# The existence and extremal characterization of eigenvalues for an*n*-th order multiple point boundary value problem

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## Summary

The existence and extremal characterization of least positive eigenvalues is considered for the n-th order multiple point eigenvalue problem: subject to the homogeneous multiple point boundary conditions, where {a

$$P_n (t)y^{(n)} (t) + P_{n - 1} (t)y^{(n - 1)} (t) + \ldots + P_0 (y)y(t) = \lambda Q(t)y(t)$$

$$y(a_i ) = y'(a_i ) = \ldots = y^{(r_i - 1)} (a_i ) = 0,i = 1,2, \ldots ,m$$

_{i}} is a partition of [α, β] of size m N2, {r_{i}}i= 1, m are positive integers that sum to the order n. This generalizes the results for the classical self adjoint problem, n even m=2, r_{1}=*r*_{2}. The principal results, obtained using the theory of μ_{0}-positive operators, state the existence of a least positive eigenvalue λ_{0}(Ω) characterised by whereT is a specified cone in a Banach space and*is an integral operator that is μ***M**_{0}-positive with respect to the coneT.## Keywords

Boundary Condition Banach Space Positive Integer Eigenvalue Problem Integral Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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