【作者】 张亚飞；

【导师】 钱有华；

【作者基本信息】 浙江师范大学 ， 基础数学， 2017， 硕士

【摘要】 随着科学技术的发展及实际生活的需要,非线性问题愈发成为人们研究的热点问题.然而,不动点理论作为解决非线性问题强有力的工具,已经被人们广泛应用到各个领域中.如变分不等式、积分方程、微分方程、经济学、博弈论、平衡论等等.然而,近年来很多专家、学者提出了耦合不动点及耦合重合点的概念,并且得到了迅速的发展.一些学者在此基础上进行了推广,主要有三大方向:(1)将不动点理论推广至更广泛的空间,比如偏序度量空间、完备的b-度量空间、完备的G-度量空间、模糊度量空间、准偏度量空间等等(2)将偏序度量空间中的映射推广至二维、三维、四维、甚至N维空间,得到高维耦合不动点及重合点定理(3)通过改变压缩条件,得到相应的不动点定理.本硕士论文主要研究完备的b-度量空间上的不动点定理及其应用.第一章介绍了不动点理论的研究背景以及现状;第二章第一节介绍了二元关系、序的相关概念;第二节给出了度量空间和相关的不动点定理的有关知识;第三章得到了一般度量空间上的几个新的不动点定理和耦合不动点定理,并且给出了具体的例子来验证理论的正确性;第四章由于受到前辈们的启发,为了拓展不动点定理的发展空间.第一节我们将一般度量空间中的耦合不动点定理推广到了完备的b-度量空间,同样得到了新的不动点定理;第二节进一步对b-度量空间上的耦合不动点定理作了推广,得到了耦合重合点定理.它是对不动点定理的推广及延伸.本文中得到的不动点定理都是在序结构的基础上得出的,我们发现,通过加上序结构,得到的不动点定理的压缩条件相对变弱,而且能够解决的问题也更加的广泛.最后,通过解决几类典型的积分方程来验证理论的正确性和有效性.更多还原

【Abstract】 With the development of technology and the actual need,nonlinear problems be come increasingly one of hot spot,which people are researching.However,fixed point theory is regarded as a powerful tool to resolve nonlinear problems and have been used widely in more fields.Such as variational inequalities,integral equations,differential equations,economics,game theory and equilibrium theory and so on.however,in recently years,some scholars provided coupled fixed point theorem and coupled coincident point theorem,which obtain long-term development.Many people mainly generalized three concept in three directions:(1)Promote the fixed theory to a broader space.such as,partially order metric space,complete b-metric space,complete G-metric space,fuzzy metric space,quasi-partial metric space(2)Generalize the mapping to two dimensional,three dimensional,four dimensional and even N dimensional,obtain high dimensional coupled fixed theorems and coincident point theorems in partially order metric space.(3)Get relevant fixed point theorems by changing contraction condition.The paper mainly study the fixed point theory on complete b-metric spaces and its applications.The first chapter introduces the research background and current situation of the fixed point theory.The second chapter first introduces the concepts of binary relation and order,and then the related knowledge of metric spaces and the fixed point theorem are also given.The third chapter obtain the equivalence theorem of the fixed point theorem and coupled coincidence point theory in partially order metric spaces,and a specific example is used to verify the correctness of the above theory.The four chapter inspired of the predecessors,in order to expand the developing space of the fixed point theorem.The first section,we will generalize the coupled fixed point theorem to complete b-metric space,also get the equivalence theorem.Further,in the second quarter,we generalize the coupled fixed point theorem to coupled coincidence point theorems on complete b-metric space.It is the promotion of the fixed point theorem.In the paper,the fixed point theorems are based on coherent structure.we know that by adding order relation to metric space,we notice that the contraction conditions of the fixed point theorems become weak,and be able to settle the wider problems.Finally,we end up with several examples to verify the correctness and effectiveness of the theory.更多还原