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 x0gR. 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.