Abstract
By using a Riccati transformation and inequality, we present some new oscillation theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.
Highlights
The theory of time scales, which has recently received a lot of attraction, was introduced by Hilger in his Ph.D
We are concerned with second-order nonlinear dynamic equations with damping xΔ tγΔpt xΔ tγqtf xσ t
1.1 on a time scale T; here p and q are real-valued positive rd-continuous positive functions defined on T, and γ is a quotient of odd positive integers
Summary
The theory of time scales, which has recently received a lot of attraction, was introduced by Hilger in his Ph.D. There are few papers dealing with the oscillation of dynamic equations with damping term 14–17. Hassan 19 studied the oscillation behavior of the second-order half-linear dynamic equation r t xΔ tγΔpt xγ t 0, 1.3 and obtained several new results. Bohner et al 20 established some oscillation criteria for the second-order nonlinear dynamic equation xΔΔ t q t xΔσ t p t f ◦ xσ 0. Saker et al 17 investigated the oscillation of second-order dynamic equations with damping term of the form r t xΔ t Δ p t xΔσ tqtfxσt. Zafer 21 studied the second-order nonlinear dynamic equations on time scales yΔΔ p t yΔ q t yσ 0, t ∈ T, 1.7 and presented some oscillation and nonoscillation criteria.
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