Abstract
Abstract The purpose of this article is to modify the Halpern-Mann-type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding recent results announced by many authors. MSC:47J05, 47H09, 49J25.
Highlights
Throughout this paper, we assume that E is a real Banach space with the dual E*, C is a nonempty closed convex subset of E, and J : E → E* is the normalized duality mapping defined byJ(x) = f * ∈ E* : x, f * = x = f *, x ∈ E.Let T : C → E be a nonlinear mapping; we denote by F(T) the set of fixed points of T
The results presented in the paper improve and extend the corresponding results of Kim [ ], Suzuki [ ], Xu [ ], Chang et al [, ], Cho et al [ ], Thong [ ], Buong [ ], Mann [ ], Halpern [ ], Qin et al [ ], Nakajo et al [ ] and others
If F is bounded in C, {xn} converges strongly to F x
Summary
Throughout this paper, we assume that E is a real Banach space with the dual E*, C is a nonempty closed convex subset of E, and J : E → E* is the normalized duality mapping defined by. ( ) T is said to be a ({kn})-quasi-φ-asymptotically nonexpansive semigroup if the set F = t≥ F(T(t)) is nonempty, and there exists a sequence {kn} ⊂ [ , ∞) with kn → such that the conditions (i)-(iii) and the following condition (v) are satisfied: (v) φ(p, Tn(t)x) ≤ knφ(p, x), ∀t ≥ , p ∈ F , n ≥ , x ∈ C. ( ) T is said to be a ({νn}, {μn}, ζ )-total quasi-φ-asymptotically nonexpansive semigroup if the set F = t≥ F(T(t)) is nonempty, and there exists nonnegative real sequences {νn}, {μn} with νn → , μn → (as n → ∞) and a strictly increasing continuous function ζ : [ , ∞) → [ , ∞) with ζ ( ) = such that the conditions (i)-(iii) and the following condition (vi) are satisfied: (vi) φ(p, Tn(t)x) ≤ φ(p, x) + νnζ (φ(p, x)) + μn, ∀n ≥ , x ∈ C, p ∈ F(T). Tn(t)x – Tn(t)y ≤ L(t) x – y , ∀x, y ∈ C, ∀n ≥ , t ≥
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