Some new analytical techniques and their application to irregular cases for the third order ordinary linear boundaryvalue problem
Author:
Nathaniel R. Stanley
Journal:
Trans. Amer. Math. Soc. 101 (1961), 351376
MSC:
Primary 34.30
Erratum:
Trans. Amer. Math. Soc. 103 (1962), 559.
Erratum:
Trans. Amer. Math. Soc. 102 (1962), 545.
MathSciNet review:
0130420
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: 1. For the operator defined by and a triple of boundary conditions irregular in the sense of Birkhoff, the reduction of this triple to canonical forms is implicit in the reduction made for a more general third order operator (Theorem 1.2). 2. A new technique is developed for calculating the Green's function for the nth order ordinary linear boundaryvalue problem (Theorem 2.4), and is applied to ; a necessary and sufficient condition is given for the identification of degenerate sets of boundary conditions for (Theorem 2.6). 3. A new technique is developed for calculating asymptotic expansions for large zeros of exponential sums, and the form of the expansion, which includes a logarithmic asymptotic series, is established by induction (Theorem 3.1); expansions for the cube roots of the eigenvalues of then follow as special cases. 4. A theorem of Dunford and Schwartz (Theorem 4.0) giving a sufficient condition for completeness of eigenfunctions in terms of growth of the norm of the resolvent operator, is applied to prove that, with a possible exception, the eigenfunctions of span (Theorem 4.5).
 [1]
George
D. Birkhoff, On the asymptotic character of the
solutions of certain linear differential equations containing a
parameter, Trans. Amer. Math. Soc.
9 (1908), no. 2,
219–231. MR
1500810, http://dx.doi.org/10.1090/S00029947190815008101
 [2]
George
D. Birkhoff, Boundary value and expansion problems
of ordinary linear differential equations, Trans. Amer. Math. Soc. 9 (1908), no. 4, 373–395. MR
1500818, http://dx.doi.org/10.1090/S00029947190815008186
 [3]
Earl
A. Coddington and Norman
Levinson, Theory of ordinary differential equations,
McGrawHill Book Company, Inc., New YorkTorontoLondon, 1955. MR 0069338
(16,1022b)
 [4]
R.
Courant and D.
Hilbert, Methods of mathematical physics. Vol. I, Interscience
Publishers, Inc., New York, N.Y., 1953. MR 0065391
(16,426a)
 [5]
Nelson
Dunford and Jacob
T. Schwartz, Linear operators. Part II, Wiley Classics
Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory.
Selfadjoint operators in Hilbert space; With the assistance of William G.
Bade and Robert G. Bartle; Reprint of the 1963 original; A
WileyInterscience Publication. MR 1009163
(90g:47001b)
 [6]
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. 3, New York, McGrawHill Book Co., Inc., 1955, pp. 212217.
 [7]
S. Hoffman, Second order linear differential operators defined by irregular boundary conditions, Ph.D. dissertation, Yale, 1957.
 [8]
James
W. Hopkins, Some convergent developments
associated with irregular boundary conditions, Trans. Amer. Math. Soc. 20 (1919), no. 3, 245–259. MR
1501125, http://dx.doi.org/10.1090/S00029947191915011255
 [9]
E.
L. Ince, Ordinary Differential Equations, Dover Publications,
New York, 1944. MR 0010757
(6,65f)
 [10]
Thomas
Muir, A treatise on the theory of determinants, Revised and
enlarged by William H. Metzler, Dover Publications, Inc., New York, 1960.
MR
0114826 (22 #5644)
 [11]
Frigyes
Riesz and Béla
Sz.Nagy, Functional analysis, Frederick Ungar Publishing Co.,
New York, 1955. Translated by Leo F. Boron. MR 0071727
(17,175i)
 [12]
J.
Schwartz, Perturbations of spectral operators, and applications. I.
Bounded perturbations, Pacific J. Math. 4 (1954),
415–458. MR 0063568
(16,144b)
 [13]
E. Schwengeler, Geometrisches über die Verteilung der Nullstellen spezieller ganzer Funktionen (Exponentialsummen), Ph.D. dissertation, Zürich, 1926.
 [14]
M.
H. Stone, Irregular differential systems of
order two and the related expansion problems, Trans. Amer. Math. Soc. 29 (1927), no. 1, 23–53. MR
1501375, http://dx.doi.org/10.1090/S00029947192715013752
 [15]
, Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Colloquium Publications, vol. 15, 1932.
 [16]
Lewis
E. Ward, Some thirdorder irregular boundary
value problems, Trans. Amer. Math. Soc.
29 (1927), no. 4,
716–745. MR
1501411, http://dx.doi.org/10.1090/S00029947192715014113
 [1]
 G. D. Birkhoff, On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc. vol. 9 (1908) pp. 219231. MR 1500810
 [2]
 , Boundaryvalue and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc. vol. 9 (1908) pp. 373395. MR 1500818
 [3]
 E. A. Coddington and N. Levinson, Theory of ordinary differential equations, New York, McGrawHill Book Co., Inc., 1955. MR 0069338 (16:1022b)
 [4]
 R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, New York, Interscience Publishers, Inc., 1953. MR 0065391 (16:426a)
 [5]
 N. Dunford and J. T. Schwartz, Linear operators. Part II, New York, Interscience Publishers, Inc. (forthcoming). MR 1009163 (90g:47001b)
 [6]
 A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. 3, New York, McGrawHill Book Co., Inc., 1955, pp. 212217.
 [7]
 S. Hoffman, Second order linear differential operators defined by irregular boundary conditions, Ph.D. dissertation, Yale, 1957.
 [8]
 J. W. Hopkins, Some convergent developments associated with irregular boundary conditions, Trans. Amer. Math. Soc. vol. 20 (1919), pp. 249259. MR 1501125
 [9]
 E. L. Ince, Ordinary differential equations, New York, Dover Publications, Inc., 1956. MR 0010757 (6:65f)
 [10]
 T. Muir, A treatise on the theory of determinants, New York, Dover Publications, Inc., 1960, pp. 213216. MR 0114826 (22:5644)
 [11]
 F. Riesz and B. SzNagy, Functional analysis, New York, Ungar Publishing Co., 1955, pp. 145151. MR 0071727 (17:175i)
 [12]
 J. T. Schwartz, Perturbations of spectral operators and applications. I, Pacific J. Math. vol. 4 (1954) pp. 415458. MR 0063568 (16:144b)
 [13]
 E. Schwengeler, Geometrisches über die Verteilung der Nullstellen spezieller ganzer Funktionen (Exponentialsummen), Ph.D. dissertation, Zürich, 1926.
 [14]
 M. H. Stone, Irregular differential systems of order two and the related expansion problems, Trans. Amer. Math. Soc. vol. 29 (1927) pp. 2353. MR 1501375
 [15]
 , Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Colloquium Publications, vol. 15, 1932.
 [16]
 L. E. Ward, Some thirdorder irregular boundary value problems, Trans. Amer. Math. Soc. vol. 29 (1927) pp. 716745. MR 1501411
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
34.30
Retrieve articles in all journals
with MSC:
34.30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947196101304204
PII:
S 00029947(1961)01304204
Article copyright:
© Copyright 1961
American Mathematical Society
