均衡问题和分裂变分不等式问题的收敛性

发布时间：2018-05-27 07:28

本文选题：均衡问题 + κ-渐近严格伪非延伸映射　；参考：《云南财经大学》2015年硕士论文

【摘要】：非线性泛函分析的根本任务是解决具有分析结构的无穷维空间中各种算子给出的方程的求解问题,但其精确解是很难求出的。因此,寻找近似解也就成为了非线性泛函分析研究的重要内容。通过设计有效的迭代算法研究非线性问题的解已成为一种主要手段。近年来,学者们构造了一些迭代算法逼近各种非线性泛函问题的解,但许多算法中存在可计算性不足的问题。因此,在本文,我们借助前辈们的研究经验,于Hilbert空间和Banach空间中构造容易计算的迭代格式逼近均衡问题与两种非线性映射的不动点问题的公共解,并且利用分裂可行性问题的思想在两个不同的Hilbert空间中研究分裂广义变分不等式和混合均衡问题的解的收敛性,进一步推广和改进目前国内外一些现有结果。本文从以下几个方面讨论：首先,简述均衡问题和变分不等式问题的背景及研究现状,阐述研究的意义。其次,在Hilbert空间中引入一种迭代算法,寻找经典均衡问题与k-渐近严格伪非延伸映射的不动点集的公共元,并得到强收敛定理。再次,在Banach空间中通过利用太阳广义非扩张收缩核,并引入一种迭代方法寻找混合均衡问题和全拟-φ-渐近非扩张映射的不动点集的公共解,并获得弱和强收敛定理。最后,在不同的两个Hilbert空间中引入一种迭代方法,逼近分裂广义变分不等式和分裂混合均衡问题的解,并获得强收敛定理。
[Abstract]:The fundamental task of nonlinear functional analysis is to solve the problem of solving the equations given by various operators in an infinite dimensional space with an analytical structure, but its exact solution is very difficult to obtain. Therefore, finding approximate solutions has become an important part of nonlinear functional analysis. It has become a main method to study the solution of nonlinear problems by designing effective iterative algorithms. In recent years, some iterative algorithms have been constructed to approximate the solutions of various nonlinear functional problems, but in many algorithms there is a lack of computability. Therefore, in this paper, we construct a computable iterative scheme in Hilbert space and Banach space to approximate the common solution of the equilibrium problem and the fixed point problem of two nonlinear mappings based on the previous research experience. The convergence of the solutions of split generalized variational inequalities and mixed equilibrium problems in two different Hilbert spaces is studied by using the idea of splitting feasibility problem, which further generalizes and improves some existing results at home and abroad. This paper discusses the following aspects: firstly, the background and research status of equilibrium problem and variational inequality problem are briefly introduced, and the significance of the research is expounded. Secondly, an iterative algorithm is introduced in Hilbert spaces to find common elements of the set of fixed points for classical equilibrium problems and k-asymptotically strictly pseudo-nonextension mappings, and obtain strong convergence theorems. Thirdly, by using the sun generalized nonexpansive contractive kernel in Banach space, an iterative method is introduced to find the common solutions of the fixed point set of the mixed equilibrium problem and the total quasi-蠁-asymptotically nonexpansive mapping, and the weak and strong convergence theorems are obtained. Finally, an iterative method is introduced in two different Hilbert spaces to approximate the solutions of split generalized variational inequalities and split mixed equilibrium problems, and obtain a strong convergence theorem.
【学位授予单位】：云南财经大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O178

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