Opial type inequalities involving fractional derivatives of two functions and applications. (English) Zbl 1094.26010

Anastassiou, G. A.

Comput. Math. Appl. 48, No. 10-11, 1701-1731 (2004).

Summary: A large variety of very general, but basic \(L_p (1 \leq p \leq \infty)\) form, Opial type inequalities is established involving generalized fractional derivatives of two functions in different orders and powers.
The above rely on a generalization of Taylor’s formula for generalized fractional derivatives [G. A. Anastassiou, Nonlinear Stud. 6, No. 2, 207–230 (1999; Zbl 0945.26019)]. 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 problems involving a very general system of two fractional differential equations. Also, upper bounds to various fractional derivatives of the solutions that are involved in the above systems are presented.

MSC:

26D10	Inequalities involving derivatives and differential and integral operators
26D15	Inequalities for sums, series and integrals
26A33	Fractional derivatives and integrals
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Keywords:

Opial type inequality; fractional derivative; system of fractional differential equations; uniqueness of solution; upper bound of solution

Citations:

Zbl 0945.26019

References:

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