Strong convergence theorems of k-strict pseudo-contractions in Hilbert spaces

Abstract

Let K be a nonempty closed convex subset of a real Hilbert space H such that K ± K ⊂ K, T: K → H a k-strict pseudo-contraction for some 0 ⩽ k < 1 such that F(T) = {x ∈ K: x = Tx} ≠ \( \not 0 \). Consider the following iterative algorithm given by $$ \forall x_1 \in K,x_{n + 1} = \alpha _n \gamma f(x_n ) + \beta _n x_n + ((1 - \beta _n )I - \alpha _n A)P_K Sx_{n,} n \geqslant 1, $$ where S: K → H is defined by Sx = kx + (1 − k)Tx, PK is the metric projection of H onto K, A is a strongly positive linear bounded self-adjoint operator, f is a contraction. It is proved that the sequence {xn} generated by the above iterative algorithm converges strongly to a fixed point of T, which solves a variational inequality related to the linear operator A. Our results improve and extend the results announced by many others.