We first establish some existence results concerning approximate coincidence point properties and approximate fixed point properties for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde‐Berinde′s fixed point theorem, Mizoguchi‐Takahashi′s fixed point theorem, and some well‐known results in the literature.