Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.     

带有模糊集值映象的一般混合拟变分包含的扰动近似点算法

本文引入一类新的带有模糊集值映象的一般混合拟变分包.在Hilbert空间中,利用极大η-单调映象的预解算子技巧,建立了这类变分包与不动点的等价性.利用这种等价性,构造了一些新的扰动近似点算法,并证明了由此算法所产生的迭代序列的收敛性.这些定理改进,统一和推广了近期文献中许多重要结果.     

A new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings

In this paper, we introduce and study a new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings. An existence theorem of solutions is proved without compactness assumptions. A new iterative algorithm is suggested and analysed. The convergence of iterative sequences generated by the new algorithm are also given. As special cases, some known results are also discussed.     

Coincidences of Nonself Operators by a Simpler Algorithm

Abstract

In this article, we introduce a new class of contractive mappings and study analytical and computational aspects of a special case of Jungck-Khan iterative algorithm generated by this class of mappings. In particular, we improve upon strong convergence, rate of convergence and data dependence results existing in the current literature. Analytical as well numerical illustrative examples are given to support the new results.     

Banach空间的上带松弛共强制的广义隐式变分包含组

本文在Banach空间上引入和研究了一类带松弛共强制的广义隐式变分包含组（SNSIVI）．使用M增值算子的预解算子技术，我们构造了一类新的迭代算法逼近这类隐式变分包含组，且在q-一致平滑Banach空间上证明了这类迭代算法的收敛性．我们的结果推广和改进了最近的相关工作．     

Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings

In this paper, we introduce a new mapping and a Hybrid iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common fixed point of a finite family of nonexpansive mappings which is a solution of the generalized equilibrium problem. The results obtained in this paper extend the recent ones of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025–1033].     

Strong convergence of an iterative method for pseudo-contractive and monotone mappings

In this paper, we introduce an iterative process which converges strongly to a common element of fixed points of pseudo-contractive mapping and solutions of variational inequality problem for monotone mapping. As a consequence, we provide an iteration scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudo-contractive mappings and solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.     

A new iterative algorithm for solving a system of generalized mixed equilibrium problems for a kind of multi-valued nonlinear mappings

By using a new technique, namely, a specific way of choosing the indexes of the involved mappings, we introduce a new iterative algorithm for approximating common fixed points of a countable family of multi-valued totally quasi-?-asymptotically nonexpansive mappings and obtain a strong convergence theorem under some suitable conditions. As application, an iterative solution to a system of generalized mixed equilibrium problems is studied. The results extend those of other authors, in which the involved mapping consists of just a single one.     

一类广义变分包含的迭代解

介绍一类新的涉及集值映射的变分包含问题，构造其迭代序列，并证明迭代序列收敛于变分包含问题的解，给出迭代序列与解的误差估计。     

模糊(Fuzzy)映象的广义补余问题*

本文引入一类关于模糊(Fuzzy)映象的新的广义补余问题,构造出一类新的迭代算法.我们讨论这类广义补余问题解的存在性及迭代序列的收敛性.     

Banach空间中一类广义混合非线性隐拟变分包含解的三步迭代

本文引入并研究了实Banach空间中一类新的广义混合非线性隐拟变分包含 ,通过对实Banach空间中的m 增生映象运用Nadler定理和Michael选择定理 ,构建了这类新的广义变分包含解的三步迭代算法 ,并证明了其解的存在性和由迭代算法生成的迭代序列的收敛性     

Generalized mixed variational-like inequality for random fuzzy mappings

In this paper, we introduce and study a new class of generalized mixed variational-like inequality for random fuzzy mappings(GMVLIP). An existence theorem for auxiliary problem of the GMVLIP is established. Further, by exploiting the theorem, we construct and analyze a new iterative algorithm for finding the solution of the GMVLIP. Furthermore, we prove the existence of a unique solution of the GMVLIP and discuss the convergence analysis of iterative sequence generated by the iterative algorithm.     

Fixed-Point Methods for a Certain Class of Operators

We introduce in this paper a new class of nonlinear operators, which contains, among others, the class of operators with semimonotone additive inverse and also the class of nonexpansive mappings. We study this class and discuss some of its properties. Then we present iterative procedures for computing fixed points of operators in this class, which allow for inexact solutions of the subproblems and relative error criteria. We prove weak convergence of the generated sequences in the context of Hilbert spaces. Strong convergence is also discussed.     

Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces

In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

     

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.     

General Strongly Nonlinear Quasivariational Inequalities with Relaxed Lipschitz and Relaxed Monotone Mappings

In this paper, we introduce and study a new class of general strongly nonlinear quasivariational inequalities and construct a general iterative algorithm by using the projection method. We establish the existence of a unique solution for general strongly nonlinear quasivariational inequalities involving relaxed Lipschitz, relaxed monotone, and strongly monotone mappings; we obtain the convergence and stability of the iterative sequences generated by the algorithm. Our results extend, improve, and unify many known results due to Bose, Noor, Siddiqi-Ansari, Verma, Yao, Zeng, and others.     

A new system of generalized nonlinear co-complementarity problems

In this paper, we introduce and study a new system of generalized nonlinear co-complementarity problems with set-valued mappings and construct an iterative algorithm for approximating the solutions of the system of generalized co-complementarity problems. We prove the existence of the solutions for the system of generalized co-complementarity problems with set-valued mappings without compactness and the convergence of iterative sequences generated by the algorithm in Hilbert spaces. We also study a new perturbed iterative algorithm for approximating a system of generalized co-complementarity problems with single-valued mappings in Hilbert spaces.     

On random variational inclusions with random fuzzy mappings and random relaxed cocoercive mappings

In this paper, we introduce and study the random variational inclusions with random fuzzy and random relaxed cocoercive mappings. We define an iterative algorithm for finding the approximate solutions of this class of variational inclusions and establish the convergence of iterative sequences generated by proposed algorithm. Our results improve and generalize many known corresponding results.     

A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces

We introduce a hybrid proximal point algorithm and establish its strong convergence to a common solution of a proximal point of a lower semi-continuous mapping and a fixed point of a demicontractive mapping in the framework of a CAT(0) space. As applications of our new result, we solve variational inequality problems for these mappings on a Hilbert space. Illustrative example is given to validate theoretical result obtained herein.     

Strong Convergence Theorems for Variational Inequalities and Fixed Point Problems of Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce a modified hybrid Mann iterative scheme with perturbed mapping which is based on well-known CQ method, Mann iteration method and hybrid (or outer approximation) method. We establish a strong convergence theorem for three sequences generated by this modified hybrid Mann iterative scheme with perturbed mapping. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings.