這篇論文裡，我們在特別的賦距空間 LC-metric spaces 中考慮平衡問題 EP(f,K) 解的存在性。首先，我們透過一個 HKKM 多值函數在抽象的拓樸空間 C-spaces 中建立一些交集非空定理。在定義域為一個 compact 下，我們證明了 EP(f,K) 解的存在性。如果把 compact 的條件拿掉的話，透過 coercive condition，可以改善我們的主要定理，也會得到 EP(f,K) 解的存在性。另外， 若應用 C-property 和 g-monotonicity 的觀念，分別取代 HKKM 多值函數和函數 f 的連續性，則我們可以得到其它的存在定理。最後，在對偶問題 DEP(f,K) 的基礎下，我們找了一些條件可以確保 DEP(f,K) 的解就是 EP(f,K) 的解。此外，在函數 f 本身具有 pseudomonotonicity 或 quasimonotonicity 之特性下，我們進一步比較了兩個 coercive conditions C_s 和 C_w 並探討其分別對 EP(f,K) 解集合的影響性。

In this paper, we consider the existence theorem of solutions for the well-known equilibrium problem EP(f,K) in LC-metric spaces. First, we establish an intersection theorem on C-spaces via a generalized HKKM mapping. Then we prove our main existence result for EP(f,K), where the constraint region K is compact. Without compactness, we also improve our main theorem under an usual coercive condition. As well, we can substitute HKKM mapping and continuity of f by the concept of C-property and g-monotonicity, respectively. Finally, based on the dual equilibrium problem DEP(f,K), we try to find a link to make sure of the existence of solutions to EP(f,K). In addition, when the bifunction f is equipped with pseudomonotonicity or quasimonotonicity, we further compare a kind of coercive condition C_s and a weaker condition C_w to see the influence on the solution set of EP(f,K).

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