Existence and uniqueness of endpoints of closed set-valued asymptotic contractions in metric spaces (Q864668)

The authors study closed set-valued maps \(T\) on a complete metric space \(M\). They assume that there is a bounded sequence \((u_n)\) in \(M\) such that \(u_{n+1}\in T^n(u_n)\) for each \(n\in\mathbb{N}\) and a highly technical assumption on \(T\) which amounts to some kind of asymptotic contractivity in order to conclude that \(T\) must have a unique endpoint, i.e., a point \(u\) such that \(T(u)=\{u\}\).