關於一致點與變分不等式的探討

我們首先推廣了許多早期有關於一致點的結果。事實上，我們藉由Gorniewicz固定點定理得到一個在G-space上的一致點定理，以及它們的一些推論。這些是很強的結果，因為在G-space上不需要有線性或凸性結構。接著我們藉由Nikaido的一致點定理，導出一些關鍵性結果。西元1959年Nikaido建立了在緊緻Hausdorff拓樸空間上一個很著名的一致點定理，並推廣了Gale的一些有關經濟平衡點（economic equilibrium）存在性與競賽問題（game problems）的結果。的確，我們簡化並改良了某些一般的擬變分不等式（quasi-variational inequalities）與關於非循環（acyclic）多值函數的 Lefschetz-type的固定點定理，以及一些相關定理。在函數不必是單調（monotonicity）且約束集合不必是賦距空間之下，我們的結果統整了許多有關古典優選問題的定理。順道一提，就某種意義下我們也證明了Nikaido一致點定理是這些結果的推論。

關鍵字

一致點；固定點；變分不等式；非循環；單位分解；局部交性質；幾乎凸；推廣型凸空間

並列摘要

The paper focuses on two important folds in nonlinear optimization; namely, coincidence theory and variational inequalities. We first extend and generalize many earlier results about coincidences. Indeed, by using Gorniewicz fixed point theorem, we obtain a coincidence theorem on G-space and some corollaries from it. Those are very strong results, because in G-space there is not necessarily any linear and convex structure. Secondly, we shall deduce several generalized key results based on a very powerful result from Nikaido. In 1959, Nikaido established a remarkable coincidence theorem in a compact Hausdorff topological space, to generalize and to give a unified treatment to the results of Gale regarding the existence of economic equilibrium and the theorems in game problems. Indeed, we shall simplify and reformulate a few generalized variational inequalities, quasi-variational inequalities, and a Lefschetz-type fixed point theorem on acyclic multifunctions, as well as some related theorems. Beyond the realm of monotonicity nor metrizability, the results derived here generalize and unify various earlier ones from classic optimization. In the sequel, we shall deduce that Nikaido's coincidence theorem becomes a consequence of our result.

並列關鍵字

coincidence ； fixed point ； variational inequality ； acyclic ； partition of unity ； local intersection property ； almost convex ； G-space

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關於一致點與變分不等式的探討

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