Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 40, pp. 1-16.

Title: Nonexistence results for fractional differential inequalities

Author: Jeffrey R. L. Webb (Univ. of Glasgow, Glasgow, UK)

Abstract: 
We prove nonexistence of global solution of fractional differential inequalities of the form
$D^{\alpha}u(t) \ge \lambda t^{\beta}|u(t)|^p$ when $p>1$ for each of the Riemann-Liouville and Caputo fractional derivatives. This is motivated by work of Laskri and Tatar (Comput. Math. Appl. (2010)) and
Shan and Lv (Filomat (2024)). The result of Laskri-Tatar was claimed to be false by  Zhang, Liu,  Wu and  Cui (J. Funct. Spaces (2017)) with a correction and counter-example. We show that the counter-example and the claims are not accurate. We use a different method to that of Laskri and Tatar, our result supports the one of Laskri and Tatar. We also improve on the result in Shan and Lv paper by considering a more general problem and giving a more precise conclusion.

Submitted March 25, 2024. Published July 30, 2024.
Math Subject Classifications: 34A08, 34A40, 45D05.
Key Words: Fractional differential equations; non-existence; Volterra integral equation.