Eva Brestovanská

In this paper, the tempered $\mathsf{\Psi }$–Riemann–Liouville fractional derivative and the tempered $\mathsf{\Psi }$–Caputo fractional derivative of order $n-1<\alpha <n\in \mathbb{N}$ are introduced for $C^{n-1}$–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered $\mathsf{\Psi }$–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered $\mathsf{\Psi }$–Caputo fractional differential equations are proved. Illustrative examples are given.

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered $\Psi$-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered $\Psi$-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered $\Psi$-Caputo fractional derivative of order $0<\alpha <1$, where the right side may depend on time, the solution, or its tempered $\Psi$-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the $\Psi$-exponential stability of trivial solutions are proven for these kinds of differential equations.