Nonlinear integral inequalities involving <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi mathvariant="normal">Ψ</a:mi></a:math>-Hilfer fractional integrals and iterated fractional integrals, with applications to <c:math xmlns:c="http://w

In this paper, modifications of the well-known desingularization method proposed by the first author are used to derive nonlinear versions of the Henry--Gronwall inequality for integral inequalities with the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi mathvariant="normal">Ψ</a:mi> </a:math>-Hilfer fractional integral, as well as nonlinear integral inequalities with iterated <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi mathvariant="normal">Ψ</c:mi> </c:math>-Hilfer fractional integrals. The results are applied to obtain bounds for solutions of initial value problems for nonlinear <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi mathvariant="normal">Ψ</e:mi> </e:math>-Caputo fractional differential equations, and sufficient conditions for the non-existence of blowing-up solutions.