Implicit Mann fixed point iterations for pseudo-contractive mappings

Abstract

Let K be a compact convex subset of a real Hilbert space H and T : K → K a continuous hemi-contractive map. Let { a n } , { b n } and { c n } be real sequences in [0, 1] such that a n + b n + c n = 1 , and { u n } and { v n } be sequences in K . In this paper we prove that, if { b n } , { c n } and { v n } satisfy some appropriate conditions, then for arbitrary x 0 ∈ K , the sequence { x n } defined iteratively by x n = a n x n − 1 + b n T v n + c n u n ; n ≥ 1 , converges strongly to a fixed point of T .