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.
Highlights
Our method is motivated by the maximum principle of Degla [4] and a result on the principal eigenvalue of multi-point Boundary Value Problems (BVP’s) of Degla [5] which allow the use of cone theoretic arguments and of the well-known general result on bifurcation from infinity; see Coyle [1], Mawhin [6] and Rabinowitz [7]
In our abstract setting, the nonlinear Krein-Rutman Theorem resets an important result on the simplicity of positive eigenvalues [8] by avoiding some inconclusive argument [8] misused in [9]
The above theorem is readily applicable to any positively 1 -homogeneous, compact and continuous operators that are strongly positive on the cone of an ordered Banach space
Summary
We generalize and improve a result of Coyle et al [1] about the bifurcation from infinity after stating in the line of Nussbaum [2], Schmitt [3], etc., a type of nonlinear Krein-Rutman theorem for a class of positively 1 -homogeneous, compact and continuous operators in Banach spaces leaving invariant cones. Our method is motivated by the maximum principle of Degla [4] and a result on the principal eigenvalue of multi-point Boundary Value Problems (BVP’s) of Degla [5] which allow the use of cone theoretic arguments and of the well-known general result on bifurcation from infinity; see Coyle [1], Mawhin [6] and Rabinowitz [7].
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