In this paper, some nonlinear Gronwall–Bellman type inequalities are established. Then, the obtained results are applied to study the Hyers–Ulam stability of a fractional differential equation and the boundedness of solutions to an integral equation, respectively.

Recommended by Narendra Kumar Govil We investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivatives , and some well-known integral operators. In special cases, the results presented in this paper provide new approaches to several previously known results.

We prove non local Hardy inequalities on Carnot groups and Riemannian manifolds, relying on integral representations of fractional Sobolev norms. AMS numbers 2000: Primary: 46E35. Secondary: 35R11, 42B35, 58J35.

We develop a stochastic calculus for the fractional Brownian motion with Hurst parameter H > 2 using the techniques of the Malliavin calclulus. We establish estimates in Lp, maximal inequalities and a continuity criterion for the stochastic integral. Finally, we derive an Itô’s formula for integral processes.

The aim of the present paper is to establish some new fractional q-integral inequalities on the specific time scale: Tt0 = {t : t = t0q, n ∈ N} ∪ {0}, where t0 ∈ R, and 0 < q < 1.