Banach fixed point theorem for fractional integral contraction

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 $\alpha = 1$.