On Bifurcation from Infinity and Multipoint Boundary Value Problems

Guy Degla

1African University of Science and Technology (AUST), Abuja, Nigeria 2Institut de Mathematiques et de Sciences Physiques (IMSP), Porto-Novo, Benin.

DOI: 10.4236/apm.2014.44018                

Abstract

We generalize a result on bifurcation from infinity of high order ordinary differential equations with multi-point boundary conditions. Our abstract setting represents a variant of Nonlinear Krein-Ruthman theorems. Furthermore, an analysis of this abstract setting raises an open question motivated by some misunderstanding and inconclusive proofs about the simplicity of principal eigenvalues in some articles in the literature.

Keywords

Krein-Rutman, Positivity, Cone, Homogeneity, Compactness, Eigenvalue, Simplicity, Bifurcation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Coyle, J., Eloe, P.W. and Henderson, J. (1995) Bifurcation from Infinity and Higher Order Ordinary Differential Equations. Journal of Mathematical Analysis and Applications, 195, 32-43. http://dx.doi.org/10.1006/jmaa.1995.1340
[2]	Nussbaum, R. (1980) Eigenvectors of Nonlinear Positive Operators and the Linear Krein-Rutman Theorem. Lecture Notes in Mathematics, 886, 309-330.
[3]	Schmitt, K. (2005) Positivity, Continuation and Nonlinear Eigenvalue Problems. World Science Publiching, Hackensack, 119-127.
[4]	Degla, G. (2001) A Unifying Maximum Principle for Conjugate Boundary Value Problems. Advanced Nonlinear Studies, 1, 121-131.
[5]	Degla, G. (2002) On the Principal Eigenvalue of Disconjugate BVPs with L1-Coefficients. Advanced Nonlinear Studies, 2, 19-39.
[6]	Mawhin, J. (1999) Leray-Schauder Degree: A Half Century of Extensions and Applications. Topological Methods in Nonlinear Analysis. Journal of the Juliusz Schauder Center, 14, 195-228.
[7]	Rabinowitz, P. (1983) On Bifurcation from Infinity. Journal of Differential Equations, 14, 462-475. http://dx.doi.org/10.1016/0022-0396(73)90061-2
[8]	Mahadevan, R. (2007) A Note on a Nonlinear Krein-Rutman Theorem. Nonlinear Analysis, 67, 3084-3090. http://dx.doi.org/10.1016/j.na.2006.09.062
[9]	Chang, K.C. (2009) A Nonlinear Krein-Rutman Theorem. Journal of Systems Science and Complexity, 22, 542-554. http://dx.doi.org/10.1007/s11424-009-9186-2
[10]	Elias, U. (1997) Oscillation Theory of Two-Term Differential Equations. In: Mathematics and Its Applications, Vol. 396, Kluwer Academic Publishers, Dodrecht, 179 p.
[11]	Coppel, W.A. (1971) Disconjugacy. In: Lecture Notes in Mathematics, Vol. 220, Springer, New York, 156 p.
[12]	Amann, H. (1976) Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces. Society for Industrial and Applied Mathematics Review, 18, 620-709.