設X是一個非空集合，Y是一個非空G-凸空間，Z是一個拓樸空間，F屬於G-s-KKM(X,Y,Z)，Q是一個從Z映到2的Y次方的Φ-函數。在某些假設條件之下，我們證得F與Q的一些同質點定理。我們也證明了一些推
廣型G-s-KKM定理，並利用這些推廣型G-s-KKM定理證明一些變分不等式的存在性定理。本文的結果推廣了許多學者的研究結果。

Let X be a nonempty set, let Y be a nonempty G-convex space, let Z be a topological space, let F in G-s-KKM(X,Y,Z) , and let Q is a set-valued mapping from Z into Y be a Φ-mapping. In this paper, we establish some coincidence theorems of F and Q under some assumptions. We also establish some generalized G-s-KKM theorems and apply these generalized G-s-KKM theorems to establish the existence theorems concerning variational inequalities. Our results generalize many other authors’ results (for example, see, [6,12,18,21]).

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