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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
Article
  • 23 Downloads

Summary

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.

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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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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