Abstract
We consider the well-known classes of functions mathcal{U}_{1}(mathbf{v},mathtt{k}) and mathcal{U}_{2}(mathbf{v},mathtt{k}), and those of Opial inequalities defined on these classes. In view of these indices, we establish new aspects of the Opial integral inequality and related inequalities, in the context of fractional integrals and derivatives defined using nonsingular kernels, particularly the Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) models of fractional calculus.
Highlights
Fractional calculus, or the subject of fractional differential equations, is usually considered as a generalization of ordinary differential equations
Many mathematical inequalities are simulated via the fractional calculus that lead to fractional integral inequalities
They have been used in finding the uniqueness of solutions for a certain fractional differential equations and in providing bounds to solve certain fractional boundary value problems [18, 19]
Summary
Fractional calculus, or the subject of fractional differential equations, is usually considered as a generalization of ordinary differential equations. Our results yield some of the recent integral inequalities of Opial type and offer new estimates on such types of inequalities Definition 2.1 ([18]) For a function f, f(n) ∈ H1( 1, 2) and n < ν < n + 1, we have the higher order left and right CF-fractional derivatives, respectively, defined by CFC 1. We can obtain the same results for the right-sided higher order CF-fractional integral on the class of functions U2(ð ◦ v, k) (see Definition 1.2).
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