Abstract:
We introduce a new iterative method for finding a common element of the set of solutions of a generalized equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space and then prove that the sequence converges strongly to a common element of the two sets. Using this result, we prove several new strong convergence theorems in fixed point problems, variational inequalities, and equilibrium problems.

Abstract:
We introduce a new iterative method for finding a common element of the set of solutions of a generalized equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space and then prove that the sequence converges strongly to a common element of the two sets. Using this result, we prove several new strong convergence theorems in fixed point problems, variational inequalities, and equilibrium problems.

Abstract:
The authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for two countable families of hemirelatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemirelatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool (2010) and many others.

Abstract:
We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.

Abstract:
We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.

Abstract:
A new general iterative method for finding a common element of the set of solutions of variational inequality and the set of common fixed points of a countable family of nonexpansive mappings is introduced and studied. A strong convergence theorem of the proposed iterative scheme to a common fixed point of a countable family of nonexpansive mappings and a solution of variational inequality of an inverse strongly monotone mapping are established. Moreover, we apply our main result to obtain strong convergence theorems for a countable family of nonexpansive mappings and a strictly pseudocontractive mapping, and a countable family of uniformly k-strictly pseudocontractive mappings and an inverse strongly monotone mapping. Our main results improve and extend the corresponding result obtained by Klin-eam and Suantai (J Inequal Appl 520301, 16 pp, 2009). Mathematics Subject Classification (2000): 47H09, 47H10

Abstract:
We introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al. (2007) and many others.

Abstract:
We introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al. (2007) and many others.

Abstract:
The purpose of this paper is to prove strong convergence theorems for common fixed points of two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized -projection operator. In order to get the strong convergence theorems, a new iterative scheme by monotone hybrid method is presented and is used to approximate the common fixed points. Then, two examples of countable families of uniformly closed nonlinear mappings are given. The results of this paper modify and improve the results of Li et al. (2010), the results of Takahashi and Zembayashi (2008), and many others.

Abstract:
The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of a countable family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Moreover, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. Our result improve and extend the corresponding results announced by Takahashi and Zembayashi (2008 and 2009), and many others.