{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:14:54Z","timestamp":1760192094850,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2019,10,24]],"date-time":"2019-10-24T00:00:00Z","timestamp":1571875200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider linear differential equations with variable delay of the form      x \u2032   ( t )  + p  ( t )  x  ( t \u2212 \u03c4  ( t )  )   = 0  ,  t \u2265  t 0  ,     where     p :  [  t 0  , \u221e )  \u2192  [ 0 , \u221e )      and     \u03c4 :  [  t 0  , \u221e )  \u2192  ( 0 , \u221e )      are continuous functions, such that     t \u2212 \u03c4 ( t ) \u2192 \u221e     (as     t \u2192 \u221e    ). It is well-known that, for the oscillation of all solutions, it is necessary that     B : =  lim sup  t \u2192 \u221e   A  ( t )  \u2265  1 e    holds ,   where  A : =  ( t )   \u222b  t \u2212 \u03c4 ( 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    \u03c4   , condition     B &gt; 1 \/ e     implies that all solutions of the above delay differential equation are oscillatory.<\/jats:p>","DOI":"10.3390\/sym11111332","type":"journal-article","created":{"date-parts":[[2019,10,25]],"date-time":"2019-10-25T04:41:27Z","timestamp":1571978487000},"page":"1332","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9693-1923","authenticated-orcid":false,"given":"\u00c1bel","family":"Garab","sequence":"first","affiliation":[{"name":"Institute of Mathematics, University of Klagenfurt, Universit\u00e4tsstra\u00dfe 65\u201367, 9020 Klagenfurt am W\u00f6rthersee, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hale, J.K., and Verduyn Lunel, S. 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[3rd ed.]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1332\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:29:00Z","timestamp":1760189340000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1332"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,24]]},"references-count":18,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2019,11]]}},"alternative-id":["sym11111332"],"URL":"https:\/\/doi.org\/10.3390\/sym11111332","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,10,24]]}}}