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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.
Khojasteh et al. (Thu,) studied this question.