Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales

and Applied Analysis 3 provided this limit exists. A function f : T → R is said to be rd-continuous provided f is continuous at right-dense points and there exists a finite left limit at all left-dense points in T .The set of all such rd-continuous functions is denoted by Crd(T).The derivativef Δ off and the forward jump operator σ are related by the formula f σ (t) = f (σ (t)) = f (t) + μ (t) f Δ (t) . (9) Also, we will use x which is shorthand for (x) to denote x(t)+μ(t)x(t). We will make use of the following product and quotient rules for the derivative of two differentiable functions f and g: (fg) Δ (t) = f Δ (t) g (t) + f σ (t) g Δ (t) = f (t) g Δ (t) + f Δ (t) g σ (t) , ( f g ) Δ (t) = f (t) g (t) − f (t) g Δ (t) g (t) g (t) , if ggσ ̸ = 0. (10) The integration by parts formula reads