本文選題：Ishikawa型算法 + Mann型算法��； 參考：《上海師范大學(xué)》2017年博士論文

【摘要】：本學(xué)位論文在無(wú)限維實(shí)Hilbert空間或Banach空間框架下研究了變分不等式問(wèn)題、非線性算子不動(dòng)點(diǎn)問(wèn)題及分裂可行性問(wèn)題.為了解決這些問(wèn)題,本文利用投影算子技巧、半閉原理等工具改進(jìn)了之前文獻(xiàn)中的外梯度方法、投影收縮方法、阻尼方法、混合方法、粘性迭代方法,并證明了修正算法的收斂性.其結(jié)果改進(jìn)、推廣與補(bǔ)充了之前文獻(xiàn)中的相應(yīng)結(jié)果.全文共分六章.第一章,介紹了分裂可行問(wèn)題的研究背景與現(xiàn)狀,并簡(jiǎn)述了本文的主要工作與結(jié)構(gòu)安排.第二章,回顧了文中將要用到的一些基本概念和預(yù)備知識(shí).第三章,研究涉及偽壓縮映象的分裂可行問(wèn)題與不動(dòng)點(diǎn)問(wèn)題.在Hilbert空間中,研究外梯度方法,用以解決涉及偽壓縮映象的分裂可行問(wèn)題與不動(dòng)點(diǎn)問(wèn)題.我們構(gòu)造一個(gè)Ishikawa型外梯度算法來(lái)逼近涉及Lipschitz偽壓縮映象的分裂可行問(wèn)題與不動(dòng)點(diǎn)問(wèn)題之公共解,進(jìn)一步,我們也構(gòu)造了一個(gè)Mann型外梯度算法來(lái)逼近涉及非Lipschitz偽壓縮映象的分裂可行問(wèn)題與不動(dòng)點(diǎn)問(wèn)題之公共解.在一定條件下,我們證得,由構(gòu)造的算法產(chǎn)生的序列弱收斂于分裂可行問(wèn)題與不動(dòng)點(diǎn)問(wèn)題的公共解.本章得到的結(jié)果推廣和改進(jìn)了一些文獻(xiàn)中相應(yīng)的結(jié)果.數(shù)值試驗(yàn)說(shuō)明了理論結(jié)果的可行性.第四章,在p-一致凸、一致光滑實(shí)Banach空間框架下,通過(guò)對(duì)Bregman擬嚴(yán)格偽壓縮映象定義以及混合投影的研究,構(gòu)造新的混合投影算法,用來(lái)逼近Banach空間中分裂可行問(wèn)題與不動(dòng)點(diǎn)問(wèn)題的解.并證明了所構(gòu)造的算法產(chǎn)生的序列強(qiáng)收斂于分裂可行問(wèn)題與不動(dòng)點(diǎn)問(wèn)題的公共解.本章得到的結(jié)果推廣和改進(jìn)了一些文獻(xiàn)中相應(yīng)的結(jié)果.數(shù)值試驗(yàn)說(shuō)明了理論結(jié)果的可行性.第五章,本章主要目的是研究尋求鄰近分裂可行問(wèn)題、不動(dòng)點(diǎn)問(wèn)題及變分不等式問(wèn)題之公共解的收縮投影方法.我們借助收縮投影技巧,構(gòu)造恰當(dāng)?shù)牡惴ㄈケ平徑至芽尚袉?wèn)題、不動(dòng)點(diǎn)問(wèn)題及變分不等式問(wèn)題的公共解,并證明了所構(gòu)造算法的強(qiáng)收斂性.第六章,本章主要目的是研究鄰近分裂可行問(wèn)題的范數(shù)最小解,我們證得,在Hilbert空間中由構(gòu)造的算法生成的序列強(qiáng)收斂于鄰近分裂可行問(wèn)題的范數(shù)最小解.本章得到的結(jié)果推廣和改進(jìn)了一些文獻(xiàn)中相應(yīng)的結(jié)果.數(shù)值試驗(yàn)說(shuō)明了理論結(jié)果的可行性.
[Abstract]:In this paper, we study the variational inequality problem, the fixed point problem of nonlinear operator and the splitting feasibility problem under the frame of infinite dimensional real Hilbert space or Banach space. In order to solve these problems, this paper improves the external gradient method, projection contraction method, damping method, hybrid method, viscous iteration method and so on by means of projection operator technique and semi-closed principle. The convergence of the modified algorithm is proved. The results are improved to extend and supplement the corresponding results in previous literatures. The full text is divided into six chapters. In the first chapter, the research background and present situation of splitting feasible problem are introduced, and the main work and structure arrangement of this paper are briefly described. In the second chapter, we review some basic concepts and preliminary knowledge that will be used in this paper. In chapter 3, the splitting feasible problem and fixed point problem of pseudo contractive mappings are studied. In Hilbert space, the external gradient method is studied to solve the splitting feasible problem and fixed point problem involving pseudo contractive mappings. We construct an Ishikawa type external gradient algorithm to approximate the common solutions of split feasible problems and fixed point problems involving Lipschitz pseudo contractive mappings. We also construct an Mann type external gradient algorithm to approximate the common solutions of split feasible problems and fixed point problems involving non- pseudo contractive mappings. Under certain conditions, we prove that the sequence generated by the constructed algorithm converges weakly to the common solutions of split feasible problems and fixed point problems. The results obtained in this chapter extend and improve the corresponding results in some literatures. Numerical experiments show the feasibility of the theoretical results. In chapter 4, under the framework of p-uniformly convex and uniformly smooth real Banach spaces, a new hybrid projection algorithm is constructed by studying the definition of Bregman quasi-strictly pseudo-contractive mappings and the mixed projection. It is used to approximate the solutions of split feasible problem and fixed point problem in Banach space. It is proved that the sequence generated by the proposed algorithm converges strongly to the common solutions of split feasible problems and fixed point problems. The results obtained in this chapter extend and improve the corresponding results in some literatures. Numerical experiments show the feasibility of the theoretical results. In chapter 5, the main purpose of this chapter is to study the constrictive projection method for finding common solutions of adjacent splitting problems, fixed point problems and variational inequality problems. By means of the technique of contraction projection, we construct an appropriate iterative algorithm to approximate the common solutions of the adjacent splitting feasible problem, fixed point problem and variational inequality problem, and prove the strong convergence of the constructed algorithm. In chapter 6, the main purpose of this chapter is to study the norm minimum solution of the adjacent splitting feasible problem. We prove that the sequence generated by the constructed algorithm converges strongly to the norm minimum solution of the adjacent splitting feasible problem in Hilbert space. The results obtained in this chapter extend and improve the corresponding results in some literatures. Numerical experiments show the feasibility of the theoretical results.
【學(xué)位授予單位】：上海師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O177.91

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