Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 137, pp. 1-18.
Title: Integral representation of solutions to boundary-value problems
on the half-line for linear ODEs with singularity of the first kind
Authors: Yulia Horishna (National Taras Shevchenko Univ, Kyiv, Ukraine)
Igor Parasyuk (National Taras Shevchenko Univ, Kyiv, Ukraine)
Lyudmyla Protsak (National Pedagogical Dragomanov Univ, Kyiv, Ukraine)
Abstract:
We study the existence of solutions
to a non-homogeneous system of linear ODEs which
has the pole of first order at $x=0$; these solutions should
vanish at infinity and be continuously differentiable on $[0,\infty)$.
The resonant case where the corresponding homogeneous problem
has nontrivial solutions is of great interest to us.
Under the conditions that the homogeneous system is exponentially
dichotomic on $[1,\infty)$ and the residue of system's
operator at $x=0$ does not have eigenvalues with real part 1, we
construct the so-called generalized Green function. We also establish
conditions under which the main non-homogeneous problem can be
reduced to the Noetherian problem with nonzero index.
Submitted May 19, 2008. Published October 09, 2008.
Math Subject Classifications: 34B16, 34B05, 34B27.
Key Words: Singular boundary-value problem on the half-line;
generalized Green function; exponential dichotomy;
Noetherian operator.