Strong convergence of an iterative method for nonexpansive mappings with new control conditions

The purpose of this paper is to study the convergence problem of an iterative method for nonexpansive mappings in Banach spaces under some new control conditions on parameters.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

The purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of variational inequality for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Our results extend the recent results of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515], Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52], Combettes and Hirstoaga [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 486–491], Iiduka and Takahashi, [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and many others.     

On the method of alternating resolvents

The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35-61] has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. In this paper, we present several algorithms based on alternating resolvents of two maximal monotone operators, A and B, that can be used to approximate common zeros of A and B. In particular, we prove that the sequences generated by our algorithms converge strongly. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals.     

Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method

Strong convergence theorems are obtained for a finite family of nonexpansive mappings and semigroups by the hybrid method.     

On the steepest descent approximation method for the zeros of generalized accretive operators

In this paper we present certain characteristic conditions for the convergence of the generalized steepest descent approximation process to a zero of a generalized strongly accretive operator, defined on a uniformly smooth Banach space. Our study is based on an important result of Reich [S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978) 85–92] and given results extend and improve some of the earlier results which include the steepest descent approximation method.     

A general iterative method for a finite family of nonexpansive mappings

We study the approximation of common fixed points of a finite family of nonexpansive mappings and suggest a modification of the iterative algorithm without the assumption of any type of commutativity. Also we show that the convergence of the proposed algorithm can be proved under some types of control conditions.     

A von Neumann alternating method for finding common solutions to variational inequalities

By modifying von Neumann’s alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert space.     

Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces

In this paper, we introduce a modified Ishikawa iterative process for approximating a fixed point of nonexpansive mappings in Hilbert spaces. We establish some strong convergence theorems of the general iteration scheme under some mild conditions. The results improve and extend the corresponding results of many others.     

A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α$\alpha$-inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].     

Convergence of Ishikawa’s iteration method for pseudocontractive mappings

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T_i:C→C,i=1,2,…,N, be a finite family of Lipschitz pseudocontractive mappings. It is our purpose, in this paper, to prove strong convergence of Ishikawa’s method to a common fixed point of a finite family of Lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on T or on C. Moreover, computation of the closed convex set C_n for each n≥1 is not required. The results obtained in this paper improve on most of the results that have been proved for this class of nonlinear mappings.     

The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces

A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved.     

Convergence and certain control conditions for hybrid viscosity approximation methods

Very recently, Yao, Chen and Yao [20] proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition β_n→0 is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space.     

Approximations for nonlinear mappings by the hybrid method in Hilbert spaces

The hybrid method in mathematical programming was introduced by Haugazeau (1968) [1] and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001) [2] proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009) [10] extended Bauschke and Combettes’s results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) [10]. And we get strong convergence theorems for a family of asymptotically κ-strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings.     

Note on the cubic decreasing region of the Chebyshev method

In this paper we reduce the two-dimensional cubic decreasing region considered in Hernandez and Salanova (2000) [1], [2] into one-dimensional region or interval for the Chebyshev method. It means that we find a simple sufficient condition for the semilocal convergence of the method.     

An iterative method of solution for equilibrium and optimization problems

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets. The results of this paper extended and improved the results of H. Iiduka and W. Takahashi [Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. Therefore, by using the above result, an iterative algorithm for the solution of a optimization problem was obtained.     

A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces

In this paper, we prove strong convergence theorems by the hybrid method for a family of hemi-relatively nonexpansive mappings in a Banach space. Our results improve and extend the corresponding results given by Qin et al. [Xiaolong Qin, Yeol Je Cho, Shin Min Kang, Haiyun Zhou, Convergence of a modified Halpern-type iteration algorithm for quasi-?-nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051-1055], and at the same time, our iteration algorithm is different from the Kimura and Takahashi algorithm, which is a modified Mann-type iteration algorithm [Yasunori Kimura, Wataru Takahashi, On a hybrid method for a family of relatively nonexpansive mappings in Banach space, J. Math. Anal. Appl. 357 (2009) 356-363]. In addition, we succeed in applying our algorithm to systems of equilibrium problems which contain a family of equilibrium problems.     

Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups

Strong convergence theorems are obtained for a finite family of asymptotically nonexpansive mappings and semigroups by the modified Mann method.     

The Ulm method under mild differentiability conditions

The Ulm method is considered to approximate a solution of a nonlinear operator equation F(x) = 0. We study the convergence of this method when F′ is ω-conditioned and prove that the R-order of convergence is at least 1 + p if ω is quasi-homogeneous of type ω(tz)≤ t ^p ω(z), for z > 0, tϵ[0,1] and pϵ[0,1]. Preparation of this paper was partly supported by the Ministry of Education and Science (MTM 2005-03091).     

Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces

Our purpose in this paper is to introduce an iterative scheme for solving multiple-set split feasiblity problems in p-uniformly convex Banach spaces which are also uniformly smooth using Bregman distance techniques. We further obtain a strong convergence result for approximating solutions of multiple-set split feasiblity problems in the framework of p-uniformly convex Banach spaces which are also uniformly smooth.     

Strong convergence studied by a hybrid type method for monotone operators in a Banach space

In this paper, we study a strong convergence for monotone operators. We first introduce the hybrid type algorithm for monotone operators. Next, we obtain a strong convergence theorem (Theorem 3.3) for finding a zero point of an inverse-strongly monotone operator in a Banach space. Finally, we apply our convergence theorem to the problem of finding a minimizer of a convex function.