In this article, we introduce a new extension of b-metric spaces, called controlled metric type spaces, by employing a control function $\alpha (x,y)$ of the right-hand side of the b-triangle inequality. Namely, the triangle inequality in the new defined extension will have the form, $d(x,y)\le \alpha (x,z)d(x,z)+\alpha (z,y)d(z,y),\mathrm{for}\mathrm{all}x,y,z\in X.$ Examples of controlled metric type spaces that are not extended b-metric spaces in the sense of Kamran et al. are given to show that our extension is different. A Banach contraction principle on controlled metric type spaces and an example are given to illustrate the usefulness of the structure of the new extension.

In this article, we give a fixed point theorem for set-valued quasi-contraction maps in b-metric spaces. This theorem extends, unifies and generalizes several well known comparable results in the existing literature.

By giving a counter-example, we point out a gap in the paper by Karapinar (Adv. Theory Nonlinear Anal. Its Appl. 2018, 2, 85–87) where the given fixed point may be not unique and we present the corrected version. We also state the celebrated fixed point theorem of Reich–Rus–Ćirić in the framework of complete partial metric spaces, by taking the interpolation theory into account. Some examples are provided where the main result in papers by Reich (Can. Math. Bull. 1971, 14, 121–124; Boll. Unione Mat. Ital. 1972, 4, 26–42 and Boll. Unione Mat. Ital. 1971, 4, 1–11.) is not applicable.

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions $\alpha (x,y)$ and $\mu (x,y)$ on the right-hand side of the $b-$ triangle inequality, that is, $$d(x,y)\le \alpha (x,z)d(x,z)+\mu (z,y)d(z,y),\mathrm{for}\mathrm{all}x,y,z\in X.$$ Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and $\varphi$ -nonlinear type contractions in the setting of double controlled metric type spaces.

By using an interpolative approach, we recognize the Hardy-Rogers fixed point theorem in the class of metric spaces. The obtained result is supported by some examples. We also give the partial metric case, according to our result.