Uniform normal structure and fixed points of nonexpansive maps in a general topological space \((X,\tau)\) with a \(\tau\)-symmetric function (Q2566572)

A fixed point theorem for a nonexpansive map defined on a general topological space \((X,\tau)\) with a \(\tau\)-symmetric function is obtained. Namely, let \((X,\tau)\) be a topological space with a \(\tau\)-symmetric \(p\) such that for each \(x\) in \(X\), the function \(p(x, .)\) is lower semicontinuous. Assume that \(X\) is \(S\)-complete, \(p\)-bounded and has uniform \(p\)-normal structure. Let \(T\) be a nonexpansive self-mapping of \(X\). Then \(T\) has a fixed point. Some results are derived as corollaries. Recall that for a given topological space \((X,\tau)\), a function \(p: X\times X\to \mathbb R^+\) is called a \(\tau\)-symmetric function if \noindent (1) for all \(x,y\in X\), \(p(x,y)=p(y,x)\), \noindent (2) for each \(x\in X\) and any neighborhood \(V\) of \(x\), there exists \(\epsilon>0\) with \(B_p(x,\epsilon):=\{y\in X : p(x,y)<\epsilon\}\subset V\). Further, a topological space \((X,\tau)\) is said to be \(S\)-complete if for every \(p\)-Cauchy sequence \(\{x_n\}\), there exists \(x\) in \(X\) with \(lim_n p(x_n, x)=0\).