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

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If T:C→HT:C→H is a non-self and k-strict pseudocontractive mapping, we can define a map v:C→ℝv:C→R by v(x):=inf{λ≥0:λx+(1−λ)Tx∈C}.v(x):=inf{λ≥0:λx+(1−λ)Tx∈C}. Then, for a fixed x0∈Cx0∈C and for α0:=max{k,v(x0)},α0:=max{k,v(x0)}, we define the Krasnoselskii–Mann algorithm xn+1=αnxn+(1−αn)Txn,xn+1=αnxn+(1−αn)Txn, where αn+1=max{αn,v(xn+1)}.αn+1=max{αn,v(xn+1)}. So, here the coefficients αnαn 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.