EXISTENCE FOR A NONLINEAR INTEGRO-DIFFERENTIAL EQUATION WITH HILFER FRACTIONAL DERIVATIVE

We study the uniqueness of solutions to a new nonlinear Hilfer integro-differential equation with an initial condition and arbitrary numbers of the Riemann–Liouville fractional integral operators. Our investigation is based on an equivalent implicit integral equation in series obtained from Babenko’s approach, the multivariate Mittag-Leffler function as well as Banach’s contractive principle in a new Banach space. The technique used clearly opens up new directions for studying other types of initial or boundary value problems with different fractional derivatives and variable coefficients. An illustrative example is provided to demonstrate applications of the key theorem.