333 research outputs found

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

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    Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly GĂąteaux 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 {ÎČn} 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→∞ÎČn≀lim⁥sup⁥n→∞ÎČn<1. For arbitrary x0∈C, let the sequence {xn} be defined iteratively by yn=αnf(xn)+(1−αn)Txn, n≄0, xn+1=ÎČnxn+(1−ÎČn)yn, n≄0. Then {xn} converges strongly to a fixed point of T

    KRASNOSELSKI–MANN ITERATION FOR HIERARCHICAL FIXED POINTS AND EQUILIBRIUM PROBLEM

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    AbstractWe give an explicit Krasnoselski–Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: where C is a closed convex subset of a Hilbert space H, G:C×C→ℝ is an equilibrium function, T:C→C is a nonexpansive mapping with Fix(T) its set of fixed points and f:C→C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, 'Weak and strong convergence of Krasnosel'skiĭ–Mann iteration for hierarchical fixed point problems', Inverse Problems24 (2008), 501–508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction f was considered. The paper follows the lines of research of corresponding results of Moudafi and ThĂ©ra

    An iterative method for fixed point problems and variational inequality problems

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    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 Regularized Gradient Projection Method for the Minimization Problem

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    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)

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

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    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

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    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
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