【摘要】：均衡問(wèn)題為優(yōu)化問(wèn)題、變分不等式問(wèn)題、不動(dòng)點(diǎn)問(wèn)題、鞍點(diǎn)問(wèn)題、非合作博弈問(wèn)題等提供了統(tǒng)一的數(shù)學(xué)結(jié)構(gòu)。作為均衡問(wèn)題的原始模型,經(jīng)典變分不等式問(wèn)題被廣泛地運(yùn)用于物理學(xué)、工程學(xué)、經(jīng)濟(jì)學(xué)等眾多學(xué)科領(lǐng)域。隨著研究范疇的不斷拓展,越來(lái)越多的研究學(xué)者將經(jīng)典變分不等式延伸到了集值變分不等式,而求解集值變分不等式問(wèn)題的方法也是數(shù)不勝數(shù)。其中,與數(shù)學(xué)規(guī)劃問(wèn)題中約束集的凸性類(lèi)似,集值映射的單調(diào)性在求解集值變分不等式過(guò)程中也起到了至關(guān)重要的作用。本文主要研究了集值映射的單調(diào)性,并構(gòu)造了二元函數(shù)和二元集值函數(shù),利用它們的性質(zhì)來(lái)刻畫(huà)集值映射的單調(diào)性。然后再將這些條件應(yīng)用于集值變分不等式問(wèn)題解的存在理論中。本文主要內(nèi)容如下:第一章,介紹研究背景、國(guó)內(nèi)外狀況以及本文所要做的工作。第二章,回顧一些相關(guān)概念和結(jié)論作為本文研究的主要工具。第三章,舉出具體的實(shí)例來(lái)驗(yàn)證集值映射、二元函數(shù)以及二元集值函數(shù)的六種單調(diào)性之間的蘊(yùn)涵關(guān)系。第四章,構(gòu)造二元函數(shù)和二元集值函數(shù)來(lái)刻畫(huà)集值映射的單調(diào)性,并詳細(xì)證明了二元函數(shù)和二元集值函數(shù)的單調(diào)性與集值映射的單調(diào)性之間的一些等價(jià)條件。第五章,利用二元函數(shù)的偽單調(diào)性與集值映射的偽單調(diào)性的等價(jià)關(guān)系,在歐式空間中得到集值變分不等式問(wèn)題解的存在性理論。
[Abstract]:Equilibrium problem provides a unified mathematical structure for optimization problem variational inequality problem fixed point problem saddle point problem non-cooperative game problem and so on. As the original model of equilibrium problem, classical variational inequality problem is widely used in many fields such as physics, engineering, economics and so on. With the continuous expansion of the scope of research, more and more researchers extend classical variational inequalities to set-valued variational inequalities, and the methods for solving set-valued variational inequalities are innumerable. Similar to the convexity of constraint set in mathematical programming, the monotonicity of set-valued mapping plays an important role in solving set-valued variational inequalities. In this paper, we mainly study the monotonicity of set-valued mappings, construct binary functions and binary set-valued functions, and use their properties to characterize the monotonicity of set-valued mappings. Then these conditions are applied to the existence theory of solutions to set-valued variational inequality problems. The main contents of this paper are as follows: chapter one introduces the research background, domestic and foreign situation and the work to be done in this paper. In the second chapter, some related concepts and conclusions are reviewed as the main tools of this paper. In chapter 3, concrete examples are given to verify the implication relations among the six monotonicity of set-valued mapping, binary function and binary set-valued function. In chapter 4, we construct binary functions and bivariate set-valued functions to characterize monotonicity of set-valued mappings, and prove in detail some equivalent conditions between monotonicity of binary functions and bivariate set-valued functions and monotonicity of set-valued mappings. In chapter 5, we obtain the existence theory of solutions to set-valued variational inequality problems in Euclidean space by using the equivalent relation between pseudo-monotonicity of binary functions and pseudo-monotonicity of set-valued mappings.
【學(xué)位授予單位】：西華師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O176

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相關(guān)期刊論文 前2條

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2 ;Strict Feasibility of Variational Inequalities in Reflexive Banach Spaces[J];Acta Mathematica Sinica(English Series);2007年03期

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本文編號(hào)：2443043