Weak convergence to a fixed point of the sequence of Mann type iterates (Q1331004)

The authors prove the following theorem: Let \(T\) be a quasi-nonexpansive self-mapping of a closed convex subset of a uniformly convex Banach space satisfying Opial's condition with \(I-T\) demiclosed with respect to zero. Then the sequence \(\{x_ n\}_{n=1}^ \infty\) defined by \(x_{n+1}= (1- \alpha_ n) x_ n+ \alpha_ n Tx_ n\) converges weakly to some fixed point of \(T\). A similar result is obtained for continuous generalized nonexpansive mappings.