Three positive solutions for second-order periodic boundary value problems with sign-changing weight
Abstract In this paper, we study the global structure of positive solutions of periodic boundary value problems {−u″(t)+q(t)u(t)=λh(t)f(u(t)),t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π), $$\textstyle\begin{cases} -u''(t)+q(t)u(t)=\lambda h(t)f(u(t)), \quad t\in (0,2\pi ), \\ u(0)=u(2\pi ), \quad\quad...
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doaj-bbed5d7e657b476498d69b48b80b87192020-11-25T01:57:01ZengSpringerOpenBoundary Value Problems1687-27702018-06-012018111710.1186/s13661-018-1011-1Three positive solutions for second-order periodic boundary value problems with sign-changing weightZhiqian He0Ruyun Ma1Man Xu2Department of Mathematics, Northwest Normal UniversityDepartment of Mathematics, Northwest Normal UniversityDepartment of Mathematics, Northwest Normal UniversityAbstract In this paper, we study the global structure of positive solutions of periodic boundary value problems {−u″(t)+q(t)u(t)=λh(t)f(u(t)),t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π), $$\textstyle\begin{cases} -u''(t)+q(t)u(t)=\lambda h(t)f(u(t)), \quad t\in (0,2\pi ), \\ u(0)=u(2\pi ), \quad\quad u'(0)=u'(2\pi ), \end{cases} $$ where q∈C([0,2π],[0,+∞)) $q\in C([0,2\pi ], [0, +\infty ))$ with q≢0 $q\not \equiv 0$, f∈C(R,R) $f\in C(\mathbb{R},\mathbb{R})$, the weight h∈C[0,2π] $h\in C[0,2\pi ]$ is a sign-changing function, λ is a parameter. We prove the existence of three positive solutions when h(t) $h(t)$ has n positive humps separated by n+1 $n+1$ negative ones. The proof is based on the bifurcation method.http://link.springer.com/article/10.1186/s13661-018-1011-1Three positive solutionsPeriodic boundary value problemBifurcation |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Zhiqian He Ruyun Ma Man Xu |
spellingShingle |
Zhiqian He Ruyun Ma Man Xu Three positive solutions for second-order periodic boundary value problems with sign-changing weight Boundary Value Problems Three positive solutions Periodic boundary value problem Bifurcation |
author_facet |
Zhiqian He Ruyun Ma Man Xu |
author_sort |
Zhiqian He |
title |
Three positive solutions for second-order periodic boundary value problems with sign-changing weight |
title_short |
Three positive solutions for second-order periodic boundary value problems with sign-changing weight |
title_full |
Three positive solutions for second-order periodic boundary value problems with sign-changing weight |
title_fullStr |
Three positive solutions for second-order periodic boundary value problems with sign-changing weight |
title_full_unstemmed |
Three positive solutions for second-order periodic boundary value problems with sign-changing weight |
title_sort |
three positive solutions for second-order periodic boundary value problems with sign-changing weight |
publisher |
SpringerOpen |
series |
Boundary Value Problems |
issn |
1687-2770 |
publishDate |
2018-06-01 |
description |
Abstract In this paper, we study the global structure of positive solutions of periodic boundary value problems {−u″(t)+q(t)u(t)=λh(t)f(u(t)),t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π), $$\textstyle\begin{cases} -u''(t)+q(t)u(t)=\lambda h(t)f(u(t)), \quad t\in (0,2\pi ), \\ u(0)=u(2\pi ), \quad\quad u'(0)=u'(2\pi ), \end{cases} $$ where q∈C([0,2π],[0,+∞)) $q\in C([0,2\pi ], [0, +\infty ))$ with q≢0 $q\not \equiv 0$, f∈C(R,R) $f\in C(\mathbb{R},\mathbb{R})$, the weight h∈C[0,2π] $h\in C[0,2\pi ]$ is a sign-changing function, λ is a parameter. We prove the existence of three positive solutions when h(t) $h(t)$ has n positive humps separated by n+1 $n+1$ negative ones. The proof is based on the bifurcation method. |
topic |
Three positive solutions Periodic boundary value problem Bifurcation |
url |
http://link.springer.com/article/10.1186/s13661-018-1011-1 |
work_keys_str_mv |
AT zhiqianhe threepositivesolutionsforsecondorderperiodicboundaryvalueproblemswithsignchangingweight AT ruyunma threepositivesolutionsforsecondorderperiodicboundaryvalueproblemswithsignchangingweight AT manxu threepositivesolutionsforsecondorderperiodicboundaryvalueproblemswithsignchangingweight |
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1724976902925451264 |