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New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators

APPLIED MATHEMATICS AND COMPUTATION 2011年第12期217卷 5458-5465页

作者： Su, Yongfu Li, Mengqin Zhang, Hong Tianjin Polytech Univ Dept Math Tianjin 300160 Peoples R China

The purpose of this article is to prove the strong convergence theorems for hemi-relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems for hemi-relatively nonexpansive mappings, a new monotone hybrid iteration algorithm is presented and is used to approximate the fixed point of hemi-relatively nonexpansive mappings. Noting that, the general hybrid iteration algorithm can be used for relatively nonexpansive mappings but it can not be used for hemi-relatively nonexpansive mappings. However, this new monotone hybrid algorithm can be used for hemi-relatively nonexpansive mappings. In addition, a new method of proof has been used in this article. That is, by using this new monotone hybrid algorithm, we firstly claim that, the iterative sequence is a Cauchy sequence. The results of this paper modify and improve the results of Matsushita and Takahashi, and some others. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Hemi-relatively nonexpansive mapping Generalized projection monotone hybrid algorithm Cauchy sequence Maximal monotone operator

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Convergence theorems of a hybrid algorithm for equilibrium problems

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NONLINEAR ANALYSIS-hybrid SYSTEMS 2009年第4期3卷 386-394页

作者： Kumam, Poom Wattanawitoon, Kriengsak King Mongkuts Univ Technol Thonburi Fac Sci Dept Math Bangkok 10140 Thailand

In this paper, we introduce a hybrid iterative scheme for finding a common element of the set of common fixed points of two hemi-relatively non-expansive mappings and the set of solutions of an equilibrium problem by the CQ hybrid method in Banach spaces. Our results improve and extend the corresponding results announced by Cheng and Tian [Y. Cheng, M. Tian, Strong convergence theorem by monotone hybrid algorithm for equilibrium problems, hemi-relatively nonexpansive mappings and maximal monotone operators, Fixed Point Theory Appl. 2008 (2008) 12 pages, doi:10.1155/2008/617248], Takahashi and Zembayashi [W. Takahashi, K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively non-expansive mappings, Fixed Point Theory Appl. (2008) doi:10.1155/2008/528476] and some others. (c) 2009 Elsevier Ltd. All rights reserved.

关键词： hybrid method Hemi-relatively non-expansive mapping Equilibrium problem monotone hybrid algorithm

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Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2010年第12期73卷 3890-3906页

作者： Su, Yongfu Xu, Hong-kun Zhang, Xin Tianjin Polytech Univ Dept Math Tianjin 300160 Peoples R China Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 80424 Taiwan King Saud Univ Dept Math Coll Sci Riyadh 11451 Saudi Arabia

The purpose of this article is to prove strong convergence theorems for common fixed points of two countable families of weak relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems, the monotone hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article modify and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266] and the results of Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space, J. Approx. Theory 149 (2007) 103-115] and the results of Su et al. [Y. Su, Z. Wang and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009) 5616-5628], and many others. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Relatively nonexpansive mapping Weak relatively nonexpansive mapping Generalized projection Common fixed point monotone hybrid algorithm

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Convergence theorems for modified generalized f-projections and generalized nonexpansive mappings

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2014年第1期2014卷 1-12页

作者： Cheng, Qingqing Su, Yongfu Zhang, Jingling Tianjin Polytech Univ Dept Math Tianjin 300160 Peoples R China

The purpose of this paper is to study a sequence of modified generalized f-projections in a reflexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalized f-projection onto the limit set. Furthermore, we prove a strong convergence theorem for a countable family of alpha-nonexpansive mappings in a uniformly convex and smooth Banach space using the properties of a modified generalized f-projection operator. Our main results generalize the results of Ziming Wang, Yongfu Su, and Jinlong Kang and enrich the research contents of alpha-nonexpansive mappings.

关键词： alpha-nonexpansive mappings monotone hybrid algorithm modified generalized f-projection operator Mosco convergence common fixed point

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Simple projection algorithm for a countable family of weak relatively nonexpansive mappings and applications

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FIXED POINT THEORY AND APPLICATIONS 2012年第1期2012卷 1-17页

作者： Zhang, Jingling Su, Yongfu Cheng, Qingqing Tianjin Polytech Univ Dept Math Tianjin 300387 Peoples R China

Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let {T-n} : C -> C be a countable family of weak relatively nonexpansive mappings such that F = boolean AND(infinity)(n=1) F(T-n) not equal emptyset. For any given gauss x(0) is an element of C, define a sequence {x(n)} in C by the following algorithm: {C-0 = C, Cn+1 = {z is an element of C-n : phi(z, T(n)x(n)) = phi(z,x(n))}, n = 0,1,2,3,..., x(n+1) = Pi C(n+1)x(0). Then {x(n)} converges strongly to q = Pi(F)x(0).

关键词： relatively nonexpansive mapping weak relatively nonexpansive mapping generalized projection projection algorithm monotone hybrid algorithm

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