{U},E)$ iff $G$ is essential (Definition 5) in $A{C}_{\partial U}(\overline{U},E)$.

From Theorem 2 we present very general Leray-Schauder type results.

Definition 6.

Write $F\in AC(E,E)$ if $F:E\to K\left(E\right)$ is an acyclic compact map.

Theorem 3.

Let $E$ be a topological vector space, $U$ an open convex subset of $E$, $\left(1\right)$ holds and $F\in A{C}_{\partial U}(\overline{U},E)$. Assume $J\in AC(E,E)$ with

$$\begin{array}{c}\hfill z\notin J\left(z\right)\mathit{for}z\in E\backslash U\end{array}$$

(2)

and

$$\begin{array}{c}\hfill \mathit{for}\mathit{any}\mathit{map}\mathsf{\Phi }\in AC(E,E)\mathit{there}\mathit{exists}y\in E\mathit{with}y\in \mathsf{\Phi }\left(y\right).\end{array}$$

(3)

Suppose $F\cong J$ in $A{C}_{\partial U}(\overline{U},E)$. Then $F$ is essential (Definition 5) in $A{C}_{\partial U}(\overline{U},E)$.

Proof. 

We show J is essential (Definition 5) in $A{C}_{\partial U}(\overline{U},E)$. If this is true then $F$ is essential (Definition 5) in $A{C}_{\partial U}(\overline{U},E)$ from Theorem 2. Let $G\in A{C}_{\partial U}(\overline{U},E)$ with ${G|}_{\partial U}{=J|}_{\partial U}$. We need to show that G has a fixed point in U. Let

$$\mathsf{\Phi }\left(x\right)=\left\{\begin{array}{c}G\left(x\right),x\in \overline{U}\hfill \\ J\left(x\right),x\in E\backslash \overline{U}.\hfill \end{array}\right.$$

Note $\mathsf{\Phi }\in AC(E,E)$ so from $\left(3\right)$ we have a $z\in E$ with $z\in \mathsf{\Phi }\left(z\right)$. Now $\left(2\right)$ yields $z\in U$ so $z\in G\left(z\right)$. □

Corollary 1.

Let $E$ be a topological vector space, $U$ an open convex subset of $E$, $\left(1\right)$ holds, $F\in A{C}_{\partial U}(\overline{U},E)$, $u_0\in U$ and suppose $\left(3\right)$ holds. Suppose $F\cong \left\{u_0\right\}$ in $A{C}_{\partial U}(\overline{U},E)$. Then $F$ is essential (Definition 5) in $A{C}_{\partial U}(\overline{U},E)$.

Proof. 

Let $J\left(x\right)=\left\{u_0\right\}$ for $x\in E$ and the result follows from Theorem 3. □

Remark 1.

(i). For spaces E which satisfy $\left(3\right)$ we refer the reader to [11].

(ii). Without loss of generality, take $u_0=0$. Note if E is a completely metrizable locally convex space and $x\notin tF\left(x\right)$ for $x\in \partial U$ and $t\in (0,1)$ then one homotopy 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 u.s.c. compact (see [12], 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 ${\mathsf{\Psi }}_t\in A{C}_{\partial U}(\overline{U},E)$. [Note E being a completely metrizable locally convex space can be replaced by any (Hausdorff) topological vector space E which has the property that the closed convex hull of a compact set in E is compact].

Approach 2.

Here we do not assume E is a topological vector space and we do not assume $\left(1\right)$. In this approach, instead of concentrating on homotopies, we will consider essential maps and spaces in general.

From Theorem 1, we present very general Leray-Schauder type results.

Definition 7.

Consider two maps $F,J\in AC(E,E)$ and they are said to be homotopic in $AC(E,E)$, written $F\cong J$ in $AC(E,E)$, if there exists a u.s.c. compact map $R:E\times [0,1]\to K\left(E\right)$ with $R_t\in AC(E,E)$ for each $t\in [0,1]$, $R_1=F$ and $R_0=J$ (where $R_t\left(x\right)=R(x,t)$).

Theorem 4.

Let $E$ be a completely regular topological space, $U$ an open subset of $E$ and suppose $F\in A{C}_{\partial U}(\overline{U},E)$. Assume $J\in AC(E,E)$ with $\left(2\right)$ holding and suppose

$$\begin{array}{c}\hfill \mathit{there}\mathit{exists}y\in U\mathit{with}y\in J\left(y\right)\end{array}$$

(4)

and

$$\begin{array}{c}\hfill \begin{array}{c}\mathit{for}\mathit{any}\mathsf{\Phi }\in AC(E,E)\mathit{with}\mathsf{\Phi }\cong J\mathit{in}AC(E,E)\hfill \\ \mathit{there}\mathit{exists}z\in E\mathit{with}z\in \mathsf{\Phi }\left(z\right).\hfill \end{array}\end{array}$$

(5)

Suppose $F\cong J$ in $A{C}_{\partial U}(\overline{U},E)$. Then $F$ is essential (Definition 4) in $A{C}_{\partial U}(\overline{U},E)$.

Proof. 

We show J is essential (Definition 4) in $A{C}_{\partial U}(\overline{U},E)$ (so then $F$ is essential (Definition 4) in $A{C}_{\partial U}(\overline{U},E)$ from Theorem 1). Let $G\in A{C}_{\partial U}(\overline{U},E)$ with ${G|}_{\partial U}{=J|}_{\partial U}$ and with $G\cong J$ in $A{C}_{\partial U}(\overline{U},E)$. We need to show G has a fixed point in U. There exists (see Definition 3) a u.s.c. compact map $\mathsf{\Theta }:\overline{U}\times [0,1]\to K\left(E\right)$ with ${\mathsf{\Theta }}_t\in A{C}_{\partial U}(\overline{U},E)$ for each $t\in [0,1]$, ${\mathsf{\Theta }}_0=J$ and ${\mathsf{\Theta }}_1=G$ and let

$$\mathsf{\Omega }=\left\{x\in \overline{U}:x\in {\mathsf{\Theta }}_t\left(x\right)\mathrm{for}\mathrm{some}t\in [0,1]\right\}.$$

Notice $\mathsf{\Omega }\ne \varnothing$ (see $\left(4\right)$) is closed and compact and $\mathsf{\Omega }\cap (E\backslash U)=\varnothing$ (note ${\mathsf{\Theta }}_t\in A{C}_{\partial U}(\overline{U},E)$ for $t\in [0,1]$). Thus, there exists a continuous map $\sigma :E\to [0,1]$ with $\sigma \left(\mathsf{\Omega }\right)=1$ and $\sigma (E\backslash U)=0$. Define $\mathsf{\Psi }:E\times [0,1]\to K\left(E\right)$ by

$$\mathsf{\Psi }(x,t)=\left\{\begin{array}{c}\mathsf{\Theta }(x,t\sigma \left(x\right)),x\in \overline{U}\hfill \\ J\left(x\right),x\in E\backslash U.\hfill \end{array}\right.$$

Note $\mathsf{\Psi }:E\times [0,1]\to K\left(E\right)$ is an upper semicontinuous compact map with ${\mathsf{\Psi }}_t\in AC(E,E)$ for each $t\in [0,1]$, so as a result ${\mathsf{\Psi }}_1\cong {\mathsf{\Psi }}_0=J$ in $AC(E,E)$. From $\left(5\right)$ we have a $x\in E$ with $x\in {\mathsf{\Psi }}_1\left(x\right)$. If $x\in E\backslash \overline{U}$ then $x\in J\left(x\right)$ which contradicts $\left(2\right)$. Consequently $x\in U$ so $x\in \mathsf{\Lambda }(x,\sigma (x\left)\right)$ and as a result $x\in \mathsf{\Omega }$ which implies $\sigma \left(x\right)=1$ and so $x\in \mathsf{\Lambda }(x,1)=G\left(x\right)$. □

Corollary 2.

Let $E$ be a completely regular topological space, $U$ an open subset of $E$, $u_0\in U$ and suppose $F\in A{C}_{\partial U}(\overline{U},E)$. Assume

$$\begin{array}{c}\hfill \begin{array}{c}\mathit{for}\mathit{any}\mathsf{\Phi }\in AC(E,E)\mathit{with}\mathsf{\Phi }\cong \left\{u_0\right\}\mathit{in}AC(E,E)\hfill \\ \mathit{there}\mathit{exists}z\in E\mathit{with}z\in \mathsf{\Phi }\left(z\right).\hfill \end{array}\end{array}$$

(6)

Suppose $F\cong \left\{u_0\right\}$ in $A{C}_{\partial U}(\overline{U},E)$. Then $F$ is essential (Definition 4) in $A{C}_{\partial U}(\overline{U},E)$.

Proof. 

Let $J\left(x\right)=\left\{u_0\right\}$ for $x\in E$ and apply Theorem 4. □

Of course if $\left(3\right)$ holds then automatically $\left(6\right)$ holds. We now give a result where $\mathsf{\Phi }\cong \left\{u_0\right\}$ in $AC(E,E)$ plays a major role.

Theorem 5.

Let $E$ be a (metrizable) ANR, $U$ an open subset of $E$, $u_0\in U$, $F\in A{C}_{\partial U}(\overline{U},E)$ and suppose $F\cong \left\{u_0\right\}$ in $A{C}_{\partial U}(\overline{U},E)$. Then $F$ is essential (Definition 4) in $A{C}_{\partial U}(\overline{U},E)$.

Proof. 

It follows immediately from Corollary 2. once we show $\left(6\right)$. Let $\mathsf{\Phi }\in AC(E,E)$ with $\mathsf{\Phi }\cong \left\{u_0\right\}$ in $AC(E,E)$, so (see Definition 7) there exists a u.s.c. compact map $R:E\times [0,1]\to K\left(E\right)$ with $R_t\in AC(E,E)$ for each $t\in [0,1]$,$R_1=\mathsf{\Phi }$ and $R_0=\left\{u_0\right\}$. Note $E$ can be regarded as a closed subset of a normed space $X$ (see the Arens-Eells theorem). Since $E\in ANR$ there is an open neighborhood $V$ of $E$ in $X$ and a retraction (continuous) $r:\overline{V}\to E$. Let $\lambda :X\to [0,1]$ be a (continuous) function with $\lambda (X\backslash V)=0$ and $\lambda \left(E\right)=1$ and let

$$Q\left(x\right)=\left\{\begin{array}{c}R(r\left(x\right),\lambda \left(x\right)),x\in \overline{V}\hfill \\ \left\{u_0\right\},x\in X\backslash V\hfill \end{array}\right.$$

(note if $x\in \partial V$ then $Q\left(x\right)=R(r\left(x\right),\lambda \left(x\right))=R(r\left(x\right),0)=R_0\left(r\left(x\right)\right)=\left\{u_0\right\}$). For fixed $x\in X$ note $Q\left(x\right)$ is acyclic valued and $Q:X\to K\left(X\right)$ is a u.s.c. compact map i.e., $Q\in AC(X,X)$. Now from [3] there exists a $x_0\in X$ with $x_0\in Q\left(x_0\right)$. If $x_0\in X\backslash V$ then $x_0\in \left\{u_0\right\}$, a contradiction (note $u_0\in E$ and $E\subseteq V$). If $x_0\in \overline{V}\backslash E$ then since $Q:X\to K\left(E\right)$ (note $R:E\times [0,1]\to K\left(E\right)$) and $x_0\in Q\left(x_0\right)$ one has $x_0\in E$, a contradiction. Thus, $x_0\in E$, $r\left(x_0\right)=x_0$, $\lambda \left(x_0\right)=1$ so $x_0\in R(x_0,1)=\mathsf{\Phi }\left(x_0\right)$ i.e., $\left(6\right)$ holds. □

Remark 2.

From the proof above, please note that one could replace E is a (metrizable) ANR with any space provided the following hold: (i). $E$ can be regarded as a closed subset of a normal space $X$, (ii). there exists an open neighborhood $V$ of $E$ in $X$ and a retraction $r:\overline{V}\to E$, and (iii). any map $\mathsf{\Psi }\in AC(X,X)$ has a fixed point in X.

One can extend the above ideas to many other natural situations. In the remainder of this section, we will consider several extensions. Let $X$ be a (Hausdorff) topological vector space (so automatically Tychonoff), $Y$ a topological vector space, and $U$ an open subset of $X$. Also $L:domL\subseteq X\to Y$ is a linear (not necessarily continuous) single valued map where $domL$ is a vector subspace of $X$ and finally let $T:X\to Y$ be a linear, continuous single valued map with $L+T:domL\to Y$ an isomorphism (i.e., a linear homeomorphism) and for convenience we say $T\in H_L(X,Y)$.

A map $F:\overline{U}\to 2^Y$ is said to be $(L,T)$ upper semicontinuous ($(L,T)$ u.s.c.) if ${(L+T)}^{-1}(F+T):\overline{U}\to K\left(X\right)$ is u.s.c. Now $F:\overline{U}\to 2^Y$ is said to be $(L,T)$ compact if ${(L+T)}^{-1}(F+T):\overline{U}\to 2^X$ is a compact map.

Definition 8.

Write $F\in AC(\overline{U},Y;L,T)$ if ${(L+T)}^{-1}(F+T)\in AC(\overline{U},X)$.

Definition 9.

Write $F\in A{C}_{\partial U}(\overline{U},Y;L,T)$ if $F\in AC(\overline{U},Y;L,T)$ with $Lx\notin F\left(x\right)$ for $x\in \partial U\cap domL$.

Definition 10.

Consider two maps $F,G\in A{C}_{\partial U}(\overline{U},Y;L,T)$ and they are said to be homotopic in $A{C}_{\partial U}(\overline{U},Y;L,T)$, written $F\cong G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$, if there exists a $(L,T)$ u.s.c., $(L,T)$ compact mapping $N:\overline{U}\times [0,1]\to 2^Y$ such that $N_t\in A{C}_{\partial U}(\overline{U},Y;L,T)$ for each $t\in [0,1]$ and $N_0=F$ with $N_1=G$ (where $N_t\left(u\right)=N(u,t)$).

Definition 11.

A map $F\in A{C}_{\partial U}(\overline{U},Y;L,T)$ is L-essential in $A{C}_{\partial U}(\overline{U},Y;L,T)$ if for any $G\in A{C}_{\partial U}(\overline{U},Y;L,T)$ with ${G|}_{\partial U}{=F|}_{\partial U}$ and with $F\cong G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$ there exists a $x\in U\cap domL$ with $Lx\in G\left(x\right)$.

In [2,4,5] we established the topological transversality theorem.

Theorem 6.

Consider maps $F$ and $G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$ with $F\cong G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. Now $F$ is L-essential in $A{C}_{\partial U}(\overline{U},Y;L,T)$ if and only if $G$ is L-essential in $A{C}_{\partial U}(\overline{U},Y;L,T)$.

We present the analogue of Theorem 2. Suppose $\left(1\right)$ holds and U is convex. Let $F,G$ be in $A{C}_{\partial U}(\overline{U},Y;L,T)$ with ${G|}_{\partial U}{=F|}_{\partial U}$. Then $F\cong G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. To see this let $F^{\star }$, $\mathsf{\Lambda }$ and $\mathsf{\Theta }$ be as before Definition 5 Note $\mathsf{\Lambda }$ and $\mathsf{\Theta }$ are $(L,T)$ u.s.c. and $(L,T)$ compact mappings and $G\cong F^{\star }$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$ (if $x\in \partial U\cap domL$ and $\lambda \in \left[0,\frac{1}{2}\right]$ with $Lx\in {\mathsf{\Lambda }}_{\lambda }\left(x\right)$ then $Lx\in G(2\lambda x+(1-2\lambda \left)x\right)=G\left(x\right)$, a contradiction) and $F^{\star }\cong F$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. Combining gives $F\cong G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$.

In this situation we replace Definition 11 with:

Definition 12.

A map $F\in A{C}_{\partial U}(\overline{U},Y;L,T)$ is L-essential in $A{C}_{\partial U}(\overline{U},Y;L,T)$ if for any $G\in A{C}_{\partial U}(\overline{U},Y;L,T)$ with ${G|}_{\partial U}{=F|}_{\partial U}$ there exists a $x\in U\cap domL$ with $Lx\in G\left(x\right)$.

From Theorem 6 we have:

Theorem 7.

Let U be convex and suppose $\left(1\right)$ holds. Consider two maps $F$ and $G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$ with $F\cong G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. Now $F$ is L-essential (Definition 12) in $A{C}_{\partial U}(\overline{U},Y;L,T)$ if and only if $G$ is L-essential (Definition 12) in $A{C}_{\partial U}(\overline{U},Y;L,T)$.

Now we present the analogue of Theorem 4.

Definition 13.

Write $F\in AC(X,Y;L,T)$ if ${(L+T)}^{-1}(F+T)\in AC(X,X)$.

Definition 14.

Consider two maps $F,J\in AC(X,Y;L,T)$ and they are said to be homotopic in $AC(X,Y;L,T)$, written $F\cong J$ in $AC(X,Y;L,T)$, if there exists a $(L,T)$ u.s.c., $(L,T)$ compact mapping $R:X\times [0,1]\to 2^Y$ with $R_t\in AC(X,Y;L,T)$ for each $t\in [0,1]$, $R_1=F$ and $R_0=J$ (where $R_t\left(x\right)=R(x,t)$).

Theorem 8.

Let $X$, $Y$, $U$, $L$ and $T$ be as above and suppose $F\in A{C}_{\partial U}(\overline{U},Y;L,T)$. Assume $J\in AC(X,Y;L,T)$ and the following hold:

$$\begin{array}{c}\hfill Lz\notin J\left(z\right)\mathit{for}z\in X\backslash (U\cap domL)\end{array}$$

(7)

$$\begin{array}{c}\hfill \mathit{there}\mathit{exists}y\in U\cap domL\mathit{with}Ly\in J\left(y\right)\end{array}$$

(8)

$$\begin{array}{c}\hfill \begin{array}{c}\mathit{for}\mathit{any}\mathsf{\Phi }\in AC(X,Y;L,T)\mathit{with}\mathsf{\Phi }\cong J\mathit{in}AC(X,Y;L,T)\hfill \\ \mathit{there}\mathit{exists}z\in X\mathit{with}z\in {(L+T)}^{-1}(\mathsf{\Phi }+T)\left(z\right).\hfill \end{array}\end{array}$$

(9)

Suppose $F\cong J$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. Then $F$ is L-essential (Definition 11) in $A{C}_{\partial U}(\overline{U},Y;L,T)$.

Proof. 

We show J is L-essential (Definition 11) in $A{C}_{\partial U}(\overline{U},Y;L,T)$ (and then apply Theorem 6). Let $G\in A{C}_{\partial U}(\overline{U},Y;L,T)$ with ${G|}_{\partial U}{=J|}_{\partial U}$ and with $G\cong J$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. We need to show there exists a $x\in U\cap domL$ with $Lx\in G\left(x\right)$. There exists (see Definition 10) a $(L,T)$ u.s.c., $(L,T)$ compact mapping $\mathsf{\Lambda }:\overline{U}\times [0,1]\to 2^Y$ with ${\mathsf{\Lambda }}_t\in A{C}_{\partial U}(\overline{U},Y;L,T)$ for each $t\in [0,1]$, ${\mathsf{\Lambda }}_0=J$ and ${\mathsf{\Lambda }}_1=G$ and let

$$\begin{array}{ccc}\hfill \mathsf{\Omega }& =& \left\{x\in \overline{U}\cap domL:Lx\in {\mathsf{\Lambda }}_t\left(x\right)\mathrm{for}\mathrm{some}t\in [0,1]\right\}\hfill \\ & =& \left\{x\in \overline{U}:x\in {(L+T)}^{-1}({\mathsf{\Lambda }}_t+T)\left(x\right)\mathrm{for}\mathrm{some}t\in [0,1]\right\}.\hfill \end{array}$$

Now $\mathsf{\Omega }\ne \varnothing$ (see $\left(8\right)$) is compact, $\mathsf{\Omega }\cap (X\backslash U)=\varnothing$, and since X is Tychonoff there exists a (continuous) map $\sigma :X\to [0,1]$ with $\sigma \left(\mathsf{\Omega }\right)=1$ and $\sigma (X\backslash U)=0$. Let $\mathsf{\Psi }:X\times [0,1]\to 2^Y$ be

$$\mathsf{\Psi }(x,t)=\left\{\begin{array}{c}\mathsf{\Lambda }(x,t\sigma \left(x\right)),x\in \overline{U}\hfill \\ J\left(x\right),x\in X\backslash U.\hfill \end{array}\right.$$

Now $\mathsf{\Psi }:X\times [0,1]\to 2^Y$ is a $(L,T)$ u.s.c., $(L,T)$ compact mapping and ${\mathsf{\Psi }}_t\in AC(X,Y;L,T)$ for each $t\in [0,1]$, so ${\mathsf{\Psi }}_1\cong {\mathsf{\Psi }}_0=J$ in $AC(X,Y;L,T)$. Now from $\left(9\right)$ there exists $x\in X$ with $x\in {(L+T)}^{-1}({\mathsf{\Psi }}_1+T)\left(x\right)$. If $x\in X\backslash (U\cap domL)$ then $Lx\in J\left(x\right)$ which contradicts $\left(7\right)$. Consequently $x\in U\cap domL$ so $Lx\in \mathsf{\Lambda }(x,\sigma (x\left)\right)$ and so $x\in \mathsf{\Omega }$, $\sigma \left(x\right)=1$ and $Lx\in \mathsf{\Lambda }(x,1)=G\left(x\right)$. □

Next we consider a generalization of essential maps, namely the d-essential maps. Let $E$ be a completely regular topological space and $U$ an open subset of $E$.

Consider $F\in AC(\overline{U},E)$ and write $F^{\star }=I\times F:\overline{U}\to K(\overline{U}\times E)$, here $I:\overline{U}\to \overline{U}$ is $I\left(x\right)=x$, and let

$$\begin{array}{c}\hfill d:\left\{{\left(F^{\star }\right)}^{-1}\left(B\right)\right\}\cup \{\varnothing \}\to D\end{array}$$

(10)

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

Definition 15.

Let $F\in A{C}_{\partial U}(\overline{U},E)$ and write $F^{\star }=I\times F$. We write $F^{\star }:\overline{U}\to K(\overline{U}\times E)$ is d-essential if for any $J\in A{C}_{\partial U}(\overline{U},E)$ (write $J^{\star }=I\times J$) and ${J|}_{\partial U}{=F|}_{\partial U}$ and $J\cong F$ in $A{C}_{\partial U}(\overline{U},E)$ we have $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$.

Remark 3.

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

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

so there exists a $x\in U$ with $(x,x)\in F^{\star }\left(x\right)$.

In [6] we established the topological transversality theorem.

Theorem 9.

Consider two maps $\mathsf{\Phi }$ and $\mathsf{\Psi }$ in $A{C}_{\partial U}(\overline{U},E)$ (write ${\mathsf{\Phi }}^{\star }=I\times \mathsf{\Phi }$ and ${\mathsf{\Psi }}^{\star }=I\times \mathsf{\Psi }$) with $\mathsf{\Phi }\cong \mathsf{\Psi }$ in $A{C}_{\partial U}(\overline{U},E)$. Now ${\mathsf{\Phi }}^{\star }$ is d-essential if and only if ${\mathsf{\Psi }}^{\star }$ is d-essential.

We present the analogue of Theorem 2. Suppose E is a (Hausdorff) topological vector space, U is convex and assume $\left(1\right)$ holds. Let $F,G$ be in $A{C}_{\partial U}(\overline{U},E)$ with ${G|}_{\partial U}{=F|}_{\partial U}$. Then before Definition 5 we showed $F\cong G$ in $A{C}_{\partial U}(\overline{U},E)$. In this situation, we can replace Definition 15 with:

Definition 16.

Let $F\in A{C}_{\partial U}(\overline{U},E)$ and write $F^{\star }=I\times F$. We write $F^{\star }:\overline{U}\to K(\overline{U}\times E)$ is d-essential if for any $J\in A{C}_{\partial U}(\overline{U},E)$ (write $J^{\star }=I\times J$) and ${J|}_{\partial U}{=F|}_{\partial U}$ we have $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$.

From Theorem 9 we have:

Theorem 10.

Let $E$ be a topological vector space, $U$ an open convex subset of $E$, and suppose $\left(1\right)$ holds. Consider two maps $\mathsf{\Phi }$ and $\mathsf{\Psi }$ in $A{C}_{\partial U}(\overline{U},E)$ (write ${\mathsf{\Phi }}^{\star }=I\times \mathsf{\Phi }$ and ${\mathsf{\Psi }}^{\star }=I\times \mathsf{\Psi }$) with $\mathsf{\Phi }\cong \mathsf{\Psi }$ in $A{C}_{\partial U}(\overline{U},E)$. Now ${\mathsf{\Phi }}^{\star }$ is d-essential (Definition 16) if and only if ${\mathsf{\Psi }}^{\star }$ is d-essential (Definition 16).

Now we present the analogue of Theorem 4.

Consider $F\in AC(E,E)$ and write $F^{\star }=I\times F:E\to K(E\times E)$, here $I:E\to E$ is $I\left(x\right)=x$, and let

$$\begin{array}{c}\hfill d:\left\{{\left(F^{\star }\right)}^{-1}\left(\tilde{B}\right)\right\}\cup \{\varnothing \}\to D\end{array}$$

(11)

be any map with values in the nonempty set $D$ where $\tilde{B}=\left\{(x,x):x\in E\right\}$.

Theorem 11.

Let $E$ be a completely regular topological space, $U$ an open subset of $E$, $B=\{(x,x):x\in \overline{U}\}$, $\tilde{B}=\left\{(x,x):x\in E\right\}$ and d is the map defined in $\left(11\right)$. Suppose $F\in A{C}_{\partial U}(\overline{U},E)$ (write $F^{\star }=I\times F$), $J\in AC(E,E)$ (write $J^{\star }=I\times J$) and $\left(2\right)$ and $\left(4\right)$ hold. Also suppose

$$\begin{array}{c}\hfill \begin{array}{c}\mathit{for}\mathit{any}\mathsf{\Phi }\in AC(E,E)(\mathit{write}{\mathsf{\Phi }}^{\star }=I\times \mathsf{\Phi })\mathit{with}\mathsf{\Phi }\cong J\hfill \\ \mathit{in}AC(E,E)\mathit{we}\mathit{have}d\left({\left({\mathsf{\Phi }}^{\star }\right)}^{-1}\left(\tilde{B}\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(\tilde{B}\right)\right)\ne d(\varnothing ),\hfill \end{array}\end{array}$$

(12)

and $F\cong J$ in $A{C}_{\partial U}(\overline{U},E)$. Then $F$ is d-essential (Definition 15).

Proof. 

We show J is d-essential (Definition 15) (and then $F$ is d-essential (Definition 15) from Theorem 9). Let $G\in A{C}_{\partial U}(\overline{U},E)$ (write $G^{\star }=I\times G$), ${G|}_{\partial U}{=J|}_{\partial U}$ with $G\cong J$ in $A{C}_{\partial U}(\overline{U},E)$. We need to show $d\left({\left(G^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$. There exists (see Definition 3) a u.s.c. compact map $\mathsf{\Lambda }:\overline{U}\times [0,1]\to K\left(E\right)$ with ${\mathsf{\Lambda }}_t\in A{C}_{\partial U}(\overline{U},E)$ for each $t\in [0,1]$, ${\mathsf{\Lambda }}_0=J$ and ${\mathsf{\Lambda }}_1=G$. Let ${\mathsf{\Lambda }}^{\star }:\overline{U}\times [0,1]\to K(\overline{U}\times E)$ be ${\mathsf{\Lambda }}^{\star }(x,t)=(x,\mathsf{\Lambda }(x,t))$ and let

$$\mathsf{\Omega }=\left\{x\in \overline{U}:(x,x)\in {\mathsf{\Lambda }}_t^{\star }\left(x\right)\mathrm{for}\mathrm{some}t\in [0,1]\right\}.$$

Notice $\mathsf{\Omega }\ne \varnothing$ (see $\left(4\right)$) is compact and $\mathsf{\Omega }\cap (E\backslash U)=\varnothing$. Thus, there exists a continuous function $\sigma :E\to [0,1]$ with $\sigma \left(\mathsf{\Omega }\right)=1$ and $\sigma (E\backslash U)=0$. Let $\mathsf{\Psi }:E\times [0,1]\to K\left(E\right)$ be

$$\mathsf{\Psi }(x,t)=\left\{\begin{array}{c}\mathsf{\Lambda }(x,t\sigma \left(x\right)),x\in \overline{U}\hfill \\ J\left(x\right),x\in E\backslash U.\hfill \end{array}\right.$$

Note $\mathsf{\Psi }:E\times [0,1]\to K\left(E\right)$ is a u.s.c. compact map with ${\mathsf{\Psi }}_t\in AC(E,E)$ for each $t\in [0,1]$, so ${\mathsf{\Psi }}_1\cong {\mathsf{\Psi }}_0=J$ in $AC(E,E)$. Write ${\mathsf{\Psi }}_1^{\star }=I\times {\mathsf{\Psi }}_1$ and $\left(12\right)$ implies

$$d\left({\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(\tilde{B}\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(\tilde{B}\right)\right)\ne d(\varnothing ).$$

Note from $\left(2\right)$ that

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

and also from $\left(2\right)$ (note ${\mathsf{\Psi }}_1\left(x\right)=J\left(x\right)$ for $x\in E\backslash U$) that

$$\begin{array}{ccc}\hfill {\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(\tilde{B}\right)& =& \left\{x\in E:(x,x)\cap (x,{\mathsf{\Psi }}_1\left(x\right))\ne \varnothing \right\}=\left\{x\in \overline{U}:(x,x)\cap (x,{\mathsf{\Psi }}_1\left(x\right))\ne \varnothing \right\}\hfill \\ & =& {\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(B\right)\hfill \end{array}$$

so

$$d\left({\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing ).$$

Finally, note $\sigma \left(\mathsf{\Omega }\right)=1$ (note $(x,\mathsf{\Lambda }(x,\sigma \left(x\right)))={\mathsf{\Lambda }}_{\sigma \left(x\right)}^{\star }\left(x\right)$) so

$$\begin{array}{ccc}\hfill {\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(\tilde{B}\right)& =& \left\{x\in \overline{U}:(x,x)\cap (x,\mathsf{\Lambda }(x,\sigma \left(x\right)))\ne \varnothing \right\}=\left\{x\in \overline{U}:(x,x)\cap (x,\mathsf{\Lambda }(x,1))\ne \varnothing \right\}\hfill \\ & =& {\left(G^{\star }\right)}^{-1}\left(B\right)\hfill \end{array}$$

and as a result

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

 □

Next we consider a generalization of L-essential maps, namely the d-L-essential maps. Let $X,Y,U,L$ and T be as described after Remark 2.

Consider $F\in AC(\overline{U},Y;L,T)$ and write $F^{\star }=I\times {(L+T)}^{-1}(F+T):\overline{U}\to K(\overline{U}\times X)$, here $I:\overline{U}\to \overline{U}$ is $I\left(x\right)=x$, and let

$$\begin{array}{c}\hfill d:\left\{{\left(F^{\star }\right)}^{-1}\left(B\right)\right\}\cup \{\varnothing \}\to D\end{array}$$

(13)

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

Definition 17.

Let $F\in A{C}_{\partial U}(\overline{U},Y;L,T)$ and write $F^{\star }=I\times {(L+T)}^{-1}(F+T)$. We write $F^{\star }:\overline{U}\to K(\overline{U}\times X)$ is d-L-essential if for any $J\in A{C}_{\partial U}(\overline{U},Y;L,T)$ (write $J^{\star }=I\times {(L+T)}^{-1}(J+T)$) with ${J|}_{\partial U}{=F|}_{\partial U}$ and $J\cong F$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$ we have $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$.

Remark 4.

If $F^{\star }$ is $d$-L-essential then

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

so there exists a $x\in U\cap domL$ with $(x,x)\in F^{\star }\left(x\right)$.

In [5] we established the topological transversality theorem.

Theorem 12.

Consider maps $\mathsf{\Phi }$ and $\mathsf{\Psi }$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$ (write ${\mathsf{\Phi }}^{\star }=I\times {(L+T)}^{-1}(\mathsf{\Phi }+T)$ and ${\mathsf{\Psi }}^{\star }=I\times {(L+T)}^{-1}(\mathsf{\Psi }+T)$) with $\mathsf{\Phi }\cong \mathsf{\Psi }$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. Now ${\mathsf{\Phi }}^{\star }$ is d-L-essential if and only if ${\mathsf{\Psi }}^{\star }$ is d-L-essential.

We present the analogue of Theorem 2. Suppose $\left(1\right)$ holds and U is convex. Let $F,G$ be in $A{C}_{\partial U}(\overline{U},Y;L,T)$ with ${G|}_{\partial U}{=F|}_{\partial U}$. Then after Theorem 6 we showed $F\cong G$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. In this situation we can replace Definition 17 with:

Definition 18.

Let $F\in A{C}_{\partial U}(\overline{U},Y;L,T)$ and write $F^{\star }=I\times {(L+T)}^{-1}(F+T)$. We write $F^{\star }:\overline{U}\to K(\overline{U}\times X)$ is d-L-essential if for any $J\in A{C}_{\partial U}(\overline{U},Y;L,T)$ (write $J^{\star }=I\times {(L+T)}^{-1}(J+T)$) with ${J|}_{\partial U}{=F|}_{\partial U}$ we have $d\left({\left(F^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing )$.

From Theorem 12 we have:

Theorem 17.

Let U be convex and suppose $\left(1\right)$ holds. Consider two maps $\mathsf{\Phi }$ and $\mathsf{\Psi }$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$ (write ${\mathsf{\Phi }}^{\star }=I\times {(L+T)}^{-1}(\mathsf{\Phi }+T)$ and ${\mathsf{\Psi }}^{\star }=I\times {(L+T)}^{-1}(\mathsf{\Psi }+T)$) with $\mathsf{\Phi }\cong \mathsf{\Psi }$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. Now ${\mathsf{\Phi }}^{\star }$ is d-L-essential (Definition 18) if and only if ${\mathsf{\Psi }}^{\star }$ is d-L-essential (Definition 18).

Finally, we present the analogue of Theorem 4. Consider $F\in AC(X,Y;L,T)$ and write $F^{\star }=I\times {(L+T)}^{-1}(F+T):X\to K(X\times X)$, here $I:X\to X$ given is $I\left(x\right)=x$, and let

$$\begin{array}{c}\hfill d:\left\{{\left(F^{\star }\right)}^{-1}\left(\tilde{B}\right)\right\}\cup \{\varnothing \}\to D\end{array}$$

(14)

be any map with values in the nonempty set $D$ where $\tilde{B}=\left\{(x,x):x\in X\right\}$.

Theorem 18.

Let $X$, $Y$, $U$, $L$ and $T$ be as above, $B=\{(x,x):x\in \overline{U}\}$, $\tilde{B}=\left\{(x,x):x\in X\right\}$ and d is the map defined in $\left(14\right)$. Suppose $F\in A{C}_{\partial U}(\overline{U},Y;L,T)$ (write $F^{\star }=I\times {(L+T)}^{-1}(F+T)$), $J\in AC(X,Y;L,T)$ (write $J^{\star }=I\times {(L+T)}^{-1}(J+T)$) and $\left(7\right)$ and $\left(8\right)$ hold. In addition assume

$$\begin{array}{c}\hfill \begin{array}{c}\mathit{for}\mathit{any}\mathsf{\Phi }\in AC(X,Y;L,T)(\mathit{write}{\mathsf{\Phi }}^{\star }=I\times {(L+T)}^{-1}(\mathsf{\Phi }+T))\hfill \\ \mathit{with}\mathsf{\Phi }\cong J\mathit{in}AC(X,Y;L,T)\mathit{we}\mathit{have}\hfill \\ d\left({\left({\mathsf{\Phi }}^{\star }\right)}^{-1}\left(\tilde{B}\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(\tilde{B}\right)\right)\ne d(\varnothing ),\hfill \end{array}\end{array}$$

(15)

and $F\cong J$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. Then $F$ is d-L-essential (Definition 17).

Proof. 

We show J is d-L-essential (Definition 17) (and then apply Theorem 12). Let $G\in A{C}_{\partial U}(\overline{U},Y;L,T)$ and write $G^{\star }=I\times {(L+T)}^{-1}(G+T)$, ${G|}_{\partial U}{=J|}_{\partial U}$ with $G\cong J$ in $A{C}_{\partial U}(\overline{U},Y;L,T)$. Now there exists (Definition 10) a $(L,T)$ u.s.c., $(L,T)$ compact map $\mathsf{\Lambda }:\overline{U}\times [0,1]\to 2^Y$ with ${\mathsf{\Lambda }}_t\in A{C}_{\partial U}(\overline{U},Y;L,T)$ for each $t\in [0,1]$, ${\mathsf{\Lambda }}_0=J$ and ${\mathsf{\Lambda }}_1=G$. Let ${\mathsf{\Lambda }}^{\star }:\overline{U}\times [0,1]\to K(\overline{U}\times X)$ be ${\mathsf{\Lambda }}^{\star }(x,t)=(x,{(L+T)}^{-1}({\mathsf{\Lambda }}_t+T)\left(x\right))$ and let

$$\mathsf{\Omega }=\left\{x\in \overline{U}:(x,x)\in {\mathsf{\Lambda }}_t^{\star }\left(x\right)\mathrm{for}\mathrm{some}t\in [0,1]\right\}.$$

Note $\mathsf{\Omega }\ne \varnothing$ (see $\left(8\right)$) is compact, $\mathsf{\Omega }\cap (X\backslash U)=\varnothing$ and since X is Tychonoff there exists a (continuous) map $\sigma :X\to [0,1]$ with $\sigma \left(\mathsf{\Omega }\right)=1$ and $\sigma (X\backslash U)=0$. Let $\mathsf{\Psi }:X\times [0,1]\to 2^Y$ by

$$\mathsf{\Psi }(x,t)=\left\{\begin{array}{c}\mathsf{\Lambda }(x,t\sigma \left(x\right)),x\in \overline{U}\hfill \\ J\left(x\right),x\in X\backslash U.\hfill \end{array}\right.$$

Note $\mathsf{\Psi }:X\times [0,1]\to 2^Y$ is a $(L,T)$ u.s.c., $(L,T)$ compact map with ${\mathsf{\Psi }}_t\in AC(X,Y;L,T)$ for each $t\in [0,1]$, so ${\mathsf{\Psi }}_1\cong {\mathsf{\Psi }}_0=J$ in $AC(X,Y;L,T)$. Write ${\mathsf{\Psi }}_1^{\star }=I\times {(L+T)}^{-1}({\mathsf{\Psi }}_1+T)$ and $\left(15\right)$ implies

$$d\left({\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(\tilde{B}\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(\tilde{B}\right)\right)\ne d(\varnothing ).$$

Note from $\left(7\right)$ that

$$\begin{array}{ccc}\hfill {\left(J^{\star }\right)}^{-1}\left(\tilde{B}\right)& =& \left\{x\in X:(x,x)\cap (x,{(L+T)}^{-1}(J+T)\left(x\right))\ne \varnothing \right\}\hfill \\ & =& \left\{x\in \overline{U}:(x,x)\cap (x,{(L+T)}^{-1}(J+T)\left(x\right))\ne \varnothing \right\}\hfill \\ & =& {\left(J^{\star }\right)}^{-1}\left(B\right)\hfill \end{array}$$

and also from $\left(7\right)$ (note ${\mathsf{\Psi }}_1\left(x\right)=J\left(x\right)$ for $x\in X\backslash U$) we have

$$\begin{array}{ccc}\hfill {\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(\tilde{B}\right)& =& \left\{x\in X:(x,x)\cap (x,{(L+T)}^{-1}({\mathsf{\Psi }}_1+T)\left(x\right))\ne \varnothing \right\}\hfill \\ & =& \left\{x\in \overline{U}:(x,x)\cap (x,{(L+T)}^{-1}({\mathsf{\Psi }}_1+T)\left(x\right))\ne \varnothing \right\}\hfill \\ & =& {\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(B\right)\hfill \end{array}$$

so

$$d\left({\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(B\right)\right)=d\left({\left(J^{\star }\right)}^{-1}\left(B\right)\right)\ne d(\varnothing ).$$

Finally, note $\sigma \left(\mathsf{\Omega }\right)=1$ (note $(x,{(L+T)}^{-1}({\mathsf{\Lambda }}_{\sigma \left(x\right)}+T)\left(x\right)))={\mathsf{\Lambda }}_{\sigma \left(x\right)}^{\star }\left(x\right)$) so

$$\begin{array}{ccc}\hfill {\left({\mathsf{\Psi }}_1^{\star }\right)}^{-1}\left(\tilde{B}\right)& =& \left\{x\in \overline{U}:(x,x)\cap (x,{(L+T)}^{-1}({\mathsf{\Lambda }}_{\sigma \left(x\right)}+T)\left(x\right)))\ne \varnothing \right\}\hfill \\ & =& \left\{x\in \overline{U}:(x,x)\cap (x,{(L+T)}^{-1}({\mathsf{\Lambda }}_1+T)\left(x\right)))\ne \varnothing \right\}\hfill \\ & =& {\left(G^{\star }\right)}^{-1}\left(B\right)\hfill \end{array}$$

and so

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

 □