On the existence of eigenvalues of differential operators dependent on a parameter
Authors:
Sh. Strelitz and S. Abramovich
Journal:
Trans. Amer. Math. Soc. 258 (1980), 407429
MSC:
Primary 34B10; Secondary 30E25, 34A20
DOI:
https://doi.org/10.1090/S00029947198005581817
MathSciNet review:
558181
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: In this paper we obtain results about the existence of eigenvalues for a system which depends polynomially on $\lambda$, \[ \begin {array}{*{20}{c}} {{{u’}_k}(x) = \sum \limits _{j = 1}^n {{b_{kj}}(x, \lambda ){u_j}(x),} } & {\sum \limits _{i = 0}^p {\sum \limits _{j = 1}^N {a_{kj}^i{u_j}({x_i}) = 0,} } } \\ \end {array} \] , $k = 1,..., N$. In order to get these results we prove that this system can be reduced to a standard system of the form \[ \begin {array}{*{20}{c}} {{{y’}_k}(x) = \sum \limits _{j = 1}^n {{a_{kj}}(x, \lambda ) {y_j}(x)} ,} & {{y_k}(0) = {a_k}(\lambda ),} & {{y_n}(1) = 0,} \\ \end {array} \] $k = 1,..., n$.

G. A. Bliss, Algebraic functions, Amer. Math. Soc. Colloq. Publ., vol. 16, Amer. Math. Soc., Providence, R. I., 1933; Chapter 2.
 Ju. È. Degutis and Š. I. Strelic, The existence of eigenvalues for a certain differential operator that depends on a parameter, Litovsk. Mat. Sb. 11 (1971), 535–556 (Russian, with Lithuanian and English summaries). MR 0298469 F. R. Gantmacher, The theory of matrices, vol. 2, Chelsea, New York, 1959.
 M. V. Keldyš, On the characteristic values and characteristic functions of certain classes of nonselfadjoint equations, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 11–14 (Russian). MR 0041353
 B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975
 M. A. Naĭmark, Spectral analysis of nonselfadjoint operators, Amer. Math. Soc. Transl. (2) 20 (1962), 55–75. MR 0137004
 I. G. Petrovski, Ordinary differential equations, PrenticeHall, Inc., Englewood Cliffs, N.J., 1966. Revised English edition. Translated from the Russian and edited by Richard A. Silverman. MR 0193298
 Š. I. Strelic and Ju. È. Degutis, A certain method of proof for the existence of eigenvalues for a certain differential operator with boundary conditions that depend on a parameter, Litovsk. Mat. Sb. 11 (1971), 683–690 (Russian, with Lithuanian and English summaries). MR 0299866
 Š. Strelic and V. Nekrašas, The existence of eigenvalues for a system of linear differential operators that depend on a parameter, Litovsk. Mat. Sb. 13 (1973), no. 3, 191–209, 237 (Russian, with Lithuanian and English summaries). MR 0328197
Retrieve articles in Transactions of the American Mathematical Society with MSC: 34B10, 30E25, 34A20
Retrieve articles in all journals with MSC: 34B10, 30E25, 34A20
Additional Information
Keywords:
Eigenvalues,
algebraic functions,
order of entire functions,
asymptotic expansion
Article copyright:
© Copyright 1980
American Mathematical Society