On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings

In this paper, several weak and strong convergence theorems are established for a modified Noor iterative scheme with errors for three asymptotically nonexpansive mappings in Banach spaces. Mann-type, Ishikawa-type, and Noor-type iterations are covered by the new iteration scheme. Our results extend and improve the recently announced ones [B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444-453; Y.J. Cho, H. Zhou, G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004) 707-717] and many others.     

Implicit Iterations of Two Finite Families for Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to study the weak and strong convergence of an implicit iteration process to a common fixed point for two finite families of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Xu and Ori, Numer. Funct. Anal. Optim. 2001; 22:767–773, Zhou and Chang, Numer. Fund. Anal. Optim. 2002; 23:911–921, Chidume and Shahzad, Nonlinear. Anal. 2005; 62: 1149–1156.     

Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces

In this paper, we establish some existence results for the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in reflexive Banach spaces. Using the concept of the stable $f$ -quasimonotonicity, the properties of Clarke’s generalized directional derivative, Clarke’s generalized gradient and KKM technique, some existence theorems of solutions are proved when the constrained set is nonempty, bounded (or unbounded), closed and convex. Our main results extend various results existing in the current literatures.     

The general iterative methods for nonexpansive mappings in Banach spaces

In this paper, we introduce a general iterative approximation method for finding a common fixed point of a countable family of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The main result extends various results existing in the current literature.     

Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce an iterative scheme based on the extragradient approximation method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of the variational inequality problem for a monotone L-Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Applications to optimization problems are given. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications (2008) 17. doi:10.1155/2008/134148. Article ID 134148], Kumam and Katchang [P. Kumam, P. Katchang, A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings, Nonlinear Anal. Hybrid Syst. (2009) doi:10.1016/j.nahs.2009.03.006] and many others.     

Existence theorems and iterative approximation methods for generalized mixed equilibrium problems for a countable family of nonexpansive mappings

In this paper, we introduce the new generalized mixed equilibrium problem basing on hemicontinuous and relaxed monotonic mapping. Using the KKM technique, we obtain the existence of solutions for the generalized mixed equilibrium problem in a Banach space. Furthermore, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of a countable family of nonexpansive mappings. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.     

Well-posedness for general parametric quasi-variational inclusion problems

In this paper, we aim to suggest the new concept of well-posedness for the general parametric quasi-variational inclusion problems (QVIP). The corresponding concepts of well-posedness in the generalized sense are also introduced and investigated for QVIP. Some metric characterizations of well-posedness for QVIP are given. We prove that under suitable conditions, the well-posedness is equivalent to the existence of uniqueness of solutions. As applications, we obtain immediately some results of well-posedness for the parametric quasi-variational inclusion problems, parametric vector quasi-equilibrium problems and parametric quasi-equilibrium problems.     

Levitin–Polyak well-posedness for set optimization problems involving set order relations

Positivity - In this paper set optimization problems with three types of set order relations are concerned. We introduce various types of Levitin–Polyak (LP) well-posedness for set...     

Strong convergence of modified Mann iterations for a countable family of nonexpansive mappings

In this paper we propose a new modified Mann iteration for computing common fixed points of nonexpansive mappings in a Banach space. We give certain different control conditions for the modified Mann iteration. Then, we prove strong convergence theorems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These results improve and extend results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], Yao, et al. [Y. Yao, R. Chen and J. Yao, Strong convergence and certain control conditions for modified Mann iteration, Nonlinear Anal. 68 (2008) 1687–1693], Qin and Su [X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007) 415–424], and many others.     

Characterizing the solution set of convex optimization problems without convexity of constraints

Some characterizations of solution sets of a convex optimization problem with a convex feasible set described by tangentially convex constraints are given.