SOME GENERALIZATIONS OF OPIAL’S INEQUALITIES ON TIME SCALES

Abstract

The Opial inequality is of great interest in differential and difference equations, and other areas of mathematics. The purpose of this paper is to generalize the Opial inequality to some time scale versions. One of these results says: \begin{align*} & \int_a^b h(x)\{|g(x)|^p|f^{\Delta^n}(x)|^q+|f(x)|^p|g^{\Delta^n}(x)|^q\}\Delta x\\ &\quad \leq{2q\over{p+q}}[({{b-a}\over {2}})^p]^n\int_a^b h(x)\{|f^{\Delta^n}(x)|^{p+q}+|g^{\Delta^n}(x)|^{p+q}\}\Delta x, \end{align*} if $p \ge 1$, $q \ge 1$ and $f,\ g \in C_{rd}([a,b], \Bbb {R})$ satisfy some suitable conditions.