In [Nonlinear Anal. 68, 2303–2308 (2008; Zbl 1151.46006)], the author proved that \((\ell_1,\|\cdot\|_1)\) could be renormed with an equivalent norm \(|||\cdot|||\) so that \((\ell_1, |||\cdot|||)\) has the fixed point property for nonexpansive self-mappings of closed, bounded, convex subsets of \(\ell_1\). In this article, the author uses the ideas introduced in the previous article to prove that a class of renormings of \(\ell_1\) have the fixed point property. In particular, for \(k\in{\mathbb{N}}\), let \(P_k\) denote the natural projections on \(\ell_1\) and let \(|||\cdot|||\) be an equivalent norm on \(\ell_1\) satisfying the following four conditions: 6mm

(1)

The unit ball of \((\ell_1, |||\cdot|||)\) is \(\sigma(\ell_1,c_0)\)-closed.

(2)

For any \(k\in{\mathbb{N}}\), \(\| I-P_k \| = 1\).

(3)

There exist \(\alpha>4\) and a decreasing sequence \((\alpha_k)\) in \((0,1)\) such that, for any normalized block basis \((f_n)\) of \((\ell_1, |||\cdot|||)\) and \(x\in\ell_1\) with \(P_{k-1}(x)=x\) and \(|||x|||<\alpha_k\), \(\limsup_n |||f_n+x||| \leq 1 + |||x|||/\alpha\).

(4)

There exist strictly decreasing sequences \((\beta_k)\) and \((\gamma_k)\) with \(\lim_k \beta_k = 0\) and \(\lim_k \gamma_k = 1\) such that, for any normalized block basis \((f_n)\) of \((\ell_1, |||\cdot|||)\) and any \(x\in\ell_1\) with \(P_{k-1}(x)=x\), \(\liminf_n |||f_n+x|||\geq 1 - \beta_k + |||x|||/\gamma_k\).

Then \((\ell_1, |||\cdot|||)\) has the fixed point property.