Abstract
AbstractThe purpose of this paper is to introduce and study the strong convergence problem of the explicit iteration process for a Lipschitzian and demicontraction semigroups in arbitrary Banach spaces. The main results presented in this paper not only extend and improve some recent results announced by many authors, but also give an affirmative answer for the open questions raised by Suzuki (2003) and Xu (2005).
Highlights
Introduction and PreliminariesThroughout this paper, we assume that E is a real Banach space, E∗ is the dual space of E, C is a nonempty closed convex subset of E, R is the set of nonnegative real numbers, and J : E → 2E∗ is the normalized duality mapping defined byJ x f ∈ E∗ : x, f x · f, x f1.1 for all x ∈ E
1 The one-parameter family T : {T t : t ≥ 0} of mappings from C into itself is called a nonexpansive semigroup if the following conditions are satisfied: Fixed Point Theory and Applications a T 0 x x for each x ∈ C; bTtsxTtTsx for any t, s ∈ R and x ∈ C; c for any x ∈ C, the mapping t → T t x is continuous; d for any t ∈ R, T t is a nonexpansive mapping on C, that is, for any x, y ∈ C, T t x−T t y ≤ x−y 1.2 for any t > 0
3 A pseudocontraction semigroup T : {T t : t ≥ 0} of mappings from C into itself is said to be Lipschitzian if the conditions a – c, e, and the following condition f are satisfied: f there exists a bounded measurable function L : 0, ∞ → 0, ∞ such that, for any x, y ∈ C, T t x−T t y ≤L t x−y
Summary
Introduction and PreliminariesThroughout this paper, we assume that E is a real Banach space, E∗ is the dual space of E, C is a nonempty closed convex subset of E, R is the set of nonnegative real numbers, and J : E → 2E∗ is the normalized duality mapping defined byJ x f ∈ E∗ : x, f x · f , x f1.1 for all x ∈ E.
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