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

#### 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.

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.

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