Abstract
AbstractWe study the convergence of Ishikawa iteration process for the class of asymptotically "Equation missing"-strict pseudocontractive mappings in the intermediate sense which is not necessarily Lipschitzian. Weak convergence theorem is established. We also obtain a strong convergence theorem by using hybrid projection for this iteration process. Our results improve and extend the corresponding results announced by many others.
Highlights
Introduction and PreliminariesThroughout this paper, we always assume that H is a real Hilbert space with inner product ·, · and norm · . and → denote weak and strong convergence, respectively. ωw xn denotes the weak ω-limit set of {xn}, that is, ωw xn {x ∈ H : ∃xnj x}
Let C be a nonempty subset of a Hilbert space H and T : C → C an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn}
Let C be a nonempty subset of a Hilbert space H and T : C → C a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn}
Summary
Introduction and PreliminariesThroughout this paper, we always assume that H is a real Hilbert space with inner product ·, · and norm · . and → denote weak and strong convergence, respectively. ωw xn denotes the weak ω-limit set of {xn}, that is, ωw xn {x ∈ H : ∃xnj x}. Recall that T is said to be an asymptotically κ-strict pseudocontraction in the intermediate sense with sequence {γn} if there exist a constant κ ∈ 0, 1 and a sequence {γn} ⊂ 0, ∞ with γn → 0 as n → ∞, such that lim sup sup T nx − T ny 2 − 1 γn x − y 2 − κ I − T n x − I − T n y 2 ≤ 0.
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