Mann iterates of directionally nonexpansive mappings in hyperbolic spaces
Let \((X, \rho)\) be a metric space. It is assumed that \((X,\rho)\) contains a nonempty family \(M\) of metric lines such that for each pair of distinct points \(x\) and \(y\) in \(X\) and for each \(t \in [0, 1]\) there is a unique point \(z\) on the line segment joining \(x\) and \(y, [x, y]\), satisfying \[ \rho(x, z) = t\rho(x, y), \qquad \rho(y, z) = (1 - t)\rho(x, y).\tag{1} \] The unique point satisfying (1) is denoted by \((1-t)x\oplus ty\). The authors define a hyperbolic space in terms of \(\oplus\) and establish the convergence of a Mann-type iteration scheme of the form \[ x_0 := x \in X, \qquad x_{n+1} := (1 - \lambda_n)x_n\oplus \lambda_nf(x_n),\tag{2} \] where \(f\) is a directional nonexpansive selfmap of a nonempty convex subset of \(X\). They show that the iteration scheme converges to a fixed point of \(T\) and obtain error bounds. While of theoretical interest, using (2) in a computational situation would be very challenging.
scientific article; zbMATH DE number 1929992