Extension theorems and minimax inequalities with applications to mathematical economics.
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文摘
This thesis presents some results in nonlinear analysis and their applications to mathematical economics.;Chapters 2 and 3 are the most important and technical part of this thesis. In Chapter 2, duals of two important results due to Gale and Mas-Colell in 1975 and Shafer and Sonnenschein in 1975 respectively in equilibrium theory are presented. Two examples of how to use these results to resolve equilibrium problems with closed preferences are given. Chapter 3 is devoted to establishing a random version of a Tietze type extension theorem together with its applications. The proof of the theorem is very complicated and technical. We think that the work of Mas-Colell et al can be classified as "selection" type in the sense that the proofs almost always invoke Michael's selection theorem or similar ones. Our work can be classified as "extension" type in the sense that we need Tietze type extension theorems to do the proofs, although in a related result, we still have to use the powerful Ky Fan minimax inequality. It is worthwhile to point out that as far as we know, this is the first time that Tietze type extension theorems have been used to study economic problems with closed preferences.;In Chapter 4, we generalize many minimax inequalities. These include Fan's, Sion's, Granas and Liu's and Tan and Yu's results. Applications to fixed point theory, variational inequalities, complementarity problems and abstract economies are given.;Chapter 5 presents a fixed point theorem for correspondences on (0,1). This is an interesting result for its simplicity and elementary proof. Some examples comparing it to related work and also some simple applications to game theory are included.