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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 2
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Research Article

Relative Morse index and multiple homoclinic orbits for a nonperiodic Hamiltonian system

Pages 524-541 | Received 25 Apr 2021, Accepted 11 Jul 2021, Published online: 20 Jul 2021

References

  • Coti-Zelati V, Ekeland I, Séré E. A variational approach to homoclinic orbits in Hamiltonian systems. Math Ann. 1990;288:133–160.
  • Séré E. Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math Z. 1992;209:27–42.
  • Hofer H, Wysocki K. First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math Ann. 1990;288:483–503.
  • Tanaka K. Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits. J Differ Eq. 1991;94:315–339.
  • Ding YH. Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms. Commun Contemp Math. 2006;8:453–480.
  • Ding YH, Girardi M. Infinitely many homoclinic orbits of a Hamiltonian system with symmetry. Nonlinear Anal. 1999;38:391–415.
  • Ding YH, Willem M. Homoclinic orbits of a Hamiltonian system. Z Angew Math Phys. 1999;50:759–778.
  • Sun J, Chu J, Feng Z. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete Contin Dyn Syst Ser. 2013;33:3807–3824.
  • Szulkin A, Zou W. Homoclinic orbits for asymptotically linear Hamiltonian systems. J Funct Anal. 2001;187:25–41.
  • Wang J, Xu J, Zhang F. Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition. Discrete Cont Dyn Syst. 2010;27:1241–1257.
  • Ding YH, Li SJ. Homoclinic orbits for first order Hamiltonian systems. J Math Anal Appl. 1995;189:585–601.
  • Bartsch T, Ding YH. Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math Nachr. 2006;279:1267–1288.
  • Ding YH, Jeanjean L. Homoclinic orbits for a nonperiodic Hamiltonian system. J Differ Eq. 2007;237:473–490.
  • Qin W, Zhang J, Zhao F. Homoclinic orbits for a class of nonperiodic Hamiltonian systems. Abstr Appl Anal. 2012;2012:769232.
  • Wang Q, Zhang Q. Homoclinic orbits for a class of nonperiodic Hamiltonian systems with some twisted conditions. Abstr Appl Anal. 2013;2013:1–11.
  • Ekeland I. Une theorie de Morse pour les systemes hamiltoniens convexes. Ann Inst H Poincaré Anal Non Linéaire. 1984;1:19–78.
  • Conley C, Zehnder E. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm Pure Appl Math. 1984;37:207–253.
  • Long Y. Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Sci China Math. 1990;33:1409–1419.
  • Long Y. A Maslov-type index theory for symplectic paths. Topol Methods Nonlinear Anal. 1997;10:47–78.
  • Long Y, Zehnder E. Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, in: S. Alberverio, et al. editor, Stock, Process. Phys. Geom. Teaneck (NJ): World Scientific; 1990. p. 528–563.
  • Long Y, Zhu C. Maslov type index theory for symplectic paths and spectral flow (II). Chin Ann Math. 2000;21B:89–108.
  • Zhu C, Long Y. Maslov type index theory for symplectic paths and spectral flow (I). Chin Ann Math. 1999;20B:413–424.
  • Liu C. Maslov-type index theory for symplectic paths with Lagrangian boundary conditions. Adv Nonlinear Stud. 2007;7:131–161.
  • Dong Y. Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations. Calc Var Partial Differ Eq. 2010;38:75–109.
  • Chen C, Hu X. Maslov index for homoclinic orbits of Hamiltonian systems. Ann Inst H Poincaré Anal Non Linéaire. 2007;24:589–603.
  • Ekeland I. Convexity methods in Hamiltonian mechanics. Berlin: Springer; 1990.
  • Long Y. Index theory for symplectic paths with applications. Basel: Birkhäuser; 2002. (Progr. Math.; 207).
  • Liu H, Wang C, Zhang D. Elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in R2n. Calc Var Partial Differ Eq. 2020;59:1–20.
  • Wang Q, Liu C. A new index theory for linear self-adjoint operator equations and its applications. J Differ Eq. 2016;260:3749–3784.
  • Dolbeault J, Esteban M, Séré E. On the eigenvalues of operators with gaps. application to Dirac operators. J Funct Anal. 2000;174:208–226.
  • Dong Y, Shan Y. Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations (II), e-print arXiv:1104.1670v1.
  • Shan Y. Homoclinic orbits for first order Hamiltonian systems with some twist conditions. Acta Math Sin Engl Ser. 2015;31:1725–1738.
  • Abbondandolo A, Molina J. Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems. Calc Var Partial Differ Eq. 2000;11:395–430.
  • Liu C. A note on the relations between the various index theories. J Fixed Point Theory Appl. 2017;19:617–648.
  • Wang Q, Liu C. The relative morse index for infinite dimensional Hamiltonian systems with applications. J Math Anal Appl. 2015;427:17–30.
  • Chang KC. Critical point theory and its applications. Shanghai: Shanghai Science and Technical Press; 1986. (in Chinese).
  • Chang KC. Infinite dimensional morse theory and multiple solution problems. Basel: Birkhauser; 1993.
  • Mawhin J, Willem M. Critical point theory and Hamiltonian systems. Berlin: Springer; 1998.

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