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  • General alternative regularization methods for split equality common Fixed-Point Problem
    Optimization, 2017
    Co-Authors: Jing Zhao, Yunnuan Jia, Hang Zhang
    Abstract:

    AbstractThe purpose of this paper is to propose a general alternative regularization algorithm for split equality common Fixed-Point Problem of nonexpansive operator in the framework of infinite-dimensional Hilbert spaces. We prove the strong convergence of the proposed algorithm with the stepsizes chosen by two ways. As a consequence, we obtain strong convergence theorems for split equality common Fixed-Point Problem of firmly-nonexpansive operator and split equality Problem. The efficiency of the proposed algorithms is illustrated by some numerical tests.

  • Solving the General Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings without Prior Knowledge of Operator Norms
    Filomat, 2017
    Co-Authors: Jing Zhao
    Abstract:

    Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$ be two bounded linear operators. The general split common Fixed-Point Problem under consideration in this paper is to  $$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$  where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$ and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are quasi-nonexpansive mappings with nonempty common Fixed-Point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above Problem (1) allows asymmetric and partial relations between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common Fixed-Point Problem (1) reduces to the general split common Fixed-Point Problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common Fixed-Point Problems (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. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility Problems. Our results improve and extend the corresponding results announced by many others.

  • viscosity approximation methods for the split equality common Fixed Point Problem of quasi nonexpansive operators
    Acta Mathematica Scientia, 2016
    Co-Authors: Jing Zhao, Shengnan Wang
    Abstract:

    Abstract Let H 1 , H 2 , H 3 be real Hilbert spaces, let A : H 1 → H 3 , B : H 2 → H 3 be two bounded linear operators. The split equality common Fixed Point Problem (SECFP) in the infinite-dimensional Hilbert spaces introduced by Moudafi (Alternating CQ-algorithm for convex feasibility and split Fixed-Point Problems. Journal of Nonlinear and Convex Analysis) is (1) to find x ∈ F ( U ) , y ∈ F ( T ) such that A x = B y , where U : H 1 → H 1 and T : H 2 → H 2 are two nonlinear operators with nonempty Fixed Point sets F(U) = { x ∈ H 1 : Ux = x } and F ( T ) = { x ∈ H 2 : Tx = x }. Note that, by taking B = I and H 2 = H 3 in (1), we recover the split Fixed Point Problem originally introduced in Censor and Segal. Recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms with weak convergence for the SECFP (1) of firmly quasi-nonexpansive operators. In this paper, we introduce two viscosity iterative algorithms for the SECFP (1) governed by the general class of quasi-nonexpansive operators. We prove the strong convergence of algorithms. Our results improve and extend previously discussed related Problems and algorithms.

  • solving split equality Fixed Point Problem of quasi nonexpansive mappings without prior knowledge of operators norms
    Optimization, 2015
    Co-Authors: Jing Zhao
    Abstract:

    Let , and be real Hilbert spaces, let and be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common Fixed-Point Problem:Section.Display where and are two firmly quasi-nonexpansive operators with nonempty Fixed-Point sets and . Note that, by taking and , 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 ...

  • Simultaneous iterative algorithms for the split common Fixed-Point Problem of generalized asymptotically quasi-nonexpansive mappings without prior knowledge of operator norms
    Fixed Point Theory and Applications, 2014
    Co-Authors: Jing Zhao
    Abstract:

    Let , , be real Hilbert spaces, let , be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms (Trans. Math. Program. Appl. 1:1-11, 2013) with weak convergence for the following split common Fixed-Point Problem: 1 where and are two firmly quasi-nonexpansive operators with nonempty Fixed-Point sets and . Note that by taking and , we recover the split common Fixed-Point Problem originally introduced in Censor and Segal (J. Convex Anal. 16:587-600, 2009). In this paper, we will continue to consider the split common Fixed-Point Problem (1) governed by the general class of generalized asymptotically quasi-nonexpansive mappings. To estimate the norm of an operator is a very difficult, if it is not an impossible task. The purpose of this paper is to propose a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information as regards the operator norms. MSC:47H09, 47H10, 47J05, 54H25.

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