English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider linear differential equations with variable delay of the form x ′ ( t ) + p ( t ) x ( t − τ ( t ) ) = 0 , t ≥ t 0 , where p : [ t 0 , ∞ ) → [ 0 , ∞ ) and τ : [ t 0 , ∞ ) → ( 0 , ∞ ) are continuous functions, such that t − τ ( t ) → ∞ (as t → ∞ ). It is well-known that, for the oscillation of all solutions, it is necessary that B : = lim sup t → ∞ A ( t ) ≥ 1 e holds , where A : = ( t ) ∫ t − τ ( t ) t p ( s ) d s . Our main result shows that, if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions on p and τ , condition B > 1 / e implies that all solutions of the above delay differential equation are oscillatory.

A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay