关于伪压缩映象不动点迭代的一点注记

设$K$是自反的并且具有一致Gateaux可微范数的Banach空间$E$的非空有界闭凸子集.设$T:K\rightarrow K$是一致连续的伪压缩映象.假设$K$的每一非空有界闭凸子集对非扩张映象具有不动点性质.设$\{\lambda_n\}$是$(0,\frac{1}{2}]$中序列满足: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$.任给$x_1\in K$,定义迭代序列$\{x_n\}$为:$x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1),n\geq 1.$若$\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$, 则上述迭代产生的$\{x_n\}$强收敛到$T$的不动点.     

两个$l^p(\Gamma)$型空间单位球面之间的1-Lipschitz映射的延拓($1

方习年 《数学研究及应用》2009,29(4):687-692

Weak compactness is equivalent to the fixed point property in

A nonempty, closed, bounded, convex subset of has the fixed point property if and only if it is weakly compact.

     

自反Banach空间中非扩张非自映射的粘滞迭代逼近方法

主要在自反和严格凸的且具有一致G(a)teaux可微范数的Banach空间中研究了非扩张非自映射的粘滞迭代逼近过程,证明了此映射的隐格式与显格式粘滞迭代序列均强收敛到它的某个不动点.     

ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUT CONVEXITY

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＨｂｅａＨｉｌｂｅｒｔｓｐａｃｅｗｉｔｈｎｏｒｍ‖·‖ａｎｄｉｎｎｅｒｐｒｏｄｕｃｔ（·，·）ａｎｄｌｅｔＣｂｅａｎｏｎｅｍｐｔｙｓｕｂｓｅｔｏｆＨ．ＡｍａｐｐｉｎｇＴ：Ｃ｜→ＣｉｓｓａｉｄｔｏｂｅＬｉｐｓｃｈｉｔ...     

Banach空间中的平均非扩张映象:不动点的存在定理

<正> 设X是Banach空间,E是X中的集合,T是映集合E到自身的映象.若T满足条件(称为平均非扩张条件)其中x,y∈E,a,b,c≥0且a+2b+2c≤1,则称T是平均非扩张映象. 文[1]概括了近年来研究关于平均非扩张映象不动点的一些主要结果.本文进     

A note on fixed point sets in CAT(0) spaces

We show that the fixed point set of a quasi-nonexpansive selfmap of a nonempty convex subset of a CAT(0) space is always closed, convex and contractible. Moreover, we give a construction of a continuous selfmap of a CAT(0) space whose fixed point set is prescribed.     

Fixed point results for multimaps in CAT(0) spaces

Common fixed point results for families of single-valued nonexpansive or quasi-nonexpansive mappings and multivalued upper semicontinuous, almost lower semicontinuous or nonexpansive mappings are proved either in CAT(0) spaces or R-trees. It is also shown that the fixed point set of quasi-nonexpansive self-mapping of a nonempty closed convex subset of a CAT(0) space is always nonempty closed and convex.     

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ?₁ can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.     

关于渐近非扩张映象不动点迭代的一点注记

设E是一致凸Banach空间,C是E的非空闭凸子集,T:C→C是具有不动点的渐近非扩张映象．该文证明了在某些适当的条件下,由下列修改了的Ishikawa迭代程序所定义的序列{xn}=xn 1=rpn,pn=(1-an)xn anTmn ryn un,yn=(1-bn)xn bnTkn xn vn, (n≥1)弱收敛到t的不动点．     

Approximation of fixed points and variational solutions for pseudo-contractive mappings in banach spaces

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Assume that every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive mappings which is also a unique solution to variational inequality problem involving ?-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, convex optimization problems, and split feasibility problems. Our result extends many recent important results.     

The point of continuity property in Banach spaces not containing ℓ<Subscript>1</Subscript>

We obtain a local characterization of the point of continuity property for bounded subsets in Banach spaces not containing basic sequences equivalent to the standard basis of ℓ₁ and, as a consequence, we deduce that, in Banach spaces with a separable dual, every closed, bounded, convex and nonempty subset failing the point of continuity property contains a further subset which can be seen inside the set of Borel regular probability measures on the Cantor set in a weak-star dense way. Also, we characterize in terms of trees the point of continuity property of Banach spaces not containing ℓ₁, by proving that a Banach space not containing ℓ₁ satis- fies the point of continuity property if, and only if, every seminormalized weakly null tree has a boundedly complete branch.     

A note on non-support points, negligible sets, Gâteaux differentiability and Lipschitz embeddings

This paper first presents a characterization of three classes of negligible closed convex sets (i.e., Gauss null sets, Aronszajn null sets and cube null sets) in terms of non-support points; then gives a generalization of Gâteaux differentiability theorems of Lipschitz mapping from open sets to those closed convex sets admitting non-support points; and as their application, finally shows that a closed convex set in a separable Banach space X can be Lipschitz embedded into a Banach space Y with the Radon–Nikodym property if and only if the closure of its linear span is linearly isomorphic to a closed subspace of Y.     

A weak Grothendieck compactness principle for Banach spaces with a symmetric basis

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In Dowling et al. (J Funct Anal 263(5):1378–1381, 2012), an analogue of the Grothendieck compactness principle for the weak topology was used to characterize Banach spaces with the Schur property. Using a different analogue of the Grothendieck compactness principle for the weak topology, a characterization of the Banach spaces with a symmetric basis that are not isomorphic to $\ell ^1$ and do not contain a subspace isomorphic to $c_0$ is given. As a corollary, it is shown that, in the Lorentz space $d(w,1)$ , every weakly compact set is contained in the closed convex hull of the rearrangement invariant hull of a norm null sequence.     

Generic Existence of Fixed Points for Set-Valued Mappings

We first consider a complete metric space of nonexpansive set-valued mappings acting on a closed convex subset of a Banach space with a nonempty interior, and show that a generic mapping in this space has a fixed point. We then establish analogous results for two complete metric spaces of set-valued mappings with convex graphs.     

代数κ-拟-A类算子的Weyl定理

若T或T~*是无穷维可分的Hilbert空间H上的代数κ-拟-A类算子,则Weyl定理对任意的f∈H(σ(T))成立,其中H(σ(T))为σ(T)的开邻域上解析函数的全体.若T~*是代数κ-拟-A类算子,则a-Weyl定理对f(T)成立。还证明了若T或T~*是代数κ-拟-A类算子,则Weyl谱与本质近似点谱的谱映射定理对f(T)成立.     

Geometrical coefficients and the structure of the fixed-point set of asymptotically regular mappings in Banach spaces

It is shown that if E is a separable and uniformly convex Banach space with Opial’s property and C is a nonempty bounded closed convex subset of E, then for some asymptotically regular self-mappings of C the set of fixed points is not only connected but even a retract of C. Our results qualitatively complement, in the case of a uniformly convex Banach space, a corresponding result presented in [T. Domínguez, M.A. Japón, G. López, Metric fixed point results concerning measures of noncompactness mappings, in: W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer Acad. Publishers, Dordrecht, 2001, pp. 239-268].     

On a Parameterized System of Nonlinear Equations with Economic Applications

We study a parameterized system of nonlinear equations. Given a nonempty, compact, and convex set, an affine function, and a point-to-set mapping from the set to the Euclidean space containing the set, we constructively prove that, under certain (boundary) conditions on the mapping, there exists a connected set of zero points of the mapping, i.e., the origin is an element of the image for every point in the connected set, such that the connected set has a nonempty intersection with both the face at which the affine function is minimized and the face at which that function is maximized. This result generalizes and unifies several well-known existence theorems including Browder??s fixed point theorem and Ky Fan??s coincidence theorem. An economic application with constrained equilibria is also discussed.     

逼近Banach空间中渐近非扩张映象的不动点

设E是一致凸Banach空间,C是E的非空闭凸子集, T:C→C是具有不动点的渐近非扩张映象. 该文证明了, 在某些适当的条件下, 由下列修改了的Ishikawa迭代程序所定义的序列{x\-n},\$\$x\-\{n+1\}=t\-nT\+n(s\-nT\+nx\-n+(1-s\-n)x\-n)+(1-t\-n)x\-n,\$\$弱收敛到T的不动点, 其中{t\-n},{s\-n}是区间\[0,1\]中满足某些限制的实数列.     

ITERATION OF FIXED POINTS ON HYPERSPACES

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