Opial integral inequalities for generalized fractional operators with nonsingular kernel
Pshtiwan Othman Mohammed, Thabet Abdeljawad
Abstract
Abstract We consider the well-known classes of functions $\mathcal{U}_{1}(\mathbf{v},\mathtt{k})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>U</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:math> and $\mathcal{U}_{2}(\mathbf{v},\mathtt{k})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>U</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:math> , 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.