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In this paper, we suggest and analyze four inertial algorithms for solving fixed point problems of Bregman quasi-nonexpansive mappings in the framework of reflexive Banach spaces. Our first two algorithms, we propose inertial-like methods based on Mann-type and Halpern-type iterations, and in the others, we propose relaxed inertial-like methods based on Mann-type and Halpern-type iterations. The weak and strong convergence of the algorithms are established under some appropriate conditions on the parameters. As an application, we utilize our main results to find a zero of the sum of Bregman inverse strongly monotone mappings and maximal monotone operators in real reflexive Banach spaces. Also, we provide several numerical experiments to show the convergence behaviour of our algorithms in both finite-dimensional and infinite-dimensional spaces. Finally, we further utilize our algorithms to numerically solve the data classification problems of lung cancer. [ABSTRACT FROM AUTHOR]

In this paper, we find the common solution of a split variational inclusion problem and a fixed point problem of a finite collection of demicontractive mappings in real Hilbert spaces. We introduce two modified inertial-type iterative methods based on Mann and viscosity schemes to solve the considered problem. We prove weak and strong convergence results for the proposed methods under reasonable conditions on the control parameters. With the help of some numerical examples, we show that our proposed methods give better convergence results than some existing methods in the literature. [ABSTRACT FROM AUTHOR]

In 1643 P. de Fermat introduced the problem of finding a point in the plane such that the sum of its Euclidean distances to the three given points is minimal. Recently, Reich and Tuyen (J Optim Theory Appl 196(1): 78–97, 2023) extended this problem in Hilbert space setting and named it ‘generalized Fermat–Torricelli problem’. They introduced some iterative methods for approximating the solution of this problem. This paper aims to formulate some novel iterative methods with inertial effect for approximating the solution of this generalized Fermat–Torricelli problem in real Hilbert spaces. First, we prove the weak and strong convergence of the proposed methods with the mild conditions on the control parameters. Then, we establish the weak and strong convergence results for approximating the solution of a variant of the split feasibility problem with multiple output sets. Finally, we provide some numerical examples to justify the validity and efficiency of the proposed iterative methods. The results in this paper improve and generalize the work of Reich and Tuyen (J Optim Theory Appl 196(1): 78–97, 2023). [ABSTRACT FROM AUTHOR]

A modular string averaging procedure (MSA, for short) for a finite number of operators was first introduced by Reich and Zalas in 2016. The MSA concept provides a flexible algorithmic framework for solving various feasibility problems such as common fixed point and convex feasibility problems. In 2001 Bauschke and Combettes introduced the notion of coherence and applied it to proving weak and strong convergence of many iterative methods. In 2019 Barshad, Reich and Zalas proposed a stronger variant of coherence which provides a more convenient sufficient convergence condition for such methods. In this paper we combine the ideas of both modular string averaging and coherence. Focusing on extending the above MSA procedure to an infinite sequence of operators with admissible controls, we establish strong coherence of its output operators. Various applications of these concepts are presented with respect to weak and strong convergence. They also provide important generalizations of known results, where the weak convergence of sequences of operators generated by the MSA procedure with intermittent controls was considered. [ABSTRACT FROM AUTHOR]

In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two modified Tseng's extragradient algorithms with a new generalized adaptive stepsize for solving monotone variational inequalities (VI) in reflexive Banach spaces. The advantage of our methods is that stepsizes do not require prior knowledge of the Lipschitz constant of the cost mapping. Based on Bregman projection-type methods, we prove weak and strong convergence of the proposed algorithms to a solution of VI. Some numerical experiments to show the efficiency of our methods including a comparison with related methods are provided. [ABSTRACT FROM AUTHOR]

This article analyzes the JK iteration process with the class of mappings that are essentially endowed with a condition called condition (E). The convergence of the iteration toward a fixed point of a specific mapping satisfying the condition (E) is obtained under some possible mild assumptions. It is worth mentioning that the iteration process JK converges better toward a fixed point compared to some prominent iteration processes in the literature. This fact is confirmed by a numerical example. Furthermore, it has been shown that the iterative scheme JK is stable in the setting of generalized contraction. The data dependence result is also established. Our results are new in the iteration theory and extend some recently announced results of the literature.

We come up with a new type of forward–backward–forward algorithms for monotone inclusion problems based on a self-adaptive technique to avoid the selection of Lipschitz assumption and also double inertial extrapolations to increase the convergence performance of our presented algorithm. We also prove its weak convergence theorem under mild hypothesis. Additionally, we provide numerical test in image deblurring and signal recovery as applications. The results show that our algorithm outperforms some known algorithms in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two algorithms for finding a common solution of the monotone inclusion problem and the fixed point problem for a relatively nonexpansive mapping in reflexive Banach spaces. The weak convergence results for both algorithms are established without the prior knowledge of the Lipschitz constant of the mapping. An application to the variational inequality problem is considered. Finally, some numerical experiments of the proposed algorithms including comparisons with other algorithms are provided. [ABSTRACT FROM AUTHOR]

Projection methods have been one of the popular methods for solving variational inequality problems in the literature. Several projection methods have been studied for variational inequality problems and most of these methods require at least two evaluations of the cost operator and (or) two projections per iteration. One of the major interests of researchers in variational inequalities is to study projection methods with less computational complexity in terms of reduced number of evaluations of the cost operator and reduced number of projections onto feasible set per iteration especially when the cost operator is non-monotone. In this paper, we propose a simple projection method which requires only one evaluation of the cost operator and one projection onto the feasible set per iteration. We obtain weak convergence results when the cost operator is quasimonotone and Lipschitz continuous with self-adaptive line search strategy. We also give weak convergence analysis when the cost operator is non-monotone and Lipschitz continuous. Some numerical implementations using standard tests are given to illustrate our method and compare with other recently related methods. [ABSTRACT FROM AUTHOR]

Let G : H → H and K : H → H be monotone mappings that are either sequentially weakly continuous or continuous, where H is a real Hilbert space. In this work, we introduce two new iterative methods for approximating solutions of the Hammerstein equation u + G K u = 0 , if they exist. The first iterative method is shown to always converge weakly to an element in the solution set of the Hammerstein equation if this solution set is nonempty. The second iterative method is a modification of the first method to upgrade weak convergence to strong convergence. Convergence results are obtained without requiring the maps to be bounded. Numerical examples are provided to demonstrate the convergence of one of these methods. Comparisons with some existing methods show that the method is cost effective in terms of the number of iterations required to obtain a solution and the computational time. [ABSTRACT FROM AUTHOR]

The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a modification of the forward–backward splitting method with an adaptive step size rule for inclusion problems in real Hilbert spaces. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish weak convergence of the proposed algorithm. Moreover, if the single-valued operator is cocoercivity, then the proposed algorithm strongly converges to the unique solution of the problem with an R-linear rate. Finally, we give several numerical experiments to illustrate the convergence of the proposed algorithm and also to compare them with others. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study quasi-monotone variational inequality problems in infinite dimensional Hilbert spaces and to prove that the iterative sequence generated by an extragradient algorithm converges weakly to a solution. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of ϕ -nonexpansive sequences and maps in a Banach space E, and prove an ergodic theorem, a fixed point theorem, and some weak and strong convergence theorems for such sequences and maps in E, with rather mild assumptions on the Banach space E, weaker than those assumed by previous authors for nonexpansive maps. These results give a partial answer to the still open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose two modified forward-backward splitting algorithms for zeros of the sum of a maximal monotone operator and a Bregman inverse strongly monotone operator in reflexive Banach spaces. We prove weak and strong convergence theorems of the generated sequences by the proposed methods under some suitable conditions. We apply our results to study the variational inequality problem and the equilibrium problem. Finally, a numerical example is given to illustrate the proposed methods. The results presented in this paper improve and generalize many known results in recent literature. [ABSTRACT FROM AUTHOR]

In this work, we investigate pseudomonotone variational inequality problems in a real Hilbert space and propose two projection-type methods with inertial terms for solving them. The first method does not require prior knowledge of the Lipschitz constant and the second one does not require the Lipschitz continuity of the mapping which governs the variational inequality. A weak convergence theorem for our first algorithm is established under pseudomonotonicity and Lipschitz continuity assumptions, and a weak convergence theorem for our second algorithm is proved under pseudomonotonicity and uniform continuity assumptions. We also establish a nonasymptotic O(1/n) convergence rate for our proposed methods. In order to illustrate the computational effectiveness of our algorithms, some numerical examples are also provided. [ABSTRACT FROM AUTHOR]

We propose and study new projection-type algorithms for solving pseudomonotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the cost operators. We prove weak and strong convergence theorems for the sequences generated by these new methods. The numerical behavior of the proposed algorithms when applied to several test problems is compared with that of several previously known algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two inertial accelerated algorithms for solving the split common fixed-point problem of directed operators in real Hilbert space. The proposed iterative algorithms combine the primal-dual method and the inertial method with the self-adaptive stepsizes such that the implementation of our algorithms does not need any prior information about bounded linear operator norms. Under suitable conditions, the weak and strong convergence results of the algorithms are obtained. Numerical results which involve image restoration problems are reported to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A convergence rate of Mann iteration has been studied by many mathematicians. Recently, its rate was improved in a Hilbert space. In this paper, we consider a convergence rate in a geodesic space and generalize the known result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative algorithm from primal-dual methods for solving the split equality common fixed-point problem of quasi-nonexpansive mappings in real Hilbert space. Our algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. We use a way of selecting the stepsizes such that the implementation of our algorithm does not need any prior information about bounded linear operator norms. It avoids the difficult task of estimating the operator norms. Under suitable conditions, we get the weak convergence of the proposed algorithm. The performance of the proposed algorithm is also illustrated by preliminary numerical experiments. The results presented in the paper improve and extend some corresponding results. [ABSTRACT FROM AUTHOR]

In this paper, an improvement of the extragradient algorithm is considered for finding a solution of pseudo-monotone equilibrium problems in real Hilbert spaces. We replace the second minimization problem onto a closed convex set in the extragradient method, with a subgradient projection onto some constructible half-space and propose a new method of getting the step-size. A convergence theorem for the generated sequence is proved and applied for finding a solution of variational inequality problem. [ABSTRACT FROM AUTHOR]

In this paper, we use the dual variable to propose a self-adaptive iterative algorithm for solving the split common fixed point problems of averaged mappings in real Hilbert spaces. Under suitable conditions, we get the weak convergence of the proposed algorithm and give applications in the split feasibility problem and the split equality problem. Some numerical experiments are given to illustrate the efficiency of the proposed iterative algorithm. Our results improve and extend the corresponding results announced by many others. [ABSTRACT FROM AUTHOR]

In this article, we first propose an extended split equality problem which is an extension of the convex feasibility problem, and then introduce a parameter w to establish the fixed point equation system. We show the equivalence of the extended split equality problem and the fixed point equation system. Based on the fixed point equation system, we present a simultaneous iterative algorithm and obtain the weak convergence of the proposed algorithm. Further, by introducing the concept of a G-mapping of a finite family of strictly pseudononspreading mappings {Ti}i=1N, we consider an extended split equality fixed point problem for G-mappings and give a simultaneous iterative algorithm with a way of selecting the stepsizes which do not need any prior information about the operator norms, and the weak convergence of the proposed algorithm is obtained. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze a Krasnoselski-Mann type iterative method to approximate a common element of solution sets of a hierarchical fixed point problem for nonexpansive mappings and a split mixed equilibrium problem. We prove that sequences generated by the proposed iterative method converge weakly to a common element of solution sets of these problems. Further, we derive some consequences from our main result. Furthermore, we extend the considered iterative method to a split monotone variational inclusion problem and deduce some consequences. Finally, we give a numerical example to justify the main result. The method and results presented in this paper generalize and unify the corresponding known results in this area. [ABSTRACT FROM AUTHOR]

In this paper, we present a proximal split feasibility algorithm with an additional inertial extrapolation term for solving a proximal split feasibility problem under weaker conditions on the step sizes. The two convex and lower semi continuous objective functions are assumed to be non-smooth. Some applications to split inclusion problem and split equilibrium problem are given. We demonstrate the efficiency of the proposed algorithm with numerical experiments. [ABSTRACT FROM AUTHOR]

The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. The behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. However, more than a decade ago, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and its weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Our proof relies on a new convergence principle for Fejér monotone sequences. Numerous examples illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the algorithm proposed in recent years by Censor, Gibali and Reich, which solves split variational inequality problem, and Korpelevich's extragradient method, which solves variational inequality problems. As our main result, we propose an iterative method for finding an element to solve a class of split variational inequality problems under weaker conditions and get a weak convergence theorem. As applications, we obtain some new weak convergence theorems by using our weak convergence result to solve related problems in nonlinear analysis and optimization. [ABSTRACT FROM AUTHOR]

We know that variational inequality problem is very important in the nonlinear analysis. The main purpose of this paper is to propose an iterative method for finding an element of the set of solutions of a variational inequality problem with a monotone and Lipschitz continuous mapping in Hilbert space. This iterative method is based on the extragradient method. We get a weak convergence theorem. Using this result, we obtain three weak convergence theorems for the equilibrium problem, the constrained convex minimization problem, and the split feasibility problem. [ABSTRACT FROM AUTHOR]

Let H be a Hilbert space and let C be a closed convex nonempty subset of H and $$T : C\rightarrow H$$ a non-self nonexpansive mapping. A map $$h : C\rightarrow R$$ defined by $$h(x) := \inf \{\lambda \ge 0 : \lambda x+(1-\lambda )Tx \in C \}$$ . Then, for a fixed $$x_0 \in C$$ and for $$\begin{aligned}{\alpha _0} = \max \left\{ {\frac{1}{2},h({x_0})} \right\}\end{aligned}$$ , Krasnoselskii-Mann algorithm is defined by $$x_{n+1}=\alpha _n+(1-\alpha _n)Tx_n,$$ where $$\alpha _{n+1}=\max \{\alpha _n, h(x_{x_{n+1}})\}$$ . Recently, Colao and Marino (Fixed Point Theory Appl 2015:39, 2015) have proved both weak and strong convergence theorems when C is a strictly convex set and T is an inward mapping. Meanwhile, they proposed a open question for a countable family of non-self nonexpansive mappings. In this article, authors will give an answer and will prove the further generalized results with the examples to support them. [ABSTRACT FROM AUTHOR]

Let,,be real Hilbert spaces, let,be two bounded linear operators. The multiple-set split equality common fixed-point problem (MSECFP) under consideration in this paper is to(Section.Display) whereare integers,andare quasi-nonexpansive mappings with nonempty fixed-point sets. Note that, the above problem (1) allows asymmetric and partial relations between the variablesand. Ifand, then the MSECFP (1) reduces to the multiple-set split common fixed-point problem proposed by Censor et al. In this paper, we introduce mixed cyclic and simultaneous iterative algorithms for the MSECFP (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. Weak convergence results are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new modified Ishikawa iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of generalized hybrid mappings in a Hilbert space. Our results generalize, extend and enrich some existing results in the literature. [ABSTRACT FROM AUTHOR]

Let,andbe real Hilbert spaces, letandbe two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem:Section.Displaywhereandare two firmly quasi-nonexpansive operators with nonempty fixed-point setsand. Note that, by takingand, we recover the split common fixed-point problem originally introduced by Cesnor and Segal. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. The weak convergence result of algorithm is obtained and some applied nonlinear analysis examples are stated. [ABSTRACT FROM AUTHOR]

Abstract: Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in Kurokawa and Takahashi [Y. Kurokawa, W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 1562–1568]. We explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of Halpern’s type for our class of mappings and hence resolved in the affirmative an open problem posed by Kurokawa and Takahashi in their final remark for the case where the mapping is averaged. [ABSTRACT FROM AUTHOR]

Abstract: Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm — one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) and later developed by Xu (2006) . The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive. [Copyright &y& Elsevier]

The purpose of this paper is to prove weak and strong convergences of a modified implicit iteration process to common fixed points for a finite family of asymptotically nonexpansive mappings. The results presented in this paper improve and extend some well known results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce an implicit iterative scheme for finding a common element of the set of common fixed points of nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The implicit iterative scheme is based on two well-known methods: extragradient and approximate proximal. We obtain a weak convergence theorem for three sequences generated by this implicit iterative scheme. On the basis of this theorem, we also construct an implicit iterative process for finding a common fixed point of mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mappings are nonexpansive. [Copyright &y& Elsevier]

Let G be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banach space E with a Frechet differentiable norm, and T = { T t : t ∈ G} be a continuous representation of G as nearly asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F(T) of T in C is nonempty. It is shown that if G is right reversible, then for each almost–orbit u(•) of T, $$ \cap _{{s \in ^{G} }} \overline{{{\text{co}}}} {\left\{ {u{\left( t \right)}:t \geqslant s} \right\}} \cap F{\left( {\text{T}} \right)} $$ consists of at most one point. Furthermore, $$ \cap _{{s \in ^{G} }} \overline{{{\text{co}}}} {\left\{ {T_{t} x:t \geqslant s} \right\}} \cap F{\left( {\text{T}} \right)} $$ is nonempty for each x ∈ C if and only if there exists a nonlinear ergodic retraction P of C onto F(T) such that PT s = T s P = P for all s ∈ G and $$ Px \in \overline{{co}} {\left\{ {T_{s} x:s \in G} \right\}} $$ for each x ∈ C. This result is applied to study the problem of weak convergence of the net { u( t) : t ∈ G} to a common fixed point of T. [ABSTRACT FROM AUTHOR]

We extend the notion of uniformly asymptotically almost nonexpansive curves which was first introduced in Rouhani and Kim (J. Nonlinear Convex Anal. 4 (2003) 175) for a commutative semigroup, to a general semitopological semigroup, study the asymptotic behavior of such curves in a Hilbert space, and apply our results to study the asymptotic behavior for almost-orbits of semigroups of asymptotically nonexpansive-type mappings over a general semitopological semigroup in a Hilbert space. Since in this case the semigroup is not directed, the proofs in Rouhani and Kim (J. Nonlinear Convex Anal. 4 (2003) 175) do not extend to this case, and new methods have to be used for the proofs. In addition to Rouhani (J. Math. Anal. Appl. 147 (1990) 465; J. Math. Anal. Appl. 151 (1990) 226; Nonlinear Anal. 44 (2001) 627; Nonlinear Anal. 51 (2002) 735) and Rouhani and Kim (J. Nonlinear Convex Anal. 4 (2003) 175), our results extend and unify many previously known results (J. Math. Anal. Appl. 127 (1987) 206; Proc. Amer. Math. Soc. 104 (1988) 431; Nonlinear Anal. 29 (1997) 539; Topol. Methods Nonlinear Anal. 5 (1995) 305; J. Math. Anal. Appl. 105 (1985) 514; Nonlinear Anal. 26 (1996) 1411; Pacific J. Math. 126 (1987) 277; J. Math. Anal. Appl. 206 (1997) 451; Chinese Ann. Math. 6 (1996) 729; Nonlinear Anal. 14 (1990) 69; J. Math. Anal. Appl. 85 (1982) 172; Proc. Amer. Math. Soc. 81 (1981) 253; Proc. Amer. Math. Soc. 97 (1986) 55; Canad. J. Math. 44 (1992) 880; J. Math. Anal. Appl. 142 (1989) 242; Proc. Amer. Math. Soc. 117 (1993) 385). [Copyright &y& Elsevier]

Recently, Hundal has constructed a hyperplane H, a cone K, and a starting point y0 in ℓ2 such that the sequence of alternating projections ((PKPH)ny0)n∈N converges weakly to some point in H∩K, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections; hence, we give an affirmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of firmly nonexpansive maps (a` la Genel and Lindenstrauss) and for the proximal point algorithm (a` la Gu¨ler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Censor et al. Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well. [Copyright &y& Elsevier]