Survey on Metric Fixed Point Theory and Applications
Fixed Point Theory is divided into the following three major areas: Topological Fixed Point Theory, which came from Brouwer’s fixed point theorem in 1912; Metric Fixed Point Theory, which came from Banach’s fixed point theorem in 1922; Discrete Fixed Poin
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Abstract Fixed Point Theory is divided into the following three major areas: • Topological Fixed Point Theory, which came from Brouwer’s fixed point theorem in 1912; • Metric Fixed Point Theory, which came from Banach’s fixed point theorem in 1922; • Discrete Fixed Point Theory, which came from Tarski’s fixed point theorem in 1955. In this chapter, we focus on recent topics on metric fixed point theory and its applications, which will be very helpful to beginners and specialists of metric fixed point theory and its applications. In fact, since Banach’s fixed point theorem in metric spaces, because of its simplicity, usefulness and applications, it has become a very popular tools in solving the existence problems in many branches of mathematical analysis and applied sciences. Recently, Banach’s fixed point theorem has been applied to Economics, Chemical Engineering Sciences, Medicine, Image Recovery, Electric Engineering, Game Theory and other applied sciences by many authors. Keywords Picard’s convergence theorem · Banach’s fixed point theorem · Contractive mapping · Multiplicative metric space · Weakly commuting mapping and compatible mapping · (C L Rg )-property · Posedness · Stability · The limit shadowing property · Picard iteration · Mann iteration · Best proximity point · n-cyclically monotone mapping · Maximal n-cyclically monotone mapping · Cyclically firmly nonexpansive mapping Mathematics Subject Classification 47H09 · 47H10 · 54H25 · 37C25
Y.J. Cho (B) Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Korea e-mail: [email protected] Y.J. Cho Center for General Education, China Medical University, Taichung 40402, Taiwan © Springer Nature Singapore Pte Ltd. 2017 M. Ruzhansky et al. (eds.), Advances in Real and Complex Analysis with Applications, Trends in Mathematics, DOI 10.1007/978-981-10-4337-6_9
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1 Introduction In 1912, Brouwer [1] proved the following fixed point theorem, which is called Brouwer’s Fixed Point Theorem: Theorem B. Every continuous mapping from the unit ball of Rn into itself has a fixed point. Since Brouwer’s fixed point theorem, some authors, Schauder [2], Tychonoff [3], Kakutani [4] and many others have improved and generalized this theorem in several ways. In fact, Schauder’s fixed point theorem is an extension of Brouwer’s fixed point theorem to topological vector spaces and, also, there are several entirely different ways to prove Brouwer’s fixed point theorem by some authors. In 1955, Tarski [5] proved the following fixed point theorem, which is called Tarski’s Fixed Point Theorem: Theorem T. If F is a monotone function on a nonempty complete lattice, then the set of fixed points of F forms a nonempty complete lattice. Note that the least fixed point of the mapping f is the least element x such that f (x) = x or, equivalently, such that f (x) ≤ x and the greatest fixed point is the greatest element x such that f (x) = x or, equivalently, such that f (x) ≤ x. Consequently, from Theorem T, f has the greatest fixed po
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