Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems
UDC 517.9 We consider the problem of existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems:\begin{align}\begin{aligned}& -_{t}D^{\alpha}_{\infty}\left(_{-\infty}D^{\alpha}_{t}x(t)\right)-L(t)x(t)+\nabla W(t,x(t))=0,\\& x\in H^{\alp...
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| Date: | 2023 |
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| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2023
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/328 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860506986914775040 |
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| author | Benhassine, A. Benhassine, Abderrazek Benhassine, A. |
| author_facet | Benhassine, A. Benhassine, Abderrazek Benhassine, A. |
| author_sort | Benhassine, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-03-06T14:26:57Z |
| description | UDC 517.9
We consider the problem of existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems:\begin{align}\begin{aligned}& -_{t}D^{\alpha}_{\infty}\left(_{-\infty}D^{\alpha}_{t}x(t)\right)-L(t)x(t)+\nabla W(t,x(t))=0,\\& x\in H^{\alpha}\left(\mathbb{R},\mathbb{R}^{N}\right),\end{aligned}\tag{FHS}\end{align}where $\alpha\in\left(\dfrac{1}{2},1\right],$ $t\in\mathbb{R},$ $x\in\mathbb{R}^N,$ and $_{-\infty}D^{\alpha}_{t}$  and $_{t}D^{\alpha}_{\infty}$ are the left and right Liouville\,--\,Weyl fractional derivatives of order $\alpha$ on the whole axis $\mathbb{R},$ respectively. The novelty of our results is that, under the assumption that  the nonlinearity $W\in C^{1}\big(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}\big)$ involves a combination of superquadratic and subquadratic terms, for the first time, we show that (FHS) possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of (FHS) goes to infinity and zero, respectively. Some recent results available in the literature are generalized and significantly improved. |
| doi_str_mv | 10.37863/umzh.v75i2.328 |
| first_indexed | 2026-03-24T02:02:08Z |
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| id | umjimathkievua-article-328 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:08Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-3282023-03-06T14:26:57Z Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems TWO DIFFERENT SEQUENCES OF INFINITELY MANY HOMOCLINIC SOLUTIONS FOR A CLASS OF FRACTIONAL HAMILTONIAN SYSTEMS Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems Benhassine, A. Benhassine, Abderrazek Benhassine, A. DIFFERENT SEQUENCES UDC 517.9 We consider the problem of existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems:\begin{align}\begin{aligned}& -_{t}D^{\alpha}_{\infty}\left(_{-\infty}D^{\alpha}_{t}x(t)\right)-L(t)x(t)+\nabla W(t,x(t))=0,\\& x\in H^{\alpha}\left(\mathbb{R},\mathbb{R}^{N}\right),\end{aligned}\tag{FHS}\end{align}where $\alpha\in\left(\dfrac{1}{2},1\right],$ $t\in\mathbb{R},$ $x\in\mathbb{R}^N,$ and $_{-\infty}D^{\alpha}_{t}$  and $_{t}D^{\alpha}_{\infty}$ are the left and right Liouville\,--\,Weyl fractional derivatives of order $\alpha$ on the whole axis $\mathbb{R},$ respectively. The novelty of our results is that, under the assumption that  the nonlinearity $W\in C^{1}\big(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}\big)$ involves a combination of superquadratic and subquadratic terms, for the first time, we show that (FHS) possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of (FHS) goes to infinity and zero, respectively. Some recent results available in the literature are generalized and significantly improved. УДК 517.9 Дві різні нескінченні послідовності  гомоклінічних розв’язків для класу дробових гамільтонових систем Розглянуто питання про існування нескінченної кількості гомоклінічних розв’язків для таких дробових гамільтонових систем:\begin{align}\begin{aligned}& -_{t}D^{\alpha}_{\infty}\left(_{-\infty}D^{\alpha}_{t}x(t)\right)-L(t)x(t)+\nabla W(t,x(t))=0,\\& x\in H^{\alpha}\left(\mathbb{R},\mathbb{R}^{N}\right),\end{aligned}\tag{FHS}\end{align}де $\alpha\in\left(\dfrac{1}{2},1\right],$ $t\in\mathbb{R},$ $x\in\mathbb{R}^N,$ а $_{-\infty}D^{\alpha}_{t}$ і $_{t}D^{\alpha}_{\infty}$ – відповідно ліві та праві дробові похідні Ліувілля–Вейля порядку $\alpha$ на всій осі $\mathbb{R}$. Новизна отриманих результатів полягає в тому, що у випадку, коли нелінійність $W\in C^{1}\big(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}\big)$ містить комбінацію суперквадратичних і субквадратичних членів, уперше показано за допомогою теореми Фонтена та дуальної теореми Фонтена, що (FHS) містить дві різні нескінченні послідовності  гомоклінічних розв’язків такі, що відповідний енергетичний функціонал (FHS) прямує до нескінченності та нуля, відповідно. Деякі останні результати, відомі з літератури, узагальнено та значно покращено. Institute of Mathematics, NAS of Ukraine 2023-03-02 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/328 10.37863/umzh.v75i2.328 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 2 (2023); 155 - 167 Український математичний журнал; Том 75 № 2 (2023); 155 - 167 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/328/9793 Copyright (c) 2023 Abderrazek Benhassine |
| spellingShingle | Benhassine, A. Benhassine, Abderrazek Benhassine, A. Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems |
| title | Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems |
| title_alt | TWO DIFFERENT SEQUENCES OF INFINITELY MANY HOMOCLINIC SOLUTIONS FOR A CLASS OF FRACTIONAL HAMILTONIAN SYSTEMS Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems |
| title_full | Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems |
| title_fullStr | Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems |
| title_full_unstemmed | Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems |
| title_short | Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systems |
| title_sort | two different sequences of infinitely many homoclinic solutions for a class of fractional hamiltonian systems |
| topic_facet | DIFFERENT SEQUENCES |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/328 |
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