Abstract:
We prove a strong convergence theorem by using a hybrid algorithm in order to find a common fixed point of Lipschitz pseudocontraction and κ-strict pseudocontraction in Hilbert spaces. Our results extend the recent ones announced by Yao et al. (2009) and many others.

Abstract:
Let $T$ be an expanding Markov map with a countable number of inverse branches and a repeller $\Lambda$ contained within the unit interval. Given $\alpha \in \R_+$ we consider the set of points $x \in \Lambda$ for which $T^n(x)$ hits a shrinking ball of radius $e^{-n\alpha}$ around $y$ for infinitely many iterates $n$. Let $s(\alpha)$ denote the infimal value of $s$ for which the pressure of the potential $-s\log|T'|$ is below $s \alpha$. Building on previous work of Hill, Velani and Urba\'{n}ski we show that for all points $y$ contained within the limit set of the associated iterated function system the Hausdorff dimension of the shrinking target set is given by $s(\alpha)$. Moreover, when $\bar{\Lambda}=[0,1]$ the same holds true for all $y \in [0,1]$. However, given $\beta \in (0,1)$ we provide an example of an expanding Markov map $T$ with a repeller $\Lambda$ of Hausdorff dimension $\beta$ with a point $y\in \bar{\Lambda}$ such that for all $\alpha \in \R_+$ the dimension of the shrinking target set is zero.

Abstract:
The aim of this paper is to obtain some existence theorems related to a hybrid projection method and a hybrid shrinking projection method for firmly nonexpansive-like mappings (mappings of type (P)) in a Banach space. The class of mappings of type (P) contains the classes of resolvents of maximal monotone operators in Banach spaces and firmly nonexpansive mappings in Hilbert spaces.

Abstract:
The aim of this paper is to obtain some existence theorems related to a hybrid projection method and a hybrid shrinking projection method for firmly nonexpansive-like mappings (mappings of type (P)) in a Banach space. The class of mappings of type (P) contains the classes of resolvents of maximal monotone operators in Banach spaces and firmly nonexpansive mappings in Hilbert spaces.

Abstract:
In this paper, for an $\lambda$-strict pseudocontraction $T$, we prove strong convergence of the modified Mann's iteration defined by $$x_{n+1}=\beta_{n}u+\gamma_nx_n+(1-\beta_{n}-\gamma_n)[\alpha_{n}Tx_n+(1-\alpha_{n})x_n],$$ where $\{\alpha_{n}\}$, $ \{\beta_{n}\}$ and $\{\gamma_n\}$ in $(0,1)$ satisfy: (i) $0 \leq \alpha_{n}\leq \frac{\lambda}{K^2}$ with $\liminf\limits_{n\to\infty}\alpha_n(\lambda-K^2\alpha_n)> 0$; (ii) $\lim\limits_{n\to\infty}\beta_n= 0$ and $\sum\limits_{n=1}^\infty\beta_n=\infty$; (iii) $\limsup\limits_{n\to\infty}\gamma_n<1$.Our results unify and improve some existing results.

Abstract:
We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Fréchet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010) is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.

Abstract:
We modify the iterative method introduced by Kim and Xu (2006) for a countable family of Lipschitzian mappings by the hybrid method of Takahashi et al. (2008). Our results include recent ones concerning asymptotically nonexpansive mappings due to Plubtieng and Ungchittrakool (2007) and Zegeye and Shahzad (2008, 2010) as special cases. 1. Introduction Let be a nonempty closed convex subset of a real Hilbert space . A mapping is said to be Lipschitzian if there exists a positive constant such that In this case, is also said to be -Lipschitzian. Clearly, if is -Lipschitzian and , then is -Lipschitzian. Throughout the paper, we assume that every Lipschitzian mapping is -Lipschitzian with . If , then is known as a nonexpansive mapping. We denote by the set of fixed points of . If is nonempty bounded closed convex and is a nonexpansive of into itself, then (see [1]). There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [2] introduced the iteration as follows: a sequence defined by where the initial guess element is arbitrary and is a real sequence in . Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Reich [3]. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak convergence [4]. Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.2): where denotes the metric projection from onto a closed convex subset of . They prove that if the sequence bounded above from one, then defined by (1.3) converges strongly to . Takahashi et al. [6] modified (1.3) so-called the shrinking projection method for a countable family of nonexpansive mappings as follows: and prove that if the sequence bounded above from one, then defined by (1.4) converges strongly to . Recently, the present authors [7] extended (1.3) to obtain a strong convergence theorem for common fixed points of a countable family of -Lipschitzian mappings by where as and prove that defined by (1.5) converges strongly to . In this paper, we establish strong convergence theorems for finding common fixed points of a countable family of Lipschitzian mappings in a real Hilbert space. Moreover, we also apply our results for asymptotically nonexpansive mappings. 2. Preliminaries Let be a real Hilbert space with inner product and norm . Then, for all and . For any points in , the following generalized identity holds: where

Abstract:
We further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their Picard iterates. We use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of result on the asymptotic behavior of Picard iterates for firmly nonexpansive mappings proved by Reich and Shafrir. From this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings. Besides this, we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting.

Abstract:
The demiclosedness principle is one of the key tools in nonlinear analysis and fixed point theory. In this note, this principle is extended and made more flexible by two mutually orthogonal affine subspaces. Versions for finitely many (firmly) nonexpansive operators are presented. As an application, a simple proof of the weak convergence of the Douglas-Rachford splitting algorithm is provided.

Abstract:
We study nearly equal and nearly convex sets, ranges of maximally monotone operators, and ranges and fixed points of convex combinations of firmly nonexpansive mappings. The main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings. A striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular. Throughout, examples are provided to illustrate the theory. We also obtain detailed information on the domain and range of the resolvent average.