Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable
In this paper, we study unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity for nonlinear Sturm--Liouville problems of the form \begin{equation} \left\{ \begin{array}{l} -\left(pu'\right)'+qu=\lambda au+af\left(x,u,u',\lambda\right)+g\left(x...
| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2013-11-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2637 |
| _version_ | 1797830627988340736 |
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| author | Guowei Dai |
| author_facet | Guowei Dai |
| author_sort | Guowei Dai |
| collection | DOAJ |
| description | In this paper, we study unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity for nonlinear Sturm--Liouville problems of the form
\begin{equation}
\left\{
\begin{array}{l}
-\left(pu'\right)'+qu=\lambda au+af\left(x,u,u',\lambda\right)+g\left(x,u,u',\lambda\right)\,\,\text{for}\,\, x\in(0,1),\\
b_0u(0)+c_0u'(0)=0,\\
b_1u(1)+c_1u'(1)=0,
\end{array}
\right.\nonumber
\end{equation}
where $a\in C([0, 1], [0,+\infty))$ and $a(x)\not\equiv 0$ on any subinterval of $[0, 1]$,
$f,g\in C([0,1]\times\mathbb{R}^3,\mathbb{R})$. Suppose that $f$ and $g$ satisfy
\begin{equation}
\vert f(x,\xi,\eta,\lambda)\vert\leq M_0\vert \xi\vert+M_1\vert \eta\vert,\,\, \forall x\in [0,1]\,\,\text{and}\,\,\lambda \in\mathbb{R}, \nonumber
\end{equation}
\begin{equation}
g(x,\xi,\eta,\lambda)=o(\vert \xi\vert+\vert \eta\vert),\,\, \text{uniformly in}\,\, x\in [0,1]\,\,\text{and}\,\,\lambda \in \Lambda,\nonumber
\end{equation}
as either $\vert \xi\vert+\vert \eta\vert\rightarrow 0$ or $\vert \xi\vert+\vert \eta\vert\rightarrow +\infty$, for some constants $M_0$, $M_1$, and any bounded
interval $\Lambda$. |
| first_indexed | 2024-04-09T13:39:08Z |
| format | Article |
| id | doaj.art-23238e4f6b524781a908002d9821725d |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:39:08Z |
| publishDate | 2013-11-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-23238e4f6b524781a908002d9821725d2023-05-09T07:53:03ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752013-11-012013651710.14232/ejqtde.2013.1.652637Global bifurcation from intervals for Sturm-Liouville problems which are not linearizableGuowei Dai0Dalian University of TechnologyIn this paper, we study unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity for nonlinear Sturm--Liouville problems of the form \begin{equation} \left\{ \begin{array}{l} -\left(pu'\right)'+qu=\lambda au+af\left(x,u,u',\lambda\right)+g\left(x,u,u',\lambda\right)\,\,\text{for}\,\, x\in(0,1),\\ b_0u(0)+c_0u'(0)=0,\\ b_1u(1)+c_1u'(1)=0, \end{array} \right.\nonumber \end{equation} where $a\in C([0, 1], [0,+\infty))$ and $a(x)\not\equiv 0$ on any subinterval of $[0, 1]$, $f,g\in C([0,1]\times\mathbb{R}^3,\mathbb{R})$. Suppose that $f$ and $g$ satisfy \begin{equation} \vert f(x,\xi,\eta,\lambda)\vert\leq M_0\vert \xi\vert+M_1\vert \eta\vert,\,\, \forall x\in [0,1]\,\,\text{and}\,\,\lambda \in\mathbb{R}, \nonumber \end{equation} \begin{equation} g(x,\xi,\eta,\lambda)=o(\vert \xi\vert+\vert \eta\vert),\,\, \text{uniformly in}\,\, x\in [0,1]\,\,\text{and}\,\,\lambda \in \Lambda,\nonumber \end{equation} as either $\vert \xi\vert+\vert \eta\vert\rightarrow 0$ or $\vert \xi\vert+\vert \eta\vert\rightarrow +\infty$, for some constants $M_0$, $M_1$, and any bounded interval $\Lambda$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2637interval bifurcation; sturm-liouville problem; unilateral global bifurcation |
| spellingShingle | Guowei Dai Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable Electronic Journal of Qualitative Theory of Differential Equations interval bifurcation; sturm-liouville problem; unilateral global bifurcation |
| title | Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable |
| title_full | Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable |
| title_fullStr | Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable |
| title_full_unstemmed | Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable |
| title_short | Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable |
| title_sort | global bifurcation from intervals for sturm liouville problems which are not linearizable |
| topic | interval bifurcation; sturm-liouville problem; unilateral global bifurcation |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2637 |
| work_keys_str_mv | AT guoweidai globalbifurcationfromintervalsforsturmliouvilleproblemswhicharenotlinearizable |