Let (X, d) be a metric space and T : X → X be a mapping. In this work, we introduce the mapping ζ : [0, ∞) × [0, ∞) → R, called the simulation function and the notion of Z-contraction with respect to ζ 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.

The concept of rectangular b-metric space is introduced as a generalization of metric space, rectangular metric space and b-metric space. An analogue of Banach contraction principle and Kannan's fixed point theorem is proved in this space. Our result generalizes many known results in fixed point theory.

In this paper, we first present some elementary results concerning cone metric spaces over Banach algebras. Next, by using these results and the related ones about c-sequence on cone metric spaces we obtain some new fixed point theorems for the generalized Lipschitz mappings on cone metric spaces over Banach algebras without the assumption of normality. As a consequence, our main results improve and generalize the corresponding results in the recent paper by Liu and Xu (Fixed Point Theory Appl. 2013:320, 2013). MSC: 54H25; 47H10

Suzuki's fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861Soc. 136: -1869Soc. 136: , 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. MSC: 47H10; 54H25

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