On the $\mathrm{\Psi }-$fractional integral and applications

Abstract

Motivated by the $\mathrm{\Psi }$-Riemann-Liouville $(\mathrm{\Psi }-\mathrm{RL})$ fractional derivative and by the $\mathrm{\Psi }$-Hilfer $(\mathrm{\Psi }-\mathrm{H})$ fractional derivative, we introduced a new fractional operator the so-called $\mathrm{\Psi }-$fractional integral. We present some important results by means of theorems and in particular, that the $\mathrm{\Psi }-$fractional integration operator is limited. In this sense, we discuss some examples, in particular, involving the Mittag-Leffler $(\mathrm{M}-\mathrm{L})$ function, of paramount importance in the solution of population growth problem, as approached. On the other hand, we realize a brief discussion on the uniqueness of nonlinear $\Psi$-fractional Volterra integral equation ($\mathrm{VIE}$) using $\beta -$distance functions.

27 pages, 3 figures