Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type
Abstract In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , $$ \bigl(x(t)-cx(t-\sigma)\bigr)...
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doaj-8e545c1ae18c464a9fbd6e3894779c142020-11-25T03:34:50ZengSpringerOpenBoundary Value Problems1687-27702020-10-012020111010.1186/s13661-020-01462-wExistence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive typeYu Zhu0School of Tencent Cloud Big Data, Ma’anshan UniversityAbstract In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , $$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$ where f : ( 0 , + ∞ ) → R $f:(0,+\infty)\rightarrow R$ , φ ( t ) > 0 $\varphi(t)>0$ and α ( t ) > 0 $\alpha(t)>0$ are continuous functions with T-periodicity in the t variable, c, γ are constants with | c | < 1 $|c|<1$ , γ ≥ 1 $\gamma\geq1$ . Many authors obtained the existence of periodic solutions under the condition 0 < μ ≤ 1 $0<\mu\leq1$ , and we extend the result to μ > 1 $\mu>1$ 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.http://link.springer.com/article/10.1186/s13661-020-01462-wPeriodic solutionNeutral equationMawhin’s continuation theoremAttractive singularity |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Yu Zhu |
spellingShingle |
Yu Zhu Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type Boundary Value Problems Periodic solution Neutral equation Mawhin’s continuation theorem Attractive singularity |
author_facet |
Yu Zhu |
author_sort |
Yu Zhu |
title |
Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type |
title_short |
Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type |
title_full |
Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type |
title_fullStr |
Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type |
title_full_unstemmed |
Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type |
title_sort |
existence of positive periodic solutions for super-linear neutral liénard equation with a singularity of attractive type |
publisher |
SpringerOpen |
series |
Boundary Value Problems |
issn |
1687-2770 |
publishDate |
2020-10-01 |
description |
Abstract In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , $$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$ where f : ( 0 , + ∞ ) → R $f:(0,+\infty)\rightarrow R$ , φ ( t ) > 0 $\varphi(t)>0$ and α ( t ) > 0 $\alpha(t)>0$ are continuous functions with T-periodicity in the t variable, c, γ are constants with | c | < 1 $|c|<1$ , γ ≥ 1 $\gamma\geq1$ . Many authors obtained the existence of periodic solutions under the condition 0 < μ ≤ 1 $0<\mu\leq1$ , and we extend the result to μ > 1 $\mu>1$ 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. |
topic |
Periodic solution Neutral equation Mawhin’s continuation theorem Attractive singularity |
url |
http://link.springer.com/article/10.1186/s13661-020-01462-w |
work_keys_str_mv |
AT yuzhu existenceofpositiveperiodicsolutionsforsuperlinearneutrallienardequationwithasingularityofattractivetype |
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1724557165091356672 |