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Opial type inequalities involving Riemann-Liouville fractional derivatives of two functions with applications. (English) Zbl 1145.44300

Summary: A large variety of very general but basic ${L}_{p}\left(1\le p\le \infty \right)$ form Opial type inequalities [Z. Opial, Ann. Pol. Math. 8, 29–32 (1960; Zbl 0089.27403)], is established involving Riemann-Liouville fractional derivatives [G. A. Anastassiou, Nonlinear Stud. 6, No. 2, 207–230 (1999; Zbl 0945.26019); V. S. Kiryakova, Generalized fractional calculus and applications. New York: John Wiley & Sons (1994; Zbl 0882.26003); K. S. Miller and B. Ross, “An introduction to the fractional calculus and fractional differential equations” (1993; Zbl 0789.26002); K. B. Oldham and J. Spanier, “The fractional calculus. Theory and applications of differentiation and integration to arbitrary order” (1974; Zbl 0292.26011)] 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.

##### MSC:
 44A15 Special transforms (Legendre, Hilbert, etc.) 26D15 Inequalities for sums, series and integrals of real functions