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- W3092751065 abstract "Abstract In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type $$ bigl(x(t)-cx(t-sigma)bigr)''+fbigl(x(t) bigr)x'(t)-varphi(t)x^{mu}(t)+ frac{alpha(t)}{x^{gamma}(t)}=e(t), $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mi>c</mml:mi><mml:mi>x</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>σ</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mo>″</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mi>φ</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>μ</mml:mi></mml:msup><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>γ</mml:mi></mml:msup><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:math> where $f:(0,+infty)rightarrow R$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mi>∞</mml:mi><mml:mo>)</mml:mo><mml:mo>→</mml:mo><mml:mi>R</mml:mi></mml:math> , $varphi(t)>0$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>φ</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math> and $alpha(t)>0$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>α</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math> are continuous functions with T -periodicity in the t variable, c , γ are constants with $|c|<1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mo>|</mml:mo><mml:mi>c</mml:mi><mml:mo>|</mml:mo><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:math> , $gammageq1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>γ</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:math> . Many authors obtained the existence of periodic solutions under the condition $0<muleq1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>μ</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math> , and we extend the result to $mu>1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>μ</mml:mi><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:math> by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem." @default.
- W3092751065 created "2020-10-22" @default.
- W3092751065 creator A5035761574 @default.
- W3092751065 date "2020-10-16" @default.
- W3092751065 modified "2023-09-25" @default.
- W3092751065 title "Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type" @default.
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- W3092751065 doi "https://doi.org/10.1186/s13661-020-01462-w" @default.
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