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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
Approximating fixed points of -firmly nonexpansive mappings in Banach spaces  [PDF]
Gang-Eun Kim
International Journal of Mathematics and Mathematical Sciences , 2000, DOI: 10.1155/s0161171200001654
Abstract: We study the convergence of the Ishikawa iteration methods to fixed points for the result of Smarzewski (1991). Our theorems also improve recent theorems due to Sharma and Sahu (1996).
Hybrid Iteration Method for Fixed Points of Nonexpansive Mappings in Arbitrary Banach Spaces  [cached]
Osilike MO,Isiogugu FO,Nwokoro PU
Fixed Point Theory and Applications , 2007,
Abstract: We prove that recent results of Wang (2007) concerning the iterative approximation of fixed points of nonexpansive mappings using a hybrid iteration method in Hilbert spaces can be extended to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator.
Hybrid Iteration Method for Fixed Points of Nonexpansive Mappings in Arbitrary Banach Spaces  [cached]
M. O. Osilike,F. O. Isiogugu,P. U. Nwokoro
Fixed Point Theory and Applications , 2008, DOI: 10.1155/2007/64306
Abstract: We prove that recent results of Wang (2007) concerning the iterative approximation of fixed points of nonexpansive mappings using a hybrid iteration method in Hilbert spaces can be extended to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator.
Two General Algorithms for Computing Fixed Points of Nonexpansive Mappings in Banach Spaces
Shuang Wang
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/658905
Abstract: Recently, Yao et al. (2011) introduced two algorithms for solving a system of nonlinear variational inequalities. In this paper, we consider two general algorithms and obtain the extension results for computing fixed points of nonexpansive mappings in Banach spaces. Moreover, the fixed points solve the same system of nonlinear variational inequalities.
Approximation of fixed points for a representation of nonexpansive mappings in Banach spaces  [PDF]
Ebrahim Soori
Mathematics , 2015,
Abstract: The purpose of this paper is to study an implicit scheme for a representation of nonexpansive mappings on a closed convex subset of a smooth and uniformly convex Banach space with respect to a left regular sequence of means defined on an appropriate space of bounded real valued functions of the semigroup. This algorithm extends the algorithm that introduced in [N. Hussain, M. L. Bami and E. Soori, An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications 2014, 2014:238].
Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces
Wei-Shih Du,Young-Ye Huang,Chi-Lin Yen
International Journal of Mathematics and Mathematical Sciences , 2002, DOI: 10.1155/s0161171202107113
Abstract: It is shown that every asymptotically regular or λ-firmly nonexpansive mapping T:C→C has a fixed point whenever C is a finite union of nonempty weakly compact convex subsets of a Banach space X which is uniformly convex in every direction. Furthermore, if {T i}i∈I is any compatible family of strongly nonexpansive self-mappings on such a C and the graphs of T i, i ∈I, have a nonempty intersection, then T i, i∈I, have a common fixed point in C.
Approximation of Fixed Points of Weak Bregman Relatively Nonexpansive Mappings in Banach Spaces  [PDF]
Jiawei Chen,Zhongping Wan,Liuyang Yuan,Yue Zheng
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/420192
Abstract: We introduce a concept of weak Bregman relatively nonexpansive mapping which is distinct from Bregman relatively nonexpansive mapping. By using projection techniques, we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. The main results in this paper develop, extend, and improve the corresponding results of Matsushita and Takahashi (2005) and Qin and Su (2007). 1. Introduction Throughout this paper, without other specifications, we denote by the set of real numbers. Let be a real reflexive Banach space with the dual space . The norm and the dual pair between and are denoted by and , respectively. Let be proper convex and lower semicontinuous. The Fenchel conjugate of is the function defined by We denote by the domain of , that is, . Let be a nonempty closed and convex subset of a nonlinear mapping. Denote by , the set of fixed points of . is said to be nonexpansive if for all . In 1967, Brègman [1] discovered an elegant and effective technique for the using of the so-called Bregman distance function (see, Section 2, Definition 2.1) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman's technique is applied in various ways in order to design and analyze iterative algorithms for solving not only feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, for computing fixed points of nonlinear mappings, and so on (see, e.g., [1–25], and the references therein). Nakajo and Takahashi [26] introduced the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space as follows: where and is the metric projection from onto a closed and convex subset of . They proved that generated by (1.2) converges strongly to a fixed point of under some suitable assumptions. Motivated by Nakajo and Takahashi [26], Matsushita and Takahashi [27] introduced the following modification of the Mann iteration method for a relatively nonexpansive mapping in a Banach space as follows: where , for all , is the duality mapping of and is the generalized projection (see, e.g., [2, 3, 28]) from onto a closed and convex subset of
Convergence Theorems of Fixed Points for a Finite Family of Nonexpansive Mappings in Banach Spaces  [cached]
Cho YeolJe,Kang ShinMin,Qin Xiaolong
Fixed Point Theory and Applications , 2008,
Abstract: We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Our results improve the results announced by many others.
Convergence Theorems of Fixed Points for a Finite Family of Nonexpansive Mappings in Banach Spaces  [cached]
Yeol Je Cho,Shin Min Kang,Xiaolong Qin
Fixed Point Theory and Applications , 2008, DOI: 10.1155/2008/856145
Abstract: We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Our results improve the results announced by many others.
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