Cleon S. Barroso
For a finite and positive measure space $(Ω,Σ,μ)$ and any weakly compact convex subset of $L\sp\infty(Ω,Σ,mu)$, a fixed point theorem for a class of nonexpansive self-mappings is proved. An analogous result is obtained for the space $C(Ω)$. An illustrative example is given.
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We have derived that on certain Banach spaces having a graph structure $G$, the iterations for asymptotically $G$-nonexpansive map will converge weakly towards a fixed point. This result unifies and extends several theorems on fixed points proved by various authors for class of nonexpansive and asymptotically nonexpansive maps. As an application of this result, we derive that for maps satisfying the nonexpansive condition locally on special Banach spaces, the successive approximations converge weakly towards a fixed point.
Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^*$. We present a novel hybrid method for finding a common solution of a family of equilibrium problems, a common solution of a family of variational inequality problems and a common element of fixed points of a family of a general class of nonlinear nonexpansive maps. The sequence of this new method is proved to converge strongly to a common element of the families. Our theorem and its applications complement, generalize, and extend various results in literature.
Bounded fixed point sets and Krasnoselskii iterates of Thompson metric nonexpansive maps
本文研究实Banach空间正规闭锥上关于Thompson度量非扩张的映射，利用Krasnoselskii迭代和Collatz-Wielandt数给出不动点集非空有界及迭代收敛的充要条件，并应用于正定Hermitian矩阵锥上的非线性矩阵方程。
On $L$-$ω$-nonexpansive maps
本文研究了 Banach 空间凸子集上的 $L$-$ω$-非扩张映射，分析了其 AFPP 和 FPP 性质，给出了映射为常数或满足 FPP 的模条件，并证明了包含 $\co$ 副本的空间在某些条件下不满足 FPP。
$\ell_1$ spreading models and the FPP for Cesàro mean nonexpansive maps
本文研究 Cesàro 平均非扩张映射的不动点性质，通过建立 𝔠𝔪-非扩张性、ℓ₁ 展开模型与 Schauder 基常数之间的联系，证明了具有弱 Banach-Saks 性质的 Banach 空间对该类映射具有不动点性质。
This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of $\ell^{1}$. We also point out some inaccurate assertions appearing in the literature on this topic.
Fixed points for uniformly local asymptotic nonexpansive maps are discussed in this article. An approximate fixed point sequence for such a map over a uniformly convex Banach space is derived. At the end, we study the unique fixed point for uniformly local asymptotic contraction.
We investigate necessary and sufficient conditions for a nonexpansive map $f$ on a Banach space $X$ to have surjective displacement, that is, for $f - \mathrm{id}$ to map onto $X$. In particular, we give a computable necessary and sufficient condition when $X$ is a finite dimensional space with a polyhedral norm. We give a similar computable necessary and sufficient condition for a fixed point of a polyhedral norm nonexpansive map to be unique. We also consider applications to nonlinear Perron-Frobenius theory and suggest some additional computable sufficient conditions for surjective displace…
We give necessary and sufficient conditions for a nonexpansive map on a finite dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if $f : V \rightarrow V$ is a nonexpansive map on a finite dimensional normed space $V$, then the fixed point set of $f$ is nonempty and bounded if and only if there exist $w_1, \ldots , w_m$ in $V$ such that $\{f(w_i) - w_i : i = 1, \ldots, m \}$ illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite dimensional spaces. We also discuss application…
Real analytic nonexpansive maps on polyhedral normed spaces
本文证明多面体范数空间上实解析非扩张映射的不动点集为仿射子空间的等距像，并由此证实Nussbaum的$2^n$猜想在实解析情形下的正确性。
Let $S$ be a semitopological semigroup, and let $C$ be a nonempty closed convex subset of a reflexive Banach space. Under some amenability conditions on $S$, we provide existence results of fixed points for several Bregman nonexpansive type actions $S\times C\to C$, $(s,x)\mapsto T_s x$, of $S$ on $C$. The mappings $T_s$ we discuss include those being Bregman generalized hybrid, Bregman nonspreading, and Bregman left asymptotically nonexpansive.
In this paper, we prove first that the iterates of a mean nonexpansive map defined on a weakly compact, convex set converge weakly to a fixed point in the presence of Opial's property and asymptotic regularity at a point. Next, we prove the analogous result for closed, convex (not necessarily bounded) subsets of uniformly convex Opial spaces. These results generalize the classical theorems for nonexpansive maps of Browder and Petryshyn in Hilbert space and Opial in reflexive spaces satisfying Opial's condition.
We consider semidifferentiable (possibly nonsmooth) maps, acting on a subset of a Banach space, that are nonexpansive either in the norm of the space or in the Hilbert's or Thompson's metric inherited from a convex cone. We show that the global uniqueness of the fixed point of the map, as well as the geometric convergence of every orbit to this fixed point, can be inferred from the semidifferential of the map at this point. In particular, we show that the geometric convergence rate of the orbits to the fixed point can be bounded in terms of Bonsall's nonlinear spectral radius of the semidiffer…
Let $(A, B)$ be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and $T: A\cup B \rightarrow A\cup B$ be a continuous and asymptotically relatively nonexpansive map. We prove that there exists $x \in A\cup B$ such that $\|x - Tx\| = \emph{dist}(A, B)$ whenever $T(A) \subseteq B$, $T(B) \subseteq A$. Also, we establish that if $T(A) \subseteq A$ and $T(B) \subseteq B$, then there exist $x \in A$ and $y\in B$ such that $Tx = x$, $Ty = y$ and $\|x - y\| = \emph{dist}(A, B)$. We prove the aforesaid results when the pair $(A, B)$ has the rectangle pr…
We establish sharp estimates for the convergence rate of the Kranosel'ski\vı-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of Mathematics 199(2), 757-772, 2014) with constant $1/\sqrtπ$ is sharp and cannot be improved. To this end we consider the recursive bounds introduced in [3] (Proceedings of the 2nd International Conference on Fixed Point Theory and Applications, World Scientific Press, London, 27-66, 1992) which we reinterpret in terms of a nested family of optimal transport proble…
Let LUC$(S)$ be the space of left uniformly continuous functions on a semitopological semigroup $S$. Suppose that $S$ is right reversible and $\operatorname{LUC}(S)$ has a left invariant mean. Let $(X,d)$ be a Fréchet space. Let $τ$ be a locally convex topology of $X$ weaker than the $d$-topology such that the metric $d$ is $τ$-lower semicontinuous. Let $K$ be a $d$--separable and $τ$--compact convex subset of $X$. We show that every jointly $τ$-continuous and super asymptotically $d$-nonexpansive action $S\times K\mapsto K$ of $S$ has a common fixed point. Similar results in the locally conve…
On $α$-Firmly Nonexpansive Operators in $r$-Uniformly Convex Spaces
本文在 r-一致凸 Banach 空间中引入并研究 α-稳固非扩张算子及其拟形式，建立复合与凸组合等演算规则，证明拟 α-稳固非扩张算子具有渐近正则性，并利用半闭性原理得到迭代弱收敛到不动点集的充分条件，同时给出神经网络、半群理论和 L_p 空间中压缩投影的应用实例。
Let $ \sc=\{T_{s}:s\in S\} $ be a representation of a semigroup $S$. First, we prove that the mapping $T_μ$ introduced by a mean on a subspace of $l^{\infty}(S)$ has many properties of the mappings in the representation $ \sc$, in Banach spaces. Then we consider a directed graph and then we define a $Q$-$G$-nonexpansive mapping in locally convex spaces and show that $T_μ$ is a $Q$-$G$-nonexpansive mapping if $T_{s}$ is a $Q$-$G$-nonexpansive mapping for each $s\in S$. Then we define $Q$-$G$-attractive point of $ \sc$ and show if a point $a$ is a $Q$-$G$-attractive point of $ \sc$ then $a$ is a…
Let $X$ be a partially ordered set with the property that each family of order intervals of the form $[a,b],[a,\rightarrow )$ with the finite intersection property has a nonempty intersection. We show that every directed subset of $X$ has a supremum. Then we apply the above result to prove that if $X$ is a topological space with a partial order $\preceq $ for which the order intervals are compact, $\mathcal{F}$ a nonempty commutative family of monotone maps from $X$ into $X$ and there exists $c\in X$ such that $c\preceq Tc$ for every $T\in \mathcal{F}$, then the set of common fixed points of $…
A hybrid scheme for fixed points of a countable family of generalized non-expansive-type maps and generalized mixed equilibrium problems
提出一种混合算法，在一致光滑且一致凸的实Banach空间中寻找广义混合平衡问题解集与一族广义非扩张型映射公共不动点的公共元，并应用于优化问题。
Logically Contractive Mappings: Fixed Points and Event-Indexed Rates
本文提出“逻辑压缩映射”，证明扩展 Banach 原理的不动点定理，导出事件索引及迭代计数收敛速率，并推广至变因子事件，统一多种广义压缩现象。
Quantization Errors, Human--AI Interaction, and Approximate Fixed Points in $L^1(μ)$
本文提出了一个严格的测度论框架分析 $L^1(μ)$ 中非膨胀映射的不动点，考虑定点算术量化误差，证明了测度紧集的不动点性质及近似不动点存在性，并应用于人机协同编辑系统验证稳定共识状态。
Previously only two examples of Banach space quotient maps which do not admit uniformly continuous right inverses were known: one due to Aharoni and Lindenstrauss and one due to Kalton ($\ell^\infty\to\ell^\infty/c_{0}$). We show through an application of Kalton's Monotone Transfinite Sequence Theorem that a quotient map of a subspace of $\ell^\infty$ of sequences that converge to zero along an ideal in $\mathbb{N}$ toward another such subspace, provided one of the ideals is `much larger' than the other, cannot have a uniformly continuous right inverse. We show in general that pairs of ideals …
On the Fixed Point Property in Reflexive Banach Spaces
本文探讨了自反 Banach 空间的不动点性质，提出了基于直径 l1 压力与加权选择泛函的定量框架，证明若不变量一致为正则空间必含 l1 副本且非自反，并提供了有限维证书、正反例及计算例程。
For a compact Hausdorff space $S$, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on $S$, denoted by $C(S)$, satisfies property-$(P_1)$ (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of $C(S)$. Various stability results related to property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if $Y$ is a proximinal finite co-dimensional subspace of $c_0$ then the closed unit ball of $Y$ sat…
For a bounded closed convex set $K$, in this note, we study the FPP for $α$-Hölder nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^α$ for all $x, y\in K$, $α\in (0,1)$. First, we note that only finite-dimensional spaces have the Hölder-FPP. Moreover, the unit ball $B_X$ of any infinite-dimensional space fails the FPP for Hölder maps with $\mathrm{d}(T, B_X)>0$, where $\mathrm{d}(T, K)$ denotes the minimal displacement of $T$. We further show that reflexivity and weak sequential continuity are sufficient conditions to capture fixed points of Hölder-Lipsch…
Let $H^\infty(\mathbb D\times\N)$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk $\mathbb D\subset\mathbb C$. We show that the dense stable rank of $H^\infty(\mathbb D\times\N)$ is one and using this fact prove some nonlinear Runge-type approximation theorems for $H^\infty(\mathbb D\times\N)$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for algebra $H^\infty(\mathbb D)$.
Let M be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec-Pelczynski subsequence decomposition of bounded sequences in L^p[0,1] to the case of the Haagerup L^p-spaces (1\le p<\infty). In particular, we prove that if (ϕ_n)_n is a bounded sequence in the predual M_* of M, then there exist a subsequence (ϕ_{n_k})_k of (ϕ_n)_n, a decomposition ϕ_{n_k}= y_k+ z_k such that {y_k, k\ge 1} is relatively weaklycompact and the support projections s(z_k)\downarrow_k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexi…
We provide a description of certain invariance properties of completely bounded bimodule maps in terms of their symbols. If $\mathbb{G}$ is a locally compact quantum group, we characterise the completely bounded $L^{\infty}(\mathbb{G})'$-bimodule maps that send $C_0(\hat{\mathbb{G}})$ into $L^{\infty}(\hat{\mathbb{G}})$ in terms of the properties of the corresponding elements of the normal Haagerup tensor product $L^{\infty}(\mathbb{G}) \otimes_{σ{\rm h}} L^{\infty}(\mathbb{G})$. As a consequence, we obtain an intrinsic characterisation of the normal completely bounded $L^{\infty}(\mathbb{G})'…
For k=1,2,... infty and a Frolicher-Kriegl order k Lipschitz differentiable map f:E supseteq U to E having derivative at x_0 in U a linear homeomorphism E to E and satisfying a Colombeau type tameness condition, we prove that x_0 has a neighborhood V subseteq U with f|V a local order k Lipschitz diffeomorphism. As a corollary we obtain a similar result for Keller C_c^{\infty} maps with E in a class including Frechet and Silva spaces. We also indicate a procedure for verifying the tameness condition for maps of the type x mapsto varphi circ [id,x] and spaces E=C^{\infty}(Q) when Q is compact by…
Linear maps in $\mathcal{L}(\ell_{\MakeLowercase{p}},\mathcal{Y}) $ preserving parallel and TEA pairs
研究从 $\ell_p$ 到 Banach 空间 $\mathcal{Y}$ 的有界线性映射保持平行对和三角形等式达到（TEA）对的条件，给出了不同 $p$ 和 $\mathcal{Y}$ 下的刻画。
Brezis and Mironescu have announced several years ago that for a compact manifold $N^n \subset \mathbb{R}^ν$ and for real numbers $0 < s < 1$ and $1 \le p < \infty$ the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{s, p}(Q^m; N^n)$ when the homotopy group $π_{\lfloor sp \rfloor}(N^n)$ of order $\lfloor sp \rfloor$ is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that $N^n$ is $\lfloor sp \rfloor$ simply connected, we give a s…
Linear maps on $\mathcal{L}(\ell_p^n,\ell_p^m)$, $(p\in \{1,\infty\})$ preserving parallel pairs
研究 $\mathcal{L}(\ell_p^n,\ell_p^m)$ ($p\in\{1,\infty\}$) 上保持平行对和 TEA 对的线性映射，证明非零映射保持 TEA 对、保持平行对且秩大于1或可逆、为等距标量倍数四者等价。
If (A_0,A_1) and (B_0,B_1) are Banach couples and a linear operator T from A_0 + A_1 to B_0 + B_1 maps A_0 compactly into B_0 and maps A_1 boundedly into B_1, does T necessarily also map [A_0,A_1]_s compactly into [B_0,B_1]_s for s in (0,1)? After 42 years this question is still not answered, not even in the case where T is also compact from A_1 to B_1. But affirmative answers are known for many special choices of (A_0,A_1) and (B_0,B_1). Furthermore it is known that it would suffice to resolve this question in the special case where (B_0,B_1) is the special couple (l^\infty(FL^\infty), l^\inf…
Given a complete noncompact Riemannian manifold $N^n$, we investigate whether the set of bounded Sobolev maps $(W^{1, p} \cap L^\infty) (Q^m; N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1, p} (Q^m; N^n)$ for $1 \le p \le m$. The density always holds when $p$ is not an integer. When $p$ is an integer, the density can fail, and we prove that a quantitative trimming property is equivalent with the density. This new condition is ensured for example by a uniform Lipschitz geometry of $N^n$. As a byproduct, we give necessary and sufficient conditions for the strong density of …
Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps
本文证明Hilbert空间中每个元素可表示为无条件收敛级数时，存在非零标量使两个序列均为框架，并等价于秩一有界线性映射的完全有界性。
In this paper we generalize to unbounded convex subsets C of hyperbolic spaces results obtained by W.A. Kirk and R. Espinola on approximate fixed points of nonexpansive mappings in product spaces $(C\times M)_\infty$, where M is a metric space and C is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families $(C_u)_{u\in M}$ of unbounded convex subsets of a hyperbolic space. The key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to Borwein, Reich and Shafrir, obtained by the authors…
We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $Ψ^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of regularizing processes we produce an algebra of generalized functions and a diffeomorphism equivariant embedding of distributions into this algebra. We provide examples invariant under certain group actions. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frechét spaces. This notion of regularity impl…
Metric constructions and fixed point theorems in product spaces
本文通过连续凸函数构造乘积空间的度量，并研究其在不动点理论中的应用及现有结果的统一框架。
Linear maps preserving $\ell_p$-norm parallel vectors
本文刻画了赋范空间中保持平行对与三角形等式达到（TEA）对的线性映射，给出了 $\ell_p$-范数下各类情形的具体刻画，并将其推广至无限维空间。
Consider probabilistic distributions on the space of infinite Hermitian matrices $Herm(\infty)$ invariant with respect to the unitary group $U(\infty)$. We describe the closure of $U(\infty)$ in the space of spreading maps (polymorphisms) of $Herm(\infty)$, this closure is a semigroup isomorphic to the semigroup of all contractive operators.
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