Open Access

# On the completeness of general boundary value problems for $2 \times 2$ first-order systems of ordinary differential equations

### Abstract

Let $B={\rm diag} (b_1^{-1}, b_2^{-1}) \not = B^*$ be a $2\times 2$ diagonal matrix with \break $b_1^{-1}b_2 \notin{\Bbb R}$ and let $Q$ be a smooth $2\times 2$ matrix function. Consider the system $$-i B y'+Q(x)y=\lambda y, \; y= {\rm col}(y_1,y_2), \; x\in[0,1],$$ of ordinary differential equations subject to general linear boundary conditions $U_1(y) = U_2(y) = 0.$ We find sufficient conditions on $Q$ and $U_j$ that guaranty completeness of root vector system of the boundary value problem. Moreover, we indicate a condition on $Q$ that leads to a completeness criterion in terms of the linear boundary forms $U_j,\ j\in \{1,2\}.$

### Article Information

 Title On the completeness of general boundary value problems for $2 \times 2$ first-order systems of ordinary differential equations Source Methods Funct. Anal. Topology, Vol. 18 (2012), no. 1, 4-18 MathSciNet MR2953328 zbMATH 1249.34249 Copyright The Author(s) 2012 (CC BY-SA)

### Authors Information

A. V. Agibalova
Donets'k National University, 24 Universytets'ka, Donets'k, 83055, Ukraine

M. M. Malamud
Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine, 74 R. Luxemburg, Donets'k, 83114, Ukraine

L. L. Oridoroga
Donets'k National University, 24 Universytets'ka, Donets'k, 83055, Ukraine

### Citation Example

A. V. Agibalova, M. M. Malamud, and L. L. Oridoroga, On the completeness of general boundary value problems for $2 \times 2$ first-order systems of ordinary differential equations, Methods Funct. Anal. Topology 18 (2012), no. 1, 4-18.

### BibTex

@article {MFAT625,
AUTHOR = {Agibalova, A. V. and Malamud, M. M. and Oridoroga, L. L.},
TITLE = {On the completeness of general boundary value problems for $2 \times 2$ first-order systems of ordinary differential  equations},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {18},
YEAR = {2012},
NUMBER = {1},
PAGES = {4-18},
ISSN = {1029-3531},
MRNUMBER = {MR2953328},
ZBLNUMBER = {1249.34249},
URL = {http://mfat.imath.kiev.ua/article/?id=625},
}