A limit-point criterion for a class of Sturm-Liouville operators defined in spaces

Author:
R. C. Brown

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2273-2280

MSC (2000):
Primary 47E05, 34C11, 34B24; Secondary 34C10

Published electronically:
March 25, 2004

MathSciNet review:
2052403

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using a recent result of Chernyavskaya and Shuster we show that the maximal operator determined by on , , where and the mean value of computed over all subintervals of of a fixed length is bounded away from zero, shares several standard ``limit-point at " properties of the case. We also show that there is a unique solution of that is in all , .

**1.**R. J. Amos and W. N. Everitt,*On integral inequalities and compact embeddings associated with ordinary differential expressions*, Arch. Rational Mech. Anal.**71**(1979), no. 1, 15–40. MR**522705**, 10.1007/BF00250668**2.**R. C. Brown,*The operator theory of generalized boundary value problems*, Canad. J. Math.**28**(1976), no. 3, 486–512. MR**0412899****3.**N. Chernyavskaya and L. Shuster,*A criterion for correct solvability of the Sturm-Liouville equation in the space 𝐿_{𝑝}(𝐑)*, Proc. Amer. Math. Soc.**130**(2002), no. 4, 1043–1054. MR**1873778**, 10.1090/S0002-9939-01-06145-7**4.**W. N. Everitt,*A note on the Dirichlet condition for second-order differential expressions*, Canad. J. Math.**28**(1976), no. 2, 312–320. MR**0430391****5.**Seymour Goldberg,*Unbounded linear operators: Theory and applications*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0200692****6.**Philip Hartman,*Ordinary differential equations*, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR**658490****7.**Tosio Kato,*Perturbation theory for linear operators*, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR**0407617****8.**M. A. Naĭmark,*Linear differential operators. Part II: Linear differential operators in Hilbert space*, With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968. MR**0262880****9.**Thomas T. Read,*Exponential solutions of 𝑦′′+(𝑟-𝑞)𝑦=0 and the least eigenvalue of Hill’s equation*, Proc. Amer. Math. Soc.**50**(1975), 273–280. MR**0377184**, 10.1090/S0002-9939-1975-0377184-2**10.**G. C. Rota,*Extension theory of differential operators. I*, Comm. Pure Appl. Math.**11**(1958), 23–65. MR**0096852****11.**Walter Rudin,*Functional analysis*, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR**0365062**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47E05,
34C11,
34B24,
34C10

Retrieve articles in all journals with MSC (2000): 47E05, 34C11, 34B24, 34C10

Additional Information

**R. C. Brown**

Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350

Email:
dbrown@gp.as.ua.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07471-4

Keywords:
Second-order differential operators of symmetric form in $L^p$ spaces,
correct solvability,
limit-point,
$L^p$ solutions

Received by editor(s):
December 18, 2002

Published electronically:
March 25, 2004

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2004
American Mathematical Society