Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications”
In the recent paper W. Shen and T. He and G. Dai and X. Han established unilateral global bifurcation result for a class of nonlinear fourth-order eigenvalue problems. They show the existence of two families of unbounded continua of nontrivial solutions of these problems bifurcating from the points...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2017/1024950 |
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| _version_ | 1832556768084885504 |
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| author | Ziyatkhan Aliyev |
| author_facet | Ziyatkhan Aliyev |
| author_sort | Ziyatkhan Aliyev |
| collection | DOAJ |
| description | In the recent paper W. Shen and T. He and G. Dai and X. Han established unilateral global bifurcation result for a class of nonlinear fourth-order eigenvalue problems. They show the existence of two families of unbounded continua of nontrivial solutions of these problems bifurcating from the points and intervals of the line trivial solutions, corresponding to the positive or negative eigenvalues of the linear problem. As applications of this result, these authors study the existence of nodal solutions for a class of nonlinear fourth-order eigenvalue problems with sign-changing weight. Moreover, they also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight. In the present comment, we show that these papers of above authors contain serious errors and, therefore, unfortunately, the results of these works are not true. Note also that the authors used the results of the recent work by G. Dai which also contain gaps. |
| format | Article |
| id | doaj-art-a5d3302349194ce4a48ddedbf4240463 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a5d3302349194ce4a48ddedbf42404632025-02-03T05:44:30ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2017-01-01201710.1155/2017/10249501024950Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications”Ziyatkhan Aliyev0Baku State University, AZ1148 Baku, AzerbaijanIn the recent paper W. Shen and T. He and G. Dai and X. Han established unilateral global bifurcation result for a class of nonlinear fourth-order eigenvalue problems. They show the existence of two families of unbounded continua of nontrivial solutions of these problems bifurcating from the points and intervals of the line trivial solutions, corresponding to the positive or negative eigenvalues of the linear problem. As applications of this result, these authors study the existence of nodal solutions for a class of nonlinear fourth-order eigenvalue problems with sign-changing weight. Moreover, they also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight. In the present comment, we show that these papers of above authors contain serious errors and, therefore, unfortunately, the results of these works are not true. Note also that the authors used the results of the recent work by G. Dai which also contain gaps.http://dx.doi.org/10.1155/2017/1024950 |
| spellingShingle | Ziyatkhan Aliyev Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications” Discrete Dynamics in Nature and Society |
| title | Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications” |
| title_full | Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications” |
| title_fullStr | Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications” |
| title_full_unstemmed | Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications” |
| title_short | Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications” |
| title_sort | comment on unilateral global bifurcation from intervals for fourth order problems and its applications |
| url | http://dx.doi.org/10.1155/2017/1024950 |
| work_keys_str_mv | AT ziyatkhanaliyev commentonunilateralglobalbifurcationfromintervalsforfourthorderproblemsanditsapplications |