Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kind

• Main Authors: Lyudmyla Protsak, Igor Parasyuk, Yulia Horishna
• Format: Article
• Language: English
• Published: Texas State University 2008-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: Singular boundary-value problem on the half-line; generalized Green function; exponential dichotomy; Noetherian operator
• Online Access: http://ejde.math.txstate.edu/Volumes/2008/137/abstr.html
• ISSN: 1072-6691

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.