Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term
In this paper, we first obtain three inequalities and two of them, in some sense, generalize Sobolev's inequality and Wirtinger's inequality from periodic case to quasi-periodic case, respectively. Then by using the least action principle and the saddle point theorem, under subquadratic ca...
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| Format: | Article |
| Language: | English |
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University of Szeged
2014-12-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=3146 |
| _version_ | 1797830642170331136 |
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| author | Xingyong Zhang |
| author_facet | Xingyong Zhang |
| author_sort | Xingyong Zhang |
| collection | DOAJ |
| description | In this paper, we first obtain three inequalities and two of them, in some sense, generalize Sobolev's inequality and Wirtinger's inequality from periodic case to quasi-periodic case, respectively. Then by using the least action principle and the saddle point theorem, under subquadratic case, we obtain two existence results of weak quasi-periodic solutions for the second order Hamiltonian system:
$$\frac{d[P(t)\dot{u}(t)]}{dt}=\nabla F(t,u(t))+ e(t),$$
which generalize and improve the corresponding results in recent literature [J. Kuang, Abstr. Appl. Anal. 2012, Art. ID 271616]. Moreover, when the assumptions $F(t,x)=F(t,-x)$ and $e(t)\equiv 0$ are also made, we obtain two results on existence of infinitely many weak quasi-periodic solutions for the second order Hamiltonian system under the subquadratic case.} |
| first_indexed | 2024-04-09T13:39:21Z |
| format | Article |
| id | doaj.art-5043fa8b4bae453088e1db549c3c4c69 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:39:21Z |
| publishDate | 2014-12-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-5043fa8b4bae453088e1db549c3c4c692023-05-09T07:53:04ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752014-12-0120146311910.14232/ejqtde.2014.1.633146Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing termXingyong Zhang0Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, P. R. ChinaIn this paper, we first obtain three inequalities and two of them, in some sense, generalize Sobolev's inequality and Wirtinger's inequality from periodic case to quasi-periodic case, respectively. Then by using the least action principle and the saddle point theorem, under subquadratic case, we obtain two existence results of weak quasi-periodic solutions for the second order Hamiltonian system: $$\frac{d[P(t)\dot{u}(t)]}{dt}=\nabla F(t,u(t))+ e(t),$$ which generalize and improve the corresponding results in recent literature [J. Kuang, Abstr. Appl. Anal. 2012, Art. ID 271616]. Moreover, when the assumptions $F(t,x)=F(t,-x)$ and $e(t)\equiv 0$ are also made, we obtain two results on existence of infinitely many weak quasi-periodic solutions for the second order Hamiltonian system under the subquadratic case.}http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=3146second order hamiltonian systemweak quasi-periodic solutionvariational methodsubquadratic case |
| spellingShingle | Xingyong Zhang Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term Electronic Journal of Qualitative Theory of Differential Equations second order hamiltonian system weak quasi-periodic solution variational method subquadratic case |
| title | Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term |
| title_full | Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term |
| title_fullStr | Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term |
| title_full_unstemmed | Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term |
| title_short | Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term |
| title_sort | existence and multiplicity of weak quasi periodic solutions for second order hamiltonian system with a forcing term |
| topic | second order hamiltonian system weak quasi-periodic solution variational method subquadratic case |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=3146 |
| work_keys_str_mv | AT xingyongzhang existenceandmultiplicityofweakquasiperiodicsolutionsforsecondorderhamiltoniansystemwithaforcingterm |