Journal of Applied Mathematics

Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term

Yuefeng Han, Xinguang Zhang, Lishan Liu, and Yonghong Wu

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Abstract

By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation L u = p ( t ) f ( t , u ( t ) , u ( t ) ) - g ( t , u ( t ) , u ( t ) ) , 0 < t < 1 , α 1 u ( 0 ) - β 1 u ' ( 0 ) = 0 , γ 1 u ( 1 ) + δ 1 u ' ( 1 ) = 0 , α 2 u ( 0 ) - β 2 u ( 0 ) = 0 , γ 2 u ( 1 ) + δ 2 u ( 1 ) = 0 , with α i , β i , γ i , δ i 0 and β i γ i + α i γ i + α i δ i > 0 ,       i = 1,2 , where L denotes the linear operator L u : = ( r u ) ' - q u , r C 1 ( [ 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 ) {\times} [ 0 , + ) {\times} ( - , + ) ( - , + ) 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.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 160891, 23 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1355495024

Digital Object Identifier
doi:10.1155/2012/160891

Mathematical Reviews number (MathSciNet)
MR2874980

Zentralblatt MATH identifier
1241.34026

Citation

Han, Yuefeng; Zhang, Xinguang; Liu, Lishan; Wu, Yonghong. Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term. J. Appl. Math. 2012 (2012), Article ID 160891, 23 pages. doi:10.1155/2012/160891. https://projecteuclid.org/euclid.jam/1355495024


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