Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C with a fixed point, for some 0?κ<1. Given an initial guess x₀∈C and given also a real sequence {αn} in (0,1). The Mann's algorithm generates a sequence {xn} by the formula: xn+1=αnxn+(1−αn)Txn, n?0. It is proved that if the control sequence {αn} is chosen so that κ<αn<1 and , then {xn} converges weakly to a fixed point of T. However this convergence is in general not strong. We then modify Mann's algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372-379] from nonexpansive mappings to strict pseudo-contractions.     

Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces

In the present paper, we establish a demiclosedness principle for asymptotically pseudo-contractions in Hilbert spaces. As its applications, we obtain a new fixed point theorem, some weak and strong convergence theorems for asymptotically pseudo-contractions. The results of this paper are interesting extensions of those known results.     

关于实李普希兹映射的lshikawa迭代的收敛率和加速收敛问题

本文研究使用Ｉｓｈｉｋａｗａ迭代格式求实李普希兹映射的不动点，指出该迭代格式仅具线性收敛率；对于参变量序列｛αｎ｝、｛βｎ｝所取的不同的值，比较了迭代格式的收敛速度．在给出加速因子定义的基础上，本文给出了加速收敛的一个充分条件．     

Strong convergence theorems for strict pseudo-contractions in Hilbert spaces

In this paper, we suggest and analyze some iterative algorithms for strict pseudo-contractions in the sense of Browder-Petryshyn in a real Hilbert space. We prove that the proposed iterative algorithms converge strongly to some fixed point of a strict pseudo-contraction.     

Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces

In this paper, we continue to study weak convergence problems for the implicit iteration process for a finite family of Lipschitzian continuous pseudocontractions in general Banach spaces. The results presented in this paper improve and extend the corresponding ones of Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773], Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81], Chen et al. [R. Chen, Y.S. Song, H.Y. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709] and others.     

Convergence theorems for common fixed points of asymptotically quasi-nonexpansive mappings in convex metric spaces

In this paper, we consider an implicit iteration process to approximate the common fixed points of two finite families of asymptotically quasi-nonexpansive mappings in convex metric spaces. As a consequence of our result, we obtain some related convergence theorems. Our results generalize some recent results of Khan and Ahmed [4], Khan et al. [6], Sun [12], Wittmann [14] and Xu and Ori [15].     

Characterizations of common fixed points of one-parameter nonexpansive semigroups, and convergence theorems to common fixed points

We first prove characterizations of common fixed points of one-parameter nonexpansive semigroups. We next present convergence theorems to common fixed points.     

Strong convergence theorems for a family of quasi-asymptotic pseudo-contractions in Hilbert spaces

In the present paper, we propose two kinds of new algorithms for a family of quasi-asymptotic pseudo-contractions in real Hilbert spaces. By using the proposed algorithms, we prove several strong convergence theorems for a family of quasi-asymptotic pseudo-contractions. The results of this paper are interesting extensions of those known results.     

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 path for a family of strict pseudo-contractions in Hilbert spaces

In this paper, we propose an implicit viscosity approximation method for finding a common fixed points of infinite countable families of strict pseudo-contractions in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by Maingé (J. Math. Anal. Appl. 325:469–479, 2007) and some others.     

Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces

In this paper, we consider an Ishikawa type iteration process with errors to approximate the fixed point of two uniformly quasi-Lipschitzian mappings in convex metric spaces. Some results have been obtained which generalize and unify many important known results in recent literature.     

Weak convergence theorems for strict pseudo-contractions in Banach spaces

In this paper, it is shown that there is a gap in the paper [Chidume, C. E., Shahzad, N.: Weak convergence theorems for a finite family of strict pseudo-contractions. Nonlinear Anal., 72, 1257–1265(2010)], consequently, the main results of the paper do not hold in uniformly smooth Banach spaces. Meanwhile, it is also shown that the main results(Lemma 3.4, Theorems 3.5–3.6, 3.8–3.9) in the paper [Cholamjiak, P., Suantai, S.: Weak convergence theorems for a countable family of strict pseudo-contractions in Banach spaces. Fixed Point Theory Appl., 2010, Article ID 632137, 16 pages(2010)] do not hold in Lpfor p 3. Finally, some modified results are presented in the setting of uniformly smooth and uniformly convex Banach spaces which include Lpfor p ≥ 2 as special cases. Furthermore, our arguments are also different from the ones given by the authors above.     

Extreme points and isometries on vector-valued Lipschitz spaces

For a Banach space E and a compact metric space (X,d), a function F:X→E is a Lipschitz function if there exists k>0 such that

     

Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces

In this paper, we modify the normal Mann’s iterative process to have strong convergence for a k$k$-strictly pseudo-contractive non-self mapping in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others.     

Extreme points of unit balls in Lipschitz function spaces

We give a new characterization of the set of all extreme points of the unit ball in the Banach space of all Lipschitz functions on a metric space This result is applied to get a total variation characterization of in the particular case when is a convex subset of a Banach space.

     

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

Convergence theorems of fixed points for asymptotically hemi-pseudocontractive mappings

In this paper, the concept of asymptotically hemi-pseudocontractive mapping in Banach space, which is a proper generalization of asymptotically pseudocontractive mapping with nonempty fixed point set as shown by an example, is introduced. Some sufficient and necessary conditions on the strong convergence of the modified Ishikawa and the modified Mann iteration sequences with mean errors for uniformly L-Lipshitzian asymptotically hemi-pseudocontractive mappings are presented.