Fractional Integral Inequalities via Hadamard’s Fractional Integral

We establish new fractional integral inequalities, via Hadamard’s fractional integral. Several new integral inequalities are obtained, including a Grüss type Hadamard fractional integral inequality, by using Young and weighted AM-GM inequalities. Many special cases are also discussed

A Gronwall inequality via the generalized proportional fractional derivative with applications

Abstract In this paper, we provide a new version for the Gronwall inequality in the frame of the generalized proportional fractional derivative. Prior to the main results, we introduce the generalized proportional fractional derivative and expose some of its features. As an application, we accommodate the newly defined derivative to prove the uniqueness and obtain a bound in terms of Mittag-Leffler function for the solutions of a nonlinear delay proportional fractional system. An example is presented to demonstrate the applicability of the theory

Some New Riemann-Liouville Fractional Integral Inequalities

In this paper, some new fractional integral inequalities are established

Existence of Solutions for Second-Order Impulsive q-Difference Equations with Integral Boundary Conditions

Recently the authors introduced in [1] the notions of qk-derivative and qk-integral of a function on finite intervals. As applications existence and uniqueness results for initial value problems for first and second order impulsive qk-difference equations was proved. In this paper, we study the existence and uniqueness of solutions for a boundary value problem of nonlinear second-order impulsive qk-difference equations with integral boundary conditions. Two results are obtained by applying Banach contraction principle and Leray-Schauder’s Nonlinear Alternative. Some examples are presented to illustrate the results