Abstract
We establish some new oscillation criteria for nonlinear dynamic equation of the form on an arbitrary time scale with , where are positive rd-continuous functions. An example illustrating the importance of our result is included.
Highlights
A time scale T is an arbitrary nonempty closed set of real numbers R with the topology and ordering inherited from R
On a time scale T, where λ is the ratio of odd positive integers, q is a positive real-valued rd-continuous function defined on T
On a time scale T with sup T = ∞, where λ is the ratio of odd positive integers, r, p, q are positive real-valued rdcontinuous functions defined on T, r(t) − μ(t)p(t) ≠ 0, τ ∈ Crd(T, T), τ(t) ≤ t, and τ(t) → ∞ as t → ∞
Summary
A time scale T is an arbitrary nonempty closed set of real numbers R with the topology and ordering inherited from R. On a time scale T, where γ ≥ 1 is the quotient of odd positive integers, a and r are positive rd-continuous functions on T, and the so-called delay function τ : T → T satisfies τ(t) ≤ t for t ∈ T and limt → ∞τ(t) = ∞ and f ∈ C(T × R, R) and obtained some oscillation criteria, which improved and extended the results that have been established in [10–12]. On a time scale T with sup T = ∞, where λ is the ratio of odd positive integers, r, p, q are positive real-valued rdcontinuous functions defined on T, r(t) − μ(t)p(t) ≠ 0, τ ∈ Crd(T, T), τ(t) ≤ t, and τ(t) → ∞ as t → ∞ They establish some new oscillation criteria of (7). A solution x(t) of (9) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory
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