On generalized fractional operators and a gronwall type inequality with applications

Abstract

In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

Minkowski’s inequality for the AB-fractional integral operator

“…Khan [15] studied inequalities for a class of n functions by means of Saigo fractional calculus. Jarad et al [16] presented a Gronwall-type inequality for the analysis of the fractional-order Atangana-Baleanu differential equation and in [17] for generalized fractional derivatives. [18] proved some Hermite-Hadamard-type inequalities for a class of s-convex functions and studied special means.…”

Some new inequalities for generalized fractional conformable integral operators

“…Remark 5 In a similar way, we can get the inequalities for the generalized right FCIO (2) and special cases for integrals (4) and (6).…”

“…Some new inequalities involving fractional conformable integrals are found in the work of Nisar et al [18]. Adjabi et al [6] presented generalized fractional integral operators and Gronwall type inequalities with applications. In [2], Abdeljawad established a Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel.…”

Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications

“…Rahman et al [45] defined the generalized proportional Hadamard fractional integrals and established certain new integral inequalities for convex functions. In [46][47][48][49][50], certain remarkable inequalities, properties, and applications can be found.…”