$c_0$ can be renormed to have the fixed point property for affine nonexpansive mappings

Abstract

P.K. Lin gave the first example of a non-reflexive Banach space
$(X,\|\cdot\|)$ with the fixed point property for nonexpansive mappings
and showed this fact for $(\ell^1, \|\cdot\|_1)$ with the equivalent norm
$\|\cdot\|$ given by
$$\|x\| = \sup_{k\in\N}\,\frac{8^k}{1+8^k}\,\sum_{n=k}^{\ii}\,|x_n|,\
\text{for all} \ x=(x_n)_{n\in\N}\in \ell^1\ .$$
We wonder $(c_0,\|\cdot\|_{\ii})$ analogue of P.K. Lin's work and we
give positive answer if functions are affine nonexpansive.
In our work, for $x={(\xi _{k})}_{k}\in c_0$, we define
\beq
\vvvert x \vvvert:&=&\lim_{p \rightarrow \ii}\sup_{k\in \N}\gamma_k
{\left(\mathop{\sum }\limits_{j=k}^{\ii}\frac{{\left\vert \xi
_{j}\right\vert}^p}{2^j} \right)}^{\frac{1}{p}}\;\text{
where}\;\gamma_k\uparrow_k 1,\, \gamma_k\,\text{is strictly increasing},
\eeq
then we prove that when $\gamma_1>\frac{2}{3}$, $\left(
c_{0},\; \vvvert\cdot \vvvert \right) $ has the fixed point property
for affine ${\vvvert\cdot\vvvert}$-nonexpansive self-mappings.

Next, we generalize this result and show that if $\gamma_k\uparrow_k 1,\, \gamma_k\,\text{is strictly increasing}$, $\gamma_1>\frac{2}{3}$ and $\rho(\cdot)$ is an equivalent norm to the usual norm on $c_0$ such that $$\limsup_{n} \rho\left(\frac{1}{n}\sum_{m=1}^{n}\;x_m+x\right)=\limsup_{n} \rho\left(\frac{1}{n}\sum_{m=1}^{n}\;x_m\right)+\rho(x)$$ for every weakly null sequence ${(x_n)}_{n}$ and for all $x\in c_0$, then for every $\lambda>0$, $\;c_0$ with the norm ${\Vert\cdot\Vert}_\rho=\rho(\cdot)+\lambda\vvvert\cdot\vvvert$ has the FPP for affine ${\Vert\cdot\Vert}_\rho$-nonexpansive self-mappings.