August  2013, 33(8): 3807-3824. doi: 10.3934/dcds.2013.33.3807

Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero

1. 

School of Science, Shandong University of Technology, Zibo 255049, China

2. 

Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China

3. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received  September 2012 Revised  November 2012 Published  January 2013

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first order periodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we establish some new criteria to guarantee that Hamiltonian systems, with asymptotically quadratic terms and spectrum point zero, have at least one and infinitely many homoclinic orbits under certain conditions.
Citation: Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
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show all references

References:
[1]

Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.  Google Scholar

[2]

J. Funct. Anal., 234 (2006), 277-320. doi: 10.1016/j.jfa.2005.11.010.  Google Scholar

[3]

J. Differential Equations, 158 (1999), 291-313. doi: 10.1006/jdeq.1999.3639.  Google Scholar

[4]

Math. Nachr., 279 (2006), 1267-1288. doi: 10.1002/mana.200410420.  Google Scholar

[5]

in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. II, Elsevier B. V., Amsterdam, (2005), 77-146.  Google Scholar

[6]

J. Math. Anal. Appl., 363 (2010), 627-638. doi: 10.1016/j.jmaa.2009.09.025.  Google Scholar

[7]

Ann. I. H. Poincaré Anal. Linéaire, 24 (2007), 589-603. doi: 10.1016/j.anihpc.2006.06.002.  Google Scholar

[8]

Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 423-443. doi: 10.3934/dcdsb.2011.16.423.  Google Scholar

[9]

Math. Ann., 228 (1990), 133-160. doi: 10.1007/BF01444526.  Google Scholar

[10]

J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.2307/2939286.  Google Scholar

[11]

Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.  Google Scholar

[12]

Interdiscip. Math. Sci., 7, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709639.  Google Scholar

[13]

Commun. Contemp. Math., 8 (2006), 453-480. doi: 10.1142/S0219199706002192.  Google Scholar

[14]

Nonlinear Anal., 38 (1999), 391-415. doi: 10.1016/S0362-546X(98)00204-1.  Google Scholar

[15]

J. Differential Equations, 237 (2007), 473-490. doi: 10.1016/j.jde.2007.03.005.  Google Scholar

[16]

J. Differential Equations, 246 (2009), 2829-2848. doi: 10.1016/j.jde.2008.12.013.  Google Scholar

[17]

J. Math. Anal. Appl., 189 (1995), 585-601. doi: 10.1006/jmaa.1995.1037.  Google Scholar

[18]

Z. Angew. Math. Phys., 50 (1999), 759-778. doi: 10.1007/s000330050177.  Google Scholar

[19]

Z. Angew. Math. Phys., 60 (2009), 363-375. doi: 10.1007/s00033-007-7102-y.  Google Scholar

[20]

Commun. Pure Appl. Anal., 6 (2007), 429-440. doi: 10.3934/cpaa.2007.6.429.  Google Scholar

[21]

J. Differential Equations, 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[22]

Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989.  Google Scholar

[23]

Discrete Contin. Dyn. Syst., 20 (2008), 53-79. doi: 10.3934/dcds.2008.20.53.  Google Scholar

[24]

Discrete Contin. Dyn. Syst., 25 (2009), 883-913. doi: 10.3934/dcds.2009.25.883.  Google Scholar

[25]

Differ. Equ. Dyn. Syst., 20 (2012), 93-109. doi: 10.1007/s12591-012-0107-9.  Google Scholar

[26]

Nonlinear Anal., 72 (2010), 4575-4586. doi: 10.1016/j.na.2010.02.034.  Google Scholar

[27]

J. Math. Anal. Appl., 373 (2011), 20-29. doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

[28]

J. Math. Anal. Appl., 378 (2011), 117-127. doi: 10.1016/j.jmaa.2010.12.044.  Google Scholar

[29]

J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798.  Google Scholar

[30]

Discrete Contin. Dyn. Syst., 27 (2010), 1241-1257. doi: 10.3934/dcds.2010.27.1241.  Google Scholar

[31]

J. Math. Anal. Appl., 247 (2000), 645-652. doi: 10.1006/jmaa.2000.6839.  Google Scholar

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