Convergence to common fixed point of nonexpansive semigroups

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Abstract

Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, C be closed convex subset of E, S={T(s):s0} be a nonexpansive semigroup on C such that the set of common fixed points of {T(s):s0} is nonempty. Let f:CC be a contraction, {αn},{βn},{tn} be real sequences such that 0<αn,βn1,limnαn=0,limnβn=0 and limntn=,y0C. In this paper, we show that the two iterative sequence as follows: xn=αnf(xn)+(1-αn)1tn0tnT(s)xnds,yn+1=βnf(yn)+(1-βn)1tn0tnT(s)yndsconverge strongly to a common fixed point of {T(s):s0} which solves some variational inequality when {αn},{βn} satisfy some appropriate conditions.

Keywords

Nonexpansive semigroup
Uniformly convex Banach space
Uniformly Gâteaux differentiable
Common fixed point

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This work is supported by the National Science Foundation of China, Grant (10471033) and Grant (10271011).