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Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems
Kamonrat Nammanee,Suthep Suantai,Prasit Cholamjiak
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/804538
Abstract: We introduce hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.
A New Hybrid Iterative Scheme for Countable Families of Relatively Quasi-Nonexpansive Mappings and System of Equilibrium Problems  [PDF]
Yekini Shehu
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/131890
Abstract: We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalized -projection operator. Using this result, we discuss strong convergence theorem concerning variational inequality and convex minimization problems in Banach spaces. Our results extend many known recent results in the literature. 1. Introduction Let be a real Banach space with dual and a nonempty, closed, and convex subset of . A mapping is called nonexpansive if A point is called a fixed point of if . The set of fixed points of is denoted by . We denote by the normalized duality mapping from to defined by The following properties of are well known (the reader can consult [1–3] for more details). (1)If is uniformly smooth, then is norm-to-norm uniformly continuous on each bounded subset of . (2) .(3)If is reflexive, then is a mapping from onto .(4)If is smooth, then is single valued. Throughout this paper, we denote by the functional on defined by It is obvious from (1.3) that Definition 1.1. Let be a nonempty subset of , and let be a mapping from into . A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . The set of asymptotic fixed points of is denoted by . We say that a mapping is relatively nonexpansive (see, e.g., [4–9]) if the following conditions are satisfied:(R1) ,(R2) ,(R3) . If satisfies (R1) and (R2), then is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings. Many authors have studied the methods of approximating the fixed points of relatively quasi-nonexpansive mappings (see, e.g., [10–12] and the references cited therein). Clearly, in Hilbert space , relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same, for , and this implies that The examples of relatively quasi-nonexpansive mappings are given in [11]. Let be a bifunction of into . The equilibrium problem (see, e.g., [13–25]) is to find such that for all . We will denote the solutions set of (1.6) by . Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.6). The equilibrium problems include fixed point problems, optimization problems, and variational inequality
The Modified Block Iterative Algorithms for Asymptotically Relatively Nonexpansive Mappings and the System of Generalized Mixed Equilibrium Problems
Kriengsak Wattanawitoon,Poom Kumam
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/395760
Abstract: The propose of this paper is to present a modified block iterative algorithm for finding a common element between the set of solutions of the fixed points of two countable families of asymptotically relatively nonexpansive mappings and the set of solution of the system of generalized mixed equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results extend many known recent results in the literature.
Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi- -Asymptotically Nonexpansive Mappings  [PDF]
Siwaporn Saewan,Poom Kumam
Abstract and Applied Analysis , 2010, DOI: 10.1155/2010/357120
Abstract: We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi- -asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results. 1. Introduction and Preliminaries The convex feasibility problem (CFP) is the problem of computing points laying in the intersection of a finite family of closed convex subsets , of a Banach space This problem appears in various fields of applied mathematics. The theory of optimization [1], Image Reconstruction from projections [2], and Game Theory [3] are some examples. There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [4]. The advantage of a Hilbert space is that the projection onto a closed convex subset of is nonexpansive. So projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [5] deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in a uniformly convex Banach space. It is known that if is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space, then the generalized projection (see, Alber [6] or Kamimura and Takahashi [7]) from onto is relatively nonexpansive, whereas the metric projection from onto is not generally nonexpansive. We note that the block iterative method is a method which is often used by many authors to solve the convex feasibility problem (CFP) (see, [8, 9], etc.). In 2008, Plubtieng and Ungchittrakool [10] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Let be a nonempty closed convex subset of a real Banach space with and being the dual space of . Let be a bifunction of into and a monotone mapping. The generalized equilibrium problem, denoted by , is to find such that The set of solutions for the problem (1.1) is denoted by , that is If , the problem (1.1) reducing into the equilibrium problem for , denoted by , is to find such that If , the problem (1.1) reducing into the
A New Iterative Scheme for Countable Families of Weak Relatively Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems  [PDF]
Yekini Shehu
Abstract and Applied Analysis , 2010, DOI: 10.1155/2010/861031
Abstract: We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth using the properties of generalized -projection operator. Using this result, we discuss strong convergence theorem concerning general -monotone mappings and system of generalized mixed equilibrium problems in Banach spaces. Our results extend many known recent results in the literature. 1. Introduction Let be a real Banach space with dual , and let be nonempty, closed and convex subset of . A mapping is called nonexpansive if A point is called a fixed point of if . The set of fixed points of is denoted by . We denote by the normalized duality mapping from to defined by The following properties of are well known (the reader can consult [1–3] for more details). (1)If is uniformly smooth, then is norm-to-norm uniformly continuous on each bounded subset of . (2) , ? . (3)If is reflexive, then is a mapping from onto .(4)If is smooth, then is single valued. Throughout this paper, we denote by , the functional on defined by From [4], in uniformly convex and uniformly smooth Banach spaces, we have Definition 1.1. Let be a nonempty subset of and let be a countable family of mappings from into . A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . The set of asymptotic fixed points of is denoted by . One says that is countable family of relatively nonexpansive mappings (see, e.g., [5]) if the following conditions are satisfied:(R1) ;(R2) , for all , ? , ? ;(R3) . Definition 1.2. A point is said to be a strong asymptotic fixed point of if contains a sequence which converges strongly to and . The set of strong asymptotic fixed points of is denoted by . One says that a mapping is countable family of weak relatively nonexpansive mappings (see, e.g., [5]) if the following conditions are satisfied:(R1) ;(R2) , for all , ? , ? ;(R3) . Definition 1.3. Let be a nonempty subset of and let be a mapping from into . A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . The set of asymptotic fixed points of is denoted by . We say that a mapping is relatively nonexpansive (see, e.g., [6–11]) if the following conditions are satisfied:(R1) ;(R2) , for all , ? ;(R3) . Definition 1.4. A point is said to be an strong asymptotic fixed
Convergence Analysis for a System of Equilibrium Problems and a Countable Family of Relatively Quasi-Nonexpansive Mappings in Banach Spaces  [PDF]
Prasit Cholamjiak,Suthep Suantai
Abstract and Applied Analysis , 2010, DOI: 10.1155/2010/141376
Abstract: We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space. 1. Introduction Let be a real Banach space, and let be the dual of . Let be a closed and convex subset of . Let be bifunctions from to , where is the set of real numbers and is an arbitrary index set. The system of equilibrium problems is to find such that If is a singleton, then problem (1.1) reduces to find such that The set of solutions of the equilibrium problem (1.2) is denoted by . Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element in the solutions set of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. In 2004, Matsushita and Takahashi [2] introduced the following algorithm for a relatively nonexpansive mapping in a Banach space : for any initial point , define the sequence by where is the duality mapping on , is the generalized projection from onto , and is a sequence in . They proved that the sequence converges weakly to fixed point of under some suitable conditions on . In 2008, Takahashi and Zembayashi [3] introduced the following iterative scheme which is called the shrinking projection method for a relatively nonexpansive mapping and an equilibrium problem in a Banach space : They proved that the sequence converges strongly to under some appropriate conditions. 2. Preliminaries and Lemmas Let be a real Banach space, and let be the unit sphere of . A Banach space is said to be strictly convex if, for any , It is also said to be uniformly convex if, for each , there exists such that, for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. The function which is called the modulus of convexity of is defined as follows: The space is uniformly convex if and only if for all . A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for . The duality mapping is defined by for all . If is a Hilbert space, then , where is the identity operator. It is also known that, if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subset
Strong and Weak Convergence Theorems for Equilibrium Problems and Weak Relatively Uniformly Nonexpansive Multivalued Mappings in Banach Spaces
Zi-Ming Wang
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/489629
Abstract: Equilibrium problem and fixed point problem are considered. A general iterative algorithm is introduced for finding a common element of the set of solutions to the equilibrium problem and the common set of fixed points of two weak relatively uniformly nonexpansive multivalued mappings. Furthermore, strong and weak convergence results for the common element in the two sets mentioned above are established in some Banach space.
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
Strong Convergence Theorem for Solving Generalized Mixed Equilibrium Problems and Fixed Point Problems for Total Quasi--Asymptotically Nonexpansive Mappings in Banach Spaces
Zhaoli Ma,Lin Wang,Yunhe Zhao
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/506210
Abstract: We introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points for countable families of total quasi--asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.
Iterative Schemes for Fixed Points of Relatively Nonexpansive Mappings and Their Applications  [PDF]
Somyot Plubtieng,Wanna Sriprad
Abstract and Applied Analysis , 2010, DOI: 10.1155/2010/301305
Abstract: We present two iterative schemes with errors which are proved to be strongly convergent to a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Using the result we consider strong convergence theorems for variational inequalities and equilibrium problems in a real Hilbert space and strong convergence theorems for maximal monotone operators in a real uniformly smooth and uniformly convex Banach space. 1. Introduction Let be a real Banach space, and the dual space of . The function is denoted by for all , where is the normalized duality mapping from to . Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed points of . A point in is said to be an asymptotic fixed point of [1] if contains a sequence which converges weakly to such that the strong equals 0. The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called nonexpansive if for all and nonexpansive with respect to the Lyapunov functional [2] if for all and it is called relatively nonexpansive [3–6] if and for all and . The asymptotic behavior of relatively nonexpansive mapping was studied in [3–6]. There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [7] introduced the iteration as follows: a sequence is defined by where the initial guess element is arbitrary and is a real sequence in . Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [1]. In an infinite-dimensional Hilbert space, Mann iteration can yield only weak convergence (see [8, 9]). Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [10] proposed the following modification of Mann iteration method (1.2) for nonexpansive mapping in a Hilbert space: in particular, they studied the strong convergence of the sequence generated by where and is the metric projection from onto . Recently, Takahashi et al. [11] extended iteration (1.6) to obtain strong convergence to a common fixed point of a countable family of nonexpansive mappings; let be a nonempty closed convex subset of a Hilbert space . Let and be families of nonexpansive mappings of into itself such that and let . Suppose that satisfies the NST-condition (I) with ; that is, for each bounded
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