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 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 . 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
 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 , 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., ) 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., ) 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
 Abstract and Applied Analysis , 2010, DOI: 10.1155/2010/123027 Abstract: We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an -inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results. 1. Introduction Let be a Banach space with norm , a nonempty closed convex subset of , and let denote the dual of . Let be a bifunction, be a real-valued function, and a mapping. The generalized mixed equilibrium problem, is to find such that The set of solutions to (1.1) is denoted by , that is, If , the problem (1.1) reduces into the mixed equilibrium problem for , denoted by , which is to find such that If , the problem (1.1) reduces into the mixed variational inequality of Browder type, denoted by , which is to find such that If and the problem (1.1) reduces into the equilibrium problem for , denoted by , which is to find such that If , the problem (1.3) reduces into the minimize problem, denoted by , is to find such that The above formulation (1.4) was shown in  to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an . In other words, the is an unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of ; see, for example, [1, 2] and references therein. Some solution methods have been proposed to solve the ; see, for example, [1, 3–11] and references therein. A Banach space is said to be strictly convex if for all with and . Let be the unit sphere of . Then a Banach space is said to be smooth if the limit exists for each It is also said to be uniformly smooth if the limit exists uniformly for . Let be a Banach space. The modulus of convexity of is the function defined by A Banach space is uniformly convex if and only if
 Journal of Inequalities and Applications , 2010, Abstract: We introduce a new iterative scheme with a countable family of nonexpansive mappings for the variational inequality problems in Hilbert spaces and prove some strong convergence theorems for the proposed schemes.
 Journal of Inequalities and Applications , 2010, DOI: 10.1155/2010/687374 Abstract: We introduce a new iterative scheme with a countable family of nonexpansive mappings for the variational inequality problems in Hilbert spaces and prove some strong convergence theorems for the proposed schemes.
 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  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  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  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 . 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  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.  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
 Jung JongSoo Journal of Inequalities and Applications , 2010, Abstract: We introduce a new composite iterative scheme by the viscosity approximation method for nonexpansive mappings and monotone mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of set of fixed points of nonexpansive mapping and the set of solutions of variational inequality for an inverse-strongly monotone mappings, which is a solution of a certain variational inequality. Our results substantially develop and improve the corresponding results of [Chen et al. 2007 and Iiduka and Takahashi 2005]. Essentially a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings is provided.
 Jong Soo Jung Journal of Inequalities and Applications , 2010, DOI: 10.1155/2010/251761 Abstract: We introduce a new composite iterative scheme by the viscosity approximation method for nonexpansive mappings and monotone mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of set of fixed points of nonexpansive mapping and the set of solutions of variational inequality for an inverse-strongly monotone mappings, which is a solution of a certain variational inequality. Our results substantially develop and improve the corresponding results of [Chen et al. 2007 and Iiduka and Takahashi 2005]. Essentially a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings is provided.
 Yekini Shehu Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/251612 Abstract: We construct a new Halpern type 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 in a uniformly convex and uniformly smooth real Banach space using the properties of generalized -projection operator. Using this result, we discuss strong convergence theorem concerning general -monotone mappings. Our results extend many known recent results in the literature.
 Peng Cheng and Anshen Zhang Journal of Inequalities and Applications , 2013, DOI: 10.1186/1029-242X-2013-87 Abstract: In this paper, a monotone variational inequality, a system of equilibrium problems, and nonexpansive mappings are investigated based on an iterative algorithm. Weak convergence theorems for common solutions are established in Hilbert spaces.
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