Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces
Yisheng Song, Poom Kumam, Yeol Je Cho · Fixed Point Theory and Applications · 2016
Abstract In this paper, we prove some existence theorems of fixed points of a monotone nonexpansive mappingTin a Banach spaceEwith the partial order ‘≤’, where a such mapping may be discontinuous. In particular, in finite dimensional spaces, such a mappingThas a fixed point inEif and only if the sequence $\{T^{n}0\}$ {Tn0} is bounded inE. In order to find a fixed point of such a mappingT, we prove the weak convergence of the Mann iteration scheme under the condition $\sum_{n=1}^{\infty}\beta_{n}(1-\beta_{n})=\infty$ ∑n=1∞βn(1−βn)=∞ , which entails $\beta _{n}=\frac{1}{n+1}$ βn=1n+1 as a special case.