Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term
By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation Lu=p(t)f(t,u(t),u′′(t))-g(t,u(t),u′′(t)),0<t<1,α1u(0)-β1u'(0)=0,γ1u(1)+δ1u'(1)=0,α2u′′(0)-β2u′′′(0)=0,γ2u′′(1)+δ2u′′′...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/160891 |
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| author | Yuefeng Han Xinguang Zhang Lishan Liu Yonghong Wu |
| author_facet | Yuefeng Han Xinguang Zhang Lishan Liu Yonghong Wu |
| author_sort | Yuefeng Han |
| collection | DOAJ |
| description | By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation Lu=p(t)f(t,u(t),u′′(t))-g(t,u(t),u′′(t)),0<t<1,α1u(0)-β1u'(0)=0,γ1u(1)+δ1u'(1)=0,α2u′′(0)-β2u′′′(0)=0,γ2u′′(1)+δ2u′′′(1)=0, with αi,βi,γi,δi≥0 and βiγi+αiγi+αiδi>0, i=1,2, where L denotes the linear operator Lu:=(ru′′′)'-qu′′,r∈C1([0,1],(0,+∞)), and q∈C([0,1],[0,+∞)). This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed term g:(0,1)×[0,+∞)×(-∞,+∞)→(-∞,+∞) only satisfies the global Carathéodory conditions, which implies that the perturbed effect of g on f is quite large so that the nonlinearity can tend to negative infinity at some singular points. |
| format | Article |
| id | doaj-art-8addd6dcd71147c3bba874aa30be2396 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-8addd6dcd71147c3bba874aa30be23962025-02-03T06:00:34ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/160891160891Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed TermYuefeng Han0Xinguang Zhang1Lishan Liu2Yonghong Wu3College of International Economics and Trade, Jilin University of Finance and Economics, Jilin, Changchun 130117, ChinaSchool of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, ChinaSchool of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, ChinaDepartment of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, AustraliaBy employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation Lu=p(t)f(t,u(t),u′′(t))-g(t,u(t),u′′(t)),0<t<1,α1u(0)-β1u'(0)=0,γ1u(1)+δ1u'(1)=0,α2u′′(0)-β2u′′′(0)=0,γ2u′′(1)+δ2u′′′(1)=0, with αi,βi,γi,δi≥0 and βiγi+αiγi+αiδi>0, i=1,2, where L denotes the linear operator Lu:=(ru′′′)'-qu′′,r∈C1([0,1],(0,+∞)), and q∈C([0,1],[0,+∞)). This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed term g:(0,1)×[0,+∞)×(-∞,+∞)→(-∞,+∞) only satisfies the global Carathéodory conditions, which implies that the perturbed effect of g on f is quite large so that the nonlinearity can tend to negative infinity at some singular points.http://dx.doi.org/10.1155/2012/160891 |
| spellingShingle | Yuefeng Han Xinguang Zhang Lishan Liu Yonghong Wu Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term Journal of Applied Mathematics |
| title | Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term |
| title_full | Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term |
| title_fullStr | Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term |
| title_full_unstemmed | Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term |
| title_short | Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term |
| title_sort | multiple positive solutions of singular nonlinear sturm liouville problems with caratheodory perturbed term |
| url | http://dx.doi.org/10.1155/2012/160891 |
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