Higher-order weighted Opial-type inequalities on time scales for arbitrary endpoint values

We study higher-order weighted Beesack–Opial inequalities on time scales when the value at an endpoint is arbitrary. For the right-sided problem, only the delta derivatives of orders \(1,… ,n-1\) vanish at the terminal point. The value of the function itself remains free and appears in the main estimate as an explicit endpoint contribution. The proof uses the intrinsic generalized monomial \(gₙ₋₁(σ (τ ),σ (t))\) . On a general time scale, this kernel cannot be replaced by the corresponding ordinary power. We obtain a weighted intrinsic-kernel estimate and a two-sided companion that keeps the left and right kernels separate. A supremum-based consequence gives a simpler constant. Ordinary-power formulas are recovered only when a verified kernel-domination condition holds; this includes \(n≤ 2\) on an arbitrary time scale, every n on \(R\) , and \(n≤ 4\) on \(hZ\) . We also apply the right-sided result to a third-order nonlinear terminal-value equation on a hybrid time scale. An exact solution gives a positive lower bound for the terminal value. For the same data, the inequality fails if the endpoint contribution is omitted, and the underlying Taylor estimate fails if the intrinsic kernel is replaced by the ordinary power. The application therefore shows why both the endpoint contribution and the intrinsic kernel are necessary.