Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces

Let C be a closed and convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself, A be an α-inverse strongly-monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. We introduce an iteration scheme of finding a point of F (T)∩(A+B)⁻¹0, where F (T) is the set of fixed points of T and (A+B)⁻¹0 is the set of zero points of A+B. Then, we prove a strong convergence theorem, which is different from the results of Halpern’s type. Using this result, we get a strong convergence theorem for finding a common fixed point of two nonexpansive mappings in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a nonexpansive mapping.     

Strong Convergence Theorems for Perturbed Maximal Monotone Operators in Banach Spaces

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极大单调算子和非扩展映射的迭代收敛定理及其应用

In this paper, two iterative schemes for approximating common element of the set of zero points of maximal monotone operators and the set of fixed points of a kind of generalized nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Two strong convergence theorems are obtained and their applications on finding the minimizer of a kind of convex functional are discussed, which extend some previous work.     

Weak and Strong Convergence Theorems of Quasi-Nonexpansive Mappings in a Hilbert Spaces

In this paper, we first prove the weak convergence for the Moudafi’s iterative scheme of two quasi-nonexpansive mappings. Then we prove the weak convergence for the Moudafi’s iterative scheme of quasi-nonexpansive mapping and nonexpansive mapping. Finally, we prove the strong convergence for the Moudafi’s iterative scheme of two quasi-nonexpansive mappings. Our results generalize the recent results due to Iemoto and Takahashi.     

Strong Convergence Theorems for a Finite Family of Sequences of Nearly Nonexpansive Mappings in Hilbert Spaces

In this article, by using the hybrid projection method or the shrinking projection method, we introduce two strong convergence theorems for finding a common fixed point of a finite family of sequences of nearly nonexpansive mappings in Hilbert spaces.     

Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.     

Banach空间中一致L-Lipschitz映象的强收敛定理

在任意实Banach空间中引入了一对一致L-Lipschitz映象的具误差的新迭代序列,并证明了这些迭代序列的强收敛性定理.结果推广、完善和改进了最近的一些结果.     

Banach空间中的两个Lipschitzian伪压缩映射的强收敛定理(英文)

在Banach空间中证明了伪压缩映射具有误差项的Ishikawa迭代序列和具有误差项的Mann迭代序列的强收敛的充分必要条件.所得结果将已有结果推广到了Banach空间.     

Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space

In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim. 14 (1976), 877–898] and Kamimura and Takahashi [J. Approx. Theory 106 (2000), 226–240]. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem.     

Hilbert空间框架下严格渐近伪压缩映像的强收敛定理

The Mann iterations have no strong convergence even for nonexpansive mappings in Hilbert spaces. The aim of this paper is to propose a modification of the Mann iterations for strictly asymptotically pseudocontractive maps in Hilbert spaces to have strong convergence. Our results extend those of Kim, Xu, Nakajo, Takahashi and many others.     

Strong Convergence Theorems for Quasi-Nonexpansive Mappings and Uniformly L-Lipschitzian Asymptotically Pseudo-Contractive Mappings in Banach Spaces

A new 𝒮-generated Ishikawa iteration with errors is proposed for a pair of quasi-nonexpansive mapping and uniformly L-Lipschitzian asymptotically pseudo-contractive mapping in real Banach spaces. We show that the proposed iterative scheme converges strongly to a common solution of quasi-nonexpansive mapping and uniformly L-Lipschitzian asymptotically pseudo-contractive mapping in real Banach spaces. A comparison table is prepared using a numeric example which shows that the proposed iterative algorithm is faster than some known iterative algorithms.     

Banach空间中均衡问题和弱相对非扩展映射的强收敛定理及其应用(英文)

本文在Banach空间中设计了一些新的杂交迭代算法用以逼近一类均衡问题解集和弱相对非扩展映射不动点集或极大单调算子零点集的公共元.得到了一些强收敛的结论,并将它们推广到逼近一类均衡问题解集和有限个弱相对非扩展映射公共不动点集或有限个极大单调算子公共零点集的公共元的情形.最后,展示了本文的迭代算法在最优化问题上的应用.     

Banach空间中的相对非扩张映射和变分包含的强收敛定理

这篇论文主要考虑了相对非扩张映射的不动点问题和关于逆强增生映射和m-增生映射的变分包含问题;我们建立了一类新的迭代方法来逼近这两个问题的公共解,证明了这一迭代序列强收敛到这两个问题的公共解.     

Strong Convergence Theorems for the Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in the Intermediate Sense in Hilbert Spaces

Abstract

The aim of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also give a numerical example to support our main results. The results obtained in this paper improve and extend many recent ones in this culture.     

半序度量空间中单调映射的不动点定理及混合单调映射的耦合不动点定理

本文在度量空间中引入半序,证明了半序度量空间中单调增加映射的不动点定理及混合单调映射的耦合不动点定理．     

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.     

Banach空间中关于非扩张映射的修改的Mann迭代算法的强收敛定理

该文在Banach空间里引进一种新的修改的Mann算法,来寻求一族无限个非扩张映射的公共元素.在适当条件下,得到了一个强收敛定理.所得结果改进和推广了许多最近的相关结果.     

Banach空间有限个极大单调算子公共零点的迭代强收敛定理

令E为实光滑、一致凸Banach空间,E为其对偶空间.令Ai E×E,i=1,2,…,m,为极大单调算子且∩mi=1Ai-10≠.将引进一个新定义、给出一种新迭代算法,并利用Lyapunov泛函与广义投影算子等技巧,证明迭代序列强收敛于Ai的公共零点,i=1,2,…,m.去掉了以往结论中过强的限定条件,是对笔者以往工作的延续。     

Banach空间中渐近非扩张映象的收敛性定理

本文在Banach空间中引入和研究渐近非扩张映象及非扩张映象的某些类型的迭代序列的收敛性,其结果改进和推广了最新的一些结果,     

Radius Theorems for Monotone Mappings

Set-Valued and Variational Analysis - For a Hilbert space X and a mapping $F: X\rightrightarrows X$ (potentially set-valued) that is maximal monotone locally around a pair $(\bar {x},\bar {y})$ in...