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Multiple Solutions for a Class of Fractional Hamiltonian Systems

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Abstract

In this paper, we establish two existence theorems for multiple solutions for the following fractional Hamiltonian system

$$\left\{ {\begin{array}{*{20}{c}} {_tD_\infty ^\alpha \left( {_{ - \infty }D_t^\alpha u\left( t \right)} \right) + L\left( t \right)u\left( t \right) = \nabla W\left( {t,u\left( t \right)} \right),} \\ {u \in {H^\alpha }\left( {R,{R^N}} \right),} \\ \end{array}} \right.$$

where α ∈ (1 / 2, 1), t∈ℝ, u= (u 1, …, uN) T∈ ℝN, and L∈C (ℝ, ℝ N 2) is a symmetric and positive definite matrix for all t∈ℝ,W∈ C 1 (ℝ× ℝN×ℝ) and ∇ W is the gradient of W about u.

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Authors and Affiliations

  1. School of Mathematics, Shandong University Jinan, 250100, Shandong, China

    Jiafa Xu

  2. School of Mathematics, Statistics and Applied Mathematics National University of Ireland, Galway, Ireland

    Donal O’Regan

  3. Department of Mathematics, Qilu Normal University Jinan, 250013, Shandong, China

    Keyu Zhang

Authors
  1. Jiafa Xu
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  2. Donal O’Regan
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  3. Keyu Zhang
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Correspondence to Jiafa Xu.

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Xu, J., O’Regan, D. & Zhang, K. Multiple Solutions for a Class of Fractional Hamiltonian Systems. FCAA 18, 48–63 (2015). https://doi.org/10.1515/fca-2015-0005

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  • DOI: https://doi.org/10.1515/fca-2015-0005

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