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Strong Convergence Theorems of a New General Iterative Process with Meir-Keeler Contractions for a Countable Family of -Strict Pseudocontractions in -Uniformly Smooth Banach Spaces  [cached]
Song Yanlai,Hu Changsong
Fixed Point Theory and Applications , 2010,
Abstract: We introduce a new iterative scheme with Meir-Keeler contractions for strict pseudocontractions in -uniformly smooth Banach spaces. We also discuss the strong convergence theorems for the new iterative scheme in -uniformly smooth Banach space. Our results improve and extend the corresponding results announced by many others.
An Iterative Shrinking Metric -Projection Method for Finding a Common Fixed Point of a Closed and Quasi-Strict -Pseudocontraction and a Countable Family of Firmly Nonexpansive Mappings and Applications in Hilbert Spaces  [PDF]
Kasamsuk Ungchittrakool,Duangkamon Kumtaeng
Abstract and Applied Analysis , 2013, DOI: 10.1155/2013/589282
Abstract: We create some new ideas of mappings called quasi-strict -pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric -projection, we propose an iterative shrinking metric -projection method for finding a common fixed point of a quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to find a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results. 1. Introduction It is well known that the metric projection operators in Hilbert spaces and Banach spaces play an important role in various fields of mathematics such as functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problem (see, e.g., [1, 2]). In 1994, Alber [3] introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized -projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by [3] and proved some properties of the generalized -projection operator. Fan et al. [5] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces. Let be a real Hilbert space; a mapping with domain and range in is called firmly nonexpansive if nonexpansive if Throughout this paper, stands for an identity mapping. The mapping is said to be a strict pseudocontraction if there exists a constant such that In this case, may be called a -strict pseudocontraction. We use to denote the set of fixed points of (i.e. . is said to be a quasi-strict pseudocontraction if the set of fixed point is nonempty and if there exists a constant such that Construction of fixed
Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces  [cached]
Cholamjiak Prasit,Suantai Suthep
Fixed Point Theory and Applications , 2010,
Abstract: We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Fréchet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010) is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.
An Iterative Shrinking Projection Method for Solving Fixed Point Problems of Closed and -Quasi-Strict Pseudocontractions along with Generalized Mixed Equilibrium Problems in Banach Spaces
Kasamsuk Ungchittrakool
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/536283
Abstract: We provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Moreover, we also find the significant inequality related to the mappings mentioned in the paper and the mappings defined from generalized mixed equilibrium problems on Banach spaces. We propose an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and -quasi-strict pseudo-contractions. Our results hold in reflexive, strictly convex, and smooth Banach spaces with the property (). The results of this paper improve and extend the corresponding results of Zhou and Gao (2010) and many others.
Strong Convergence Theorems of a New General Iterative Process with Meir-Keeler Contractions for a Countable Family of λi-Strict Pseudocontractions in q-Uniformly Smooth Banach Spaces  [cached]
Yanlai Song,Changsong Hu
Fixed Point Theory and Applications , 2010, DOI: 10.1155/2010/354202
Abstract:
An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions  [cached]
Peng Jian-Wen,Liou Yeong-Cheng,Yao Jen-Chih
Fixed Point Theory and Applications , 2009,
Abstract: We introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.
An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions  [cached]
Jian-Wen Peng,Yeong-Cheng Liou,Jen-Chih Yao
Fixed Point Theory and Applications , 2009, DOI: 10.1155/2009/794178
Abstract: We introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.
A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces  [cached]
Kamraksa Uthai,Wangkeeree Rabian
Journal of Inequalities and Applications , 2010,
Abstract: The purpose of this paper is to introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pseudo-contraction mappings.
A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces  [cached]
Uthai Kamraksa,Rabian Wangkeeree
Journal of Inequalities and Applications , 2010, DOI: 10.1155/2010/190126
Abstract: The purpose of this paper is to introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pseudo-contraction mappings.
σ-Porosity of the set of strict contractions in a space of non-expansive mappings  [PDF]
Christian Bargetz,Michael Dymond
Mathematics , 2015,
Abstract: We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uni- form convergence. We show that the set of strict contractions is a {\sigma}-porous subset.
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