A new approach to the study of fixed point theory for simulation functions

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.