On the approximation of fixed points of non-self strict pseudocontractions

Abstract

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If $T:C\rightarrow H$ is a non-self and k-strict pseudocontractive mapping, we can define a map $v:C\rightarrow \mathbb {R}$ by $\, v(x):=\inf \{\lambda \ge 0:\lambda x+(1-\lambda )Tx\in C\}.$ Then, for a fixed $x_{0}\in C$ and for $\alpha _{0}:=\max \{k,v(x_{0})\},$ we define the Krasnoselskii–Mann algorithm $x_{n+1}=\alpha _{n}x_{n}+(1-\alpha _{n})Tx_{n},$ where $\alpha _{n+1}=\max \{\alpha _{n},v(x_{n+1})\}.$ So, here the coefficients $\alpha _{n}$ are not chosen a priori, but built step by step. We prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.