I know every contraction is a BW map, so I suspect the BW class of mappings is strictly larger. Can someone help me construct an example of a mapping that is not a contraction but is a BW map?
Edit: Let $X$ be a complete metric space, $P = \{d(x, y) : x, y \in X\}$. Then $T : X \to X$ is a BW mapping if there is a function $\phi : P \to \mathbb{R^+}$ such that
i) $\phi(t) < t$ for all $t \in P-\{0\}$ and $\phi(0)=0$,
ii) $d(Tx, Ty) \le \phi(d(x, y))$ for all $x, y \in X$,
iii) $\phi$ is an upper semi continuous function.
$T : X \to X$ is a contraction when $d(Tx, Ty) \le k d(x, y)$, for all $x, y \in X$ and some fixed $k \in [0, 1)$.