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

DOI:
https://doi.org/10.1090/S0002-9947-1976-0404796-5

MathSciNet review:
0404796

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the equation

() |

where is a Banach space valued function taking values in a dense subdomain of the Banach space

*B*. Here

*A*is a closed (possibly unbounded) linear operator on 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 as well as sufficient conditions on the point spectrum to insure uniqueness of weak solutions.

*u*is only required to satisfy (a) as if , (b) as , (c) as . 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

() |

The proofs rest upon an application of the Hahn-Banach Theorem and the consequent separation properties of , 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