Positive solutions for a system of $n$th-order nonlinear boundary value problems
In this paper, we investigate the existence, multiplicity and uniqueness of positive solutions for the following system of $n$th-order nonlinear boundary value problems \[\begin{cases} u^{(n)}(t)+f(t,u(t),v(t))=0,0<t<1,\\v^{(n)}(t)+g(t,u(t),v(t))=0, 0<t<1,\\ u(0)=u'(0)=\ldots=u^{(n-...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2011-02-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=578 |
| _version_ | 1797830697656778752 |
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| author | Jiafa Xu Zhilin Yang |
| author_facet | Jiafa Xu Zhilin Yang |
| author_sort | Jiafa Xu |
| collection | DOAJ |
| description | In this paper, we investigate the existence, multiplicity and uniqueness of positive solutions for the following system of $n$th-order nonlinear boundary value problems
\[\begin{cases}
u^{(n)}(t)+f(t,u(t),v(t))=0,0<t<1,\\v^{(n)}(t)+g(t,u(t),v(t))=0, 0<t<1,\\
u(0)=u'(0)=\ldots=u^{(n-2)}(0)=u(1)=0,\\
v(0)= v'(0)=\ldots=v^{(n-2)}(0)=v(1)=0.
\end{cases}\]
Based on a priori estimates achieved by using Jensen's integral inequality, we use fixed point index theory to establish our main results. Our assumptions on the nonlinearities are mostly formulated in terms of spectral radii of associated linear integral operators. In addition, concave and convex functions are utilized to characterize coupling behaviors of $f$ and $g$, so that we can treat the three cases: the first with both superlinear, the second with both sublinear, and the last with one superlinear and the other sublinear. |
| first_indexed | 2024-04-09T13:40:15Z |
| format | Article |
| id | doaj.art-2ae5bfd096f241af97822640381029d3 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:40:15Z |
| publishDate | 2011-02-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-2ae5bfd096f241af97822640381029d32023-05-09T07:53:00ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752011-02-012011411610.14232/ejqtde.2011.1.4578Positive solutions for a system of $n$th-order nonlinear boundary value problemsJiafa Xu0Zhilin Yang1School of Mathematics, Shandong University, Jinan, Shandong, P. R. ChinaQingdao Technological University, Qingdao, Shandong, P. R. ChinaIn this paper, we investigate the existence, multiplicity and uniqueness of positive solutions for the following system of $n$th-order nonlinear boundary value problems \[\begin{cases} u^{(n)}(t)+f(t,u(t),v(t))=0,0<t<1,\\v^{(n)}(t)+g(t,u(t),v(t))=0, 0<t<1,\\ u(0)=u'(0)=\ldots=u^{(n-2)}(0)=u(1)=0,\\ v(0)= v'(0)=\ldots=v^{(n-2)}(0)=v(1)=0. \end{cases}\] Based on a priori estimates achieved by using Jensen's integral inequality, we use fixed point index theory to establish our main results. Our assumptions on the nonlinearities are mostly formulated in terms of spectral radii of associated linear integral operators. In addition, concave and convex functions are utilized to characterize coupling behaviors of $f$ and $g$, so that we can treat the three cases: the first with both superlinear, the second with both sublinear, and the last with one superlinear and the other sublinear.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=578 |
| spellingShingle | Jiafa Xu Zhilin Yang Positive solutions for a system of $n$th-order nonlinear boundary value problems Electronic Journal of Qualitative Theory of Differential Equations |
| title | Positive solutions for a system of $n$th-order nonlinear boundary value problems |
| title_full | Positive solutions for a system of $n$th-order nonlinear boundary value problems |
| title_fullStr | Positive solutions for a system of $n$th-order nonlinear boundary value problems |
| title_full_unstemmed | Positive solutions for a system of $n$th-order nonlinear boundary value problems |
| title_short | Positive solutions for a system of $n$th-order nonlinear boundary value problems |
| title_sort | positive solutions for a system of n th order nonlinear boundary value problems |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=578 |
| work_keys_str_mv | AT jiafaxu positivesolutionsforasystemofnthordernonlinearboundaryvalueproblems AT zhilinyang positivesolutionsforasystemofnthordernonlinearboundaryvalueproblems |