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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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Uniqueness criteria for solutions of singular boundary value problems
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by D. R. Dunninger and Howard A. Levine PDF
Trans. Amer. Math. Soc. 221 (1976), 289-301 Request permission

Abstract:

In this paper we consider the equation \begin{equation}\tag {$1$} u''(t) + (k/t)uā€™(t) + Au(t) = 0,\quad 0 < t < T,\quad u(T) = 0,\end{equation} where $u:(0,T) \to D(A) \subset B$ is a Banach space valued function taking values in a dense subdomain $D(A)$ of the Banach space B. Here A is a closed (possibly unbounded) linear operator on $D(A)$ while k is a real constant. The differential equation is an abstract Euler-Poisson-Darboux equation. We give necessary and sufficient conditions on the point spectrum of A to insure uniqueness of the strong solution $u \equiv 0$ as well as sufficient conditions on the point spectrum to insure uniqueness of weak solutions. u is only required to satisfy (a) ${t^k}\left \| {uā€™(t)} \right \| \to {0^ + }$ as $t \to {0^ + }$ if $k > 1$, (b) ${t^k}\left \| {uā€™(t)} \right \| + {t^{k + 1}}\left \| {u(t)} \right \| \to 0$ as $t \to {0^ + },0 < k \leqslant 1$, (c) $t\left \| {uā€™(t)} \right \| + \left \| {u(t)} \right \| \to 0$ as $t \to {0^ + },k < 0$. The operator A need not possess a complete set of eigenvectors nor need one have a backward uniqueness theorem available for (1) for the Cauchy final value problem. Our techniques extend to the n-axially symmetric abstract equation \begin{equation}\tag {$2$} \sum \limits _{i = 1}^n {[{\partial ^2}u/\partial t_i^2 + ({k_i}/{t_i})\partial u/\partial {t_i}] + Au = 0.} \end{equation} The proofs rest upon an application of the Hahn-Banach Theorem and the consequent separation properties of ${B^\ast }$, the dual of B, as well as the completeness properties of the eigenfunctions of certain Bessel equations associated with (1).
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 221 (1976), 289-301
  • MSC: Primary 34G05; Secondary 35Q05, 35R20
  • DOI: https://doi.org/10.1090/S0002-9947-1976-0404796-5
  • MathSciNet review: 0404796