On the Existence of Solutions for Impulsive Duffing Dynamic Equations on Time Scales with Dirichlet Boundary Conditions

and Applied Analysis 3 Definition 2.1 see 13 . A time scale T is an arbitrary nonempty closed subset of the real setR with the topology and ordering inherited from R. The forward and backward jump operators σ, ρ : T → T, and the graininess μ, ν : T → R are defined, respectively, by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, μ t : σ t − t, ν t : t − ρ t . 2.1 The point t ∈ T is called left dense, left scattered, right dense, or right scattered if ρ t t, ρ t < t, σ t t, or σ t > t, respectively. Points that are right dense and left dense at the same time are called dense. If T has a left-scattered maximumm1, defined T T−{m1}; otherwise, set T T. If T has a right-scattered minimum m2, defined Tκ T − {m2}; otherwise, set Tκ T. Definition 2.2 see 13 . For f : T → R and t ∈ T, then the delta derivative of f at the point t is defined to be the number fΔ t provided it exists with the property that for each > 0, there is a neighborhood U of t such that ∣∣f σ t − f s − fΔ t σ t − s ∣∣ ≤ |σ t − s|, ∀s ∈ U. 2.2 Definition 2.3 see 13 . A function f is rd continuous provided it is continuous at each right-dense point in T and has a left-sided limit at each left-dense point in T. The set of rdcontinuous functions f will be denoted by Crd T . Definition 2.4 see 13 . Assume that T is an arbitrary time scale. We say that a function p : T → R is regressive provided 1 μ t p t / 0, ∀t ∈ T. 2.3 Definition 2.5 see 13 . Assume that T is an arbitrary time scale and a function p : T → R is regressive, then we define the exponential function on T by