Let

$E$ be a uniformly convex Banach space,

$C$ a closed convex subset of

$E$ and

${\left\{{T}_{i}\right\}}_{i=1,N}$ a finite family of uniformly

$L$-Lipschitzian asymptotically quasi-nonexpansive self mappings of

$C$. Under some additional assumptions, it is proven that the sequence

$\left\{{x}_{n}\right\}$ defined by

${x}_{n}={\alpha}_{n}{x}_{n-1}+(1-{\alpha}_{n}){T}_{i}^{k}{x}_{n},\phantom{\rule{4pt}{0ex}}n\ge 1$, where

$n=(k-1)N+i$,

$i\in \{1,2,\xb7\xb7\xb7,N\}$, and

$\left\{{\alpha}_{n}\right\}$ is a real sequence in

$(0,1)$, converges strongly to a common fixed point of the mappings

${\left\{{T}_{i}\right\}}_{i=1,N}$ provided that one mapping in the family is semi-compact. Other related results are also considered.