New iteration process for a general class of contractive mappings

Abstract

Let K be a closed convex subset of X, and let T : K → K be a self-mapping with the set F_T of fixed points such that ‖Tx − ρ‖ ≤ δ‖x − ρ‖ for all x ∈ K, ρ ∈ F_T and some δ ∈ (0, 1). We introduce a new iteration process called Picard-hybrid iteration and show that this iteration process converges to the unique fixed point of T. It is also shown that our iteration process converges more rapidly than the Picard–Mann and Picard iteration processes. Our result improves a recent result of S. H. Khan and some other results.