In this paper, we establish a new fixed point result for Hardy-Rogers type contractions in double controlled metric-like spaces. Our result generalizes many important theorems in the literature. We will provide an example to illustrate our results.

In this publication, our objective was to introduce and establish the concepts of ${\kappa }_{G_m}$-contraction and generalized $(\alpha ,{\kappa }_{G_m})$-contraction in complete $G_m$-metric spaces, which led to the discovery of novel fixed points, coincidence points, and common fixed points. Additionally, we demonstrated the usefulness of our main results by applying it to the investigation of the integral equation. Also, we presenting a noteworthy example demonstrating the practicality of our primary hypothesis.