## Rocky Mountain Journal of Mathematics

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

Jean Mawhin

#### Article information

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

Dates
First available in Project Euclid: 5 June 2007

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

#### 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.