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, $${}^{\rho }{\mathcal{D}}_a^{\alpha }f(x)=\frac{{\rho }^{\alpha -n+1}}{\mathrm{\Gamma }(n-\alpha )}(x^{1-\rho }\frac{d}{dx}{)}^n{\int }_a^x\frac{{\tau }^{\rho -1}f(\tau )}{(x^{\rho }-{\tau }^{\rho }{)}^{\alpha -n+1}}d\tau$$ 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.