New iterative schemes for strongly relatively nonexpansive mappings and maximal monotone operators

In this paper, some new iterative schemes for approximating the common element of the set of fixed points of strongly relatively nonexpansive mappings and the set of zero points of maximal monotone operators in a real uniformly smooth and uniformly convex Banach space are proposed. Some weak convergence theorems are obtained, which extend and complement some previous work.     

Banach空间中关于平衡问题与半相对非扩张映象的强收敛定理

在Banach空间框架下,将半相对非扩张映象引入到平衡问题中,利用收缩投影迭代来近似平衡问题的解集与半相对非扩张映象的不动点集的公共点,并在适当的条件下证明了该迭代序列强收敛于某个公共点.所得的结果推广和改进了已有的相关结果.     

Banach空间中非膨胀映象和α-逆强增生算子的强收敛性

在2-一致光滑的Banach空间中,引入一种新的迭代算法研究非膨胀映象的不动点集与α-逆强增生算子的变分不等式解集的公共元素,并获得了迭代算法的强收敛性定理.而且应用这些结果考虑了非膨胀映象和严格伪压缩映象公共不动点的收敛性问题.     

Viscosity approximation methods for nonexpansive multimaps in Banach spaces

We prove strong convergence of the viscosity approximation process for nonexpansive nonself multimaps. Furthermore, an explicit iteration process which converges strongly to a fixed point of multimap T is constructed. It is worth mentioning that, unlike other authors, we do not impose the condition ＂Tz = {z}＂ on the map T.     

非线性混合似变分不等式的扰动近似点-投影算法

In this paper we introduce a new perturbed proximal-projection algorithm for finding the common element of the set of fixed points of non-expansive mappings and the set of solutions of nonlinear mixed variational-like inequalities. The convergence criteria of the iterative sequences generated by the new iterative algorithm is also given. Our approach and results generalize many known results in this field.     

Approximating the common fixed points of two sequences of uniformly quasi-Lipschitzian mappings in convex metric spaces

In this paper, a kind of Ishikawa type iterative scheme with errors for approximating a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings is introduced and studied in convex metric spaces. Under some suitable conditions, the convergence theorems concerned with the Ishikawa type iterative scheme with errors to approximate a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings were proved in convex metric spaces. The results presented in the paper generalize and improve some recent results of Wang and Liu (C. Wang, L.W. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Anal., TMA 70 (2009), 2067-2071).     

An iterative method for equilibrium problems and a finite family of relatively nonexpansive mappings in a Banach space

In this paper, we introduce a new iterative method for finding a common element of the set of fixed points of a finite family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in uniformly convex and uniformly smooth Banach spaces. Then we prove a strong convergence theorem by using the generalized projection.     

Iterative Methods for Fixed Points of Asymptotically Weakly Contractive Maps

Suppose K is a closed convex nonexpansive retract of a real uniformly smooth Banach space E with P as the nonexpansive retraction. Suppose T : K → E is an asymptotically d-weakly contractive map with sequence {k_n }, k_n ≥ 1, lim k_n = 1 and with F(T) _n int (K) ≠ ø F(T):= {x ∈ K: Tx = x}. Suppose {x _n } is iteratively defined by x _n+1 = P((l ? k_nα_n )x _n +k _n α _n T(PT) ^n?l x_n ), n = 1,2,...,x ₁ ∈ K, where α_n∈ (0,l) satisfies lim α_n = 0 and Σα_n = ∞. It is proved that {x _n } converges strongly to some x ^* ∈ F(T)∩ int K. Furthermore, if K is a closed convex subset of an arbitrary real Banach space and T is, in addition uniformly continuous, with F(T) ≠ ø, it is proved that {x_n } converges strongly to some x ^* ∈ F(T).     

Fixed Point Theorems for Mean Nonexpansive Mappings in CAT(0) Spaces

In this article, we prove the existence of fixed points and the demiclosed principle for mean nonexpansive mappings in Cartan, Alexandrov and Toponogov(0) spaces. We also obtain a Δ-convergence theorem and a strong convergence theorem of Ishikawa iteration for mean nonexpansive mappings in Cartan, Alexandrov and Toponogov(0) spaces.     

Fixed Point Theorem of {a,b,c} Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following： 1. If T is {a,b,c}-ntype mapping, then inf{ ｜｜ T（x）-x｜｜ ：x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞｜｜T（xn） - Xn｜｜ = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn （x）}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].