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

Let EE be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from EE to E∗E∗, and CC be a nonempty closed convex subset of EE. Let {T(t):t≥0}{T(t):t≥0} be a nonexpansive semigroup on CC such that F≔⋂t≥0Fix(T(t))≠0̸, and f:C→Cf:C→C be a fixed contractive mapping. When {αn},{βn},{tn}{αn},{βn},{tn} satisfy some appropriate conditions, the two iterative processes given as follows: xn=αnf(xn)+(1−αn)T(tn)xn,for n∈N.yn+1=βnf(yn)+(1−βn)T(tn)yn,for n∈N. converge strongly to q∈⋂t≥0Fix(T(t)), which is the unique solution in FF 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].