Abstract
We prove convergence theorems of modified Ishikawa iterative sequence for two nonexpansive semigroups in Hilbert spaces by the two hybrid methods. Our results improve and extend the corresponding results announced by Saejung (2008) and some others.
Highlights
Let C be a subset of real Hilbert spaces H with the inner product ·, · and the norm ·
We prove convergence theorems of modified Ishikawa iterative sequence for two nonexpansive semigroups in Hilbert spaces by the two hybrid methods
We denote by F T the set of fixed points of T, that is, F T {x ∈ C : x T x}
Summary
Let C be a subset of real Hilbert spaces H with the inner product ·, · and the norm ·. Let {T t : t ≥ 0} be a family of mappings from a subset C of H into itself We call it a nonexpansive semigroup on C if the following conditions are satisfied:. In 2007, Chen and He 6 studied the viscosity approximation process for a nonexpansive semigroup and prove another strong convergence theorem for a nonexpansive semigroup in Banach spaces, which is defined by xn 1 αnf xn 1 − αn T tn xn, ∀n ∈ N, 1.5 where f : C → C is a fixed contractive mapping. He and Chen 7 is proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. This result extends and improves the result of Saejung 8 and some others
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