Let

$E$ be a reflexive and strictly convex and smooth Banach space,

$C$ a closed convex subset of

$E$,

$T$ a selfmap of

$C$ with

$F\left(T\right)$ denoting the fixed point set of

$T$. A point

$p$ in

$C$ is said to be an asymptotic fixed point of

$T$ if

$C$ contains a sequence

$\left\{{x}_{n}\right\}$ that converges weakly to

$p$ and is such that the strong limit of

$(T{x}_{n}-{x}_{n})$ is 0. The set of asymptotic fixed points is denoted by

$\widehat{F}\left(T\right)$.

$T$ is called relatively nonexpansive if

$F\left(T\right)=\widehat{F}\left(T\right)$ and

$\phi (p,Tx)\le \phi (p,x)$ for each

$p\in F\left(T\right)$ and

$x\in C$, where

$\phi (x,y):={\parallel x\parallel}^{2}-2\langle x,j\left(y\right)\rangle +{\parallel y\parallel}^{2}$.

$T$ is called relatively asymptotically nonexpansive if

$F\left(T\right)=\widehat{F}\left(T\right)$ and

$\phi (x,{T}^{n}x)\le {k}^{n}\phi (p,x)$ for each

$x\in C$,

$p\in F\left(T\right)$. With

$E$ a uniformly convex and uniformly smooth Banach space, and using a complicated iteration scheme involving duality maps and their inverses, the authors obtain the strong convergence to a fixed point of

$T$ for

$T$ either strictly relatively nonexpansive or relatively asymptotically nonexpansive, provided that

$F\left(T\right)\ne \varnothing $.