Modified mann iterations for nonexpansive semigroups in Banach space

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)) . ., and f: C . C be a fixed contractive mapping. If {. n }, {. n }, {a n }, {b n }, {t n } satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as: {yn=.nxn+(1..n)T(tn)xn,xn=.nf(xn)+(1..n)yn. .....u0.C,vn=anun+(1+a n)T(tn)un,un+1=bnf(un)+(1.bn)vn. We prove that the approximate solutions obtained from these methods converge strongly to q . .t.0 Fix(T(t)), which is a unique solution in F to the following variational inequality: .(I.f)q,j(q.u)..0.u.F. Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133.2136 (2002)], and Kim and XU [Nonlear Analysis, 61, 51.60 (2005)] and Chen and He [Appl. Math. Lett., 20, 751.757 (2007)].