Three-point third-order problems with a sign-changing nonlinear term
In this article we study a well-known boundary value problem $$\displaylines{ u'''(t) = f(t, u(t)), \quad 0 < t < 1, \cr u(0) = u'(1/2) = u''(1)=0. }$$ With $u'(\eta)=0$ in place of $u'(1/2)=0$, many authors studied the existence of positive soluti...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2014-08-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2014/175/abstr.html |
| _version_ | 1818252647175553024 |
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| author | Johnny Henderson Nickolai Kosmatov |
| author_facet | Johnny Henderson Nickolai Kosmatov |
| author_sort | Johnny Henderson |
| collection | DOAJ |
| description | In this article we study a well-known boundary value problem
$$\displaylines{
u'''(t) = f(t, u(t)), \quad 0 < t < 1, \cr
u(0) = u'(1/2) = u''(1)=0.
}$$
With $u'(\eta)=0$ in place of $u'(1/2)=0$, many authors studied the existence
of positive solutions of both the positone problems with $\eta \geq 1/2$ and
the semi-positone problems for $\eta > 1/2$. It is well-known that the standard
method successfully applied to the semi-positone problem with $\eta > 1/2$
does not work for $\eta =1/2$ in the same setting. We treat the latter as a
problem with a sign-changing term rather than a semi-positone problem.
We apply Krasnosel'skii's fixed point theorem [4] to obtain positive solutions. |
| first_indexed | 2024-12-12T16:27:30Z |
| format | Article |
| id | doaj.art-15ab286d0fe8447c824236bcc9e8c65a |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-12T16:27:30Z |
| publishDate | 2014-08-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-15ab286d0fe8447c824236bcc9e8c65a2022-12-22T00:18:51ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912014-08-012014175,110Three-point third-order problems with a sign-changing nonlinear termJohnny Henderson0Nickolai Kosmatov1 Baylor Univ., Waco, TX, USA Univ. of Arkansas, Little Rock, AR, USA In this article we study a well-known boundary value problem $$\displaylines{ u'''(t) = f(t, u(t)), \quad 0 < t < 1, \cr u(0) = u'(1/2) = u''(1)=0. }$$ With $u'(\eta)=0$ in place of $u'(1/2)=0$, many authors studied the existence of positive solutions of both the positone problems with $\eta \geq 1/2$ and the semi-positone problems for $\eta > 1/2$. It is well-known that the standard method successfully applied to the semi-positone problem with $\eta > 1/2$ does not work for $\eta =1/2$ in the same setting. We treat the latter as a problem with a sign-changing term rather than a semi-positone problem. We apply Krasnosel'skii's fixed point theorem [4] to obtain positive solutions.http://ejde.math.txstate.edu/Volumes/2014/175/abstr.htmlGreen's functionfixed point theorempositive solutionsthird-order boundary-value problem |
| spellingShingle | Johnny Henderson Nickolai Kosmatov Three-point third-order problems with a sign-changing nonlinear term Electronic Journal of Differential Equations Green's function fixed point theorem positive solutions third-order boundary-value problem |
| title | Three-point third-order problems with a sign-changing nonlinear term |
| title_full | Three-point third-order problems with a sign-changing nonlinear term |
| title_fullStr | Three-point third-order problems with a sign-changing nonlinear term |
| title_full_unstemmed | Three-point third-order problems with a sign-changing nonlinear term |
| title_short | Three-point third-order problems with a sign-changing nonlinear term |
| title_sort | three point third order problems with a sign changing nonlinear term |
| topic | Green's function fixed point theorem positive solutions third-order boundary-value problem |
| url | http://ejde.math.txstate.edu/Volumes/2014/175/abstr.html |
| work_keys_str_mv | AT johnnyhenderson threepointthirdorderproblemswithasignchangingnonlinearterm AT nickolaikosmatov threepointthirdorderproblemswithasignchangingnonlinearterm |