Banach空间非扩张映像不动点强收敛定理

设E是具弱序列连续对偶映像自反Banach空间, C是E中闭凸集, T:C→ C是具非空不动点集F(T)的非扩张映像.给定u∈ C,对任意初值x0∈ C,实数列{αn}n∞=0,{βn}∞n=0∈ (0,1),满足如下条件:(i)sum from n=α to ∞α_n=∞, α_n→0;(ii)β_n∈[0,α) for some α∈(0,1);(iii)sun for n=α to ∞|α_(n-1) α_n|<∞,sum from n=α|β_(n-1)-β_n|<∞设{x_n}_(n_1)~∞是由下式定义的迭代序列:{y_n=β_nx_n (1-β_n)Tx_n x_(n 1)=α_nu (1-α_n)y_n Then {x_n}_(n=1)~∞则{x_n}_(n=1)~∞强收敛于T的某不动点.     

一个构造增生映像零点的带误差的迭代算法

设E是一致光滑的Banach空间,A:D(A)E→2~E是一个满足值域条件的增生算子,进一步满足线性增长条件:‖Ax‖≤C(1+‖x‖)对某个常数C0, x∈D(A).设z∈D(A)是任意固定元,x_1∈D(A), A~(-1)0≠Φ.定义序列{x_n}D(A)如下:x_(n+1)∈x_n-λ_n(Ax_n+θ_n(x_n-z+e_n)),n≥1,其中{λ_n}与{θ_n}是满足一定条件的非负数列.则x_n→x~*∈A~(-1)(0),(n→∞).作为应用,我们推出构造连续伪压缩映像的不动点的收敛定理.     

Banach空间中渐近非扩张映射逼近序列的强收敛性

该文研究了序列{x_n}的收敛性。其中x_0∈C, x_{n+1}=α_n T^n x_n+(1-α_n)x, n=0,1,2,…,这里0≤α_n≤1，T是Banach空间中非空闭凸子集C到自身的渐近非扩张映射。同时证明了：当z_n=(1-t_n/k_n)u+t_n/k_n T^n z_n且lim_{n→∞}{(k_n-1)/(1-t_n)}=0,lim‖z_n-Tz_n‖=0时，T有不动点当且仅当{z_n}有界。这时{z_n}强收敛于T的不动点。     

广义Lipschitz伪压缩映射黏滞迭代逼近方法的强收敛

K是Banach空间E的一个非空闭凸子集,T:K→K是一个广义Lipschitz伪压缩映射.对Lipschitz强伪压缩映射f:K→K和x_1∈K,序列{x_n}由下式定义:x_n+1=(1-α_n-β_n)x_n+α_nf(x_n)+β_nTx_n.在{α_n}与{β_n}满足合适条件的情况下,每当{z∈K;μ_n‖x_n-z‖~2=inf_(y∈K)μ_n‖x_n-y‖~2}∩F(T)≠φ时,{x_n}强收敛到T的某个不动点x~*.     

非线性算子具误差的隐迭代程序的强收敛性

设K是一致凸Banach空间中的非空闭凸子集,T_i:K→K(i=1,2,…,N)是有限族完全渐近非扩张映象.对任意的x_0∈K,具误差的隐迭代序列{x_n}为:x_n=α_nx_n-1+β_nT_n~kx_n+γ_nu_n,n≥1,其中{α_n},{β_n},{γ_n}■[0,1]满足α_n+β_n+γ_n=1,{u_n}是K中的有界序列.在一定的条件下,该文建立了隐迭代序列{x_n}的强收敛性.得到隐迭代序列{x_n}强收敛于有限族完全渐近非扩张映象公共不动点的充要条件.所得结果改进和推广了Shahzad与Zegeye,Zhou与Chang,Chang,Tan,Lee与Chan等人的相应结果.     

Banach空间中关于增生算子的强收敛迭代序列

主要研究求解增生算子零点问题的一类算法:x_(n+1)=α_nu+(1-α_n)((1-λ)x_n+λJ_r_nx_n),其u是固定向量,λ∈(0,1),{r_n}和{α_n}是实数列,J_r_n表示增生算子A的预解式.其中(r_n)收敛是保证算法收敛的一个充分条件,该文主要证明了此条件可减弱为limn|1-(r_n+1)/r_n|=0.     

逼近非扩张非自映像不动点（英文）

设E是一致凸Banach空间,K是E中非空闭凸集且是一个非扩张收缩核,T：K→E是具非空不动点集F（T）：={x∈K：Tx=x}的非扩张映像.设{α_n},{β_n},{γ_n},{α′_n},{β′_n},{γ′_n}是[0,1]中实数列满足α_n＋β_n＋γ_n=α′_n＋γ′_n＋γ′_n=1,对任意初值x_1∈K,定义{x_n}如下（ⅰ）如果对偶空间E＊具有Kadec-Klee性质,那么{x_n}弱收敛于T的某不动点x＊∈F（T）;（ⅱ）若T满足（A）条件,那么{x_n}强收敛于T的某不动点x＊∈F（T）.     

一类不等式

<正> 本文首先将(2)换为下面的(4),然后将(3)推广,导出一类不等式. §.2 本文采用记号如下: S_n为n元集{1,2,…,n}上的全体置换所组成的置换群,G为S_n的一个子群. x=(x_1,x_2,…,x_n),α=(α_1,α_2,…,α_n),β=(β_1,β_2,…,β_n)等均为n维欧氏空间中的点,并且不作特别申明时约定各个分量为正.     

Banach空间中k-次增生型变分包含解的存在与收敛性

在实自反Banach空间中,引入并研究一类k-次增生型变分包含问题,证明了这类变分包含解的存在与唯一性,并在去掉α_n→0,β_n→0(n→∝)以及序列{x_n)和{_η(g(x_n))}有界限制的条件下,建立了k-次增生型变分包含和变分不等式解的具有混合误差的多步迭代序列的强收敛性定理,给出了收敛率的估计式,从而改进和推广了前人的研究结果.     

求数列极限的一个定理

本文介绍一个有关求数列极限的定理,利用它可以较方便地求出一些数列的极限. 定理对于数列{x_n},若存在一个小于1的正数Υ,使不等式 |x_(n+1)-α|≤Υ|x_n-α| 对一切大于某自然数N的n都成立,则 limα_n=α.n→∞     

Viscosity approximation methods for nonexpansive mappings and monotone mappings

Viscosity approximation methods for nonexpansive mappings are studied. Consider the iteration process {xn}, where x₀∈C is arbitrary and xn+1=αnf(xn)+(1−αn)SPC(xn−λnAxn), f is a contraction on C, S is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. It is shown that {xn} converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly-monotone mapping which solves some variational inequality.     

Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points F(S) of a nonexpansive mapping S and the set of solutions Ω _A of the variational inequality for a monotone, Lipschitz continuous mapping A. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of \({F(S)\cap\Omega_{A}}\). As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.     

Strong convergence of the modified hybrid steepest-descent methods for general variational inequalities

In this paper, we consider the general variational inequality GVI(F, g, C), whereF andg are mappings from a Hilbert space into itself andC is the fixed point set of a nonexpansive mapping. We suggest and analyze a new modified hybrid steepest-descent method of type methodu _n+1=(1?α+θ _n+1)Tu _n +αu _n ?θ _n+1 g (Tu _n )?λ _n+1 μF(Tu _n ),n≥0. for solving the general variational inequalities. The sequencex _n is shown to converge in norm to the solutions of the general variational inequality GVI(F, g, C) under some mild conditions. Application to constrained generalized pseudo-inverse is included. Results proved in the paper can be viewed as an refinement and improvement of previously known results.     

无限簇非扩张非自映象公共不动点的黏性逼近法

设E是具有一致Gateaux可微范数的严格凸的自反的Banach空间,K是E的非空闭凸子集而且是E的sunny非扩张收缩核.设f:K→K是一压缩映象,P:E→K是一sunny非扩张保核收缩,{T_n}_n~∞1:K→E是一可数无限簇非扩张非自映象且■是[0,1]中的非负数列.考虑下列迭代序列■其中W_n是由P,T_n,T_(n-1),…,T_1和λ_n,λ_(n-1),…,λ_1,■n≥1生成的W-映象.该文在较弱条件下用黏性逼近方法证明了迭代序列{x_n}强收敛于p∈F且p是下列变分不等式〈(I-f)p,j(p-x~*)〉≤0,■x~*∈F的唯一解.     

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.     

Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T ₁, T ₂ and T ₃: K → E be asymptotically nonexpansive mappings with {k _n }, {l _n } and {j _n }. [1, ∞) such that Σ _n=1 ^∞ (k _n − 1) < ∞, Σ _n=1 ^∞ (l _n − 1) < ∞ and Σ _n=1 ^∞ (j _n − 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T _1x = T _2x = T ₃ x} = x} denotes the common fixed points set of T ₁, _{T 2} and T ₃. Let {α _n }, {α′ _n } and {α″ _n } be real sequences in (0, 1) and ∈ ≤ {α _n }, {α′ _n }, {α″ _n } ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x ₁ ∈ K define the sequence {x _n } by

Convergence theorems for fixed point problems and variational inequality problems in Hilbert spaces

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an α ‐inverse strongly monotone mapping in a Hilbert space. We show that the sequence converges strongly to a common element of two sets under some mild conditions on parameters (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities

Assume that F is a nonlinear operator on a real Hilbert space H which is -strongly monotone and -Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We devise an iterative algorithm which generates a sequence (x _n ) from an arbitrary initial point x ₀H. The sequence (x _n ) is shown to converge in norm to the unique solution u* of the variational inequality

Approximating a Zero of Sum of Two Monotone Operators Which Solves a Fixed Point Problem in Reflexive Banach Spaces

Abstract

The purpose of this paper is to introduce an iterative method for approximating a point in the set of zeros of the sum of two monotone mappings, which is also a solution of a fixed point problem for a Bregman strongly nonexpansive mapping in a real reflexive Banach space. With our iterative technique, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a variational inclusion problem for sum of two monotone mappings and the set of solutions of a fixed point problem for Bregman strongly nonexpansive mapping. We give applications of our result to convex minimization problem, convex feasibility problem, variational inequality problem, and equilibrium problem. Our result complements and extends some recent results in literature.