Annali di Matematica Pura ed Applicata

, Volume 126, Issue 1, pp 223–232 | Cite as

The existence and extremal characterization of eigenvalues for ann-th order multiple point boundary value problem

  • R. D. Gentry
  • C. C. Travis


The existence and extremal characterization of least positive eigenvalues is considered for the n-th order multiple point eigenvalue problem:
$$P_n (t)y^{(n)} (t) + P_{n - 1} (t)y^{(n - 1)} (t) + \ldots + P_0 (y)y(t) = \lambda Q(t)y(t)$$
subject to the homogeneous multiple point boundary conditions,
$$y(a_i ) = y'(a_i ) = \ldots = y^{(r_i - 1)} (a_i ) = 0,i = 1,2, \ldots ,m$$
where {ai} is a partition of [α, β] of size m N2, {ri}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, r1=r2. The principal results, obtained using the theory of μ0-positive operators, state the existence of a least positive eigenvalue λ0(Ω) characterised by
whereT is a specified cone in a Banach space andM is an integral operator that is μ0-positive with respect to the coneT.


Boundary Condition Banach Space Positive Integer Eigenvalue Problem Integral Operator 
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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1980

Authors and Affiliations

  • R. D. Gentry
    • 1
  • C. C. Travis
    • 2
  1. 1.Guelph
  2. 2.Oak Ridge

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