Abstract
We prove some weighted inequalities for delta derivatives acting on products and compositions of functions on time scales and apply them to obtain generalized dynamic Opial-type inequalities. We also employ these inequalities to establish some new dynamic Lyapunov-type inequalities, which are essential in studying disfocality, disconjugacy, lower bounds of eigenvalues, and distance between generalized zeros for half-linear dynamic equations. In particular, we solve an open problem posed by Saker in [Math. Comput. Modelling 58 (2013), 1777-1790]. Moreover, the results presented in this paper generalize, improve, extend, and unify most of known results not only in the discrete and continuous analysis but also on time scales.
