A Willett type criterion with the best possible constant for linear dynamic equations

Abstract

We establish oscillation criteria for the linear dynamic equation $(r(t)y^{\Delta})^{\Delta} + p(t) y^{\sigma} = 0$. These criteria can be understood as an extension of the classical Willett criterion. What is special on these new results is that the constant involved in the criteria, which is equal to the "magic" 1/4 in the differential equations case, is in fact no more constant. In general case, it depends on the asymptotic behavior of the coefficients $p$, $r$, and primarily on the asymptotic behavior of graininess. In addition, we prove that the value of this new "constant" is the best possible.