Tatsuki Ezawa
In this paper, we consider the CQ projection method and the shrinking projection method given by a finite family of nonexpansive mappings of the unit sphere of a real Hilbert space, and prove strong convergence theorems to common fixed points of the mappings.
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Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, presented here for the first time, we introduce and study the class of {\it enriched nonexpansive mappings} in Hilbert spaces. In order to approximate the fixed points of enriched nonexpansive mappings we use the Krasnoselskij iteration for which we prove strong and weak convergence theorems. Examples to illustrate the richness of the new class of contractive mappings are also given. Our results in this paper extend some classical convergence theorems established by Browder and Petryshyn in [Browder, F. …
In this paper, we give a simple proof and some generalizations of results in Falset, Llorens-Fuster, Marino, and Rugiano (2016).
In this paper by using $W_{n}$-mapping, we introduce a composite iterative method for finding a common fixed point for infinite family of nonexpansive mappings and a solution of a certain variational inequality. Furthermore, the strong convergence of the proposed iterative method is established. Finally, some simulation examples are presented. Our results improve and extend the previous results.
In this paper, we introduce a new problem called the split feasibility and fixed point equality problems (SFFPEP) and propose a new iterative algorithm for solving the problem (SFFPEP) for the class of quasi-nonexpansive mappings in Hilbert spaces. Furthermore, we study the convergence of the proposed algorithm. At the end, we give numerical example that illustrate our theoretical result. The SFFPEP is a generalization of the split feasibility problem (SFP), split feasibility and fixed point problems (SFFPP) and split equality fixed point problem (SEFPP).
Let $H$ be an infinite dimensional, reflexive, separable Hilbert space and $NA(H)$ the class of all norm-attainble operators on $H.$ In this note, we study an implicit scheme for a canonical representation of nonexpansive contractions in norm-attainable classes.
Generic nonexpansive Hilbert space mappings
本文研究了可分无限维 Hilbert 空间闭凸集上一般非扩张自映射的不动点问题，通过引入“略有界集”概念，在 Baire 范畴意义下建立了该性质与不动点存在性的拓扑 0-1 律，并证明了不动点的唯一性与迭代收敛性。
Fixed points theorems for $b$-enriched multivalued nonexpansive mappings and *-$b$-enriched nonexpansive mappings
本文推广了单值$b$-富集非扩张映射的不动点定理至多值情形，引入*- $b$-富集非扩张映射，并在Hilbert空间中证明其存在性及Krasnoselskii迭代的弱、强收敛。
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.
A Solution of variational inequality problem for a finite family of nonexpansive mappings in Hilbert spaces
本文针对Hilbert空间上有限族非扩张映射，提出显式迭代过程并证明其强收敛到公共不动点，且该不动点解决不动点集上的变分不等式。
Remarks on b-enriched nonexpansive mappings
该研究针对文献中提出的富集非扩张映射概念展开分析，指出其与经典非扩张映射概念重合，需重新审视，为度量不动点理论的相关文献研究提供参考并指明后续澄清方向。
On a novel approach to nonexpansive mappings
本文引入周长非扩张映射这一新类，证明其规模大于现有非扩张与拟非扩张映射类，并在赋范线性空间和Hilbert空间中确立了不动点存在性结论。
In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.
In this paper, we study $Δ$- convergence of iterations for a sequence of strongly quasi-nonexpansive mappings as well as the strong convergence of the Halpern type regularization of them in Hadamard spaces. Then, we give some their applications in iterative methods, convex and pseudo-convex minimization(proximal point algorithm), fixed point theory and equilibrium problems. The results extend several new results in the literature and some of them seem new even in Hilbert spaces.
We prove weak and strong convergence theorems for a double Krasnoselskij type iterative method to approximate coupled solutions of a bivariate nonexpansive operator F : C x C --> C, where C is a nonempty closed and convex subset of a Hilbert space. The new convergence theorems generalize, extend, improve and complement very important old and recent results in coupled fixed point theory. Some appropriate examples to illustrate our new results and their generalization are also given.
Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the resul…
By using the Ishikawa iterative algorithm, we approximate the fixed points and the best proximity points of a relatively non expansive mapping. Also, we use the von Neumann sequence to prove the convergence result in a Hilbert space setting. A comparison table is prepared using a numerical example which shows that the Ishikawa iterative algorithm is faster than some known iterative algorithms such as Picard and Mann iteration.
In this paper, first we introduce a new mapping for finding a common fixed point of an infinite family of nonexpansive mappings then we consider iterative method for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for $α$-inverse-strongly monotone mapping in a Hilbert space. We show that under suitable conditions, the sequence converges strongly to a common element of the above three sets. Our results presented in this paper improve and exten…
In this note, we highlight some properties of the metric projection onto a closed convex in a Hilbert space. In particular, we use some recent results on fixed points of nonexpansive potential operators.
We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert's metric or Thompson's metric. We establish several Denjoy-Wolff type theorems that confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.
Strong convergence for countable generalized Bregman nonexpansive-type mappings with equilibrium and variational inequality constraints
该研究针对含可数族不动点型映射、平衡双函数及单调变分不等式算子约束的公共解问题，提出Legendre–Bregman外逼近法，在无需NST相容性条件下获强收敛性，还推导了Hilbert空间等特殊情形。
We provide a direct proof of a result regarding the asymptotic behavior of alternating nearest point projections onto two closed and convex sets in a Hilbert space. Our arguments are based on nonexpansive mapping theory.
We give a rate of metastability for Halpern's iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth Banach spaces, extracted from a proof due to Xu. In Hilbert space, the latter is known, so we get an explicit rate of metastability. We also extract a rate of asymptotic regularity for general normed spaces. Such rates have already been extracted by Kohlenbach and Leuştean for different and incomparable conditions on $(λ_n)$. The proof analyzed in this paper is also more effective than the proofs treated by Kohlenbach and Leuştean in that i…
We introduce and analyze an abstract algorithm that aims to find the projection onto a closed convex subset of a Hilbert space. When specialized to the fixed point set of a quasi nonexpansive mapping, the required sufficient condition (termed "fixed-point closed") is less restrictive than the usual conditions based on the demiclosedness principle. A concrete example of a subgradient projector is presented which illustrates the applicability of this generalization.
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。
On the Karlsson-Nussbaum conjecture for resolvents of nonexpansive mappings
本文研究有界凸域上无不动点的非扩张映射，证明其预解式族聚点的凸包位于边界中，并由此得到椭球上预解式的Wolff-Denjoy型定理。
Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For example, the proximal mappings in the sense of Moreau are precisely the resolvents of subdifferential operators. More general notions of "resolvent", "proximal mapping" and "firmly nonexpansive" have been stu…
On the Rate of Asymptotic Regularity of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization
本文将Banach和Hilbert空间中的渐近正则性证明技术扩展到完备CAT(0)空间，并引入双曲优化的Halpern型优化器作为欧几里得HalpernSGD的非线性对应。
On nonexpansiveness of metric projection operators on Wasserstein spaces
本文研究Wasserstein空间到闭测地凸真子集的度量投影，证明一维$W_2$投影非扩张，并揭示高维中在$p$过小或过大时失效，同时证明其无处局部满足Busemann NPC与投影准则。
Cesàro means of firmly nonexpansive iterates need not converge strongly
本文通过构造无穷维Hilbert空间中的反例，证明严格非扩张算子的迭代Cesàro均值可能不收敛到不动点，揭示了平均技术在无穷维优化中的固有限制。
A well-known result of W. Ray asserts that if $C$ is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping $T$: $C\to C$ that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup $S$ of nonexpansive mappings acting on a closed convex subset $C$ of a Hilbert space, assuming that there is a point $c\in C$ with a bounded orbit and assuming that certain subspace of $C_b(S)$ has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups int…
We introduce a new extragradient iterative process, motivated and inspired by [S. H. Khan, A Picard-Mann Hybrid Iterative Process, Fixed Point Theory and Applications, doi:10.1186/1687-1812-2013-69], for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a Hilbert space. Using this process, we prove a weak convergence theorem for the class of nonexpansive mappings in Hilbert spaces. Finally, as an application, we give some theorems by using resolvent operator and strictly…
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…
On a notion of averaged operators in CAT(0) spaces
本文在CAT(0)空间中引入α-强非扩张算子作为平均算子的推广，建立拟α-强非扩张算子的演算规则，并证明其迭代弱收敛到不动点，在正则条件下投影强收敛到该弱极限。
On the plasticity of the unit spheres of $\ell_1$, $\ell_{\infty}$, $c$, and Hilbert spaces
本文研究了 $\ell_1$、$\ell_{\infty}$ 和 $c$ 的单位球的扩展-收缩可塑性，并证明了 Hilbert 空间单位球的强可塑性。
The split common fixed point problems has found its applications in various branches of mathematics both pure and applied. It provides us a unified structure to study a large number of nonlinear mappings. Our interest here is to apply these mappings and propose some iterative methods for solving the split common fixed point problems and its variant forms, and we prove the convergence results of these algorithms. As a special case of the split common fixed problems, we consider the split common fixed point equality problems for the class of finite family of quasi-nonexpansive mappings. Furtherm…
Uniform Embeddings of Banach Space Spheres into Hilbert Spaces and Property (H)
本文证明可分无穷维实 Banach 空间单位球面均匀嵌入 Hilbert 空间蕴含该空间具有 Kasparov–Yu 性质 (H)。
We use techniques of proof mining to extract a uniform rate of metastability (in the sense of Tao) for the strong convergence of approximants to fixed points of uniformly continuous pseudocontractive mappings in Banach spaces which are uniformly convex and uniformly smooth, i.e. a slightly restricted form of the classical result of Reich. This is made possible by the existence of a modulus of uniqueness specific to uniformly convex Banach spaces and by the arithmetization of the use of the limit superior. The metastable convergence can thus be proved in a system which has the same provably tot…
In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT(0) spaces stands in opposed to the monotonicity defined earlier in CAT(0) spaces by Khatibzadeh and Ranjbar [14] and Chaipunya and Kumam [8]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida app…
In this paper, we introduce and study a new extragradient iterative process for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a real Hilbert space. Also, we prove that under quite mild conditions the iterative sequence defined by our new extragradient method converges strongly to a solution of the fixed point problem for an infinite family of nonexpansive mappings and the classical variational inequality problem. In addition, utilizing this resul…
We introduce a one-parameter family of transforms, $U^t_{(m)}$, $t>0$, from the Hilbert space of Clifford algebra valued square integrable functions on the $m$--dimensional sphere, $L^2(S^{m},dσ_{m})\otimes \mathbb{C}_{m+1}$, to the Hilbert spaces, ${\mathcal M}L^2(\mathbb{R}^{m+1} \setminus \{0\},dμ_t)$, of monogenic functions on $\mathbb{R}^{m+1}\setminus \{0\}$ which are square integrable with respect to appropriate measures, $dμ_t$. We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal-Bargman coherent state transform, $U_{(1)} : L^2(…
Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, introduced in [Berinde, V., {\it Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces}, Carpathian J. Math. {\bf 35} (2019), no. 3, 277-288.], we introduce the class of enriched Chatterjea contractions and prove general fixed point theorems for such contractions in the setting of a Banach space. Examples to illustrate the richness of the new class of contractions and the relationship between enriched Banach contractions, enriched Kannan contractions an…
Connected Hamel bases in Hilbert spaces
本文旨在寻找Hilbert空间$\ell^2$中构成单位球面连通且局部连通子集的代数基。
It is proved that a discrete group $G$ is amenable if and only if for every unitary representation of $G$ in an infinite-dimensional Hilbert space $\cal H$ the maximal uniform compactification of the unit sphere $\s_{\cal H}$ has a $G$-fixed point, that is, the pair $(\s_{\cal H},G)$ has the concentration property in the sense of Milman. Consequently, the maximal $U({\cal H})$-equivariant compactification of the sphere in a Hilbert space $\cal H$ has no fixed points, which answers a 1987 question by Milman. This is a version as of November 19, 1998, incorporating some revisions.
A close look at the entropy numbers of the unit ball of the Reproducing Hilbert Space of isotropic positive definite kernels
本文研究了正交正定核再生希尔伯特空间单位球的熵数，给出了覆盖数的精确上界和下界，并分析了其与维度和核系数的关系。
The image restoration problem is one of the popular topics in image processing studied by many authors on account of its applications in various areas. The aim of this paper is to present a new algorithm by using viscosity approximation with inertial effect for finding a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space and obtaining more quality images from degenerate images. Some strong convergence theorems are proved under mild conditions. The obtained results are applied to solve monotone inclusion problems, convex minimization problems, variational inequ…
Ryll-Wojtaszczyk Formulas for bihomogeneous polynomials on the sphere
本文通过平均技术推导了双齐次多项式在球面上的投影常数公式，扩展了经典Ryll-Wojtaszczyk公式至双齐次情形。
Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all $H$-valued continuous functions on $K$, equipped with the supremum norm, satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space, every surjective isometry $Δ$ from the unit sphere of $C(K,H)$ onto the unit sphere of $Y$ admits a unique extension to a surjective real linear isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of $C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For …
In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe--Ritter to the sphere $\mathbb{S}^n$. We determine the resulting Osterwalder--Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group $G^c = \mathrm{O}_{1,n}(\mathbb{R})^{\uparrow}$ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key to…
Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $S^m$ in the Euclidean space $\mathbb{R}^{m+1}$ are known to depend upon the Mercer's expansion of the compact and self-adjoint $L^2(S^m)$-operator associated to the kernel. The estimation of the Kolmogorov $n$-th width of the unit ball of the reproducing kernel Hilbert space in $L^2(S^m)$ and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. Thi…
We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces, including that they are uniformly convex and uniformly smooth, and that they form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known r…
We establish a close link between the amenability of a unitary representation $π$ of a group $G$ (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system $(\s_π,G)$, where $\s_\H$ is the unit sphere the Hilbert space of representation. We prove that $π$ is amenable if and only if either $π$ contains a finite-dimensional subrepresentation or the maximal uniform compactification of $\s_π$ has a $G$-fixed point. Equivalently, the latter means that the $G$-space $(\s_π,G)$ has the concentration property: every finite cover of the sphe…
Metric projection operators can be defined in similar wayin Hilbert and Banach spaces. At the same time, they differ signifitiantly in their properties. Metric projection operator in Hilbert space is a monotone and nonexpansive operator. It provides an absolutely best approximation for arbitrary elements from Hilbert space by the elements of convex closed sets . This leads to a variety of applications of this operator for investigating theoretical questions in analysis and for approximation methods. Metric projection operators in Banach space do not have properties mentioned above and their ap…
The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(λ) = \exp(i(λ_1ψ_1 + ... + λ_nψ_n)) for λin R^n, gives rise to a manifold M of orthogonal projections for the subspaces u(λ)H^2(R) of L^2(R). For classes of admissible functions ψ_i the strong operator topology closures of M and M \cup M^\perp are determined explicitly as various n-balls and n-spheres. The arguments used are direct and rely on the analysis of oscillatory integrals and Hilbert space geometry. Some classes of these closed projection manifolds are classified up to unitary eq…
Upon improving and extending the concept of redundancy of frames, we introduce the notion of redundancy of fusion frames, which is concerned with the properties of lower and upper redundancies. These properties are achieved by considering the minimum and maximum values of the redundancy function which is defined from the unit sphere of the Hilbert space into the positive real numbers. In addition, we study the relationship between redundancy of frames (fusion frames) and dual frames (dual fusion frames). Moreover, we indicate some results about excess of fusion frames. We state the relationshi…
The aim of this paper is to study the $w^*$-fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of $w^*$-closed subsets of the dual sphere is equivalent to the $w^*$-fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable $w^*$-fixed point property. The last geometrical notion was introduced by Fonf and Veselý as a strengthenin…
We study Hilbert generated versions of nonseparable Banach spaces $\mathcal X$ considered by Shelah, Steprāns and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded operator on $\mathcal X$ other than a diagonal operator (or a scalar multiple of the identity) plus a separable range operator. We address the questions if these spaces admit uncountable equilateral sets and if their unit spheres admit uncountable $(1+)$-separated or $(1+\varepsilon)$-separated sets. We resolve some of the above questions for two types of these spac…
We consider the sound ranging problem, which is to find the position of the source-point from the moments when the wave-sphere of linearly, with time, increasing radius reaches the sensor-points, in the infinite-dimensional separable Euclidean space H, and describe the solving methods, for entire space and for its unit sphere. In the former case, we give some sufficient conditions for uniqueness of the solution. We also provide two examples with the sets of sensors being a basis of H: 1st, when sound ranging problem and so-called dual problem both have single solutions, and 2nd, when sound ran…
Uniform homeomorphisms between spheres of unitarily invariant ideals
本文证明非交换版本的Odell-Schlumprecht定理，通过两类广义Mazur映射（基于p-凸化和量子相对熵最小化）建立酉不变理想球面与Hilbert空间球面间的均匀同胚，提供满足性质(H)的新例子。
We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur--Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur--Ulam property.
Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T: B(K)\to B(H)$. This provides a positive answer to Tingley's problem in the setting of $B(H)$ spaces.
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