# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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## On the existence of eigenvalues of differential operators dependent on a parameterHTML articles powered by AMS MathViewer

by Sh. Strelitz and S. Abramovich
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 $\begin {array}{*{20}{c}} {{{y’}_k}(x) = \sum \limits _{j = 1}^n {{a_{kj}}(x, \lambda ) {y_j}(x)} ,} & {{y_k}(0) = {a_k}(\lambda ),} & {{y_n}(1) = 0,} \\ \end {array}$ $k = 1,..., n$.
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