Integral representations of scalar delay non-instantaneous impulsive Riemann-Liouville fractional differential equations

Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and non-instantaneous impulses are studied where initial conditions and impulsive conditions are set up in appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the case of a fixed lower limit of the fractional derivative and the case of the changeable lower limit at each impulsive time and integral representations of the solutions are obtained. These integral presentations are used to study the existence on finite time intervals of various types of initial value problems.