Convergence Theorems of The Iterative Sequences for Nonexpansive Mappings
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Abstract
In this paper, we will prove the following: Let D be anonempty subset of a normed linear space X and T:DrightarrowX be a nonexpansive mapping. Let {x_n} be a sequence in Dand {t_n}, {s_n} be real sequences such that rosteritem"{(i)}" 0le t_nle t<1 andsum_{n=1}^{infty}t_n=infty,item"{(ii)}" (a) 0le s_nle 1,;s_nrightarrow 0 asnrightarrow infty and sum_{n=1}^{infty}t_ns_n<infty or(b) s_n=s for all nge 1 and s in [0,1),item"{(iii)}" x_{n+1}=(1-t_n)x_n+t_nT(s_nTx_n+(1-s_n)x_n) forall nge 1.endrosterThen, if the sequence {x_n} is bounded, thenlim_{nrightarrow infty}|x_n-Tx_n|=0.This result improves and complements a result of Deng [2].Furthermore, we will show that certain conditions on D, X andT guarantee the weak and strong convergence of the Ishikawaiterative sequence to a fixed point of T.
참고문헌 (6)
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