Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kind
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 s...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2008-10-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2008/137/abstr.html |
| _version_ | 1818964758769958912 |
|---|---|
| author | Lyudmyla Protsak Igor Parasyuk Yulia Horishna |
| author_facet | Lyudmyla Protsak Igor Parasyuk Yulia Horishna |
| author_sort | Lyudmyla Protsak |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-20T13:06:12Z |
| format | Article |
| id | doaj.art-7d4ac321a88044fcadab0cbc5bfd74e1 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-20T13:06:12Z |
| publishDate | 2008-10-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-7d4ac321a88044fcadab0cbc5bfd74e12022-12-21T19:39:47ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912008-10-012008137,118Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kindLyudmyla ProtsakIgor ParasyukYulia HorishnaWe 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.http://ejde.math.txstate.edu/Volumes/2008/137/abstr.htmlSingular boundary-value problem on the half-linegeneralized Green functionexponential dichotomyNoetherian operator |
| spellingShingle | Lyudmyla Protsak Igor Parasyuk Yulia Horishna Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kind Electronic Journal of Differential Equations Singular boundary-value problem on the half-line generalized Green function exponential dichotomy Noetherian operator |
| title | Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kind |
| title_full | Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kind |
| title_fullStr | Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kind |
| title_full_unstemmed | Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kind |
| title_short | Integral representation of solutions to boundary-value problems on the half-line for linear ODEs with singularity of the first kind |
| title_sort | integral representation of solutions to boundary value problems on the half line for linear odes with singularity of the first kind |
| topic | Singular boundary-value problem on the half-line generalized Green function exponential dichotomy Noetherian operator |
| url | http://ejde.math.txstate.edu/Volumes/2008/137/abstr.html |
| work_keys_str_mv | AT lyudmylaprotsak integralrepresentationofsolutionstoboundaryvalueproblemsonthehalflineforlinearodeswithsingularityofthefirstkind AT igorparasyuk integralrepresentationofsolutionstoboundaryvalueproblemsonthehalflineforlinearodeswithsingularityofthefirstkind AT yuliahorishna integralrepresentationofsolutionstoboundaryvalueproblemsonthehalflineforlinearodeswithsingularityofthefirstkind |