A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

• Published in: Axioms Vol. 13; no. 5; p. 301
• Main Authors: Medved’, Milan, Pospíšil, Michal, Brestovanská, Eva
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.05.2024
• Subjects: blow-up solution; Derivatives; Differential equations; Differential equations, Nonlinear; Fractional calculus; generalized Henry–Gronwall inequality; Inequalities (Mathematics); Inequality; Integrals; Partial differential equations; tempered Ψ–Caputo fractional derivative; tempered Ψ–Hilfer fractional integral inequality; tempered Ψ–Riemann–Liouville fractional derivative; Slovakia
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms13050301

In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n−1<α<n∈N are introduced for Cn−1–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ–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 Ψ–Caputo fractional differential equations are proved. Illustrative examples are given.