Convergence of paths for pseudo-contractive mappings in Banach spaces

Abstract

Abstract. Let X be a real Banach space, let K be a closed convex subset of X, and let T , from K into X, be a pseudo-contractive mapping (i.e.. Suppose the space X has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of K enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path t → xt ∈ K, t ∈ [0, 1), defined by the equation xt = tT xt + (1 − t)x 0 is continuous and strongly converges to a fixed point of T as t → 1 − , provided that T satisfies the weakly inward condition.

Strong convergence for the modified Mann’s iteration of λ-strict pseudocontraction

“…Morales and Jung [13], in 2000, proved the following behavior for pseudocontractive mappings. Also see Song and Chen [17,18] for more details.…”

“…Lemma 2.3. ( [13,17,18]) Let C be a nonempty, closed and convex subset of a uniformly smooth Banach space E, and let T : C → C be a continuous pseudocontractive mapping with F (T ) = ∅. Suppose that for t ∈ (0, 1) and u ∈ C, x t defined by…”

Weak and Strong Convergence Theorems of Modified Projection-Type Ishikawa Iteration Scheme for Lipschitz α-Hemicontractive Mappings

“…], [? ], [20], [14], [26], [27] and the references contained in them). In [24], Voluhan introduced the modified projection-type Ishikawa iterative method in the following way: Let H be a Hilbert space, K nonempty, closed and convex subset of H and T : K −→ K be an L-Lipshitz pseudocontractive mapping.…”

The New Viscosity Approximation Methods for Nonexpansive Nonself-Mappings

“…Lemma 2.1. ( [6]) Let X be a real Banachspace, then for each , x y X ∈ , the following inequality holds:…”