S-Iterative Process for a Pair of Single Valued and Multi Valued Mappings in Banach Spaces

Naknimit Akkasriworn, Kritsana Sokhuma

Abstract

Let E be a nonempty compact convex subset of a uniformly convex
Banach space X, and t : E ! E and T : E ! KC(E) be a single valued and
a multivalued mappings , both satisfying the conditions (C). Assume in addition that Fix(t) \ Fix(T) 6= ; and Tw = {w} for all w 2 Fix(t) \ Fix(T). We prove that the sequence of the modified S-iteration method generated from an arbitrary
x0 2 E by
yn = (1 - n)xn + nzn
xn+1 = (1 - n)zn + ntyn
where zn 2 Txn and {n} , {n} are sequences of positive numbers satisfying
0 < a n, n b < 1, converges strongly to a common fixed point of t and T,
i.e., there exists x 2 E such that x = tx 2 Tx.