Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators

In this paper, the initial value problem of a nonlinear differential equation with higher order Caputo type modification of the Erdélyi-Kober fractional derivatives was studied. Based on the transmutation method, the well-posedness of initial value problem of the higher order linear model was proved and an explicit solution was presented. Then some new Gronwall type inequalities involving Erdélyi-Kober fractional integral were established. By applying these results and some fixed point theorems, the existence and uniqueness of the positive solution of the nonlinear differential equation were proved. The method is applicable to the fractional differential equation with any order $\gamma \in (n-1,n]$.