Skip to main content

The P-Index Theory

  • Chapter
  • First Online:
Index theory in nonlinear analysis

Abstract

If the Hamiltonian function \(H \in C^{2}(\mathbb {R} \times \mathbb {R}^{2n}, \mathbb {R})\) satisfying H(t + τ, Px) = H(t, x) with P ∈Sp(2n), it is natural to consider the following nonlinear Hamiltonian system with P-boundary condition.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bott, R.: On the iteration of closed geodesics and the Sturm intersection theory. Commun. Pure Appl. Math. 9, 171–206 (1956)

    Article  MathSciNet  Google Scholar 

  2. Conley, C., Zehnder, E.: Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure. Appl. Math. 37, 207–253 (1984)

    Article  MathSciNet  Google Scholar 

  3. Dong, Y.: P-index theory for linear Hamiltonian systems and multiple solutions for nonlinear Hamiltonian systems. Nonlinearity 19(6), 1275–1294 (2006)

    Article  MathSciNet  Google Scholar 

  4. Dong, Y., Long, Y.: Closed characteristics on partially symmetric convex hypersurfaces in \(\mathbb {R}^{2n}\). J. Differ. Equ. 196, 226–248 (2004)

    Google Scholar 

  5. Fei, G., Qiu, Q.: Periodic solutions of asymptotically linear Hamiltonian systems. Chin. Ann. Math. 18B(3), 359–372 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Fei, G., Qiu, Q.: Minimal period solutions of nonlinear Hamiltonian systems. Nonlinear Anal. Theory Methods Appl. 27, 821–839 (1996)

    Article  MathSciNet  Google Scholar 

  7. Liu, C.: Maslov P-index theory for a symplectic path with applications. Chin. Ann. Math. 4, 441–458 (2006)

    Article  MathSciNet  Google Scholar 

  8. Liu, C.: Relative index theories and applications. Topol. Methods Nonlinear Anal. 49(2), 587–614 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Liu, C., Long, Y.: Iteration inequalities of the Maslov-type index theory with applications. J. Differ. Equ. 165, 355–376 (2000)

    Article  MathSciNet  Google Scholar 

  10. Liu, C., Tang, S.: Maslov (P, ω)-index theory for symplectic paths. Adv. Nonlinear Stud. 15, 963–990 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Liu, C., Tang, S.: Iteration inequalities of the Maslov P-index theory with applications. Nonlinear Anal. 127, 215–234 (2015)

    Article  MathSciNet  Google Scholar 

  12. Long, Y.: Index Theory for Symplectic Path with Applications. Progress in Mathematics, vol. 207. Birkhäuser, Basel (2002)

    Google Scholar 

  13. Long, Y.: A Maslov-type index theory for symplectic paths. Top. Methods Nonlinear Anal. 10, 47–78 (1997)

    Article  MathSciNet  Google Scholar 

  14. Long, Y.: Bott formula of the Maslov-type index theory. Pacific J. Math. 187, 113–149 (1999)

    Article  MathSciNet  Google Scholar 

  15. Long, Y.: Index Theory of Hamiltonian Systems with Applications. Science Press, Beijing (1993) (in Chinese)

    Google Scholar 

  16. Long, Y., Zehnder, E.: Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In: Albeverio, S., et al. (ed.) Stochastic Processes in Physics and Geometry, pp. 528–563. World Scientific, Singapore (1990)

    Google Scholar 

  17. Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in \(\mathbb {R}^{2n}\). Ann. Math. 155, 317–368 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Liu, C. (2019). The P-Index Theory. In: Index theory in nonlinear analysis. Springer, Singapore. https://doi.org/10.1007/978-981-13-7287-2_4

Download citation

Publish with us

Policies and ethics