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Generalizations of Himmelberg type fixed point theorems in locally FC-spaces. (English) Zbl 1102.47042

Let \(\Delta_n\) be a standard \(n\)-dimensional simplex with vertices \(e_0,e_1,\dots,e_n\). \((X,\{\varphi_n\})\) is said to be a finitely continuous topological space (in short, FC-space), if \(X\) is a topological space and for each \(N=\{x_0,x_1,\dots,x_n\}\in \langle X\rangle\) where some elements in \(X\) may be equal and \(\langle X\rangle\) denotes the set of all nonempty finite subsets of \(X\), there exists a continuous mapping \(\varphi_N:\Delta_n\to X\). A subset \(D\) of \((X,\{\varphi_N\})\) is said to be an FC-subspace of \(X\) if for each \(N=\{x_0,x_1,\dots,x_n\}\in \langle X\rangle\) and for any \(\{x_{i_0},\dots,x_{i_k}\}\subset D\cap N\), \(\varphi_n(\Delta_k)\subset D\),where \(\Delta_k=\text{co}\{e_{i_j}:j=0,\dots,k\}\).
\((X,\{\varphi_N\})\) is said to be a locally finitely continuous topological space (in short, locally FC-space) if \((X,\{\varphi_N\})\) is an FC-space and \((X,{\mathcal U})\) is a uniform space with a basis \(\{V_\lambda\}_{\lambda\in\Lambda}\) of the uniformity \(\mathcal U\) such that for each \(x\in X\) and \(V\in\{V_\lambda\}_{\lambda\in\Lambda}\), the set \(V[x]=\{y\in X:(y,z)\in V\}\) is a FC-subspace of \(X\).
The notion of locally FC-spaces generalizes the concepts of locally convex \(H\)-spaces, locally \(G\)-convex spaces and locally \(L\)-convex spaces.
Let \(T:X\to 2^X\) be a set valued mapping such that if \(F:X\to 2^y\) is a generalized KKM mapping with respect to \(T\), then the family \(\{\overline {Fx}:x\in X\}\) has the finite intersection property, where \(\overline{Fx}\) denotes the closure of \(Fx\). Then \(T\) is said to have the KKM property. Write \(\text{KKM}(X,Y)=\{T:X\to 2^Y: T\) has the KKM property}.
The main results of this paper are the following theorems.
Theorem 3.1. Let \((X,{\mathcal U},\{\varphi_N\})\) be a locally FC-space. If \(T\in\text{KKM}(X,X)\) is a compact mapping, then for each open entourage \(U\in{\mathcal U}\) there exists \(x_U\in X\) such that \(F(x_U)\cap V[x_U]\neq\emptyset\).
Theorem 3.2. Let \((X,{\mathcal U},\{\varphi_N\})\) be a locally FC-space and \(T\in\text{KKM}(X,X)\) be compact upper semicontinuous mappings with closed values. Then \(T\) has a fixed point in \(X\).
These theorems improve, unify and generalize known results in the literature.

MSC:

47H10 Fixed-point theorems
47H04 Set-valued operators
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