In this article, we introduce a new extension to $J-$metric spaces, called $C_J-$metric spaces, where $\theta$ is the controlled function in the triangle inequality. We prove some fixed point results in this new type of metric space. In addition, we present some applications to systems of linear equations to illustrate our results.

In this paper, we introduced the strong-controlled Branciari $b$-distance, a generalized metric structure designed to consolidate disparate fixed point theorems scattered throughout existing literature. Through illustrative examples, we demonstrated how this framework subsumes and extends previous results. We further established the applicability of our theoretical developments by presenting a concrete problem whose solution leverages the properties of strong-controlled Branciari $b$-distance spaces.