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- W3031267909 abstract "Abstract This paper deals with the existence of positive ω -periodic solutions for n th-order ordinary differential equation with delays in Banach space E of the form $$L_{n}u(t)=fbigl(t,u(t-tau_{1}),ldots,u(t- tau_{m})bigr),quad tinmathbb{R}, $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mi>L</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace /><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo></mml:math> where $L_{n}u(t)=u^{(n)}(t)+sum_{i=0}^{n-1}a_{i} u^{(i)}(t)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mi>L</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:math> is the n th-order linear differential operator, $a_{i}inmathbb {R}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>R</mml:mi></mml:math> ( $i=0,1,ldots,n-1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math> ) are constants, $f: mathbb{R}times E^{m}rightarrow E$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>R</mml:mi><mml:mo>×</mml:mo><mml:msup><mml:mi>E</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>→</mml:mo><mml:mi>E</mml:mi></mml:math> is a continuous function which is ω -periodic with respect to t , and $tau_{i}>0$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mi>τ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math> ( $i=1,2,ldots,m$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:math> ) are constants which denote the time delays. We first prove the existence of ω -periodic solutions of the corresponding linear problem. Then the strong positivity estimation is established. Finally, two existence theorems of positive ω -periodic solutions are proved. Our discussion is based on the theory of fixed point index in cones." @default.
- W3031267909 created "2020-06-05" @default.
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- W3031267909 date "2020-04-16" @default.
- W3031267909 modified "2023-09-24" @default.
- W3031267909 title "Positive periodic solutions for high-order differential equations with multiple delays in Banach spaces" @default.
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- W3031267909 doi "https://doi.org/10.1186/s13662-020-02595-z" @default.
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