Opial Inequalities Research Articles

Multidimensional weighted Opial inequalities

In this article we produce Opial-type weighted multidimensional inequalities over balls and arbitrary smooth bounded domains. The inequalities are sharp. The functions under consideration vanish on the boundary.

Weighted Opial inequalities

This paper presents a class of very general weighted Opial type inequalities. The notivation comes from the monograph of Agarwal and Pang (Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Acad., Dordrecht 1995) and the work of Anastassiou and Pecaric (J. Math. Anal. Appl. 239 (1999), 402-418). Assuming only a very general inequality, we extend the latter paper in several directions. A new result generalizing the original Opial's inequality is obtained, and applications to fractional derivatives are given.

Opial type Lp‐inequalities for fractional derivatives

This paper presents a class of Lp‐type Opial inequalities for generalized fractional derivatives for integrable functions based on the results obtained earlier by the first author for continuous functions (1998). The novelty of our approach is the use of the index law for fractional derivatives in lieu of Taylor′s formula, which enables us to relax restrictions on the orders of fractional derivatives.

Opial inequalities on time scales

We present a version of Opial's inequality for time scales and point out some of its applications to so-called dynamic equations. Such dynamic equations contain both dieren tial equations and dierence equations as special cases. Various extensions of our inequality are oered as well.

General Weighted Opial Inequalities for Linear Differential Operators

A complete set of Lr (r≠0) form very general Opial type weighted inequalities is given for a general linear differential operator L. These involve its related initial value problem solution y, Ly, the associated Green's function H, and initial conditions point x0∈R. This work is inspired by work of R. P. Agarwal and P. Y. H. Pang (1995, “Opial Inequalities with Applications in Differential and Difference Equations,” Kluwer Academic, Dordrecht) and G. A. Anastassiou (1998, Math. Inequalities Appl.1, No. 2, 193–200) and generalizes their related results. An application to proving uniqueness in solutions of initial value problems is given at the end.

First and second order Opial inequalities

First and second order Opial inequalities