Strong convergence theorems of common elements for equilibrium problems and fixed point problems in banach spaces

In this paper,we consider hybrid algorithms for finding common elements of the set of common fixed points of two families quasi-φ-non-expansive mappings and the set of solutions of an equilibrium problem.We establish strong convergence theorems of common elements in uniformly smooth and strictly convex Banach spaces with the property (K).     

An iterative method for equilibrium problems and a finite family of relatively nonexpansive mappings in a Banach space

In this paper, we introduce a new iterative method for finding a common element of the set of fixed points of a finite family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in uniformly convex and uniformly smooth Banach spaces. Then we prove a strong convergence theorem by using the generalized projection.     

Strong Convergence Theorems for a Family of Quasi-φ-Asymptotically Nonexpansive Mappings

The purpose of this article is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for a family of quasi-φ-asymptotically nonexpansive mappings.Its results hold in reflexive,strictly convex,smooth Banach spaces with the property(K).The results of this paper improve and extend recent some relative results.     

Extragradient methods for split feasibility problems and generalized equilibrium problems in Banach spaces

The purpose of this paper is the presentation of a new extragradient algorithm in 2‐uniformly convex real Banach spaces. We prove that the sequences generated by this algorithm converge strongly to a point in the solution set of split feasibility problem, which is also a common element of the solution set of a generalized equilibrium problem and fixed points of of two relatively nonexpansive mappings. We give a numerical example to investigate the behavior of the sequences generated by our algorithm.     

Modified block iterative algorithm for Quasi-?-asymptotically nonexpansive mappings and equilibrium problem in banach spaces

The purpose of this paper is to propose a modified block iterative algorithm for find a common element of the set of common fixed points of an infinite family of quasi-?-asymptotically nonexpansive mappings and the set of an equilibrium problem. Under suitable conditions, some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. As an application, at the end of the paper a numerical example is given. The results presented in the paper improve and extend the corresponding results in Qin et al. [Convergence theorems of common elements for equilibrium problems and fixed point problem in Banach spaces, J. Comput. Appl. Math., 225, 2009, 20-30], Zhou et al. [Convergence theorems of a modified hybrid algorithm for a family of quasi-?-asymptotically nonexpansive mappings, J. Appl. Math. Compt., 17 March, 2009, doi:10.1007/s12190-009-0263-4], Takahashi and Zembayshi [Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70, 2009, 45-57], Wattanawitoon and Kumam [Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Syst., 3, 2009, 11-20] and Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory, 134, 2005, 257-266] and others.     

均衡问题、变分不等式问题和不动点问题的强收敛定理

在一致光滑与2-一致凸Banach空间里,引进一个新的混合投影算法,找到了两族半相对非扩张映射的公共不动点集,有限个一般均衡问题的解集与宽松的协合算子的有限个变分不等式问题解集的公共元.所得结果推广了许多最近成果.     

WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES

In this paper,we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces.Then we study the weak convergence of the two iterative sequences.Our results improve and extend the results announced by many others.     

Convergence theorem for equilibrium problem and Bregman strongly nonexpansive mappings in Banach spaces

In this paper, we present an iterative scheme for Bregman strongly nonexpansive mappings in the framework of Banach spaces. Furthermore, we prove the strong convergence theorem for finding common fixed points with the set of solutions of an equilibrium problem.     

Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi-?-nonexpansive mappings in Banach spaces

In this paper, we consider a hybrid projection method for finding a common element in the set of fixed points of a infinite family of asymptotically quasi-?-nonexpansive mappings and in the set of solutions of a generalized mixed equilibrium problem. Some strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property. The results presented in the paper improve and extend some recent results.     

Approximation theorems for total quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to prove some approximation theorems of common fixed points for countable families of total quasi-?-asymptotically nonexpansive mappings which contain several kinds of mappings as its special cases in Banach spaces. In order to get the approximation theorems, the hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article extend and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 134 (2005) 257-266], Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 149 (2007) 103-115], Li, Su [H. Y. Li, Y. F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72(2) (2010) 847-855], Su, Xu and Zhang [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3960], Wang et al. [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. 73 (2010) 2260-2270], Chang et al. [S.S. Chang, C.K. Chan, H.W. Joseph Lee, Modified block iterative algorithm for quasi-?-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces, Appl. Math. Comput. 217 (2011) 7520-7530], Ofoedu and Malonza [E.U. Ofoedu, D.M. Malonza, Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type, Appl. Math. Comput. 217 (2011) 6019-6030] and Yao et al. [Y.H. Yao, Y.C. Liou, S.M. Kang, Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings, Appl. Math. Comput. 208 (2009) 211-218].     

Strong convergence theorems for infinite family of relatively quasi nonexpansive mappings and systems of equilibrium problems

In this paper, we construct a new iterative scheme by hybrid method for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. Then, we prove strong convergence of the scheme to a common element of the two sets. Furthermore, we apply our results to solve convex minimization problem. Our results extend important recent results.     

Banach空间中关于平衡问题与半相对非扩张映象的强收敛定理

在Banach空间框架下,将半相对非扩张映象引入到平衡问题中,利用收缩投影迭代来近似平衡问题的解集与半相对非扩张映象的不动点集的公共点,并在适当的条件下证明了该迭代序列强收敛于某个公共点.所得的结果推广和改进了已有的相关结果.     

Strong convergence for maximal monotone operators, relatively quasi-nonexpansive mappings, variational inequalities and equilibrium problems

In this paper, we introduce a new hybrid iterative scheme for finding a common element of the set of zeroes of a maximal monotone operator, the set of fixed points of a relatively quasi-nonexpansive mapping, the sets of solutions of an equilibrium problem and the variational inequality problem in Banach spaces. As applications, we apply our results to obtain strong convergence theorems for a maximal monotone operator and quasi-nonexpansive mappings in Hilbert spaces and we consider a problem of finding a minimizer of a convex function.     

Block Iterative Methods for a Finite Family of Generalized Nonexpansive Mappings in Banach Spaces

In this paper, we study an operator generated by a finite family of generalized nonexpansive mappings in a Banach space. We first prove that the set of fixed points of this operator is identical to the set of all common fixed points of the mappings. Next, using this operator, we construct an iterative sequence to approximate a common fixed point of the family of generalized nonexpansive mappings. We finally apply our results to solve the feasibility problem in Banach spaces.     

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

The purpose of this paper is by using the hybrid iterative method to prove some strong convergence theorems for approximating a common element of the set of solutions to a system of generalized mixed equilibrium problems and the set of common fixed points for two countable families of closed and asymptotically relatively nonexpansive mappings in Banach space. The results presented in the paper improve and extend the corresponding results of Su et al. [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3906], Li and Su [H.Y. Li, Y.F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72 (2) (2010) 847-855], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. TMA 73 (2010) 2260-2270], Kang et al. [J. Kang, Y. Su, X. Zhang, Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications, Nonlinear Anal. HS 4 (4) (2010) 755-765], Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory 134 (2005) 257-266], Tan et al. [J.F. Tan, S.S. Chang, M. Liu, J.I. Liu, Strong convergence theorems of a hybrid projection algorithm for a family of quasi-?-asymptotically nonexpansive mappings, Opuscula Math. 30 (3) (2010) 341-348], Takahashia and Zembayashi [W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009) 45-57] and Wattanawitoon and Kumam [K. Wattanawitoon, P. Kumam, Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Systems 3 (2009) 11-20] and others.     

Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type

In this paper we study hybrid iterative scheme for finding a common element of set of solutions of generalized mixed equilibrium problem, set of common fixed points of finite family of weak relatively nonexpansive mapping and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which are announced recently. Application of our theorem to solution of equations of Hammerstein-type is of independent interest.     

2一致凸Banach空间中均衡问题、变分不等式问题和不动点问题的一个弱收敛定理

In this paper,we introduce a new iterative scheme for finding the common element of the set of solutions of an equilibrium problem,the set of solutions of variational inequalities for an α-inversely strongly monotone operator and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and 2-uniformly convex Banach space.Some weak convergence theorems are obtained,to extend the previous work.     

Convergence theorems for semigroup-type families of non-self mappings

Motivated by a paper Chidume and Zegeye [Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups, Applicable Analysis, 86 (2007), 353–366], we prove several strong convergence theorems for a family (not necessarily a semigroup) ℱ = {T(t): t ∈ G} of nonexpansive or pseudocontractive non-self mappings in a reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm, where G is an unbounded subset of ℝ⁺. Our results extend and improve the corresponding ones byMatsushita and Takahashi [Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,Nonlinear Analysis, 68 (2008), 412–419],Morales and Jung [Convergence of paths for pseudo-contractive mappings in Banach spaces, Proceedings of American Mathematical Society, 128 (2000), 3411–3419], Song [Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Applicable Analysis, 86 (2007), 1329–1337], Song and Xu [Strong convergence theorems for nonexpansive semigroup in Banach spaces, Journal of Mathematical Analysis and Applications, 338 (2008), 152–161], Wong, Sahu, and Yao [Solving variational inequalities involving nonexpansive type mappings, Nonlinear Analysis, (2007) doi:10.1016/j.na. 2007.11.025] in the context of a non-semigroup family of non-self mappings.      

An extragradient method for relaxed cocoercive variational inequality and equilibrium problems

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansivemappings, the set of an equilibrium problem and the set of solutions of the variational inequality problem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao, Takahashi and many others.