On the Geometric Structure for the Set of Fixed Points and Ishikawa Iterative Process of Nonexpansive Mappings
Yongfu Su · Journal of Tianjin Normal University · 2004
Firstly, it was proved that fixed point set F(T) with nonexpansive mappings T in strictly convex Banach space was a closed convex set, and then proved again that Ishikawa iterative sequence x_(n+1)=t_nT(s_nTx_n+(1-s_n)x_n)+(1-t_n)x_n,t_n,s_n∈[0,1], n≥0 for nonexpansive mapping T from any point x_0 converged to certain fixed point p as F(T) being a manifold in Hilbert space and this p must be element of the optimal approximation, e.g. ‖x_0-p‖=d(x_0,F(T)). Another result was that (lim)n→+∞y-p,x_n-p‖x_n-p‖≤0, y∈F(T),if Ishikawa iteration process x_n→p∈F(T), which was said to be obtuse angle principle for approximating fixed points.