One of the most intensively studied and generalized results in metric fixed point theory is Branciari’s fixed point theorem, which asserts that in a complete metric space $(M, \rho )$ , a mapping $T: M \to M$ satisfying \( \int _{0}^{\rho (Tx, Ty)} \omega (t) \, dt \leq c \int _{0}^{\rho (x, y)} \omega (t) \, dt \) for some $\omega : [0, \infty ) \to [0, \infty )$ , $c \in (0,1)$ , and all $x, y \in M$ , admits a unique fixed point $x^{*} \in M$ . This reduces to Banach fixed theorem when $\omega (t) = 1$ . We extend this to Riemann–Liouville fractional integral contractions, proving the existence of a fixed point for T under \( \frac{1}{\Gamma (\alpha )} \int _{0}^{\rho (Tx, Ty)} \bigl( \rho (Tx, Ty) - t \bigr)^{\alpha -1} \varphi (t) \, dt \leq c \cdot \frac{1}{\Gamma (\alpha )} \int _{0}^{\rho (x, y)} \bigl( \rho (x, y) - t \bigr)^{\alpha -1} \varphi (t) \, dt, \) which recovers Branciari’s integral condition for $\alpha = 1$ .
