A Class of Definite Boundary Problems

  • William T. Reid
Part of the Applied Mathematical Sciences book series (AMS, volume 31)


Antedating the work of Morse on the extension of the Sturmian theory to self-adjoint differential systems, Bliss [2] considered a real two-point boundary problem which in terms of an n-dimensional vector function y(t) = (yα(t)), (α = l,…, n), may be written as
$$ (a)y'(t) = A(t)y(t) + \lambda B(t)y(t),\quad t \in \left[ {a,b} \right].(b)s\left[ y \right] \equiv My(a) + Ny(b) = 0. $$
For such a system he introduced the concept of “self-adjointness under a real non-singular transformation z = T(t)y”, and considered in detail a special class of such problems which he called “definitely self-adjoint”. This class of problems included the so-called accessory boundary problem for a non-singular simple integral variational problem involving no auxiliary differential equations as restraints, but included the accessory problem for a variational problem of Lagrange or Bolza type only in case the condition of identical normality held.


Vector Function Boundary Problem Differential System Boundary Prob Normality Condition 
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Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  • William T. Reid
    • 1
  1. 1.Department of MathematicsUniversity of OklahomaUSA

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