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• W2107369564 abstract "Consider the delay differential equation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x prime left-parenthesis t right-parenthesis equals minus f left-parenthesis x left-parenthesis t minus 1 right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>x’(t)=-f(x(t-1))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f element-of upper C left-parenthesis double-struck upper R comma double-struck upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>fin C(mathbb {R}, mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is odd and satisfies <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x f left-parenthesis x right-parenthesis greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>xf(x)>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x not-equals 0> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>xne 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. When <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha equals limit Underscript x right-arrow 0 Endscripts StartFraction f left-parenthesis x right-parenthesis Over x EndFraction> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>=</mml:mo> <mml:munder> <mml:mo movablelimits=true form=prefix>lim</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>x</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:munder> <mml:mfrac> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> </mml:mfrac> </mml:mrow> <mml:annotation encoding=application/x-tex>alpha =lim _{xto 0}frac {f(x)}{x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta equals limit Underscript x right-arrow normal infinity Endscripts StartFraction f left-parenthesis x right-parenthesis Over x EndFraction> <mml:semantics> <mml:mrow> <mml:mi>β<!-- β --></mml:mi> <mml:mo>=</mml:mo> <mml:munder> <mml:mo movablelimits=true form=prefix>lim</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>x</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:munder> <mml:mfrac> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> </mml:mfrac> </mml:mrow> <mml:annotation encoding=application/x-tex>beta =lim _{xto infty }frac {f(x)}{x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exist, there is at least one Kaplan-Yorke periodic solution with period <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=4> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding=application/x-tex>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=min left-brace right-brace comma alpha comma beta greater-than StartFraction pi Over 2 EndFraction greater-than max left-brace right-brace comma alpha comma beta> <mml:semantics> <mml:mrow> <mml:mo movablelimits=true form=prefix>min</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>></mml:mo> <mml:mfrac> <mml:mi>π<!-- π --></mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>></mml:mo> <mml:mo movablelimits=true form=prefix>max</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>min {alpha ,beta }>frac {pi }{2}>max {alpha ,beta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. When this condition is not satisfied, we present several sufficient conditions on the existence/nonexistence of such periodic solutions. It is worthy of mention that some results are on the existence of at least two Kaplan-Yorke periodic solutions with period <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=4> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding=application/x-tex>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and in some cases we do not require the limits <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha> <mml:semantics> <mml:mi>α<!-- α --></mml:mi> <mml:annotation encoding=application/x-tex>alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and/or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=application/x-tex>beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to exist. Hence our results not only greatly improve but also complement existing ones. Moreover, some of the theoretical results are illustrated with examples." @default.
• W2107369564 created "2016-06-24" @default.
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• W2107369564 date "2012-08-10" @default.
• W2107369564 modified "2023-09-23" @default.
• W2107369564 title "A note on periodic solutions of the delay differential equation 𝑥’(𝑡)=-𝑓(𝑥(𝑡-1))" @default.
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• W2107369564 doi "https://doi.org/10.1090/s0002-9939-2012-11386-3" @default.
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