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Transactions of the American Mathematical Society

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

 

 

Uniqueness criteria for solutions of singular boundary value problems


Authors: D. R. Dunninger and Howard A. Levine
Journal: Trans. Amer. Math. Soc. 221 (1976), 289-301
MSC: Primary 34G05; Secondary 35Q05, 35R20
MathSciNet review: 0404796
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Abstract: In this paper we consider the equation

$\displaystyle u''(t) + (k/t)u'(t) + Au(t) = 0,\quad 0 < t < T,\quad u(T) = 0,$ ($ 1$)

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\Vert {u'(t)} \right\Vert \to {0^ + }$ as $ t \to {0^ + }$ if $ k > 1$, (b) $ {t^k}\left\Vert {u'(t)} \right\Vert + {t^{k + 1}}\left\Vert {u(t)} \right\Vert \to 0$ as $ t \to {0^ + },0 < k \leqslant 1$, (c) $ t\left\Vert {u'(t)} \right\Vert + \left\Vert {u(t)} \right\Vert \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

$\displaystyle \sum\limits_{i = 1}^n {[{\partial ^2}u/\partial t_i^2 + ({k_i}/{t_i})\partial u/\partial {t_i}] + Au = 0.}$ ($ 2$)

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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DOI: https://doi.org/10.1090/S0002-9947-1976-0404796-5
Article copyright: © Copyright 1976 American Mathematical Society