$\rho-F$-contraction fixed point theorem

In this paper, we study the question of conditions for the existence and uniqueness of a fixed point of a mapping over a complete metric space. We first discuss the concepts of $F$-contraction and $F^*$-contraction in fixed point theory. These concepts, developed respectively by Wardowski and Piri with Kumam, have catalyzed significant research in various metric spaces. We then propose a generalization of these concepts, $\rho-F$-contraction and $\rho-F^*$-contraction, and demonstrate its effectiveness in ensuring the existence and uniqueness of fixed points. This new approach provides greater flexibility by including a function $\rho$ that modulates the contraction, extending the applicability of $F$- and $F^*$-contractions. We conclude the paper with an example of a mapping that is a $\rho-F$-contraction and a $\rho-F^*$-contraction, respectively, and has a unique fixed point. However, this mapping does not satisfy the conditions of Wardowski and the conditions of Piri and Kumam.