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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Quadratic functionals of second order


Author: Walter Leighton
Journal: Trans. Amer. Math. Soc. 151 (1970), 309-322
MSC: Primary 49.00; Secondary 34.00
MathSciNet review: 0264485
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Abstract: In this paper we study the minimizing of the general second-order quadratic functional (1.3) in a class of admissible functions $ y(x)$ with fixed endpoint conditions on $ y(x)$ and its derivative at $ x = a$ and at $ x = b$. Necessary conditions and sufficient conditions are obtained. These lead, in turn, to natural extensions of the Sturm comparison theorem to fourth-order selfadjoint equations. These extensions include and are more general than previously stated comparison theorems. Finally, it is found that the present variational theory provides an orderly approach to second-order Wirtinger-like inequalities, and the results include as special cases many published results of this type.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0264485-1
PII: S 0002-9947(1970)0264485-1
Keywords: Quadratic functional of second order, selfadjoint fourth-order differential equation, second variation, minimum of a functional, Euler equation, conjugate point, admissible variation, comparison theorem, Wirtinger inequality
Article copyright: © Copyright 1970 American Mathematical Society