Coexistence of positive solutions of nonlinear three-point boundary value and its conjugate problem

In this paper, the three-point boundary value problem − u ″ ( t ) = f ( t , u ( t ) ) , 0 ⩽ t ⩽ 1 , u ′ ( 0 ) = 0 , u ( 1 ) = α u ( η ) , and its conjugate boundary value problem − v ″ ( s ) = f ( s , v ( s ) ) , 0 ⩽ s ⩽ 1 , s ≠ η , v ′ ( 0 ) = 0 , v + ′ ( η ) − v − ′ ( η ) = α v ′ ( 1 ) , v ( 1 ) =...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2007-06, Vol.330 (1), p.334-351
Main Authors: Wang, Shuli, Liu, Jinsheng
Format: Article
Language:English
Subjects:
Cone
Conjugate
Eigenvalue
Exact sciences and technology
Fixed point index
Global analysis, analysis on manifolds
Mathematical analysis
Mathematics
Ordinary differential equations
Positive solutions
Sciences and techniques of general use
Three-point boundary value problem
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:In this paper, the three-point boundary value problem − u ″ ( t ) = f ( t , u ( t ) ) , 0 ⩽ t ⩽ 1 , u ′ ( 0 ) = 0 , u ( 1 ) = α u ( η ) , and its conjugate boundary value problem − v ″ ( s ) = f ( s , v ( s ) ) , 0 ⩽ s ⩽ 1 , s ≠ η , v ′ ( 0 ) = 0 , v + ′ ( η ) − v − ′ ( η ) = α v ′ ( 1 ) , v ( 1 ) = 0 , are studied under some conditions concerning the same first eigenvalue of the both linear problems. By applying the fixed point index theory, the coexistence of single and multiple positive solutions of the above mentioned problems is verified. As an application, some examples are given to illustrate our results.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2006.07.073