The fixed point property and the Opial condition on tree-like Banach spaces

Abstract

We introduce some new tree-like Banach spaces, belonging to the class of separable Banach spaces not containing $\ell_1$ with non-separable dual, each one of which satisfies the following: $(1)$~the space has the fixed point property and $(2)$~the space does not satisfy the Opial condition. In addition, one of these spaces contains subspaces isomorphic to $c_0$, whose Banach-Mazur distance from $c_0$ becomes arbitrarily large.