Strong convergence and control condition of modified Halpern iterations in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gâteaux differentiable norm. Let T∈ΓC and f∈ΠC. Assume that {xt} converges strongly to a fixed point z of T as t→0, where xt is the unique element of C which satisfies xt=tf(xt)+(1−t)Txt. Let {αn} and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1)lim⁡n→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1. For arbitrary x0∈C, let the sequence {xn} be defined iteratively by yn=αnf(xn)+(1−αn)Txn, n≥0, xn+1=βnxn+(1−βn)yn, n≥0. Then {xn} converges strongly to a fixed point of T

A Regularized Gradient Projection Method for the Minimization Problem

We investigate the following regularized gradient projection algorithm xn+1=Pc(I−γn(∇f+αnI))xn, n≥0. Under some different control conditions, we prove that this gradient projection algorithm strongly converges to the minimum norm solution of the minimization problem minx∈Cf(x)

An iterative method for fixed point problems and variational inequality problems

In this paper, we present an iterative method for fixed point problems and variational inequality problems. Our method is based on the so-called extragradient method and viscosity approximation method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for monotone mapping

A strong convergence of a modified Krasnoselskii‐Mann method for non‐expansive mappings in Hilbert spaces

In this paper, we introduce a new method based on the well‐known Krasnoselskii‐Mann's method for non‐expansive mappings in Hilbert spaces. We show that the proposed method has strong convergence for non‐expansive mappings. Keywords: non‐expansive mapping, fixed point, modified Krasnoselskii‐Mann's method, strong convergence, Hilbert space. First published online: 09 Jun 201

Extended Extragradient Methods for Generalized Variational Inequalities

We suggest a modified extragradient method for solving the generalized variational inequalities in a Banach space. We prove some strong convergence results under some mild conditions on parameters. Some special cases are also discussed