Fixed Point Theory and Applications (Jan 2011)
Some results on a general iterative method for <it>k</it>-strictly pseudo-contractive mappings
Abstract
Abstract Let H be a Hilbert space, C be a closed convex subset of H such that C ± C ⊂ C, and T : C → H be a k-strictly pseudo-contractive mapping with F(T) ≠ ∅ for some 0 ≤ k < 1. Let F : C → C be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0 and f : C → C be a contraction with the contractive constant α ∈ (0, 1). Let , and τ < 1. Let {αn } and {βn } be sequences in (0, 1). It is proved that under appropriate control conditions on {αn } and {βn }, the sequence {xn } generated by the iterative scheme x n+1 = α n γf(x n ) + β n x n + ((1 - β n )I - α n μF)P C Sx n , where S : C → H is a mapping defined by Sx = kx + (1 - k)Tx and PC is the metric projection of H onto C, converges strongly to q ∈ F(T), which solves the variational inequality 〈μFq - γf(q), q - p〉 ≤ 0 for p ∈ F(T). MSC: 47H09, 47H05, 47H10, 47J25, 49M05
