摘要
均衡问题为研究关于经济、金融、最优化等一系列问题提供了较为系统的研究框架.近年来,许多作者(如[1-3]、[5-8]、[30-34])对该问题做了较为全面和深入的研究.受这些最新研究成果的启发,本文引入了均衡问题的若干迭代算法,并做了相应的收敛性分析.全文共分三章.
第一章为导论.本章概括了国内外专家学者在这方面的最新研究成果,介绍了本文相关的一些概念、引理.
第二章针对不同种类的映射,依次得到了相对非扩张映射、非扩张映射、严格伪压缩映射、渐近非扩张映射的不动点集与均衡问题解集的强收敛定理.
第三章通过引入新的迭代格式,研究了此格式在广义投射下,关于相对非扩张映射的强收敛性,并且作为应用得到了关于非扩张映射与凸可行性问题的一些收敛定理.
我们的结果是新的,并且可以看作文献[14]、[15]、[24]和[28]等相应结果的直接改进和推广.本文还给出了文献[13]-[15]、[21]、[22]等所讨论的问题在一些新的条件下的收敛性.在证明过程中,我们也提供了一些新的估计技巧.
Equilibrium problems provide us with a systematic framework to study a wide class of problems arising in finance economics, optimization and operation research etc.. In recent years, equilibrium problems have been deeply and thoroughly researched. See, for example, [1-3],[5-8] and [30-34]. Inspired and motivated by their work, in this paper, we introduce several iterative methods and their corresponding convergence theorems for equilibrium problems. Our paper is divided into three chapters.
The first chapter is the induction which is constituted by two parts. They introduce the recent work of the famous authors in this field and contain corresponding lemmas and definitions respectively.
In chapter two, by changing the kind of mappings, we obtain strong convergence for equilibrium problems and relatively nonexpansive mapping, nonexpansive mapping, strictly pseudocontractive mapping, asymptotically nonexpansive mapping respectively.
In the third chapter, we intrduce some new iterative methods under the generalized projection and proof the strong convergence for these methods. As application, we obtain convergence for nonexpansive mapping and convex feasibility problems.
Our results are new and can be viewed as generalizations and extensions of the corresponding results obtained in [14], [15], [24] and [28]. We also get the convergence property of the problems discussed in the papers [13]-[15], [21] and [22] etc. under mild conditions.In this paper, we also provide some new estimation techniques in the proofs of the results.
引文
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