Abstract
We establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.
Highlights
In reality, it is known that the movement in the vacuum or ideal state is rare, while the movement with damping and disturbance is extensive
The graininess function μ : T → [t0, ∞) of the time scale is defined by μ(t) = σ(t) − t
For a function f : T → R, the derivative is defined by fΔ
Summary
It is known that the movement in the vacuum or ideal state is rare, while the movement with damping and disturbance is extensive. The study of the oscillation of the second-order dynamic equations with damping on time scales is emerging; see [1–7], for example. Inspired by the above work, this paper will study the oscillatory behavior of all solutions of a more extensive second-order half-linear neutral delay dynamic equation with damping, which is given as follows:. The solution of (1) defines a nontrivial real-valued function x satisfying (1) for t ∈ T. The purpose of this paper is to establish the oscillation criteria of Philos [34] for (1). The two famous results of Philos [34] about oscillation of second-order linear differential equations are extended to (1), while it satisfies. We established four new oscillatory criteria when conditions (4) and (5) hold, respectively, for the solutions of (1) in this paper
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