Rocky Mountain Journal of Mathematics

Oscillatory Properties of Solutions and Nonlinear Differential Equations with Periodic Boundary Conditions

Jean Mawhin

Full-text: Open access

Article information

Source
Rocky Mountain J. Math., Volume 25, Number 1 (1995), 7-37.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1181072265

Digital Object Identifier
doi:10.1216/rmjm/1181072265

Mathematical Reviews number (MathSciNet)
MR1339990

Zentralblatt MATH identifier
0830.34034

Citation

Mawhin, Jean. Oscillatory Properties of Solutions and Nonlinear Differential Equations with Periodic Boundary Conditions. Rocky Mountain J. Math. 25 (1995), no. 1, 7--37. doi:10.1216/rmjm/1181072265. https://projecteuclid.org/euclid.rmjm/1181072265


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References

  • S. Ahmad, A.C. Lazer and J.L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J. 25 (1976), 933-944.
  • G.D. Birkhoff, Proof of Poincaré's geometric theorem, Trans. Amer. Math. Soc. 14 (1913), 14-22.
  • M. Brown and W.D. Neumann, Proof of the Poincaré-Birkhoff fixed point theorem, Michigan Math. J. 24 (1977), 21-31.
  • G.J. Butler, Rapid oscillation, non-extendability and the existence of periodic solutions to second order nonlinear differential equations, J. Differential Equations 22 (1976), 467-477.
  • --------, Periodic solutions of sublinear second order differential equations, J. Math. Anal. Appl. 62 (1978), 676-690.
  • --------, The Poincaré-Birkhoff `twist' theorem and periodic solutions of second-order nonlinear differential equations, in Differential equations: Proceedings of fall conference on differential equations, Academic Press, New York, 1980, 135-147.
  • A. Capietto, J. Mawhin and F. Zanolin, A continuation approach to superlinear periodic boundary value problems, J. Differential Equations 88 (1990), 347-395.
  • E.N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh 76 (1977), 283-300.
  • D.G. de Figueiredo and B. Ruf, On the periodic Fučik spectrum and a superlinear Sturm-Liouville equation, Proc. Roy. Soc. Edinburgh 123 A (1993), 95-107.
  • T. Ding and F. Zanolin, Periodic solutions of Duffing's equation with superquadratic potential, J. Differential Equations 97 (1992), 328-378.
  • --------, Subharmonic solutions of second order nonlinear equations: A time-map approach, J. Nonlinear Anal. 20 (1993), 509-532.
  • W. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations (in Chinese), Acta Math. Sinica 25 (1982), 227-235.
  • L.H. Erbe, A survey of G.J. Butler's research in the qualitative theory of ordinary differential equations, Rocky Mountain J. Math. 20 (1990), 821-838.
  • C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math. (Basel) 60 (1993), 266-276.
  • A. Fonda and A.C. Lazer, Subharmonic solutions of conservative systems with non-convex potentials, Proc. Amer. Math. Soc. 115 (1992), 183-190.
  • A. Fonda and M. Ramos, Large amplitude subharmonic oscillations for scalar second order differential equations with asymmetric nonlinearities, J. Differential Equations, to appear.
  • A. Fonda, Z. Schneider and F. Zanolin, Periodic oscillations for a nonlinear suspension bridge model, to appear.
  • S. Fučik, Solvability of nonlinear equations and boundary value problems, Reidel, Dordrecht, 1980.
  • S. Fučik and J. Mawhin, Generalized periodic solutions of nonlinear telegraph equations, J. Nonlinear Anal. 2 (1978), 609-617.
  • P. Hartman, On boundary value problems for superlinear second order differential equations, J. Differential Equations 26 (1977), 37-53.
  • H. Jacobowitz, Periodic solutions of $x^ \pp+f(t,x)=0$ via the Poincaré-Birkhoff theorem, J. Differential Equations 20 (1976), 37-52. Corrigendum ibid. 25 (1977), 148-149.
  • A.C. Lazer, On Schauder's fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl. 21 (1968), 421-425.
  • J. Mawhin, An extension of a result of A.C. Lazer on forced nonlinear oscillations, J. Math. Anal. Appl. 40 (1972), 20-29.
  • --------, Topological degree methods in nonlinear boundary value problems, CBMS Regional Confer. Nr. 40, American Math. Soc., Providence, 1979.
  • --------, Topological degree and boundary value problems for nonlinear differential equations, CIME lectures, Montecatini, 1991, Springer Lect. Notes Math., 1537 Springer-Verlag, Berlin, 1993, 74-142.
  • J. Mawhin and J.R. Ward, Periodic solutions of some Liénard differential equations at resonance, Arch. Math. 41 (1983), 337-351.
  • J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Springer, New York, 1989.
  • G.R. Morris, An infinite class of periodic solutions of $x^\pp+2x^3=p(t)$, Proc. Cambridge Phil. Soc. 61 (1965), 157-164.
  • Z. Opial, Sur les solutions périodiques de l'équation différentielle $x^\pp+g(x)=p(t)$, Bull. Acad. Pol. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 151-156.
  • H. Poincaré, Sur un théorème de géométrie, Rend. Circolo Mat. Palermo 33 (1912), 375-407.
  • P. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, in Nonlinear analysis: A collection of papers in honor of Erich Rothe, Academic Press, New York, 1978.
  • --------, On subharmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 33 (1980), 609-633.