Analysis of mixed differential equations involving the Caputo and Riemann-Liouville fractional derivatives

In this paper, we investigate the existence of positive solution as well as the uniqueness results for a fractional order differential equation involving the Caputo fractional derivative and the Riemann-Liouville fractional derivative. First of all, we show the existence and uniqueness of the positive solution by means of the fixed point theory, namely, Banach’s contraction Principle. Second of all, we convert the posed problem to a sum of two integral operators, then we apply Kronelskii’s fixed point theorem to conclude the existence of nontrivial solutions. As applications, we present examples for the demonstration of our main results.