Opial type inequalities involving Riemann–Liouville fractional derivatives of two functions with applications

Abstract

A large variety of very general but basic L p ( 1 ≤ p ≤ ∞ ) form Opial type inequalities, [Z. Opial, Sur une inégalite, Ann. Polon. Math. 8 (1960) 29–32] is established involving Riemann–Liouville fractional derivatives [G.A. Anastassiou, Opial type inequalities involving fractional derivatives of functions, Nonlinear Stud. 6 (2) (1999) 207–230; Virginia Kiryakova, Generalized Fractional Calculus and Applications, in: Pitman Research Notes in Math. Series, vol. 301, Longman Scientific and Technical, Harlow; copublished in U.S.A with John Wiley & Sons, Inc., New York, 1994; Kenneth Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc. New York, 1993; Keith Oldham, Jerome Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications, New York, 2006] of two functions in different orders and powers. From the developed results derive several other concrete results of special interest. The sharpness of inequalities is established there. Finally applications of some of these special inequalities are given in establishing uniqueness of solution and in giving upper bounds to solutions of initial value fractional problems involving a very general system of two fractional differential equations. Also upper bounds to various Riemann–Liouville fractional derivatives of the solutions that are involved in the above systems are presented.