On some problem of set-valued analysis in asymmetric normed spaces
The Schauder theorem, which claims the existence of a ﬁxed point of every mappingf : B → B, where B is a compact convex set in a normed space E, is well known. If a convexset B is closed and bounded in E, then the result remains valid for the case in which f(B) isprecompact.In recent papers [1, 2] we suggest an approach to ﬁxed-point theorems for mappings ofa bounded closed set B in a normed space without the assumption that the image f(B) isprecompact (which leads naturally to conditions of new type). This approach is based on injectiveand compact embeddings of E in some another Banach space E0. Note that such a method isapplicable if and only if the space E admits a countable total set of continuous linear functionals:T0= {`n}∞n=1⊂ E∗: ∀x, y ∈ E `n(x) = `n(y) ⇐⇒ x = y.In ([2], Section 3) new analogues of Shauder ﬁxed-point theorem were obtained for a specialclass of set-valued mappings in normed space.This paper is devoted to some generalizations of these results to the class of asymmetricnormed spaces. We note some applications of asymmetric normed spaces to theoretical computerscience and approximation theory.For asymmetric normed space E we consider the family CL(E) of bounded closed convexsets in E. CL(E) is a convex normed cone and it is injectively isometrically embedded in somelinear normed space L(E). Let E∗be a collection of linear bounded functionals ` : E → R.Generally, E∗is a convex normed cone with a normk`k∗:= supx6=0`(x)kx|.The following result holds. Об одной задаче многозначного анализа в пространствах с несимметричной нормой 83Theorem 1. For each ` ∈ E∗there is a functional ϕ`: L(E) → R: contained in the conjugatespace L∗(E) and the set {ϕ`}`∈E∗is total in L(E).Theorem 2. If the conjugate cone E∗is separable and E∗0is a countable dense subset of E∗,then Φ = {ϕ`| ` ∈ E∗0} is a total set in L(E).Let us consider one corollary of Theorem 2. We introduce following class of Φ-uniformlycontinuous mappings f : CL(E) → CL(E):∀L > 0 ∃δ > 0 : |max `(x) - max `(y)| < δ ⇒ |max `(f(x)) - max `(f (y))| < L ∀` ∈ E∗0.Note that `(x) and `(y) are segments in R.Corollary 1. If B is a convex and bounded set in L(E), E∗is a separable normed cone and amapping f : CL(E) → CL(E) is Φ-uniformly continuous, then there is a sequence {xn}∞n=1∈ Bsuch thatlimn→∞max `(xn) = limn→∞max `(f(xn)) ∀` ∈ E∗0.