Krasnoselskii-Mann method for non-self mappings

Abstract

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If T : C → H is a non-self and non-expansive mapping, we can define a map h : C → R by h(x) := inf{λ ≥ 0 : λx + (1 -λ)Tx ∈ C}. Then, for a fixed x 0 ∈ C and for α 0 := max{1/2, h(x 0 )}, we define the Krasnoselskii-Mann algorithm x n+1 = α n x n + (1 -α n )Tx n , where α n+1 = max{α n , h(x n+1 )}. We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.

Halpern–Ishikawa type iterative method for approximating fixed points of non-self pseudocontractive mappings

“…Recently, Colao et al [5] extended this result of Colao and Marino [4] to a class of kstrictly pseudocontractive mappings. We observe that these results (the results obtained in [4] and [5]) provide a way forward to avoid the use of metric projection or sunny nonexpansive mapping in constructing algorithms for approximating fixed points of a more general class of non-self mappings.…”

“…In 2015, Colao and Marino [4] introduced a new searching strategy for the coefficient α n which makes the Mann algorithm well-defined for non-self mappings in the setting of a real Hilbert space H. In fact, they studied the following scheme:…”

Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings

“…To avoid this difficulty, Colao and Marino [7] studied the following iterative scheme. Let K be a nonempty, convex and closed subset of a real Hilbert space H and let T : K → H be a mapping.…”

“…Motivated by Colao and Marino [7], it is our purpose in this paper to introduce a Mann-type iterative scheme which converges weakly or strongly to a fixed point of multi-valued nonexpansive non-self mappings in real Hilbert spaces. We also introduce Mann-type iterative scheme which converges strongly to a fixed point of multi-valued quasi-nonexpansive non-self mappings in uniformly convex Banach spaces.…”

“…If, in Theorem 3.2, we assume that T is single-valued then we get Theorem 1 of Colao and Marino [7].…”