On some strong convergence results of a new Halpern-type iterative process for quasi-nonexpansive mappings and accretive operators in Banach spaces

On the Iterations of a Sequence of Strongly Quasi-nonexpansive Mappings with Applications

In this paper, we study $Δ$- convergence of iterations for a sequence of strongly quasi-nonexpansive mappings as well as the strong convergence of the Halpern type regularization of them in Hadamard spaces. Then, we give some their applications in iterative methods, convex and pseudo-convex minimization(proximal point algorithm), fixed point theory and equilibrium problems. The results extend several new results in the literature and some of them seem new even in Hilbert spaces.

Solution of variational inequality problems on fixed point sets of nonexpansive mappings using iterative methods

In this paper, we introduce new implicit and explicit iterative schemes which converge strongly to a unique solution of variational inequality problems for strongly accretive operators over a common fixed point set of finite family of nonexpansive mappings in $q$-uniformly smooth real Banach spaces. As an application, we introduce an iteration process which converges strongly to a solution of the variational inequality which is a common fixed point of finite family of strictly pseudocontractive mappings. Our theorems extend, generalize, improve and unify the corresponding results of Xu \cite{2…

Effective metastability of Halpern iterates in CAT(0) spaces

This paper provides an effective uniform rate of metastability (in the sense of Tao) on the strong convergence of Halpern iterations of nonexpansive mappings in CAT(0) spaces. The extraction of this rate from an ineffective proof due to Saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs. This methodology is applied here for the first time to a proof that uses Banach limits and hence makes a substantial reference to the axiom of choice.

Strong convergence of modified inertial Mann algorithms for nonexpansive mappings

In this paper, we introduce two new modified inertial Mann Halpern and viscosity algorithms for solving fixed point problems. We establish strong convergence theorems under some suitable conditions. Finally, our algorithms are applied to split feasibility problem, convex feasibility problem and location theory. The algorithms and results presented in this paper can summarize and improve corresponding results previously known in this area.

可数族平衡问题和变分不等式问题的黏性-Halpern混合方案

A viscosity-Halpern hybrid scheme for countable families of equilibrium and variational inequality problems

提出一种黏性-Halpern混合投影方案，用于逼近一致光滑且一致凸实Banach空间中可数族广义非扩张型映射的不动点集、可数族变分不等式问题的解集和可数族平衡问题的解集的公共元，并证明强收敛性。

Quantitative translations for viscosity approximation methods in hyperbolic spaces

In the setting of hyperbolic spaces, we show that the convergence of Browder-type sequences and Halpern iterations respectively entail the convergence of their viscosity version with a Rakotch map. We also show that the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type sequence. Our results follow from proof-theoretic techniques (proof mining). From an analysis of theorems due to T. Suzuki, we extract a transformation of rates for the original Browder type and Halpern iterations into rates for the corresponding viscos…

The finitary content of sunny nonexpansive retractions

We use techniques of proof mining to extract a uniform rate of metastability (in the sense of Tao) for the strong convergence of approximants to fixed points of uniformly continuous pseudocontractive mappings in Banach spaces which are uniformly convex and uniformly smooth, i.e. a slightly restricted form of the classical result of Reich. This is made possible by the existence of a modulus of uniqueness specific to uniformly convex Banach spaces and by the arithmetization of the use of the limit superior. The metastable convergence can thus be proved in a system which has the same provably tot…

String-Averaging Methods for Best Approximation to Common Fixed Point Sets of Operators: The Finite and Infinite Cases

String-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm at hand requires to employ the operators in a specific order. Sequential orderings are well-known and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by subsets of the index set of all operators, to apply the operators along these strings and then to combine their end-points in some agreed manner to yield the next iterate of the algorithm. String-averaging methods were discusse…