A Class of Fan-Browder Type Fixed-Point Theorem and Its Applications in Topological Space

and Applied Analysis 3 Throughout this paper, all topological spaces are assumed to be Hausdorff. In order to prove our main theorems, we need the following results. Lemma 2.1. Let X and Y be two topological spaces and G : X → 2 a set-valued mapping with nonempty values. Then the following conditions are equivalent: I G has the compactly local intersection property, II for each nonempty compact subset K of X and for each y ∈ Y , there exists an open subset Oy of X (which may be empty) such that Oy ∩K ⊂ G−1 y and K ⋃ y∈Y Oy ∩K , III for each nonempty compact subset K of X, there exists a set-valued mapping F : X → 2 such that for each y ∈ Y , F−1 y is open or empty in X and F−1 y ∩K ⊂ G−1 y for each y ∈ Y , and K ⋃y∈Y F−1 y ∩K , IV for each nonempty compact subsetK of X and for each x ∈ K, there exists y ∈ Y such that x ∈ cintG−1 y ∩K and K ⋃y∈Y cintG−1 y ∩K ⋃ y∈Y G −1 y ∩K , V G−1 : Y → 2 is transfer compactly open-valued on X, VI X ⋃ y∈Y cintG −1 y , VII for each y ∈ Y , G−1 y {x ∈ X;y ∈ G x } contains a relatively open subset Oy of Y (Oy could be empty set for some y) such that ⋃ y∈Y Oy Y , VIII let S, T : k → 2 be two multivalued maps, co G x ⊂ T x and G x is nonempty, and G−1 is open in X. Proof. By Lemma 1.1 of Ding in 10 , I , II , III , IV , and V are equivalent. By Lemma 2.2 of Lin and Ansari in 13 , V and VI are equivalent, and by Ansari in 14 , VI , VII , and VIII are equivalent. This completes our proof. Remark 2.2. Lemma 2.1 includes Lemma 1.1 of Ding in 10 and Lemma 2.2 of Ansari in 13 as special cases. Lemma 2.3 see 15 . LetX and Y be topological spaces, letD be a nonempty closed subset ofX, and let Φ,Ψ : X → 2 be two set-valued mappings such that Φ x ⊂ Ψ x for each x ∈ X. Suppose that Φ−1,Ψ−1 : Y → 2 are both transfer compactly open-valued on Y . Then the mapping G : X → 2 defined by