Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces

The purpose of this article is to propose a splitting algorithm for finding a common zero of a finite family of inclusion problems of accretive operators in Banach space. Under suitable conditions, some strong convergence theorems of the sequence generalized by the algorithm to a common zero of the inclusion problems are proved. Some applications to the convex minimization problem, common fixed point problem of a finite family of pseudocontractive mappings, and accretive variational inequality problem in Banach spaces are presented.     

Strong convergence theorems for approximating common fixed points of families of nonexpansive mappings and applications

An implicit algorithm for finding common fixed points of an uncountable family of nonexpansive mappings is proposed. A new inexact iteration method is also proposed for countable family of nonexpansive mappings. Several strong convergence theorems based on our main results are established in the setting of Banach spaces. Both algorithms are applied for finding zeros of accretive operators and for solving convex minimization, split feasibility and equilibrium problems.     

Strong convergence theorems for variational inequality problems and fixed point problems in uniformly smooth and uniformly convex Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings and the set of common fixed points for a nonexpansive mapping in a uniformly smooth and uniformly convex Banach space. We obtain a strong convergence theorem under some suitable conditions. Our results improve and extend the recent ones announced by many others in the literature.     

φ-强增生型变分包含的Ishikawa迭代序列强收敛的充要条件

研究Banach空间中φ-强增生型变分包含问题,在实的自反的光滑Banach空间中,证明了这类变分包含问题解得存在唯一性,并给出Ishikawa迭代序列{xn}强收敛的充要条件     

Generalized viscosity implicit rules for solving quasi-inclusion problems of accretive operators in Banach spaces

The generalized viscosity implicit rules for solving quasi-inclusion problems of accretive operators in Banach spaces are established. The strong convergence theorems of the rules to a solution of quasi-inclusion problems of accretive operators are proved under certain assumptions imposed on the sequences of parameters. The results presented in this paper extend and improve the main results of Refs. (Moudafi, J Math Anal Appl. 2000;241:46–55; Xu et al., Fixed Point Theory Appl. 2015;2015:41; López et al., Abstr Appl Anal. 2012;2012; Cholamjiak, Numer Algor. DOI:10.1007/s11075-015-0030-6.). Moreover, some applications to monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem are presented.     

Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

Let H be a real Hilbert space and let T: H→2^H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit vT⁻¹0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.     

Strong convergence theorem by hybrid method for equilibrium problems,variational inequality problems and maximal monotone operators

We introduce an iterative scheme for finding a common element of the solution set of the equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-monotone operators and the solution set of a maximal monotone operator in a 2-uniformly convex and uniformly smooth Banach space, and then we present strong convergence theorems which generalize the results of many others.     

增生算子方程带误差的Noor三步迭代解与收敛率的估计

1引言与预备知识设X是实Banach空间,X^*是X的对偶空间,（·,·）表示X和X^*的广义对偶组.正规对偶映象J:X→2^X*定义为J（x）={∈X^*:（x,f）=‖x‖²=‖f‖²}.用D（T）表示     

On iterations methods for zeros of accretive operators in Banach spaces

In this paper, three iterations are designed to approach zeros of set-valued accretive operators in Banach spaces. The first one is the continuous Picard type iteration involving the resolvent, the second one is the approximate Picard type iteration involving the resolvent and the third one is the Halpern type iteration involving the resolvent. Some strong convergence theorems for three iterations are proved.     

A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping

Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point set are provided in the final section.