Existence and Uniqueness results for a class of Generalized Fractional Differential Equations

The author (Bull. Math. Anal. App. 6(4)(2014):1-15), introduced a new fractional derivative, $${}^\rho \mathcal{D}_a^\alpha f (x) = \frac{\rho^{\alpha-n+1}}{\Gamma({n-\alpha})} \, \bigg(x^{1-\rho} \,\frac{d}{dx}\bigg)^n \int^x_a \frac{\tau^{\rho-1} f(\tau)}{(x^\rho - \tau^\rho)^{\alpha-n+1}}\, d\tau$$ which generalizes two familiar fractional derivatives, namely, the Riemann-Liouville and the Hadamard fractional derivatives to a single form. In this paper, we derive the existence and uniqueness results for a generalized fractional differential equation governed by the fractional derivative in question.