Opial inequalities on time scales (Q2759181)

The authors derive a unification and extension of the continuous and discrete Opial inequalities, as presented by \textit{R. P. Agarwal} and \textit{P. Y. H. Pang} [Opial inequalities with applications in differential and difference equations, Dordrecht: Kluwer Academic Publishers (1995; Zbl 0821.26013)], to an arbitrary time scale, which is any closed subset of the reals. The original continuous and discrete results are due to \textit{Z. Opial} [Ann. Pol. Math. 8, 65-69 (1960; Zbl 0090.06602)] and \textit{A. Lasota} [Ann. Pol. Math. 20, 183-190 (1968; Zbl 0157.15602)], respectively. An application to a dynamic initial value problem and some further extensions to dynamic inequalities involving higher-order \(\Delta\)-derivatives are also provided.