Convergence to common fixed point of nonexpansive semigroups

摘要

Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, C be closed convex subset of E, S={T(s):s⩾0} be a nonexpansive semigroup on C such that the set of common fixed points of {T(s):s⩾0} is nonempty. Let f:C→C be a contraction, {αn},{βn},{tn} be real sequences such that 0<αn,βn⩽1,limn→∞αn=0,limn→∞βn=0 and limn→∞tn=∞,y0∈C. In this paper, we show that the two iterative sequence as follows: xn=αnf(xn)+(1-αn)1tn∫0tnT(s)xnds,yn+1=βnf(yn)+(1-βn)1tn∫0tnT(s)yndsconverge strongly to a common fixed point of {T(s):s⩾0} which solves some variational inequality when {αn},{βn} satisfy some appropriate conditions.