【摘要】：該文在Banach空間中提出了一種單調(diào)混雜迭代方法用于逼近半相對(duì)非擴(kuò)張映像不動(dòng)點(diǎn),證明了強(qiáng)收斂定理。文章的結(jié)果完善并改進(jìn)了Matsushita和Talahashi以及其他人的結(jié)果.在Banach空間中使用加速混雜算法,證明了有限可數(shù)族Bregman擬-利普希茨映像族和可數(shù)族擬Bregma嚴(yán)格偽壓縮映像族不動(dòng)點(diǎn)的強(qiáng)收斂定理,并將結(jié)果應(yīng)用到均衡問(wèn)題,變分不等式問(wèn)題,優(yōu)化問(wèn)題解的逼近當(dāng)中.其結(jié)果改善擴(kuò)展了目前許多學(xué)者的最新研究成果.在Hilbert空間中,用一種新的多元混雜迭代逼近算法解決了由可數(shù)族擬-利普希茨映像族的公共不動(dòng)點(diǎn)問(wèn)題和廣義分裂均衡問(wèn)題組成的公共解的逼近問(wèn)題,這種迭代能加速迭代序列的收斂速度.主要結(jié)果還應(yīng)用到含有可數(shù)族擬利普希茨映像在分裂變分不等式及分裂優(yōu)化問(wèn)題中,其結(jié)果改善擴(kuò)展了目前許多學(xué)者的最新研究成果.全文分五部分:第一部分介紹了不動(dòng)點(diǎn)理論在非線(xiàn)性泛函分析中的重要作用,以及非線(xiàn)性算子迭代算法的知識(shí)背景和研究狀況.第二部分在Banach空間中研究了半相對(duì)非擴(kuò)張映射臨近不動(dòng)點(diǎn)問(wèn)題,構(gòu)造有效的迭代算法逼近它們的不動(dòng)點(diǎn)集,得到相應(yīng)的強(qiáng)收斂定理,并給出應(yīng)用.第三部分在Banach空間中對(duì)有限可數(shù)族Bregman擬-利普希茨映像和可數(shù)族擬Bregman嚴(yán)格偽壓縮映像進(jìn)行深入的研究,構(gòu)造不同的迭代格式,得到有效的收斂定理,并給出應(yīng)用.第四部分在Hilbert空間中對(duì)可數(shù)族擬-利普希茨映像進(jìn)行深入研究,構(gòu)造出多元混合壓縮投影方法,得到有效的強(qiáng)收斂定理,并將迭代算法應(yīng)用到分裂變分不等式及分裂優(yōu)化問(wèn)題中.最后是總結(jié)與展望.
[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.
【學(xué)位授予單位】：天津工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O177.91

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