Positive solution of nonlinear fractional differential equations with Caputo-like counterpart hyper-Bessel operators

By studying a weakly singular integral whose kernel involves Mittag-Leffler functions, we obtain some new Gronwall-type integral inequalities. Applying these inequalities and fixed point theorems, existence and uniqueness of positive solution of initial value problem to nonlinear fractional differential equation with Caputo-like counterpart hyper-Bessel operators are established.