Global structure of solutions to boundary-value problems of impulsive differential equations
In this article, we study the structure of global solutions to the boundary-value problem $$\displaylines{ -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\cr \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\cr x(0)=...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2016-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2016/55/abstr.html |
| _version_ | 1798043672149753856 |
|---|---|
| author | Yanmin Niu Baoqiang Yan |
| author_facet | Yanmin Niu Baoqiang Yan |
| author_sort | Yanmin Niu |
| collection | DOAJ |
| description | In this article, we study the structure of global solutions to the
boundary-value problem
$$\displaylines{
-x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\cr
\Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad
\Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\cr
x(0)=x(1)=0,
}$$
where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$,
$\Delta x|_{t=1/2}=x(\frac{1}{2}+0)-x(\frac{1}{2})$,
$\Delta x'|_{t=1/2}=x'(\frac{1}{2}+0)-x'(\frac{1}{2}-0)$,
and $f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$
are continuous. By a comparison principle and spectral properties
of the corresponding linear equations, we prove the existence of
solutions by using Rabinowitz-type global bifurcation
theorems, and obtain results on the behavior of positive solutions
for large $\lambda$ when $f(x)=x^{p+1}$. |
| first_indexed | 2024-04-11T22:52:22Z |
| format | Article |
| id | doaj.art-14352822168f410ea62499cff7f57843 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-11T22:52:22Z |
| publishDate | 2016-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-14352822168f410ea62499cff7f578432022-12-22T03:58:32ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912016-02-01201655,123Global structure of solutions to boundary-value problems of impulsive differential equationsYanmin Niu0Baoqiang Yan1 Shandong Normal Univ., Jinan, China Shandong Normal Univ., Jinan, China In this article, we study the structure of global solutions to the boundary-value problem $$\displaylines{ -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\cr \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\cr x(0)=x(1)=0, }$$ where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$, $\Delta x|_{t=1/2}=x(\frac{1}{2}+0)-x(\frac{1}{2})$, $\Delta x'|_{t=1/2}=x'(\frac{1}{2}+0)-x'(\frac{1}{2}-0)$, and $f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$ are continuous. By a comparison principle and spectral properties of the corresponding linear equations, we prove the existence of solutions by using Rabinowitz-type global bifurcation theorems, and obtain results on the behavior of positive solutions for large $\lambda$ when $f(x)=x^{p+1}$.http://ejde.math.txstate.edu/Volumes/2016/55/abstr.htmlComparison argumentseigenvaluesglobal bifurcation theoremmultiple solutionsasymptotical behavior of solutions |
| spellingShingle | Yanmin Niu Baoqiang Yan Global structure of solutions to boundary-value problems of impulsive differential equations Electronic Journal of Differential Equations Comparison arguments eigenvalues global bifurcation theorem multiple solutions asymptotical behavior of solutions |
| title | Global structure of solutions to boundary-value problems of impulsive differential equations |
| title_full | Global structure of solutions to boundary-value problems of impulsive differential equations |
| title_fullStr | Global structure of solutions to boundary-value problems of impulsive differential equations |
| title_full_unstemmed | Global structure of solutions to boundary-value problems of impulsive differential equations |
| title_short | Global structure of solutions to boundary-value problems of impulsive differential equations |
| title_sort | global structure of solutions to boundary value problems of impulsive differential equations |
| topic | Comparison arguments eigenvalues global bifurcation theorem multiple solutions asymptotical behavior of solutions |
| url | http://ejde.math.txstate.edu/Volumes/2016/55/abstr.html |
| work_keys_str_mv | AT yanminniu globalstructureofsolutionstoboundaryvalueproblemsofimpulsivedifferentialequations AT baoqiangyan globalstructureofsolutionstoboundaryvalueproblemsofimpulsivedifferentialequations |