| Summary: | We consider linear differential equations with variable delay of the form <disp-formula> <math display="block"> <semantics> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>=</mo> <mn>0</mn> </mrow> <mo>,</mo> <mspace width="2.em"></mspace> <mi>t</mi> <mo>≥</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> </disp-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo lspace="0pt">:</mo> <mrow> <mo>[</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>,</mo> <mo>∞</mo> <mo>)</mo> </mrow> <mo>→</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mo>∞</mo> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo lspace="0pt">:</mo> <mrow> <mo>[</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>,</mo> <mo>∞</mo> <mo>)</mo> </mrow> <mo>→</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mo>∞</mo> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> are continuous functions, such that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula> (as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>). It is well-known that, for the oscillation of all solutions, it is necessary that <disp-formula> <math display="block"> <semantics> <mrow> <mi>B</mi> <mo>:</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </munder> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>≥</mo> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> <mspace width="1.em"></mspace> <mrow> <mi>holds</mi> <mo>,</mo> </mrow> <mspace width="4.pt"></mspace> <mi>where</mi> <mspace width="1.em"></mspace> <mi>A</mi> <mo>:</mo> <mo>=</mo> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>t</mi> </msubsup> <mi>p</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mspace width="0.166667em"></mspace> <mi>d</mi> <mi>s</mi> <mo>.</mo> </mrow> </semantics> </math> </disp-formula> Our main result shows that, if the function <i>A</i> is slowly varying at infinity (in additive form), then under mild additional assumptions on <i>p</i> and <inline-formula> <math display="inline"> <semantics> <mi>τ</mi> </semantics> </math> </inline-formula>, condition <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>></mo> <mn>1</mn> <mo>/</mo> <mi>e</mi> </mrow> </semantics> </math> </inline-formula> implies that all solutions of the above delay differential equation are oscillatory.
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