# 臺灣博碩士論文加值系統

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 摘要 本論文主要探討一些固定點定理以及一些推廣型KKM定理，並將這些結果應用到一些相關課題上。如：匹配定理、擬平衡點、擬變分不等式等。 本論文共分為三章，第一章探討有關向內地(inward)收縮和非擴張函數的固定點定理以及二函數的同值點定理。第二章則先引入一些非凸集合，探討其性質，再進而探討在這些非凸集合上的一些推廣型S-KKM定理、匹配定理、擬平衡點存在性定理等。在本章中並引入一個新的函數族：Q(X,Y)，探討其性質及相關定點定理。第三章則探討在均勻空間上可逼近(approachable )函數族的一些性質，及固定點定理以及二函數的同值點定理。 本論文內容主要藉由探討近年來一些學者的相關論著，並引入一些新的函數族，探討其性質，進一步推導出更廣的結果，這些結果涵蓋了許多學者的一些結果。
 Abstract The purpose of this paper is to study the fixed point theory and the KKM theory , we get some fixed point theorems and generalized KKM theorems. As applications, we use the above results to related topics, for examples, the matching theorems, the existence theorems of quasi-equilibrium, quasi-variational inequalities. This paper contains three chapters. In the first chapter, we discuss some fixed point theorems and coincidence theorems about inward contractive functions and inward nonexpansive functions. In second chapter, we introduce some conceptions of non-convexities, study their properties, and apply these properties to get some generalized KKM theorems, the matching theorems, and the existence theorems of quasi-equilibrium. In this chapter, we also introduce a new family of functions, Q(X,Y), we research its properties and get some fixed point theorems about this family. In the last chapter, we study the properties of the family of approachable functions in uniform spaces. By using these properties, we attain some fixed point theorems and coincidence theorems. The results of this paper actually extend many results of authors as in the references.
 Chapter 1 Fixed-point Theorems for the Inward Maps 1. Introduction and Preliminaries 2. Fixed point Theorems and Coincidence Theorems 2.1 Fixed point theorems for contraction and nonexpansive set-valued mappings. 2.2 Coincidence theorems for two mappings 3 Fixed point Theorems for closed inward set valued maps Chapter 2 Fixed-Point Theorems on Non-convex Constraint Regions 1. Introduction and Preliminaries 2 Fixed-point theorems on nearly-convex sets 2.1 Fixed-point theorems on nearly-convex sets for mappings 2.2 Fixed-point theorems on nearly-convex sets for 2.3 Fixed-point theorems for the class 2.4 Applications to quasi-equilibrium theorems 3 Fixed-point theorems on almost convex sets Chapter 3 Approachable and Fixed points for Non-convex Set-Valued Maps on Uniform Spaces 1. Introduction and Preliminaries 2. The class A of approachable maps on uniform spaces 3. Fixed point theorems for the class A of approachable maps
 [1] E. G. Begle, Locally connected spaces and generalizedmanifolds, Amer. Math. J (1942)553-574.[2] H. Ben-El-Mechaiekh and P. Deguire. Approachability andfixed points for non-convex set-valued maps, J. Math. Anal.Appl. 170 (1992) 477-500.[3] H. Ben-El-Mechaiekh and P. Deguire, Approximation of non- convex set-valved maps, C. R. Acad. Sci. Paris, t.312(1991) 379-384.[4] H. Ben-El-Mechaiekh and P. Deguire, General fixed pointtheorems for non-convex set-valved maps, C. R. Acad. Sci.Paris, t.312 (1991) 433-438[5] H. Ben-El-Mechaiekh, A remark concerning a matchingtheorem of Ky Fan, Chinese. Jour. Math. 17 (1989) 309-314.[6] F. E. Bowder, The fixed point theory of multivaluedmappings in topological vector spaces, Math. Ann. 177, 283-301(1968).[7] T. H. Chang, J. C, Jeng and K. W. Kuo, On S-KKM propertyand related topics, J. Math. Anal. Appl, in press.[8] T. H. Chang and C. L. Yen, KKM property and fixed pointtheorems, J. Math. Anal. Appl. 203 (1996) 224-235.[9] P. Z. Daffer and H. Kaneko, Multi-valued f-contractionmappings, Boll. U. M. I (1994) 233-241.[10] R. Engelking, General Topology, Heldremann Verlag,Berlin (1989).[11] K. Fan, A generalization of Tychonoff’s fixed pointtheorems, Math. Ann. 142(1961)305-310.[12] P. M. Fitzpatrick and W. V. Petryshyn, Fixed pointtheorems for multivalued noncompact inward mappings, J.Math. Anal. Appl. 46, 756-767(1974).[13] K. Geobel, On a fixed point theorem for multivaluednonexpansive mapping, Ann. Univ. M. Curie-Skowdska 29(1975) 70-72.[14] G. Koyhe, Topological Vector Spaces I, Springer, NewYork(1983).[15] J. L. Kelly. General Topology, van Nostrand, Princeton,NJ,1955.[16] W. A. Kirk and S. Massa, Remarks on asymptotic andChebyshev centers, Houston J. Math. 16 (1990) 357-364.[17] M. Lassonde, Fixed points for Kakutani factorizablemultifunctions, preprint, 1989.[18] T. C. Lim, Remarks on some fixed point theorems, Proc.Amer. Math. Soc. 60 (1976) 179-182.[19] G. J. Minty, On the maximal domain of a monotonefunction, Michigan Math. J. 8 (1961)135-137.[20] W. V. Petryshyn and P. M. Fitzpatrick, A degree theory,fixed point theorems, and mapping theorems for noncompactmappings, Trans. Amer. Math. Soc. (to appear)[21] T. Y. Wu, General vector quasi-variational inequalities o nonconvex constraint regions. in press.
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