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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the existence of eigenvalues of differential operators dependent on a parameter
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by Sh. Strelitz and S. Abramovich PDF
Trans. Amer. Math. Soc. 258 (1980), 407-429 Request permission

Abstract:

In this paper we obtain results about the existence of eigenvalues for a system which depends polynomially on $\lambda$, \[ \begin {array}{*{20}{c}} {{{u’}_k}(x) = \sum \limits _{j = 1}^n {{b_{kj}}(x, \lambda ){u_j}(x),} } & {\sum \limits _{i = 0}^p {\sum \limits _{j = 1}^N {a_{kj}^i{u_j}({x_i}) = 0,} } } \\ \end {array} \] , $k = 1,..., N$. In order to get these results we prove that this system can be reduced to a standard system of the form \[ \begin {array}{*{20}{c}} {{{y’}_k}(x) = \sum \limits _{j = 1}^n {{a_{kj}}(x, \lambda ) {y_j}(x)} ,} & {{y_k}(0) = {a_k}(\lambda ),} & {{y_n}(1) = 0,} \\ \end {array} \] $k = 1,..., n$.
References
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 258 (1980), 407-429
  • MSC: Primary 34B10; Secondary 30E25, 34A20
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0558181-7
  • MathSciNet review: 558181