Multiplicity results for periodic solutions

of second order ODEs with asymmetric nonlinearities

Authors:
C. Rebelo and F. Zanolin

Journal:
Trans. Amer. Math. Soc. **348** (1996), 2349-2389

MSC (1991):
Primary 34C25; Secondary 34B15

DOI:
https://doi.org/10.1090/S0002-9947-96-01580-2

MathSciNet review:
1344211

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equation , as or where is a smooth function defined on a open interval The hypotheses we assume on the nonlinearity allow us to cover the case (or ) and having superlinear growth at infinity, as well as the case (or ) and having a singularity in (respectively in ). Applications are given also to situations like (including the so-called ``jumping nonlinearities''). Our results are connected to the periodic Ambrosetti - Prodi problem and related problems arising from the Lazer - McKenna suspension bridges model.

**1.**A. Ambrosetti and G. Prodi,*On the inversion of some differentiable mappings with singularities between Banach spaces*, Ann. Mat. Pura Appl.**93**(1972), 231-247. MR**47:9377****2.**M. Berger and E. Podolak,*On the solutions of a nonlinear Dirichlet problem*, Indiana Univ. Math. J.**24**(1975), 837-846. MR**51:13447****3.**A. Castro and R. Shivaji,*Multiple solutions for a Dirichlet problem with jumping nonlinearities, II*, J. Math. Anal. Appl.**133**(1988), 509-528. MR**89e:34031****4.**R. Conti,*Soluzioni periodiche dell'equazione di Liénard generalizzata. Esistenza ed unicità*, Boll. Un. Mat. Ital.**3**(1952), 111-118. MR**14:558a****5.**D.G. Costa, D.G. De Figueiredo and P.N. Srikanth,*The exact number of solutions for a class of ordinary differential equations through Morse index computation*, J. Differential Equations**96**(1992), 195-199. MR**93c:34050****6.**E.N. Dancer,*Boundary value problems for weakly nonlinear ordinary differential equations*, Bull. Austral. Math. Soc.**15**(1976), 321-328. MR**55:3389****7.**C. De Coster,*La méthode des sur et sous solutions dans l'étude de problèmes aux limites*, Ph. D. Thesis, Université Catholique de Louvain, 1994.**8.**M. del Pino, R. F. Manásevich and A. Murua,*On the number of periodic solutions for using the Poincaré-Birkhoff theorem*, J. Differential Equations**95**(1992), 240-258. MR**93e:34062****9.**M. del Pino, R. Manásevich and A. Montero,*-periodic solutions for some second order differential equations with singularities*, Proc. Roy. Soc. of Edinburgh**120 A**(1992), 231-243. MR**93c:34091****10.**W.Y. Ding,*A generalization of the Poincaré-Birkhoff theorem*, Proc. Amer. Math. Soc.**88**(1983), 341-346. MR**84f:54053****11.**T. Ding and F. Zanolin,*Periodic solutions of Duffing's equations with superquadratic potential*, J. Differential Equations**97**(1992), 328-378. MR**93e:34059****12.**T. Ding and F. Zanolin,*Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type*, Proc. of the 1st World Congress of Nonlinear Analysts (V. Lakshmikantham, ed.), Tampa 1992 (to appear).**13.**T. Ding, R. Iannacci and F. Zanolin,*Existence and multiplicity results for periodic solutions of semilinear Duffing equations*, J. Differential Equations**105**(1993), 364-409. MR**94g:34060****14.**C. Fabry and P. Habets,*Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities*, Arch. Math.**60**(1993), 266-276. MR**93j:34055****15.**C. Fabry, J. Mawhin and M. N. Nkashama,*A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations*, Bull. London Math. Soc.**18**(1986), 173-180. MR**87e:34072****16.**D.G. Figueiredo and B. Ruf,*On the periodic Fu\v{c}ik spectrum and a superlinear Sturm - Liouville equation*, Proc. Roy. Soc. of Edinburgh**123 A**(1993), 95-107. MR**93m:34027****17.**A. Fonda and A. C. Lazer,*Subharmonic solutions of conservative systems with nonconvex potentials*, Proc. Amer. Math. Soc.**115**(1992), 183-190. MR**92h:34082****18.**A. Fonda, R. Manásevich and F. Zanolin,*Subharmonic solutions for some second-order differential equations with singularities*, SIAM J. Math. Anal.**24**(1993), 1294-1311. MR**94f:34085****19.**A. Fonda and M. Ramos,*Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities*, J. Differential Equations**109**(1994), 354-372. MR**95e:34032****20.**A. Fonda, M. Ramos and M. Willem,*Subharmonic solutions for second order differential equations*, Topological Methods in Nonlinear Analysis**1**(1993), 49-66. MR**94c:58027****21.**S. Fu\v{c}ik,*Solvability of nonlinear equations and boundary value problems*, Reidel, Dordrecht, 1980. MR**83c:47079****22.**P. Habets and L. Sanchez,*Periodic solutions of some Liénard equations with singularities*, Proc. Amer. Math. Soc.**109**(1990), 1035-1044. MR**90k:34049****23.**D. C. Hart, A. C. Lazer and P. J. McKenna,*Multiplicity of solutions of nonlinear boundary value problems*, SIAM J. Math. Anal.**17**(1986), 1332-1338. MR**87k:34027****24.**L. Hua,*Introduction to number theory*, Springer-Verlag, Berlin, 1982. MR**83f:10001****25.**A.C. Lazer and P.J. McKenna,*Large scale oscillatory behaviour in loaded asymmetric systems*, Ann. Inst. H. Poincaré Anal. Non Linéaire,**4**(1987), 243-274. MR**88m:58028****26.**A.C. Lazer and P.J. McKenna,*Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities*, Trans. Amer. Math. Soc.**315**(1989), 721-739. MR**90a:34011****27.**A.C. Lazer and P.J. McKenna,*Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis*, SIAM Review**32**(1990), 537-578. MR**92g:73059****28.**A.C. Lazer and S. Solimini,*On periodic solutions of nonlinear differential equations with singularities*, Proc. Amer. Math. Soc.**88**(1987), 109-114. MR**87k:34064****29.**D. Lupo, S. Solimini and P. N. Srikanth,*Multiplicity results for an ODE problem with even nonlinearity*, Nonlinear Anal. TMA**12**(1988), 657-673. MR**89j:34026****30.**H.G. Kaper and M.K. Kwong,*On two conjectures concerning the multiplicity of solutions of a Dirichlet problem*, SIAM J. Math. Anal.**23**(1992), 571-578. MR**93c:34055****31.**W. Massey,*Algebraic topology: an introduction*, Springer-Verlag, Berlin, 1984.**32.**J. Mawhin,*Continuation principles and boundary value problems*, Topological methods for ordinary differential equations (Furi and Zecca, eds.), C.I.M.E., Montecatini 1991, vol. 1537, LNM Springer-Verlag, Berlin, 1993, pp. 74-142. MR**94h:47121****33.**J. Mawhin and M. Willem,*Critical point theory and Hamiltonian systems*, Springer-Verlag, New York, 1989. MR**90e:58016****34.**R. Michalek and G. Tarantello,*Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems*, J. Differential Equations**72**(1988), 28-55. MR**89c:58040****35.**W. Neumann,*Generalizations of the Poincaré-Birkhoff fixed point theorem*, Bull. Austral. Math. Soc.**17**(1977), 375-389. MR**58:28435****36.**R. Ortega,*Stability of a periodic problem of Ambrosetti - Prodi type*, J. Differential and Integral Equations**3**(1990), 275-284. MR**90m:34104****37.**M. Pei,*Mather sets for twist maps and Duffing equations*, preprint.**38.**V. Pliss,*Nonlocal problems of the theory of oscillations*, Academic Press, New York, 1966. MR**33:4391****39.**C. Rebelo, Ph. D. Thesis, in preparation.**40.**K. Schmitt,*Boundary value problems with jumping nonlinearities*, Rocky Mountain J. Math.**16**(1986), 481-496. MR**87m:34016****41.**J. You,*Boundedness for solutions of superlinear Duffing equations via the twist theorem*, Science in China (A)**35**(1992), 399-412. MR**95c:34067****42.**F. Zanolin,*Continuation theorems for the periodic problem via the translation operator*, Rend. Sem. Mat. Univ. Torino (to appear).**43.**B. Zinner,*Multiplicity of solutions for a class of superlinear Sturm-Liouville problems*, J. Math. Anal. Appl.**176**(1993), 282-291. MR**94j:34026**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
34C25,
34B15

Retrieve articles in all journals with MSC (1991): 34C25, 34B15

Additional Information

**C. Rebelo**

Affiliation:
International School for Advanced Studies, via Beirut 2-4, 34013 Trieste, Italy

Address at time of publication:
Centro de Matemática e Aplicações Fundamentais, Av. Prof. Gama Pinto 2, 1699 Lisboa Codex, Portugal

Email:
carlota@ptmat.lmc.fc.ul.pt

**F. Zanolin**

Affiliation:
Dipartimento di Matematica e Informatica, Università, via delle Scienze 208 (loc. Rizzi), 33100 Udine, Italy

Email:
zanolin@dimi.uniud.it

DOI:
https://doi.org/10.1090/S0002-9947-96-01580-2

Keywords:
Periodic solutions,
subharmonics,
asymmetric nonlinearities,
Poincaré-Birkhoff fixed point theorem

Received by editor(s):
August 4, 1994

Received by editor(s) in revised form:
February 28, 1995

Additional Notes:
Work performed in the frame of the EEC project “Non linear boundary value problems: existence, multiplicity and stability of solutions”, grant ERB CHRX-CT94-0555.

The first author is on leave of absence from Faculdade de Ciências da Universidade de Lisboa with a fellowship from Programa Ciência (JNICT).

The second author’s work performed under the auspices of GNAFA-CNR and supported by MURST (40% and 60% funds).

Article copyright:
© Copyright 1996
American Mathematical Society