Existence of at least three solutions of nonlinear three point boundary value problems with super-quadratic growth

We establish the existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type − x ″ = f ( t , x , x ′ ) , 0 < t < 1 , x ′ ( 0 ) = 0 , x ( 1 ) = δ x ( η ) , 0 < η < 1 , 0 < δ <...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2007-04, Vol.328 (1), p.690-698
Main Authors: Khan, Rahmat Ali, Webb, J.R.L.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematics
Multiple solutions
Ordinary differential equations
Sciences and techniques of general use
Three-point problem
Topological degree
Upper and lower solutions
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Summary:We establish the existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type − x ″ = f ( t , x , x ′ ) , 0 < t < 1 , x ′ ( 0 ) = 0 , x ( 1 ) = δ x ( η ) , 0 < η < 1 , 0 < δ < 1 . We employ a condition weaker than the well-known Nagumo condition which allows the nonlinearity f ( t , x , x ′ ) to grow faster than quadratically with respect to x ′ in some cases. Our method uses some degree theory arguments.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2006.05.074