In the paper, we revisited the well-known fixed point theorem of Kannan under the aspect of interpolation. By using the interpolation notion, we propose a new Kannan type contraction to maximize the rate of convergence.

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.

The present paper aims to define three new notions: ${\mathsf{\Theta }}_e$ -contraction, a Hardy–Rogers-type $\mathsf{\Theta }$ -contraction, and an interpolative $\mathsf{\Theta }$ -contraction in the framework of extended b-metric space. Further, some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique.

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.