Let $(X,d)$ be a complete metric space, and let $T:X \to P_{d,bd}(X)$ be a multi-valued $\varphi$-weak or generalized $\varphi$-weak contractive mapping. Then $T$ has a unique endpoint if and only if $T$ has the approximation endpoints property. Our results extend previous results given by \'Ciri\'c (1974), Nadler (1969), Daffer-Kaneko (1995), Rhoades (2001), Rouhani and Moradi (2010), Amini-Harandi (2010) and Moradi and Khojasteh (2011).