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

In this paper, we introduce a modified version of ordered partial b-metric spaces. We demonstrate a fundamental lemma for the convergence of sequences in such spaces. Using this lemma, we prove some fixed point and common fixed point results for (ψ , ϕ)-weakly contractive mappings in the setup of ordered partial b-metric spaces.Finally, examples are presented to verify the effectiveness and applicability of our main results. MSC: 47H10; 54H25

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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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In this work, the class of b-rectangular metric spaces is introduced, some fixed point results dealing with rational type contractions and almost generalized weakly contractive mappings are obtained. Certain examples are given to support the results.

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