| From the current research situation, approximation theory has already become an important branch of modern pure mathematics and applied mathematics. It plays a crucial role in theoretical research and practical applications. Fixed point theory has been an essential branch of current mathematical research for many years, also it is the foundation of modern mathematics. Fixed point theory gets a rapid development in applied mathematics and other fields. It is worth mentioning that many of today's studies are from finding the fixed point to find the best proximity points. It is a great business to do research on best proximity points in Banach space, this paper studies the best proximity points at star-shaped set of Banach space. So what have done is valuable.This paper paid more attention to characterizing a concrete point which has the property of approximative compactness and studied a topic of best proximity points at star-shaped set. Especially considering the best proximity points under relatively u-continuous mapping at star-shaped set in Banach space. The main contents are as follows:The first chapter introduced the history of approximative compactness and fixed point, then introduced the purpose and significance of approximative compactness and fixed point and summarized the achievement of predecessors, at last, this chapter introduced my study logic and results.The second chapter introduced the preliminary knowledges of learning a point which has the property of approximative compactness in Banach space, then characterized this point, and found a connection between this point and H point, at last, this chapter discussed some connections between this point and metric projection mapping.The third chapter introduced some preliminary knowledge of nonexpansive mappings on star-shaped set firstly, then especially discussed the best proximity points under the relatively u-continuous mapping at star-shaped set and nonempty compact convex subset in uniformly convex Banach space. |
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