Об одной задаче многозначного анализа в пространствах с несимметричной нормой

The Schauder theorem, which claims the existence of a fixed point of every mapping $f:B\rightarrow B$, where $B$ is a compact convex set in a normed space $E$ , is well known. If a convex set $B$ is closed and bounded in $E$, then the result remains valid for the case in which $f(B)$ is precompact.
In recent papers [1, 2] we suggest an approach to fixed-point theorems for mappings of a bounded closed set $B$ in a normed space without the assumption that the image $f(B)$ is precompact (which leads naturally to conditions of new type). This approach is based on injective and compact embeddings of $E$ in some another Banach space $E^{'}$. Note that such a method is applicable if and only if the space $E$ admits a countable total set of continuous linear functionals:
\begin{align}$T_{0}=\left \{ l_{n} \right \} _{n=1}^{\infty }\subset E^{*}:\forall x,y \in E\ \; l_{n}(x)=l_{n}(y)\Leftrightarrow x=y$\end{align} In ([2], Section 3) new analogues of Shauder fixed-point theorem were obtained for a special class of set-valued mappings in normed space.
This paper is devoted to some generalizations of these results to the class of asymmetric normed spaces. We note some applications of asymmetric normed spaces to theoretical computer science and approximation theory.
For asymmetric normed space $E$ we consider the family $CL(E)$ of bounded closed convex sets in $E$. $CL(E)$ is a convex normed cone and it is injectively isometrically embedded in some linear normed space $L(E)$. Let $E^{*}$ be a collection of linear bounded functionals $l:E\rightarrow \mathbb{R}$. Generally, $E^{*}$ is a convex normed cone with a norm
\begin{align}
\left \| l \right \|_{*}:=\sup _{x\neq 0}\frac{l(x)}{\left \| x \right \|}.
\end{align} The following result holds.
Theorem 1.For each $l\in E^{*}$ there is a functional $\varphi _{l}: L(E)\rightarrow \mathbb{R}$ : contained in the conjugate space $L^{*}(E)$ and the set $\left \{ \varphi _{l} \right \}_{l\in E^{*}}$ is total in $L(E)$.
Theorem 2.If the conjugate cone $E^{*}$ is separable and $E_{0}^{*}$ is a countable dense subset of $E^{*}$, then $\Phi = \left \{ \varphi _{l}| l\in E_{0}^{*} \right \}$ is a total set in $L(E)$.
Let us consider one corollary of Theorem 2. We introduce following class of $\Phi$-uniformly continuous mappings $f:CL(E)\rightarrow CL(E)$:
\begin{align}\forall L>0\exists \delta >0:\left | max\, l(x)-max\, l(y) \right |<\delta \Rightarrow |max\, l(f(x))-max\, l(f(y))|>L\;\forall l\in E_{0}^{}.\end{align} Note that $l(x)$ and $l(y)$ are segments in $\mathbb{R}$.
Corollary 1.If $B$ is a convex and bounded set in $L(E)$, $E^{*}$ is a separable normed cone and a mapping $f:CL(E)\rightarrow CL(E)$ is $\Phi$-uniformly continuous, then there is a sequence $\left \{ x_{n} \right \}_{n=1}^{\infty }\in B$ such that
\begin{align}\lim_{n \to \infty }\; max\, l(x_{n})=\lim_{n \to \infty }\; max\, l(f(x_{n}))\; \forall l\in E_{0}^{*}.\end{align}Keywords: asymmetric normed space, conjugate cone, compact embedding, Hausdorff metric, Shauder theorem.