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Viscosity Approximation Methods for split common fixed-point problem of Directed Operators

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2015年第4期36卷 528-547页

作者： Zhao, Jing He, Songnian Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China Civil Aviat Univ China Tianjin Key Lab Adv Signal Proc Tianjin Peoples R China

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 common fixed-point problem (SCFP) is [GRAPHICS] where U: H (1) -> H (1) and T: H (2) -> H (2) are two nonlinear operators with nonempty fixed-point sets F(U) = {x is an element of H (1): Ux = x} and F(T) = {x is an element of H (2): Tx = x}. Recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms with weak convergence for the SCFP (1) of firmly quasi-nonexpansive operators. In this article, we introduce viscosity iterative algorithm and prove the strong convergence of algorithm for the SCFP (1) governed by the directed operators (i.e. firmly quasi-nonexpansive operators). Finally, we provide some applications.

关键词： Hilbert space Viscosity iterative algorithm Strong convergence Feasibility problem split common fixed-point problem Directed operators 54H25 47J05 47H09 47H10

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Inertial accelerated algorithms for the split common fixed-point problem of directed operators

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OPTIMIZATION 2021年第5-6期70卷 1375-1407页

作者： Zhao, Jing Zhao, Ning-ning Hou, Dingfang Civil Aviat Univ China Coll Sci Tianjin Peoples R China Civil Aviat Univ China Coll Air Traff Management Tianjin Peoples R China

In this paper, we introduce two inertial accelerated algorithms for solving the split common fixed-point problem of directed operators in real Hilbert space. The proposed iterative algorithms combine the primal-dual method and the inertial method with the self-adaptive stepsizes such that the implementation of our algorithms does not need any prior information about bounded linear operator norms. Under suitable conditions, the weak and strong convergence results of the algorithms are obtained. Numerical results which involve image restoration problems are reported to show the effectiveness of the proposed algorithms.

关键词： split common fixed-point problem directed operator inertial technique self-adaptive algorithm weak convergence

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A new method for split common fixed-point problem without priori knowledge of operator norms

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JOURNAL OF fixed point THEORY AND APPLICATIONS 2017年第4期19卷 2427-2436页

作者： Wang, Fenghui Luoyang Normal Univ Dept Math Luoyang 471022 Peoples R China

The split common fixed-point problem is an inverse problem that consists in finding an element in a fixed-point set such that its image under a linear transformation belongs to another fixed-point set. In this paper, we propose a new algorithm for the split common fixed-point problem that does not need any priori information of the operator norm. Under standard assumptions, we establish a weak convergence theorem of the proposed algorithm.

关键词： split common fixed-point problem projection variable step-size strong convergence

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Alternating mann iterative algorithms for the split common fixed-point problem of quasi-nonexpansive mappings

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fixed point THEORY AND APPLICATIONS 2013年第1期2013卷 1-12页

作者： Zhao, Jing He, Songnian Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

Very recently, Moudafi (Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal.) introduced an alternating CQ-algorithm with weak convergence for the following split common fixed-point problem. 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. Find x is an element of F(U), y is an element of F(T) such that Ax = By, (1) where U : H-1 -> H-1 and T : H-2 -> H-2 are two firmly quasi-nonexpansive operators with nonempty fixed-point sets F(U) = {x is an element of H-1 : Ux = x} and F(T) = {x is an element of H-2 : Tx = x}. Note that by taking H-2 = H-3 and B = I, we recover the split common fixed-point problem originally introduced in Censor and Segal (J. Convex Anal. 16:587-600, 2009) and used to model many significant real-world inverse problems in sensor net-works and radiation therapy treatment planning. 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 two alternating Mann iterative algorithms and prove the weak convergence of algorithms. At last, we provide some applications. Our results improve and extend the corresponding results announced by many others.

关键词： split common fixed-point problem quasi-nonexpansive mapping weak convergence Mann iterative algorithm Hilbert space

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An iterative viscosity approximation method for the split common fixed-point problem

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OPTIMIZATION 2021年第5-6期70卷 1261-1274页

作者： He, Huimin Peng, Jigen Fan, Qinwei Guangzhou Univ Sch Math & Informat Sci Guangzhou Peoples R China Xidian Univ Sch Math & Stat Xian Peoples R China Xian Polytech Univ Sch Sci Xian Peoples R China

The split common fixed point problem (SCFPP) for the demicontractive operators is studied. An iterative viscosity approximation method (VAM) for SCFPP is introduced. It is shown that the sequence generated by VAM is strongly convergent to a solution of SCFPP under certain conditions, and this solution is identified to be the unique solution of some variational inequality. The main result of this paper extends and improves some results of Yao et al. [Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. fixed point Theory Appl. 2014;183], Boikanyo [A strongly convergent algorithm for the split common fixed point problem. Appl Math Comput. 2015;265:844-853] and Cui-Wang [Iterative methods for the split common fixed point problem in Hilbert spaces. fixed point Theory Appl. 2014;78].

关键词： split common fixed-point problem demicontractive operator viscosity approximation method strong convergence

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Simultaneous iterative algorithms for the split common fixed-point problem of generalized asymptotically quasi-nonexpansive mappings without prior knowledge of operator norms

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fixed point THEORY AND APPLICATIONS 2014年第1期2014卷 73-73页

作者： Zhao, Jing He, Songnian Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

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. Moudafi introduced simultaneous iterative algorithms (Trans. Math. Program. Appl. 1:1-11, 2013) with weak convergence for the following split common fixed-point problem: find x is an element of F(U), y is an element of F(T) suchthat Ax = By, (1) where U:H-1 -> H-1 and T:H-2 -> H-2 are two firmly quasi-nonexpansive operators with nonempty fixed-point sets F(U) = {x is an element of H-1 : Ux = x} and F(T) = {x is an element of H-2 : Tx = x}. Note that by taking H-2 = H-3 and B = I, 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.

关键词： split common fixed-point problem generalized asymptotically quasi-nonexpansive mappings weak convergence simultaneous iterative algorithm Hilbert space

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Dynamical technique for split common fixed point problem in Banach spaces

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COMPUTATIONAL & APPLIED MATHEMATICS 2022年第4期41卷 1-27页

作者： Ugwunnadi, Godwin Chidi Izuchukwu, Chinedu Khan, Abdul Rahim Univ Eswatini Dept Math Private Bag 4 Kwaluseni Eswatini Sefako Makgatho Hlth Sci Univ Dept Math & Appl Math POB 60 ZA-0204 Pretoria South Africa Univ KwaZulu Natal Sch Math Stat & Comp Sci Durban South Africa Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran Saudi Arabia

We study the split common fixed point problem for Bregman demigeneralized type mappings in the context of two real Banach spaces. We propose a new self-adaptive method and prove that it converges strongly to a solution of this problem. As consequences of our results, we propose some new self-adaptive methods for solving split feasibility problem, split common null point problem and split equilibrium problem, using some dynamical stepsize techniques, which allow these methods to be easily implemented without prior knowledge of the norm of the bounded linear operator. We perform some numerical experiments to demonstrate implementation and efficiency of our methods.

关键词： Bregman distance Bregman demigeneralized map Dynamical stepsize method split common fixed-point problem split equilibrium problem split feasibility problem

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Iterative solutions of the split common fixed point problem for strictly pseudo-contractive mappings

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JOURNAL OF fixed point THEORY AND APPLICATIONS 2018年第2期20卷 1-12页

作者： Cui, Huanhuan Ceng, Luchuan Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China

In this paper, we study the the split common fixed point problem in Hilbert spaces. We establish a weak convergence theorem for the method recently introduced by Wang, which extends a existing result from firmly nonexpansive mappings to strictly pseudo-contractive mappings. Moreover, our condition that guarantees the weak convergence is much weaker than that of Wang's. A strong convergence theorem is also obtained under some additional conditions. As an application, we obtain several new methods for solving various split inverse problems and split equality problems. Numerical examples are included to illustrate the applications in signal processing of the proposed algorithm.

关键词： split common fixed-point problem strictly pseudo-contractive mappings split inverse problems split equality problem

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ITERATIVE METHODS FOR SOLVING split common fixed-point problemS WITH A WEAKENED CONDITION ON STEP SIZES

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS 2025年第2期9卷 297-308页

作者： Cui, Huanhuan Wang, Fenghui Luoyang Normal Univ Dept Math Luoyang 471934 Peoples R China

In this paper, we investigate split common fixed-point problems with multiple output subsets when the involved nonlinear mappings are demicontractive. By exploring the properties of demicontractive mappings, we further weaken the condition on the step size, which ensures the convergence of iterative methods for solving such problems. The preliminary experimental findings provide compelling empirical evidence substantiating the effectiveness of the proposed step size.

关键词： Demiclosedness principle Demicontractive mapping split common fixed-point problem split feasibility problem Step size

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Viscosity approximation methods for solving fixed-point problems and split common fixed-point problems

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JOURNAL OF fixed point THEORY AND APPLICATIONS 2017年第2期19卷 1481-1499页

作者： Duong Viet Thong Natl Econ Univ Fac Econ Math 207 Giai Phong St Hanoi Vietnam

In this paper, we introduce the strong convergence theorem for the viscosity approximation methods for solving the split common fixed-point problem in Hilbert spaces. As a consequence, we obtain strong convergence theorems for split variational inequality problems for Lipschitz continuous and monotone operators and split common null point problems for maximal monotone operators. Our results improve and extend the corresponding results announced by many others.

关键词： fixed-point theory split common fixed-point problem split feasibility problem split variational inequality problem split null point problem

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