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Multiple Solutions for a Class of Fractional Hamiltonian Systems

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Published/Copyright: February 10, 2015
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 18 Issue 1

Abstract

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

{D∞α(D∞αu(t))+L(t)u(t)=∇W(t,u(t)),u∈H(ℝ,ℝN),

where α∈(1/2,1),t∈ℝ,u=(u1,…,uN)T∈ℝN, and L∈C(ℝ,ℝN2) is a symmetric and positive definite matrix for all t∈ℝ,W∈C1(ℝ×ℝN×ℝ) and ∇W is the gradient of W about u.

MSC 2010: Primary 34C37; Secondary 35A15; 35B38
Key Words and Phrases: fractional Hamiltonian system; critical point; variational methods; multiple solutions

References

[1] M. Al-Refai, Y. Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications. Fract. Calc. Appl. Anal. 17, No 2 (2014), 483-498; DOI: 10.2478/s13540-014-0181-5, http://link.springer.com/article/10.2478/s13540-014-0181-5.Search in Google Scholar

[2] A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, No 4 (1973), 349-381.Search in Google Scholar

[3] G. Bin, Multiple solutions for a class of fractional boundary value problems. Abstr. Appl. Anal. 2012 (2012), Article ID 468980.Search in Google Scholar

[4] C. Bai, Existence of three solutions for a nonlinear fractional boundary value problem via a critical points theorem. Abstr. Appl. Anal. 2012 (2012), Article ID 963105.Search in Google Scholar

[5] J. Chen, X. Tang, Infinitely many solutions for a class of fractional boundary value problem. Bull. Malays. Math. Sci. Soc. 36, No 4 (2013), 1083-1097.Search in Google Scholar

[6] X. Chang, Ground state solutions of asymptotically linear fractional Schrodinger equations. J. Math. Phys. 54 (2013), Article ID 061504.10.1063/1.4809933Search in Google Scholar

[7] X. Chang, Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity26 (2013), 479-494.10.1088/0951-7715/26/2/479Search in Google Scholar

[8] W. Dong, J. Xu, Z. Wei, Infinitely many weak solutions for a fractional Schrodinger equation. Boundary Value Problems2014 (2014), Article ID 159, 14 p.10.1186/s13661-014-0159-6Search in Google Scholar

[9] F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, No 3 (2011), 1181-1199.Search in Google Scholar

[10] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22, No 4 (2012), Article ID 1250086.10.1142/S0218127412500861Search in Google Scholar

[11] Y. Li, H. Sun, Q. Zhang, Existence of solutions to fractional boundary value problems with a parameter. Electron. J. Diff. Equ. 141 (2013), 1-12.Search in Google Scholar

[12] Y. Luchko, Maximum principle and its application for the timefractional diffusion equations. Fract. Calc. Appl. Anal. 14, No 1 (2011), 110-124; DOI: 10.2478/s13540-011-0008-6; http://link.springer.com/article/10.2478/s13540-011-0008-6.Search in Google Scholar

[13] X. Lin, X. Tang, Existence of infinitely many solutions for p-Laplacian equations in RN. Nonlinear Anal. 92, (2013), 72-81.10.1016/j.na.2013.06.011Search in Google Scholar

[14] Q. Li, H. Su, Z. Wei, Existence of infinitely many large solutions for the nonlinear Schrodinger-Maxwell equations. Nonlinear Anal. 72, No 11 (2010), 4264-4270.Search in Google Scholar

[15] W. Liu, X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 39, No 1-2 (2012), 473-487.10.1007/s12190-012-0536-1Search in Google Scholar

[16] N. Nyamoradi, Infinitely many solutions for a class of fractional boundary value problems with Dirichlet boundary conditions. Mediterr. J. Math. 11, No 1 (2014), 75-87.Search in Google Scholar

[17] J. Sun, H. Chen, J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 373, No 1 (2011), 20-29.Search in Google Scholar

[18] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. 3rd Ed., Springer-Verlag, New York (2000).Search in Google Scholar

[19] C. Torres, Mountain pass solution for a fractional boundary value problem. J. Fract. Calc. Appl. 5 (2014), 1-10.Search in Google Scholar

[20] C. Torres, Existence of solution for a class of fractional Hamiltonian systems. arXiv:1212.5811v1.Search in Google Scholar

[21] M. Willem, Minimax Theorems. Birkhauser, Boston (1996).10.1007/978-1-4612-4146-1Search in Google Scholar

[22] Y. Ye, C. Tang, Multiple solutions for Kirchhoff-type equations in RN. J. Math. Phys. 54 (2013), Article ID 081508.10.1063/1.4819249Search in Google Scholar

[23] Z. Zhang, R. Yuan, Variational approach to solutions for a class of fractional Hamiltonian systems. Math. Methods Appl. Sci. 37, No 13 (2014), 1873-1883.Search in Google Scholar

Received: 2014-3-14
Published Online: 2015-2-10

© 2015 Diogenes Co., Sofia

Articles in the same Issue

  1. FCAA Related News, Events and Books (FCAA-Volume 18-1-2015)
  2. Electrolytic Fractional Derivative and Integral of Two-Terminal Element
  3. Initial Value Problem of Fractional Integro-Differential Equations in Banach Space
  4. Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation
  5. Multiple Solutions for a Class of Fractional Hamiltonian Systems
  6. Some Analytical and Numerical Properties of the Mittag-Leffler Functions
  7. Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions
  8. Systems of Nonlinear Fractional Differential Equations
  9. Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian
  10. Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations
  11. Filippov Lemma for a Class of Hadamard-Type Fractional Differential Inclusions
  12. New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses
  13. Several Results of Fractional Derivatives in Dʹ(R+)
  14. Modeling Extreme-Event Precursors with the Fractional Diffusion Equation
  15. Multiplicity Results for Integral Boundary Value Problems of Fractional Order with Parametric Dependence
  16. Improvements on Flat Output Characterization for Fractional Systems
  17. Nonlocal Fractional Boundary Value Problems with Slit-Strips Boundary Conditions
  18. Remarks to the Paper “On the Existence of Blow UP Solutions for a Class of Fractional Differential Equations” by Z. Bai et al., In “FCAA”, VOL. 17, NO 4 (2014), 1175-1187
https://doi.org/10.1515/fca-2015-0005
Keywords for this article
fractional Hamiltonian system; critical point; variational methods; multiple solutions

Articles in the same Issue

  1. FCAA Related News, Events and Books (FCAA-Volume 18-1-2015)
  2. Electrolytic Fractional Derivative and Integral of Two-Terminal Element
  3. Initial Value Problem of Fractional Integro-Differential Equations in Banach Space
  4. Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation
  5. Multiple Solutions for a Class of Fractional Hamiltonian Systems
  6. Some Analytical and Numerical Properties of the Mittag-Leffler Functions
  7. Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions
  8. Systems of Nonlinear Fractional Differential Equations
  9. Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian
  10. Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations
  11. Filippov Lemma for a Class of Hadamard-Type Fractional Differential Inclusions
  12. New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses
  13. Several Results of Fractional Derivatives in Dʹ(R+)
  14. Modeling Extreme-Event Precursors with the Fractional Diffusion Equation
  15. Multiplicity Results for Integral Boundary Value Problems of Fractional Order with Parametric Dependence
  16. Improvements on Flat Output Characterization for Fractional Systems
  17. Nonlocal Fractional Boundary Value Problems with Slit-Strips Boundary Conditions
  18. Remarks to the Paper “On the Existence of Blow UP Solutions for a Class of Fractional Differential Equations” by Z. Bai et al., In “FCAA”, VOL. 17, NO 4 (2014), 1175-1187
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