Let (X,d) be a metric space and T: X → X be a mapping. In this work, we introduce the mapping ζ:[0,1)x[0,1) → R, called the simulation function and the notion of Z-contraction with respect to Z which generalize the Banach contraction principle and unify several known types of contractions involving the combination of d(Tx,Ty) and d(x,y). The related fixed point theorems are also proved.

Several fixed point and common fixed point theorems are obtained in the setting of metric-type spaces introduced by M. A. Khamsi in 2010.

Suzuki’s fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861-1869, 2008) and (Suzuki, Nonlinear Anal. 71:5313-5317, 2009) are extended to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.