Picard Iteration Process For A General Class Of Contractive Mappings

Abstract

Let (E, ρ) be a metric space. Let T : E → E be a map with F(T) := {x ∈ E : T x = x} 6= ∅ such that ρ(T x, p) ≤ aρ(x, p), ∀x ∈ E, p ∈ F(T) and some a ∈ [0, 1). It is shown that the class of mappings satisfying this condition is more general than the class of contraction mappings with fixed points. Several classes of nonlinear operators studied by various authors are shown to belong to this class. Finally, it is shown that the Picard iteration process converges to the unique fixed point of T. Our theorem improves a recent result of Akewe and Olaleru (British Journal of Mathematics and Computer Science, 3(3): 437-447, 2013) and a host of other results.