Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space

TLDR: In this article, it was shown that any fixed point of a Lipschitzian, strictly pseudocontractive mapping T on a closed, convex subset K of a Banach space X is necessarily unique, and may be norm approximated by an iterative procedure.

Abstract: We show that any fixed point of a Lipschitzian, strictly pseudocontractive mapping T on a closed, convex subset K of a Banach space X is necessarily unique, and may be norm approximated by an iterative procedure. Our argument provides a convergence rate estimate and removes the boundedness assumption on K, generalizing theorems of Liu. Let (X, ‖ · ‖) be a Banach space. Let K be a non-empty closed, convex subset of X and T : K → K. We will assume that T is Lipschitzian, i.e. there exists L > 0 such that ‖T (x)− T (y)‖ ≤ L‖x− y‖, for all x, y ∈ K. Of course, we are most interested in the case where L ≥ 1. We also assume that T is strictly pseudocontractive. Following Liu [1] this may be stated as: there exists k ∈ (0, 1) for which ‖x− y‖ ≤ ‖x− y + r[(I − T − kI)x− (I − T − kI)y]‖, for all r > 0 and all x, y ∈ K. Throughout, N will denote the set of positive integers. The following results generalize Liu [1, Theorems 1 and 2], because we remove the assumption that K is bounded and we provide a general convergence rate estimate. We note in passing, however, that the proof of Theorem 2 of Liu [1] does not use the stated boundedness assumption. Our results still extend this enhanced version of Liu [1, Theorem 2], by improving the convergence rate estimate. Theorem 1. Let (X, ‖ · ‖),K, T, L and k be as described above. Let q ∈ K be a fixed point of T . Suppose that (αn)n∈N is a sequence in (0, 1] such that for some η ∈ (0, k), for all n ∈ N, αn ≤ k − η (L+ 1)(L+ 2− k) ; while ∞ ∑