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We propose an extension of a special form of gradient descent -- in the literature known as linearised Bregman iteration -- to a larger class of non-convex functions. We replace the classical (squared) two norm metric in the gradient…

Optimization and Control · Mathematics 2021-05-26 Martin Benning , Marta M. Betcke , Matthias J. Ehrhardt , Carola-Bibiane Schönlieb

In this paper, we prove the existence of fixed points of mappings satisfying the condition (Da), a kind of generalized nonexpansive mappings, on a weakly compact convex subset in a Banach space satisfying Opial's condition. And we use…

Functional Analysis · Mathematics 2020-07-07 Chang Il Rim , Jong Gyong Kim

In this paper, using generalized metric projection, we propose a new extragradient method for finding a common element of the solutions set of a generalized equilibrium problem and a variational inequality for an $\alpha$-inverse-strongly…

Functional Analysis · Mathematics 2016-11-01 Zeynab Jouymandi , Fridoun Moradlou

The aim of this paper is to establish a strong convergence theorem for a strongly nonexpansive sequence in a Banach space. We also deal with some applications of the convergence theorem.

Functional Analysis · Mathematics 2025-09-17 Koji Aoyama , Masashi Toyoda

In this paper, we first introduce an iterative process in modular function spaces and then extend the idea of a {\lambda}-firmly nonexpansive mapping from Banach spaces to modular function spaces. We call such mappings as…

Functional Analysis · Mathematics 2021-11-23 Safeer Hussain Khan

We introduce an abstract algorithm that aims to find the Bregman projection onto a closed convex set. As an application, the asymptotic behaviour of an iterative method for finding a fixed point of a quasi Bregman nonexpansive mapping with…

Functional Analysis · Mathematics 2013-09-26 Heinz H. Bauschke , Jiawei Chen , Xianfu Wang

We discuss a special form of gradient descent that in the literature has become known as the so-called linearised Bregman iteration. The idea is to replace the classical (squared) two norm metric in the gradient descent setting with a…

Optimization and Control · Mathematics 2016-12-28 Martin Benning , Marta M. Betcke , Matthias J. Ehrhardt , Carola-Bibiane Schönlieb

In this paper, we prove the strong convergence theorems for nearly nonexpansive mappings, using the modified Picard-Mann hybrid iteration process in the context of uniformly convex Banach space.

Functional Analysis · Mathematics 2021-01-15 Adrian Ghiura

We prove strong convergence theorems of some iterative algorithms in a real uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors.

Functional Analysis · Mathematics 2015-08-28 Abba Auwalu

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…

Functional Analysis · Mathematics 2014-10-21 Sattar Alizadeh , Fridoun Moradlou

In this study, we introduce a new iterative processes to approximate common fixed points of an infinite family of quasi-nonexpansive mappings and obtain a strongly convergent iterative sequence to the common fixed points of these mappings…

Functional Analysis · Mathematics 2015-02-24 K. Dogan , V. Karakaya

In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method…

Optimization and Control · Mathematics 2018-05-08 Bui Van Dinh , Nguyen Ngoc Hai , Do Sang Kim

The aim of this paper is to establish strong convergence theorems for a strongly relatively nonexpansive sequence in a smooth and uniformly convex Banach space. Then we employ our results to approximate solutions of the zero point problem…

Functional Analysis · Mathematics 2020-12-29 Koji Aoyama , Yasunori Kimura , Fumiaki Kohsaka

The purpose of this paper is to study an implicit scheme for a representation of nonexpansive mappings on a closed convex subset of a smooth and uniformly convex Banach space with respect to a left regular sequence of means defined on an…

Functional Analysis · Mathematics 2015-06-10 Ebrahim Soori

Fixed point iterations are a fundamental tool in numerical analysis and scientific computing for the approximation of solutions to nonlinear problems. Their convergence is often established via the Banach fixed point theorem, provided that…

Numerical Analysis · Mathematics 2026-04-29 Thomas P. Wihler

Our work presents a new iterative scheme to approximate the fixed points of nonexpansive mapping. The proposed algorithm is constructed to enhance convergence efficiency while preserving theoretical robustness. Under appropriate assumptions…

Functional Analysis · Mathematics 2026-01-12 Nida Izhar Mallick , Izhar Uddin

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…

Functional Analysis · Mathematics 2010-07-07 Eric U. Ofoedu

A Strong Convergence Theorem for finite families of Bregman Demimetric Mappings in a Banach Space under a New Shrinking projection Method

Functional Analysis · Mathematics 2021-09-07 Bijan Orouji , Ebrahim Soori , Donal O'Regan , Ravi P. Agarwal

In this paper, inspired by Iemoto and Takahashi [S. Iemoto, W. Takahashi, Nonlinear Analysis 71, (2009), 2082-2089], we study the Halpern's method to approximate strongly fixed points of a nonexpansive mapping and of a nonspreading mapping.…

Functional Analysis · Mathematics 2014-05-06 Filomena Cianciaruso , Giuseppe Marino , Angela Rugiano , Bruno Scardamaglia

In this paper we prove the weak and strong convergence of the implicit iterative process with errors to a common fixed point of a finite family $\{T_j\}_{i=1}^N$ of asymptotically quasi $I_j-$nonexpansive mappings as well as a family of…

Functional Analysis · Mathematics 2010-10-18 Farrukh Mukhamedov , Mansoor Saburov