Browder and Göhde fixed point theorem for monotone nonexpansive mappings (Q269984)

Suppose that \(X:=(X,d)\) is a hyperbolic space and \(\leq\) is a partial order on \(X\) such that the order intervals \(\{x\in X:a\leq x\}\) and \(\{x\in X:x\leq a\}\) are closed and convex for all \(a\in X\). Recall that \(T:C\to C\) where \(C\) is a bounded closed convex subset of \(X\), is a monotone nonexpansive mapping if \(Tx\leq Ty\) and \(d(Tx,Ty)\leq d(x,y)\) for all \(x,y\in C\) with \(x\leq y\). In this paper, the authors prove the following. Theorem. Suppose that \((X,d)\) is a uniformly convex hyperbolic space and \(C\) is a bounded closed and convex subset of \(X\). If \(T:C\to C\) is monotone nonexpansive and there exists an element \(x_0\in C\) such that \(x_0\leq Tx_0\) or \(Tx_0\leq x_0\), then \(T\) has a fixed point. The authors also discuss the result above where \(X\) is a Banach space which is uniformly convex in every direction and \(C\) is a weakly compact convex subset of \(X\).