几类映像不动点及相关问题的混杂算法及应用
发布时间：2018-12-11 19:20
【摘要】：该文在Banach空间中提出了一种单调混杂迭代方法用于逼近半相对非扩张映像不动点,证明了强收敛定理。文章的结果完善并改进了Matsushita和Talahashi以及其他人的结果.在Banach空间中使用加速混杂算法,证明了有限可数族Bregman拟-利普希茨映像族和可数族拟Bregma严格伪压缩映像族不动点的强收敛定理,并将结果应用到均衡问题,变分不等式问题,优化问题解的逼近当中.其结果改善扩展了目前许多学者的最新研究成果.在Hilbert空间中,用一种新的多元混杂迭代逼近算法解决了由可数族拟-利普希茨映像族的公共不动点问题和广义分裂均衡问题组成的公共解的逼近问题,这种迭代能加速迭代序列的收敛速度.主要结果还应用到含有可数族拟利普希茨映像在分裂变分不等式及分裂优化问题中,其结果改善扩展了目前许多学者的最新研究成果.全文分五部分:第一部分介绍了不动点理论在非线性泛函分析中的重要作用,以及非线性算子迭代算法的知识背景和研究状况.第二部分在Banach空间中研究了半相对非扩张映射临近不动点问题,构造有效的迭代算法逼近它们的不动点集,得到相应的强收敛定理,并给出应用.第三部分在Banach空间中对有限可数族Bregman拟-利普希茨映像和可数族拟Bregman严格伪压缩映像进行深入的研究,构造不同的迭代格式,得到有效的收敛定理,并给出应用.第四部分在Hilbert空间中对可数族拟-利普希茨映像进行深入研究,构造出多元混合压缩投影方法,得到有效的强收敛定理,并将迭代算法应用到分裂变分不等式及分裂优化问题中.最后是总结与展望.
[Abstract]:In this paper, a monotone hybrid iterative method for approximating fixed points of semi-relative nonexpansive mappings in Banach spaces is presented, and the strong convergence theorem is proved. The results of the article improve and improve the results of Matsushita and Talahashi and others. By using the accelerated hybrid algorithm in Banach spaces, the strong convergence theorems for fixed points of Bregman pseudo-contractive mapping family and quasi Bregma strictly pseudocontractive mapping family of finite countable families are proved, and the results are applied to equilibrium problems and variational inequality problems. In the approximation of the solution of the optimization problem. The improved results extend the latest research results of many scholars. In Hilbert spaces, a new multivariate hybrid iterative approximation algorithm is used to solve the common solution approximation problem consisting of the common fixed point problem and the generalized split equilibrium problem of the countable family of pseudo-Lipschitz mappings. This kind of iteration can accelerate the convergence rate of iterative sequence. The main results are also applied to quasi-Lipschitz maps with countable families in splitting variational inequalities and splitting optimization problems. The results improve and extend the latest research results of many scholars. The paper is divided into five parts: the first part introduces the important role of fixed point theory in nonlinear functional analysis, and the knowledge background and research status of nonlinear operator iterative algorithm. In the second part, we study the near fixed point problem of semi-relative nonexpansive mappings in Banach spaces, construct effective iterative algorithms to approximate their fixed point sets, obtain corresponding strong convergence theorems, and give applications. In the third part, the finite countable family of Bregman pseudo-contractive mappings and the countable family quasi Bregman strictly pseudo-contractive mappings are studied in Banach spaces. Different iterative schemes are constructed, and effective convergence theorems are obtained, and their applications are given. In the fourth part, the countable family pseudo-Lipschitz map is studied in Hilbert space, and the multivariate mixed contractive projection method is constructed, and the effective strong convergence theorem is obtained. The iterative algorithm is applied to split variational inequalities and split optimization problems. Finally is the summary and the prospect.
【学位授予单位】：天津工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177.91
本文编号：2373089
[Abstract]:In this paper, a monotone hybrid iterative method for approximating fixed points of semi-relative nonexpansive mappings in Banach spaces is presented, and the strong convergence theorem is proved. The results of the article improve and improve the results of Matsushita and Talahashi and others. By using the accelerated hybrid algorithm in Banach spaces, the strong convergence theorems for fixed points of Bregman pseudo-contractive mapping family and quasi Bregma strictly pseudocontractive mapping family of finite countable families are proved, and the results are applied to equilibrium problems and variational inequality problems. In the approximation of the solution of the optimization problem. The improved results extend the latest research results of many scholars. In Hilbert spaces, a new multivariate hybrid iterative approximation algorithm is used to solve the common solution approximation problem consisting of the common fixed point problem and the generalized split equilibrium problem of the countable family of pseudo-Lipschitz mappings. This kind of iteration can accelerate the convergence rate of iterative sequence. The main results are also applied to quasi-Lipschitz maps with countable families in splitting variational inequalities and splitting optimization problems. The results improve and extend the latest research results of many scholars. The paper is divided into five parts: the first part introduces the important role of fixed point theory in nonlinear functional analysis, and the knowledge background and research status of nonlinear operator iterative algorithm. In the second part, we study the near fixed point problem of semi-relative nonexpansive mappings in Banach spaces, construct effective iterative algorithms to approximate their fixed point sets, obtain corresponding strong convergence theorems, and give applications. In the third part, the finite countable family of Bregman pseudo-contractive mappings and the countable family quasi Bregman strictly pseudo-contractive mappings are studied in Banach spaces. Different iterative schemes are constructed, and effective convergence theorems are obtained, and their applications are given. In the fourth part, the countable family pseudo-Lipschitz map is studied in Hilbert space, and the multivariate mixed contractive projection method is constructed, and the effective strong convergence theorem is obtained. The iterative algorithm is applied to split variational inequalities and split optimization problems. Finally is the summary and the prospect.
【学位授予单位】：天津工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177.91
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