Browder's convergence theorem for multivalued mappings without endpoint condition (Q428818)

A geodesic path joining \(x,y\) in a metric space \((X,d)\) is an isometric mapping \(c:[0,1]\to X\) such that \(c(0)=x\) and \(c(1)=y\). The space is said to be geodesic if any two points can be joined by a geodesic path. It is said to be an \(\mathbb{R}\)-tree if for any \(x,y\in X\) there is a unique geodesic path from \(x\) to \(y\) denoted by \([x,y]\) such that the following condition holds: if \([y,x]\cap[x,z]=\{x\}\) then \([y,x]\cup[x,z]=[y,z]\). Let then \(E\) be a nonempty closed convex subset of a complete \(\mathbb{R}\)-tree \(X\) and let \(T\) be a multivalued nonexpansive mapping from \(E\) into the nonempty compact convex subsets of \(E\). Choose \(u\in E\) and define \(f:E\to E\) by defining \(f(x)\) as the nearest point to \(u\) in \(T(x)\). For \(s\in(0,1)\) denote by \(t_s(x)\) the point \(z\in[u,f(x)]\) that satisfies \(d(u,z)=s\). Finally, let \(x_s\) be the unique fixed point of \(t_s\). The authors prove that \(T\) has a fixed point if and only if \(\{x_s\}\) remains bounded as \(s\to0\). In this case \((x_s)\) converges to the unique fixed point of \(T\) that is nearest to \(u\).