分裂等式不动点问题的迭代算法

发布时间：2018-01-16 04:33

本文关键词：分裂等式不动点问题的迭代算法　出处：《渤海大学》2017年硕士论文　论文类型：学位论文

【摘要】：分裂等式不动点问题是一类重要的数学问题,是分裂可行问题的推广,作为变分不等式的相关问题,在信号处理、医学影像重塑、尤其是放射性治疗,CT断层摄影等方面有重要应用,目前引起了许多学者的特别关注.本论文主要研究分裂等式不动点问题和一类广义非线性混合拟似变分不等式组的迭代算法,所取得的主要结果概述如下:第2章主要研究分裂等式不动点问题的迭代算法及其收敛性.首先构造了Ishikawa型迭代算法,在映射是Lipschitz连续且为伪压缩映射的条件下,证明了由迭代算法生成的序列的强收敛性和弱收敛性.其次,在映射是伪压缩映射但不具有Lipschitz连续性的条件下,通过构建Mann型迭代算法,证明了由迭代算法生成的序列的弱收敛性.最后,为避免计算投影所带来的困难,利用非扩张映射替代投影映射,对分裂等式不动点问题构造了Ishikawa型迭代算法,并在适当的条件下证明了算法的收敛性.所得结果推广了相关文献的相应结果.第3章主要研究一类广义非线性混合拟似变分不等式组的辅助问题和迭代算法.首先利用辅助原理的技术以及凸函数的极小化序列在一定的条件下得到了辅助广义非线性混合拟似变分不等式组解的存在唯一性定理.其次,在辅助问题解的存在性基础之上构造了一个迭代算法,并在适当的条件下证明了原问题解的存在性以及由算法生成的迭代序列的收敛性.所得结果推广了近期文献的某些结果.
[Abstract]:The split equality fixed point problem is an important mathematical problem, which is a generalization of splitting feasible problem, as a related problem of variational inequality, in signal processing, medical image remodeling, especially radiation therapy. Ct tomography has important applications, and has attracted special attention of many scholars. In this paper, we mainly study the fixed point problem of split equality and iterative algorithms for a class of generalized nonlinear mixed quasi-variational-like inequalities. The main results obtained are summarized as follows: in chapter 2, the iterative algorithm and its convergence of split equality fixed point problem are studied. Firstly, the Ishikawa iterative algorithm is constructed. Under the condition that the map is Lipschitz continuous and pseudo-contractive mapping, the strong convergence and weak convergence of the sequence generated by the iterative algorithm are proved. Under the condition that the map is pseudo contractive mapping without Lipschitz continuity, the weak convergence of the sequence generated by the iterative algorithm is proved by constructing the Mann type iterative algorithm. In order to avoid the difficulty of calculating projection, the Ishikawa iteration algorithm is constructed for the fixed point problem of split equality by using nonexpansive mapping instead of projection mapping. The convergence of the algorithm is proved under appropriate conditions. The results generalize the corresponding results of the relevant literatures. In chapter 3, the auxiliary problems and iterative algorithms for a class of generalized nonlinear mixed quasi-variational inequalities are studied. By using the technique of auxiliary principle and the minimization sequence of convex functions, the existence and uniqueness theorems of solutions of auxiliary generalized nonlinear mixed quasi-variational-like inequalities are obtained under certain conditions. Second, the existence and uniqueness of solutions of auxiliary generalized nonlinear mixed quasi-variational-like inequalities are obtained. An iterative algorithm is constructed based on the existence of the solution of the auxiliary problem. The existence of the solution of the original problem and the convergence of the iterative sequence generated by the algorithm are proved under appropriate conditions.
【学位授予单位】：渤海大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177.91

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