A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce and analyze a new general iterative scheme by the viscosity approximation method for finding the common element of the set of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings and the set solutions of the variational inequality problems for an ξ-inverse-strongly monotone mapping in Hilbert spaces. We show that the sequence converge strongly to a common element of the above three sets under some parameters controlling conditions. The result extends and improves a recent result of Chang et al. (Nonlinear Anal. 70:3307–3319, 2009) and many others.     

A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings

In this paper, we investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for k-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of Kumam [P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turk. J. Math. 33 (2009) 85–98], 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, Article ID 134148, 17 pages, doi:10.1155/2008/134148], Yao et al. [Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an finite family of nonexpansive mappings, Nonlinear Analysis 69 (5–6) (2008) 1644–1654], Qin et al. [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis (69) (2008) 3897–3909], and 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.     

Coupled fixed point results for nonlinear integral equations

In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and ?iri?’s theorem [1] but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations.     

A new hybrid algorithm for a system of equilibrium problems and variational inclusion

The purpose of this paper is to consider a shrinking projection method of finding the common element of the set of common fixed points for a finite family of a ξ-strict pseudo-contraction, the set of solutions of a systems of equilibrium problems and the set of solutions of variational inclusions. Then, we prove strong convergence theorems of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng, Wang, Shyu and Yao (J Inequal Appl, 2008:15, Article ID 720371, 2008), Takahashi, Takeuchi and Kubota (J Math Anal Appl 341:276–286, 2008), Takahashi and Takahashi (Nonlinear Anal 69:1025–1033, 2008) and many others.     

A general composite explicit iterative scheme of fixed point solutions of variational inequalities for nonexpansive semigroups

In this paper, we introduce a composite explicit viscosity iteration method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. We prove strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions. Our result extends and improves those announced by Li et al [General iterative methods for a one-parameter nonexpansive semigroup in Hilbert spaces, Nonlinear Anal. 70 (2009) 3065–3071], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Modelling 48 (2008) 279–286], Plubtieng and Wangkeeree [S. Plubtieng, R. Wangkeeree, A general viscosity approximation method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Bull. Korean Math. Soc. 45 (4) (2008) 717–728] and many others.     

Split common fixed point problems for demicontractive operators

In this paper, first, we introduce a new iterative algorithm involving demicontractive mappings in Hilbert spaces and, second, we prove some strong convergence theorems of the proposed method with the Armijo-line search to show the existence of a solution of the split common fixed point problem. Finally, we give some numerical examples to illustrate our main results.