Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space

Abstract

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E ∗ , and C be a nonempty closed convex subset of E . Let { T ( t ) : t ≥ 0 } be a nonexpansive semigroup on C such that F ≔ ⋂ t ≥ 0 Fix ( T ( t ) ) ≠ 0̸ , and f : C → C be a fixed contractive mapping. When { α n } , { β n } , { t n } satisfy some appropriate conditions, the two iterative processes given as follows: x n = α n f ( x n ) + ( 1 − α n ) T ( t n ) x n , for n ∈ N . y n + 1 = β n f ( y n ) + ( 1 − β n ) T ( t n ) y n , for n ∈ N . converge strongly to q ∈ ⋂ t ≥ 0 Fix ( T ( t ) ) , which is the unique solution in F to the following variational inequality: 〈 ( I − f ) q , j ( q − u ) 〉 ≤ 0 ∀ u ∈ F . Our results extend and improve corresponding ones of Suzuki [T. Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131 (2002) 2133–2136] and Xu [H.K. Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72 (2005) 371–379] and Chen [R. Chen, Strong convergence to common fixed point of nonexpansive semigroups in Banach space, Comput. Math. Appl. http://www.sciencedirect.com/science/journal/03770427].