T-Stability of Picard Iteration in Metric Spaces

Let X, d be a complete metric space and T a self-map of X. Let xn 1 f T, xn be some iteration procedure. Suppose that F T , the fixed point set of T , is nonempty and that xn converges to a point q ∈ F T . Let {yn} ⊂ X and define n d yn 1, f T, yn . If lim n 0 implies that limyn q, then the iteration procedure xn 1 f T, xn is said to be T -stable. Without loss of generality, we may assume that {yn} is bounded, for if {yn} is not bounded, then it cannot possibly converge. If these conditions hold for xn 1 Txn, that is, Picard’s iteration, then we will say that Picard’s iteration is T -stable. We will obtain sufficient conditions that Picard’s iteration is T -stable for an arbitrary self-map, and then demonstrate that a number of contractive conditions are Picard T -stable. We will need the following lemma from 1 .