LAUSR

Motivated by the $$\Psi $$Ψ-Riemann–Liouville $$(\Psi -\mathrm{RL})$$(Ψ-RL) fractional derivative and by the $$\Psi $$Ψ-Hilfer $$(\Psi -\mathrm{H})$$(Ψ-H) fractional derivative, we introduced a new fractional operator the so-called $$\Psi $$Ψ-fractional integral. We present some important results by means of theorems and in particular, that the $$\Psi $$Ψ-fractional integration operator is limited. In this sense, we discuss some examples, in particular, involving the Mittag–Leffler $$(\mathrm{M-L})$$(M-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 $$\Psi $$Ψ-fractional Volterra integral equation ($$\mathrm{VIE}$$VIE) using $$\beta $$β-distance functions.