In this paper, first, we give the separation theorem which is an extension of the separation theorem due to Jachymski and Jó´zwik (J. Math. Anal. Appl. 300:147-159, 2004). Then, by using this and the related results, we prove that two generalized weak contraction multi-valued mappings have a unique common endpoint if and only if either they have the usual approximate endpoint property or they have the common approximate strict fixed point property. This result is an extension and correct version of the main result given by Khojasteh and Rakoˇcevi´c (Appl. Math. Lett. 25:289-293, 2012).
