Sharp oscillation and nonoscillation tests for delay dynamic equations

In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation xΔ(t)+p(t)x(τ(t))=0fort∈[t0,∞)T, where p∈Crd([t0,∞)T,R0+) and τ∈Crd([t0,∞)T,T) satisfy τ(t) ≤ σ(t) for all large t and limt→∞τ(t)=∞. As an important corollary, we obtain the time sc...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2019-06, Vol.42 (9), p.2993-3001
Main Author: Karpuz, Basak
Format: Article
Language:English
Subjects:
Delay
dynamic equation
nonoscillation
oscillation
time scale
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Summary:In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation xΔ(t)+p(t)x(τ(t))=0fort∈[t0,∞)T, where p∈Crd([t0,∞)T,R0+) and τ∈Crd([t0,∞)T,T) satisfy τ(t) ≤ σ(t) for all large t and limt→∞τ(t)=∞. As an important corollary, we obtain the time scale invariant integral condition for nonoscillation: ∫τ(t)σ(t)p(η)Δη≤1e for all large t. Also, with some examples, we show that newly presented results are sharp.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.5558