Skip to main content
SpringerLink
Log in
Book cover

Topological Methods in the Study of Boundary Value Problems pp 145–187Cite as

Applications

  • Chapter
  • First Online:

Part of the book series: Universitext ((UTX))

Abstract

This chapter is devoted to applications of the topological degree theory to different boundary value problems. Starting from specific examples, we obtain some general continuation theorems that can be applied in many situations. In particular, most of the sections are devoted to the study of resonant problems, in connection with which we discuss some classical results and different extensions.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Notes

  1. 1.

    In some cases the shooting method seems to be even more effective than degree theory or other methods; see, e.g., [11], where a two-dimensional shooting method is introduced to solve a Neumann problem arising in two-ion electrodiffusion.

  2. 2.

    In the delightful paper [95], an upper bound for the number of solutions is obtained by means of a classical result in complex analysis: the Jensen inequality.

References

  1. S. Ahmad, A. Lazer, J. Paul, Elementary critical point theory and perturbation of elliptic boundary value problems at resonance. Indiana Univ. Math. J., 25 (1976), 933–944.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Amster and M. Clapp: Periodic solutions of resonant systems with rapidly rotating nonlinearities. Discrete and Continuous Dynamical Systems, Series A 31 No. 2 (2011), 373–383.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Amster and A. Déboli: Existence of positive T-periodic solutions of a generalized Nicholson’s blowflies model with a nonlinear harvesting term. Applied Mathematics Letters 25 No. 9 (2012), 1203–1207.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Amster and P. De Nápoli: Non-asymptotic Lazer–Leach type conditions for a nonlinear oscillator. Discrete and Continuous Dynamical Systems, Series A 29, No. 3 (2011), 757–767.

    MathSciNet  MATH  Google Scholar 

  5. P. Amster and J. Haddad: A Hartman-Nagumo type condition for a class of contractible domains. Topological Methods in Nonlinear Analysis 41, No. 2 (2013), 287–304.

    MATH  Google Scholar 

  6. P. Amster, M. K. Kwong and C. Rogers: On a Neumann Boundary Value Problem for Painlevé II in Two Ion Electro-Diffusion. Nonlinear Analysis, TMA 74, 9 (2011), 2897–2907.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Amster and M. Maurette: Periodic solutions of systems with singularities of repulsive type. Advanced Nonlinear Studies 11 (2011), 201–220.

    MathSciNet  MATH  Google Scholar 

  8. J. Bebernes, K. Schmitt: Periodic boundary value problems for systems of second order differential equations. Journal of Differential Equations 13, no. 1 (1973) 32–47.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. do Carmo: Riemannian Geometry, Birkhäuser (1992).

    Google Scholar 

  10. C. Fabry and C. Franchetti: Nonlinear equations with growth restrictions on the nonlinear term. J. Differential Equations 20 No. 2 (1976), 283–291.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Fournier and J. Mawhin: On periodic solutions of forced pendulum-like equations. Journal of Differential Equations 60 (1985) 381–395.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Frederickson and A. Lazer: Necessary and sufficient damping in a second order oscillator, J. Differential Eqs. 5 No. 2 (1969), 262–270.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Fučík and M. Krbec: Boundary value problems with bounded nonlinearity and general null-space of the linear part. Mathematische Zeitschrift 155, 2 (1977), 129–138.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Gaines and J. Mawhin: Coincidence degree and nonlinear differential equations, Lecture Notes in Mathematics 586, Springer, Berlin (1977).

    Google Scholar 

  15. R. Gaines and J. Mawhin: Ordinary differential equations with nonlinear boundary conditions. J. Differential Equations 26 (1977) 200–222.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Hartman: On boundary value problems for systems of ordinary nonlinear second order differential equations. Trans. Amer. Math. Soc. 96 (1960), 493–509.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Hess: On a theorem by Landesman and Lazer. Indiana Univ. Math. J. 23 No. 9 (1974), 827–829

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Kannan and K. Nagle: Forced oscillations with rapidly vanishing nonlinearities. Proc. Amer. Math. Soc. 111 (1991), 385–393.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Kannan and R. Ortega: Periodic solutions of pendulum-type equations. J. Differential Equations 59 (1985), 123–144.

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Knobloch: On the existence of periodic solutions of second order vector differential equations, J. Differential Equations 9 (1971), 67–85.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Landesman and A. Lazer: Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609–623.

    MathSciNet  MATH  Google Scholar 

  22. A. Lazer: On Schauder’s Fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl. 21 (1968) 421–425.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Lazer and D. Leach: Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl. 82 (1969), 49–68.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Mawhin: Landesman-Lazer conditions for boundary value problems: A nonlinear version of resonance. Bol. de la Sociedad Española de Mat. Aplicada 16 (2000), 45–65.

    Google Scholar 

  25. J. Mawhin: Topological degree methods in nonlinear boundary value problems, NSF-CBMS Regional Conference in Mathematics no. 40, American Mathematical Society, Providence, RI (1979).

    Google Scholar 

  26. J. Mawhin: Boundary value problems for nonlinear second-order vector differential equations, J. Differential Equations 16 (1974), 257–269.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Mawhin: The Bernstein-Nagumo problem and two-point boundary value problems for ordinary differential equations. Qualitative Theory of Differential Equations, Farkas ed., Budapest (1981), 709–740.

    Google Scholar 

  28. J. Mawhin and M. Willem: Critical point theory and Hamiltonian systems. Springer, New York (1989).

    Book  MATH  Google Scholar 

  29. L. Nirenberg: Generalized degree and nonlinear problems, Contributions to nonlinear functional analysis, Ed. E. H. Zarantonello, Academic Press New York (1971), 1–9.

    Google Scholar 

  30. R. Ortega: A counterexample for the damped pendulum equation, Acad. Roy. Belg. Bull. Cl. Sci. 73 (1987), 405–409.

    MathSciNet  MATH  Google Scholar 

  31. R. Ortega: Counting periodic solutions of the forced pendulum equation. Nonlinear Analysis 42 (2000) 1055–1062.

    Article  MathSciNet  MATH  Google Scholar 

  32. R. Ortega and L. Sánchez: Periodic solutions of forced oscillators with several degrees of freedom, Bull. London Math. Soc. 34 (2002), 308–318.

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Ortega, E. Serra and M. Tarallo: Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction. Proc. of Am. Math. Soc. 128, 9 (2000), 2659–2665.

    Article  MathSciNet  MATH  Google Scholar 

  34. D. Ruiz and J. R. Ward Jr.: Some notes on periodic systems with linear part at resonance. Discrete and Continuous Dynamical Systems 11 (2004), 337–350.

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Whyburn, Topological Analysis, Princeton University Press, Princeton, NJ, (1964).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, FCEN-Universidad de Buenos Aires and IMAS-CONICET, Buenos Aires, Argentina

    Pablo Amster

Authors
  1. Pablo Amster
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Amster, P. (2014). Applications. In: Topological Methods in the Study of Boundary Value Problems. Universitext. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8893-4_6

Download citation

Publish with us

Policies and ethics

Search

Navigation