Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces

In this paper, we prove a strong convergence theorem by the hybrid method for a family of nonexpansive mappings which generalizes Nakajo and Takahashi's theorems [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372-379], simultaneously. Furthermore, we obtain another strong convergence theorem for the family of nonexpansive mappings by a hybrid method which is different from Nakajo and Takahashi. Using this theorem, we get some new results for a single nonexpansive mapping or a family of nonexpansive mappings in a Hilbert space.     

Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups

In this paper, we established strong convergence theorems for a common fixed point of two asymptotically nonexpansive mappings and for a common fixed point of two asymptotically nonexpansive semigroups by using the hybrid method in a Hilbert space. Moreover, we also proved a strong convergence theorem for a common fixed point of two nonexpansive mappings. Our results extend and improve the recent ones announced by Kim and Xu [T.W. Kim, H.W. Xu, Strong convergence of modified Mann iteration for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152], Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379], and many others.     

Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups

The Mann iterations for nonexpansive mappings have in general only weak convergence in a Hilbert space. We modify an iterative method of Mann's type introduced by Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] for nonexpansive mappings and prove strong convergence of our modified Mann's iteration processes for asymptotically nonexpansive mappings and semigroups.     

Generalized Projection Algorithms for Nonlinear Operators

In this paper, we prove three strong convergence theorems of modified Mann iteration for relatively asymptotically nonexpansive mappings, relatively nonexpansive mappings, and maximal monotone operators in a Banach space, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Matsushita, Kim, Martinez-Yanex, Xu, and some others.     

Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space

In this paper, we establish strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Our results extend and improve the recent ones announced by Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266], Matinez-yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411], and many others.     

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

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.     

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

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.     

Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings

The purpose of this article is to prove strong convergence theorems for common fixed points of two closed hemi-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. Finally, a new simplified hybrid algorithm has been proposed and relative convergence theorem has been proved by using the new method for proofs. The results of this article modify and improve the results of Matsushita, 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, 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 many others.     

Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces

In this paper, we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods. Our results improve and extend the recent ones announced by Chen et al. [J.M. Chen, L.J. Zhang, T.G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, doi:10.1016/j.jmaa.2006.12.088], Iiduka and Tahakshi [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350], Yao and Yao [Y.H. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput, doi:10.1016/j.amc.2006.08.062] and Many others.     

Strong convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258??271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140??152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400??411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372??79 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336??46 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334??345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835??842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77??8 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766??76 (2007)].     

Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups

In this paper, we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in Hilbert spaces by the hybrid method in the mathematical programming.     

弱相对非扩张映像不动点单调CQ算法与应用

Kamimura和Takahashi$^{[7]}$证明了相对非扩张映像CQ迭代算法的强收敛定理.该文构造了单调CQ算法, 用来逼近弱相对非扩张映像不动点, 证明了强收敛定理. 并将结果应用于逼近Banach空间极大单调算子的零点. 单调CQ算法比目前的CQ算法收敛速度快. 另外, 为证明弱相对非扩张映像不动点强收敛定理,该文运用了新的Cauchy列证明方法, 而不用Kadec-Klee性质, 该文结果改进了S.Matsushita 和 W.Takahashi及其它人的结果.     

On a hybrid method for a family of relatively nonexpansive mappings in a Banach space

We prove strong convergence theorems by the hybrid method given by Takahashi, Takeuchi, and Kubota for a family of relatively nonexpansive mappings under weaker conditions. The method of the proof is different from the original one and it shows that the type of projection used in the iterative method is independent of the properties of the mappings. We also deal with the problem of finding a zero of a maximal monotone operator and obtain a strong convergence theorem using this method.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

The purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of variational inequality for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Our results extend the recent results of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515], Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52], Combettes and Hirstoaga [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 486–491], Iiduka and Takahashi, [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and many others.     

Strong convergence of the CQ method for fixed point iteration processes

Strong convergence theorems are obtained for the CQ method for an Ishikawa iteration process, a contractive-type iteration process for nonexpansive mappings, and the proximal point algorithm for maximal monotone operators in Hilbert spaces.     

Approximating fixed points of Suzuki-generalized nonexpansive mappings

In this paper, we prove weak and strong convergence theorems for Ishikawa iteration of Suzuki-generalized nonexpansive mappings in uniformly convex Banach spaces. Furthermore, we extend the results to CAT(0) spaces. Our work extends the results of Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mapping, J. Math. Anal. Appl. 340 (2008) 1088–1095] and Takahashi and Kim [W. Takahashi, G.E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Jpn. 48 (1998) 1–9].     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43-52], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (1) (2007) 506-515], and many others.     

Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces

In this paper, we introduce two modifications of the Ishikawa iteration, by using the hybrid methods, for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups in a Hilbert space. Then, we prove that such two sequences converge strongly to common fixed points of two symptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our main result is connected with the results of Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups, Nonlinear. Anal. 67(2007) 2306-2315], Martinez-Yanes and Xu [C. Martinez-Yanes, H.K. Xu, Strong convergence of CQ method for fixed point iteration processes, Nonlinear. Anal. 64 (2006) 2400-2411] and many others.     

Convergence theorems of a hybrid algorithm for equilibrium problems

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.     

Approximation of fixed points of nonexpansive mapping in Banach spaces

In this paper, some iterative schemes are given to approximate a fixed point of the nonexpansive non-self-mapping and nonexpansive self-mapping. Furthermore, the strong convergence of the scheme to a fixed point is shown in a Banach space with uniformly Gâteaux differentiable norm. The theorems extend and improve some corresponding results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions, Nonlinear Anal. 68 (2008) 412–419], Chang et al. [S.S. Chang, H.W. Joseph Lee, C.K. Chan, On Reich’s strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. 66 (2007) 2364–2374], Chidume and Chidume [C.E. Chidume, C.O. Chidume, Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal. Appl. 318 (2006) 288–295] and Suzuki [T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed point of nonexpansive mappings, Proc. Amer. Math. Society 135 (1) (2007) 99–106].