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TMF, 2018, Volume 195, Number 1, Pages 81–90 (Mi tmf9371)  

This article is cited in 1 scientific paper (total in 1 paper)

Fractional Hamiltonian systems with locally defined potentials

A. B. Benhassine

Department of Mathematics, Higher Institute of Informatics and Mathematics, Monastir, Tunisia

Abstract: We study solutions of the nonperiodic fractional Hamiltonian systems
$$ - _tD^{\alpha}_{\infty}( _{-\infty} D_{t}^{\alpha}x(t))-L(t)x(t)+ \nabla W(t,x(t))=0,\quad x\in H^\alpha(\mathbb{R},\mathbb{R}^N), $$
where $\alpha\in(1/2,1]$, $t\in\mathbb R$, $L(t)\in C(\mathbb R,\mathbb R^{N^2})$, and $ _{-\infty}D^{\alpha}_{t}$ and $ _tD^{\alpha}_{\infty}$ are the respective left and right Liouville–Weyl fractional derivatives of order $\alpha$ on the whole axis $\mathbb R$. Using a new symmetric mountain pass theorem established by Kajikia, we prove the existence of infinitely many solutions for this system in the case where the matrix $L(t)$ is not necessarily coercive nor uniformly positive definite and $W(t,x)$ is defined only locally near the coordinate origin $x=0$. The proved theorems significantly generalize and improve previously obtained results. We also give several illustrative examples.

Keywords: fractional Hamiltonian system, critical point theory, symmetric mountain pass theorem

DOI: https://doi.org/10.4213/tmf9371

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English version:
Theoretical and Mathematical Physics, 2018, 195:1, 563–571

Bibliographic databases:

MSC: 34C37, 35A15, 37J45
Received: 22.03.2017
Revised: 25.08.2017

Citation: A. B. Benhassine, “Fractional Hamiltonian systems with locally defined potentials”, TMF, 195:1 (2018), 81–90; Theoret. and Math. Phys., 195:1 (2018), 563–571

Citation in format AMSBIB
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\vol 195
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\pages 563--571
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    This publication is cited in the following articles:
    1. S. Ahadpour, A. Nemati, F. Mirmasoudi, N. Hematpour, “Projective synchronization of piecewise nonlinear chaotic maps”, Theoret. and Math. Phys., 197:3 (2018), 1856–1864  mathnet  crossref  crossref  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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