Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications
We establish a unilateral global bifurcation result from interval for a class of fourth-order problems with nondifferentiable nonlinearity. By applying the above result, we firstly establish the spectrum for a class of half-linear fourth-order eigenvalue problems. Moreover, we also investigate the e...
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Wiley
2016-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2016/5956713 |
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| author | Wenguo Shen Tao He |
| author_facet | Wenguo Shen Tao He |
| author_sort | Wenguo Shen |
| collection | DOAJ |
| description | We establish a unilateral global bifurcation result from interval for a class of fourth-order problems with nondifferentiable nonlinearity. By applying the above result, we firstly establish the spectrum for a class of half-linear fourth-order eigenvalue problems. Moreover, we also investigate the existence of nodal solutions for the following half-linear fourth-order problems: x″″=αx++βx-+ratfx, 0<t<1, x(0)=x(1)=x″(0)=x″(1)=0, where r≠0 is a parameter, a∈C([0,1],(0,∞)), x+=max{x,0}, x-=-min{x,0}, α,β∈C[0,1], and f∈C(R,R), sf(s)>0, for s≠0. We give the intervals for the parameter r which ensure the existence of nodal solutions for the above fourth-order half-linear problems if f0∈[0,∞) or f∞∈[0,∞], where f0=lims→0f(s)/s and f∞=lims→+∞f(s)/s. We use the unilateral global bifurcation techniques and the approximation of connected components to prove our main results. |
| format | Article |
| id | doaj-art-25700ba51d6444a39943d0145a1ff725 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-25700ba51d6444a39943d0145a1ff7252025-02-03T01:02:17ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2016-01-01201610.1155/2016/59567135956713Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its ApplicationsWenguo Shen0Tao He1Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou 730050, ChinaDepartment of Basic Courses, Lanzhou Institute of Technology, Lanzhou 730050, ChinaWe establish a unilateral global bifurcation result from interval for a class of fourth-order problems with nondifferentiable nonlinearity. By applying the above result, we firstly establish the spectrum for a class of half-linear fourth-order eigenvalue problems. Moreover, we also investigate the existence of nodal solutions for the following half-linear fourth-order problems: x″″=αx++βx-+ratfx, 0<t<1, x(0)=x(1)=x″(0)=x″(1)=0, where r≠0 is a parameter, a∈C([0,1],(0,∞)), x+=max{x,0}, x-=-min{x,0}, α,β∈C[0,1], and f∈C(R,R), sf(s)>0, for s≠0. We give the intervals for the parameter r which ensure the existence of nodal solutions for the above fourth-order half-linear problems if f0∈[0,∞) or f∞∈[0,∞], where f0=lims→0f(s)/s and f∞=lims→+∞f(s)/s. We use the unilateral global bifurcation techniques and the approximation of connected components to prove our main results.http://dx.doi.org/10.1155/2016/5956713 |
| spellingShingle | Wenguo Shen Tao He Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications Discrete Dynamics in Nature and Society |
| title | Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications |
| title_full | Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications |
| title_fullStr | Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications |
| title_full_unstemmed | Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications |
| title_short | Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications |
| title_sort | unilateral global bifurcation from intervals for fourth order problems and its applications |
| url | http://dx.doi.org/10.1155/2016/5956713 |
| work_keys_str_mv | AT wenguoshen unilateralglobalbifurcationfromintervalsforfourthorderproblemsanditsapplications AT taohe unilateralglobalbifurcationfromintervalsforfourthorderproblemsanditsapplications |